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Talk:Right Hand Rule

2024-12-03T16:47:11Z

Dsaxena8: Created page with "No Comp. Model"

No Comp. Model

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:27:39Z

Dsaxena8: /* Examples */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''

<math>{I = \frac{V}{R}}</math>

<math>{I = \frac{3}{2}}</math>

<math>{I = 1.5 \text{ Amps}}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''

<math>{V = IR}</math>

<math>{V = 6*2}</math>

<math>{V = 12 \text{ Volts}}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''

<math>{R = \frac{I}{V}}</math>

<math>{R = \frac{18}{10}}</math>

<math>{R = 1.8 \text{ Ohms}}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''

<math>{V = 2*3 = 6}</math>

<math>{R = 4*2 = 8}</math>

<math>{I = \frac{6}{8} = 0.75 \text{ Amps}}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''

Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:27:23Z

Dsaxena8: /* Examples */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''

<math>{I = \frac{V}{R}}</math>

<math>{I = \frac{3}{2}}</math>

<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''

<math>{V = IR}</math>

<math>{V = 6*2}</math>

<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''

<math>{R = \frac{I}{V}}</math>

<math>{R = \frac{18}{10}}</math>

<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''

<math>{V = 2*3 = 6}</math>

<math>{R = 4*2 = 8}</math>

<math>{I = \frac{6}{8} = 0.75 \text{ Amps}}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''

Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:26:57Z

Dsaxena8: /* Difficult */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''

<math>{V = 2*3 = 6}</math>

<math>{R = 4*2 = 8}</math>

<math>{I = \frac{6}{8} = 0.75 \text{ Amps}}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''

Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:26:51Z

Dsaxena8: /* Middling */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''

<math>{V = 2*3 = 6}</math>

<math>{R = 4*2 = 8}</math>

<math>{I = \frac{6}{8} = 0.75 \text{ Amps}}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:26:43Z

Dsaxena8: /* Middling */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>

<math>{R = 4*2 = 8}</math>

<math>{I = \frac{6}{8} = 0.75 \text{ Amps}}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:26:31Z

Dsaxena8: /* Difficult */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>
<math>{R = 4*2 = 8}</math>
<math>{I = \frac{6}{8} = 0.75 Amps}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:26:11Z

Dsaxena8: /* Difficult */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>
<math>{R = 4*2 = 8}</math>
<math>{I = \frac{6}{8} = 0.75 Amps}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):
<math>{V = 1.5+2.25+3.75+4.5 = 12 Volts}</math>
Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:25:32Z

Dsaxena8: /* Difficult */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>
<math>{R = 4*2 = 8}</math>
<math>{I = \frac{6}{8} = 0.75 Amps}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{Volts}}</math>
Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 Volts}</math>
Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 Volts}</math>
Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 Volts}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):
<math>{V = 1.5+2.25+3.75+4.5 = 12 Volts}</math>
Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:24:54Z

Dsaxena8: /* Examples */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>
<math>{R = 4*2 = 8}</math>
<math>{I = \frac{6}{8} = 0.75 Amps}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 Volts}</math>
Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 Volts}</math>
Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 Volts}</math>
Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 Volts}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):
<math>{V = 1.5+2.25+3.75+4.5 = 12 Volts}</math>
Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:24:28Z

Dsaxena8: /* Examples */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>
<math>{R = 4*2 = 8}</math>
<math>{I = \frac{6}{8} = 0.75 Amps}</math>

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 Volts}</math>
Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 Volts}</math>
Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 Volts}</math>
Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 Volts}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):
<math>{V = 1.5+2.25+3.75+4.5 = 12 Volts}</math>
Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:23:51Z

Dsaxena8:

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''
<math>{I = \frac{V}{R}}</math>
<math>{I = \frac{3}{2}}</math>
<math>{I = 1.5 Amps}</math>

'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''
<math>{V = IR}</math>
<math>{V = 6*2}</math>
<math>{V = 12 Volts}</math>

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''
<math>{R = \frac{I}{V}}</math>
<math>{R = \frac{18}{10}}</math>
<math>{R = 1.8 Ohms}</math>

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''
<math>{V = 2*3 = 6}</math>
<math>{R = 4*2 = 8}</math>
<math>{I = \frac{6}{8} = 0.75 Amps}</math>

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''
Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 Volts}</math>
Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 Volts}</math>
Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 Volts}</math>
Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 Volts}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):
<math>{V = 1.5+2.25+3.75+4.5 = 12 Volts}</math>
Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:03:02Z

Dsaxena8: /* Computational Model */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:02:49Z

Dsaxena8: /* Computational Model */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[File:Glowscriptmodel-ezgif.com-crop.gif]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:02:20Z

Dsaxena8: /* Computational Model */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif|thumb |400px|Current Model]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T21:01:03Z

Dsaxena8: /* Computational Model */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:glowscriptmodel-ezgif.com-crop.gif|thumb |400px|Current Model]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

File:Glowscriptmodel-ezgif.com-crop.gif

2024-11-25T20:59:35Z

Dsaxena8: Current,Resistance, and Voltage: How they interact

== Summary ==
Current,Resistance, and Voltage: How they interact

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T20:50:04Z

Dsaxena8: /* Model */

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

<iframe src="https://trinket.io/embed/glowscript/922d8d312818" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T20:40:22Z

Dsaxena8: /* Model */

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T20:39:45Z

Dsaxena8: /* Model */

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T20:39:05Z

Dsaxena8:

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T20:37:35Z

Dsaxena8:

Understanding Fundamentals of Current, Voltage, and Resistance

2024-11-25T00:57:22Z

Dsaxena8:

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024. Edited by Dhruv Saxena - Fall 2024'''

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow, voltage signifies the driving force behind it, and resistance encapsulates the opposition encountered in the circuit. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Gravitational Potential Energy

2024-04-24T20:34:55Z

Dsaxena8: /* Examples */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

===== Gravitational Potential Energy Relation to Gravitational Force: =====

* Potential energy is always the integral of the force over the distance the force is applied.
* We can observe how the gravitational potential energy comes about by looking at the gravitational force

<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:EnergyAndPositionGraphs.png|frame|none|alt=|caption ]]
From https://www.glowscript.org/#/user/Dhruv3745/folder/MyPrograms/program/lab2
[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Guide on Solving Problems ==

The Law of Conservation of energy states that energy is neither created nor destroyed. This means that a system's energy is always the same at any point in time as long as it is not acted upon by an external influence.

<math> E_i = E_f </math>

<math> U_{g_{init}} + K_{init} = U_{g_{final}} + K_{final} </math>

Upon seeing any problem, you need to calculate the energy it would have at the final/initial state (depending on what information is given). Isolate the unknown and solve.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

The idea of gravitational potential energy specifically was introduced by Johann Bernoulli in 1738. The concept was later taken up by several people in several different ways. For example Euler, Lagrange, Laplace and Legendre looked at gravity as it revolved around something while Newton focused on gravitational attraction if the object is a sphere. Euler and Co later confirmed the shape does not matter for the object.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

Gravitational Potential Energy

2024-04-24T13:16:18Z

Dsaxena8: /* History */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

===== Gravitational Potential Energy Relation to Gravitational Force: =====

* Potential energy is always the integral of the force over the distance the force is applied.
* We can observe how the gravitational potential energy comes about by looking at the gravitational force

<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:EnergyAndPositionGraphs.png|frame|none|alt=|caption ]]
From https://www.glowscript.org/#/user/Dhruv3745/folder/MyPrograms/program/lab2
[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

The idea of gravitational potential energy specifically was introduced by Johann Bernoulli in 1738. The concept was later taken up by several people in several different ways. For example Euler, Lagrange, Laplace and Legendre looked at gravity as it revolved around something while Newton focused on gravitational attraction if the object is a sphere. Euler and Co later confirmed the shape does not matter for the object.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

Gravitational Potential Energy

2024-04-24T12:59:05Z

Dsaxena8: /* Computational Model */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

===== Gravitational Potential Energy Relation to Gravitational Force: =====

* Potential energy is always the integral of the force over the distance the force is applied.
* We can observe how the gravitational potential energy comes about by looking at the gravitational force

<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:EnergyAndPositionGraphs.png|frame|none|alt=|caption ]]
From https://www.glowscript.org/#/user/Dhruv3745/folder/MyPrograms/program/lab2
[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

File:EnergyAndPositionGraphs.png

2024-04-24T12:56:07Z

Dsaxena8: EnergyAndPositionGraphs

== Summary ==
EnergyAndPositionGraphs

Gravitational Potential Energy

2024-04-24T12:10:30Z

Dsaxena8: /* Main Idea */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

===== Gravitational Potential Energy Relation to Gravitational Force: =====

* Potential energy is always the integral of the force over the distance the force is applied.
* We can observe how the gravitational potential energy comes about by looking at the gravitational force

<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

Gravitational Potential Energy

2024-04-24T11:51:04Z

Dsaxena8: /* Gravitational Potential Energy Relation to Gravitational Force: */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

===== Gravitational Potential Energy Relation to Gravitational Force: =====

* Potential energy is always the integral of the force over the distance the force is applied.
* We can observe how the gravitational potential energy comes about by looking at the gravitational force

<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

Gravitational Potential Energy

2024-04-24T11:50:31Z

Dsaxena8: /* Mathematical Model */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

===== Gravitational Potential Energy Relation to Gravitational Force: =====

* Potential energy is always the integral of the force over the distance the force is applied.
* We can observe how the gravitational potential energy comes about by looking at the gravitational force

<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

Gravitational Potential Energy

2024-04-24T11:41:45Z

Dsaxena8: /* Main Idea */

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]

Gravitational Potential Energy

2024-04-24T03:49:39Z

Dsaxena8: Claiming

'''Claimed by: Dhruv Saxena (Spring 2023)'''

== Main Idea ==

Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>|U_{grav}|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>|U_{grav}|</math> will be greater.

An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Checkout this [https://www.youtube.com/watch?v=z3ujg_CkslI video] for a quick review and explanation.

=== Mathematical Model ===

-----

===== General Form for Gravitational Potential Energy: =====

<math>U_{grav} = -G\frac{m_1 m_2}{r}</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m_1</math> is the mass of the first object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>m_2</math> is the mass of the second object you are measuring <math>U_{grav}</math> between. The standard unit for mass is kilograms (<math>kg</math>).
* <math>r</math> is the distance between the center of mass <math>cm</math> of both masses <math>m_1</math> and <math>m_2</math>. (If you don't know how to find center of mass yet, you can probably assume that its the center of whatever mass you're using). The standard unit for distance is meters (<math>m</math>).
* <math>G</math> is the gravitational constant or <math>6.67408 \times 10^{-11} \frac{m^3}{kg \times s^2}</math>

-----

We can make some simplifying approximations (similar to what we did with gravitational force) when we are trying to measure gravitational potential energy due to the Earth near the surface of the Earth.

===== Gravitational Potential Energy near the Earth's surface: =====

<math>U_{grav} = mgh</math>

* <math>U_{grav}</math> is the total gravitational potential energy. Like other energy it is measured in Joules (<math>J</math>).
* <math>m</math> is the mass of the object you are measuring <math>U_{grav}</math> of. The standard unit for mass is kilograms (<math>kg</math>).
* <math>h</math> is the height above the Earth's surface that the object is (make sure that it is near the Earth's surface. A good rule of thumb is that it really shouldn't be in space :-) ). The standard unit for height is meters (<math>m</math>).

-----

=== Computational Model ===

In order to understand gravitational potential energy in a computational manner, we must be able to represent it in some way which we can visually understand it. To do this, we will use an energy graph. An ''energy graph'' is a graph which plots potential energy, kinetic energy, and total energy. There are a few easy steps to follow when making an energy graph.

# Draw <math>U_{grav}</math> vs <math>|\vec r|</math> for the interaction (<math>|\vec r|</math> is the distance the object is from what it's attracted to)
# At some distance <math>|\vec r|</math> where <math>K</math> is known, plot the point <math>(|\vec r|, K)</math>
# Add the known value of <math>K</math> to the value of <math>U</math> at that same separation <math>|\vec r|</math>
# Plot the horizontal line <math>K+U</math> across the graph
# Calculate the rest of <math>K</math> knowing that <math>K+U</math> = the same constant
# Plot the values

[[File:Totalenergy.gif|frame|none|alt=|caption ]]

The above graph shows an example of what an energy versus distance graph may look like. We can see that as the distance increases, <math>U_{grav}</math> decreases and <math>K</math> increases.

==== VPython Examples ====

Suppose that we have a spaceship that is passing close to the Earth. We can computationally model the gravitational potential energy of this spaceship and visually represent it using energy graphs. The basic structure of our program will follow what is usually done for VPython programs:

<pre class="pseudocode">Pick a delta t
Initialize our variables
while our time < time bound do
Calculate new values for our variables
Update how they are presented
increment our time + delta t</pre>

''Create/Initialize the Energy Graphs''

<pre class="pseudocode">trail = curve(color=craft.color) #This creates a trail for the spacecraft
U_graph = gcurve(color=color.blue) #A plot of the Potential energy
K_graph = gcurve(color=color.yellow) #A plot of the Kinetic energy
Energy_graph = gcurve(color=color.green)#A plot of the Total energy</pre>
''Calculate and Update the variables in our Loop''

<pre class="pseudocode">K_craft = 0.5*mcraft*mag2(pcraft/mcraft) #Kinetic Energy using equation: .5mv^2
U_craft_Earth = -G*mcraft*mEarth/mag(craft.pos-Earth.pos) #Craft + Earth interaction energy
E = K_craft + U_craft_Earth #Approximate Total energy</pre>
''Update our graphs''

<pre class="pseudocode">U_graph.plot(pos=(t,U_craft_Earth)) #Potential energy as a function of time
K_graph.plot(pos=(t,K_craft)) #Kinetic energy as a function of time
Energy_graph.plot(pos=(t,E)) #Total energy as a function of time</pre>

Here is what a slightly more complex product would like like (this example has the moon's gravity incident on the spaceship as well):

Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

== Examples ==

=== Simple ===

'''Question: '''

A ball of mass <math>100</math> grams is <math>7</math> m above the ground, initially at rest (<math>K_i = 0</math>). When the ball is <math>4</math> m above the ground what is the kinetic energy (<math>K_f</math>)? Choose the ball + Earth system.

''Hint:'' Try drawing a picture of the situation. What has changed between the ball being at a higher height versus a lower height? Specifically how does this relate to the balance of potential energy and kinetic energy?

'''Solution:'''

Start from the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Since the ball starts from rest we know that <math>K_i = 0</math>. Since the ball is also <math>7</math> meters above the ground we know that there will be some potential energy due to gravity. We can use the equation above for an estimate of gravitational potential energy for an object close to the Earth's surface: <math>mgh</math>. Therefore, we now know:

<math>0 + mgh_i = K_f + U_f</math>

where <math>h_i</math> is the initial height of the ball. Since the ball doesn't fall all the way to the ground in its final state (It's still <math>4</math> m above the ground) we know that there will still be some potential energy due to gravity.

<math>mgh_i = K_f + mgh_f</math>

We can then isolate <math>K_f</math> with some subtraction

<math>K_f = mg(h_i - h_f)</math>

If we know substitute the numbers that we know into the equation we get that

<math>K_f = .1 \cdot 9.8 \cdot (7 - 4) = 2.94</math> Joules

=== Medium ===

'''Question:'''

In February 2013 a large meteor, whose mass has been estimated to be <math>1.2 \times 10^7 kg</math>, fell to Earth near Chelyabinsk, Russia. (This meteor exploded spectacularly at height of about <math>30</math> km, doing significant damage to objects on the ground.) Consider a meteor of the same mass falling toward the Earth. Choose the Earth plus the meteor as the system. As the meteor falls from a distance of <math>1 \times 10^8</math> m from the center of the Earth to <math>1 \times 10^7</math> m, what is the change in the kinetic energy of the meteor? The mass of the Earth is about <math>6 \times 10^{24}kg</math>.

''Hint:'' Consider how far the meteor is from Earth. Would it be considered an object near the surface of Earth or would we have to use the other model of gravitational potential energy? What is the quantity that we're solving for?

'''Solution:'''

Start with the energy principle:

<math>K_i + U_i = K_f + U_f</math>

Perhaps a better organization of this equation, keeping in mind that we are solving for the ''change'' in kinetic energy would be as follows:

<math>K_i - K_f = U_f - U_i</math>

And if we want to put it in terms of changes

<math>- (K_f - K_i) = U_f - U_i</math>

<math>- \Delta K = \Delta U</math>

We can now utilize the fact that we know both the initial and final height of the meteor in relation to the Earth. Since the meteor is not quite near the Earth's surface, we will use the formula for gravitational potential energy that does not make that assumption.

We know that:

<math>\Delta U = U_f - U_i</math>

Which if we substitute in the gravitational potential energy equations would be:

<math>\Delta U = -G\frac{m_1 m_2}{r_f} - -G\frac{m_1 m_2}{r_i} = -G\frac{m_1 m_2}{r_f} + G\frac{m_1 m_2}{r_i}</math>

We can then substitute in the values that we are given in the problem to solve for potential energy.

<math> - 6.67408 \times 10^{-11} \frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^7} + 6.67408 \times 10^{-11}\frac{(6\times 10^{24})(1.2 \times 10^7)}{1\times 10^8} = \Delta U</math>

<math>\Delta U = -4.32216 \times 10^{14}J</math>

We can then use this value to solve for the change in kinetic energy using our earlier derived equation which says that:

<math>-\Delta K = \Delta U</math>

<math> - \Delta K = -4.32216 \times 10^{14}J</math>

<math>\Delta K = 4.32216 \times 10^{14}J</math>

=== Hard ===

'''Question:'''

A robot spacecraft lands on an asteroid, picks up a sample, and blasts off to return to the Earth; it's total mass is <math>1500</math> kg. When it is <math>200</math> km from the center of mass of the asteroid, its speed is <math>5.0 \frac{m}{s}</math>, and the rockets are turned off. At the moment when it has coasted to a distance <math>500</math> km from the center of the asteroid, its speed has a decreased to <math>4.1</math> m/s. Calculate the mass of the asteroid.

''Hint:'' Do you see any chance of a variable cancelling out once you start working with your algebra?

'''Solution:'''

Start with the energy principle as always:

<math>K_i + U_i = K_f + U_f</math>

We know that the beginning and end of its journey both has kinetic energy as well as potential energy due to the gravitational potential energy between the spaceship and the asteroid. Knowing this, and taking it into account, we can rewrite our energy principle with slightly more specific terms.

<math>\frac{1}{2}m_{ship}v_i^2 - G\frac{m_{ship} m_{asteroid}}{r_i} = \frac{1}{2}m_{ship}v_f^2 - G\frac{m_{ship} m_{asteroid}}{r_f}</math>

If we then factor out the <math>m_{ship}</math> we might see a relation that is helpful.

<math>m_{ship}(\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i}) = m_{ship}(\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f})</math>

We can divide both sides of the equation by <math>m_{ship}</math>.

<math>\frac{1}{2}v_i^2 - G\frac{m_{asteroid}}{r_i} =\frac{1}{2}v_f^2 - G\frac{m_{asteroid}}{r_f}</math>

We can now isolate the <math>m_{asteroid}</math> with a little bit more algebra.

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = G\frac{m_{asteroid}}{r_i} - G\frac{m_{asteroid}}{r_f}</math>

<math>\frac{1}{2}v_i^2 - \frac{1}{2}v_f^2 = m_{asteroid}(G\frac{1}{r_i} - G\frac{1}{r_f})</math>

<math> m_{asteroid} = \frac{\frac{1}{2}(v_i^2 - v_f^2)}{G(\frac{1}{r_i} - \frac{1}{r_f})}</math>

We can then substitute in the values given to us in the problem to determine what the mass of the asteroid was.

<math> m_{asteroid} = \frac{\frac{1}{2}((5)^2 -(4.1)^2)}{G(\frac{1}{200} - \frac{1}{500})}</math>

Which gives us our final answer of:

<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>

== Connectedness ==

Gravitational potential energy can be related to industrial engineering for someone who chooses to work in a manufacturing or aerospace. It is also very important to include when determining the escape velocity and other forces necessary for rockets, probes, and satellites to travel within space. It, also, has some importance in animation and game development as it is an essential part of physics engines and, without a doubt, one of the most obvious interactions visible in society.

== History ==

Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.

== See also ==

Might be interesting to explore spring potential!

http://theory.uwinnipeg.ca/physics/work/node5.html

=== Further reading ===

If you'd like to read a book regarding gravitational potential energy, be sure to check this one out.

[[https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics gravitational potential&f=false|https://books.google.com/books?id=8p0gaOL802AC&pg=PA123&dq=physics+gravitational+potential&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMIkKHT0NadyQIVBtgeCh2y8Qrm#v=onepage&q=physics%20gravitational%20potential&f=false]]

=== External links ===

'''Internet resources on this topic:'''

Simple Explanation:

http://scienceworld.wolfram.com/physics/GravitationalForce.html

Videos:

https://www.youtube.com/watch?v=a91DAsbYCo4

https://www.youtube.com/watch?v=8a4D2xqHBF4

== References ==

http://www.physicsclassroom.com/class/circles/Lesson-3/Newton-s-Law-of-Universal-Gravitation

http://matterandinteractions.org/

http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

http://weelookang.blogspot.com/2010/10/ejs-open-source-gravity-field-model.html

[[Category:Energy]]