Potential Difference at One Location - Revision history

Kless32: /* Connectedness */

2026-04-27T01:32:09Z

Connectedness

← Older revision		Revision as of 21:32, 26 April 2026
Line 140:		Line 140:

	The negative result indicates the contribution from the negative charge dominates at point <math>P</math>.		The negative result indicates the contribution from the negative charge dominates at point <math>P</math>.

			==Computational Model==
			<iframe src="https://trinket.io/embed/glowscript/fd0967ba5c6c" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

	==Connectedness==		==Connectedness==

Kless32: /* Connectedness */

2026-04-27T01:22:20Z

Connectedness

← Older revision		Revision as of 21:22, 26 April 2026
Line 148:		Line 148:
	#'''Is there an interesting industrial application?''' Electric potential has various applications in industry, particularly in the aspects of circuit design and analysis. A particular implementation in medicine would be a defibrillator.		#'''Is there an interesting industrial application?''' Electric potential has various applications in industry, particularly in the aspects of circuit design and analysis. A particular implementation in medicine would be a defibrillator.

	~~'''~~Real-World Applications~~'''~~		==Real-World Applications==

	Electric potential appears in many technologies and physical systems and serves as an important practical application of electrostatics.		Electric potential appears in many technologies and physical systems and serves as an important practical application of electrostatics.

Kless32: /* Connectedness */

2026-04-27T01:21:49Z

Connectedness

← Older revision		Revision as of 21:21, 26 April 2026
Line 204:		Line 204:

	Electric potential is therefore not only a theoretical concept, but also a practical engineering tool used across science and engineering.		Electric potential is therefore not only a theoretical concept, but also a practical engineering tool used across science and engineering.

	~~'''Common Exam Mistakes'''~~

	~~Students often confuse electric potential with electric field.~~

	~~Electric potential is~~

	~~<math> V=\frac{kq}{r} </math>~~

	~~while electric field is~~

	~~<math> E=\frac{kq}{r^2} </math>~~

	~~These are not interchangeable.~~

	~~Common mistakes include:~~

	~~Forgetting electric potential is a scalar quantity.~~
	~~Adding electric fields algebraically instead of vectorially.~~
	~~Ignoring the sign of charge when determining whether potential is positive or negative.~~

	~~if <math>q>0</math>, then <math>V>0</math>~~

	~~if <math>q<0</math>, then <math>V<0</math>~~

	~~Mixing up electric potential~~

	~~<math>V</math>~~

	~~with electric potential energy~~

	~~<math> U=qV </math>~~
	~~Forgetting electric potential is path independent and depends only on endpoints.~~

	==Common Mistakes==		==Common Mistakes==

Kless32: /* Real-World Applications */

2026-04-27T01:21:14Z

Real-World Applications

@@ Line 148: / Line 148: @@
 #'''Is there an interesting industrial application?''' Electric potential has various applications in industry, particularly in the aspects of circuit design and analysis. A particular implementation in medicine would be a defibrillator.
-==Real-World Applications==
+'''Real-World Applications'''
-Electric potential appears in many technologies and physical systems.
+Electric potential appears in many technologies and physical systems and serves as an important practical application of electrostatics.
-Capacitors store electrical energy through potential differences and are used in electronics, camera flashes, and energy storage devices.
+'''Capacitors'''
-Particle accelerators use voltage differences to accelerate charged particles for research and medical applications.
+Capacitors store electrical energy through a potential difference between two conductors.
-Electrostatic precipitators use electric potential to remove pollutants from industrial exhaust.
+Energy stored in a capacitor is
 Power transmission systems rely on high voltages to reduce resistive losses during long-distance energy transfer.
-Electric potential is therefore not only a theoretical concept, but also a practical engineering tool.
+Power is related by
-Common Exam Mistakes
+'''Common Exam Mistakes'''
 Students often confuse electric potential with electric field.
 ==Common Mistakes==

Kless32: /* Difficult Examples */

2026-04-27T01:17:43Z

Difficult Examples

← Older revision		Revision as of 21:17, 26 April 2026
Line 125:		Line 125:
	<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>		<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>

	Difficult Example 2 — Superposition of Potential		'''Difficult Example 2 — Superposition of Potential'''

	Parameters: Two point charges contribute to the electric potential at point <math>P</math>.		Parameters: Two point charges contribute to the electric potential at point <math>P</math>.

Kless32: /* Difficult Example 2 — Superposition of Potential */

2026-04-27T01:17:05Z

Difficult Example 2 — Superposition of Potential

← Older revision		Revision as of 21:17, 26 April 2026
Line 125:		Line 125:
	<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>		<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>

	==Difficult Example 2 — Superposition of Potential==		Difficult Example 2 — Superposition of Potential

	Parameters: Two point charges contribute to the electric potential at point <math>P</math>.		Parameters: Two point charges contribute to the electric potential at point <math>P</math>.

Kless32: /* Difficult Examples */

2026-04-27T01:16:50Z

Difficult Examples

← Older revision		Revision as of 21:16, 26 April 2026
Line 124:		Line 124:

	<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>		<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>

			==Difficult Example 2 — Superposition of Potential==

			Parameters: Two point charges contribute to the electric potential at point <math>P</math>.

			<math> q_1=+5\mu C,\quad r_1=0.20m </math> <math> q_2=-3\mu C,\quad r_2=0.10m </math>

			Using superposition,

			<math> V=k\left(\frac{q_1}{r_1}+\frac{q_2}{r_2}\right) </math>

			Substituting values,

			<math> V=(9\times10^9)\left(\frac{5\times10^{-6}}{0.20}-\frac{3\times10^{-6}}{0.10}\right) </math> <math> V=(9\times10^9)(25\times10^{-6}-30\times10^{-6}) </math> <math> V=(9\times10^9)(-5\times10^{-6}) </math> <math> V=-4.5\times10^4V </math>

			The negative result indicates the contribution from the negative charge dominates at point <math>P</math>.

	==Connectedness==		==Connectedness==

Kless32: /* Common mistakes include: */

2026-04-27T01:15:55Z

Common mistakes include:

← Older revision		Revision as of 21:15, 26 April 2026
Line 150:		Line 150:
	Students often confuse electric potential with electric field.		Students often confuse electric potential with electric field.

	==Common ~~mistakes include:~~==		==Common Mistakes==

	~~Using V = kq/r instead of distinguishing it from E = kq/r².~~		Students often make several common mistakes when working with electric potential.
	~~Forgetting electric potential is a scalar quantity and adds algebraically.~~
	~~Ignoring the sign of charge~~ when ~~determining whether potential is positive or negative.~~
	~~Mixing up electric potential V and~~ electric potential ~~energy U.~~
	~~Forgetting potential is path independent and depends only on endpoints~~.

	~~Exam Tip:~~		Confusing electric potential and electric field.

	~~For multiple charges,~~		Using

	~~V_net~~ = ~~Σ(kq_i~~/~~r_i)~~		<math> V=\frac{kq}{r} </math>

	~~Use superposition and add potentials algebraically.~~		instead of distinguishing it from

	~~Difficult Example~~ 2 ~~— Superposition~~ of ~~Potential~~		<math> E=\frac{kq}{r^2} </math>
			Forgetting electric potential is a scalar quantity and adds algebraically rather than vectorially.
			Ignoring the sign of the source charge when determining whether potential is positive or negative.

	~~Suppose two charges contribute to the electric potential at point P.~~		if <math>q>0</math>, then <math>V>0</math>

	~~q₁ = +5 μC at r₁ =~~ 0~~.20 m~~		if <math>q<0</math>, then <math>V<0</math>

	~~q₂ = −3 μC at r₂ = 0.10 m~~		Mixing up electric potential
			<math> V </math>

	~~Using superposition:~~		and electric potential energy

	V = ~~k(q₁~~/~~r₁ + q₂/r₂)~~		<math> U=qV </math>
			Forgetting electric potential is path independent and depends only on endpoints.

	~~Substituting:~~		'''Exam Tip'''

	~~V = (9 × 10⁹)[(5 × 10⁻⁶ / 0.20) − (3 × 10⁻⁶ / 0.10)]~~		For multiple charges, use superposition:

	V = -4		<math> V_{net}=\sum_i \frac{kq_i}{r_i} </math>

			Since potential is scalar, add potentials algebraically.

	==History==		==History==

Kless32: /* The Main Idea */

2026-04-27T01:14:30Z

The Main Idea

← Older revision		Revision as of 21:14, 26 April 2026
Line 3:		Line 3:
	==The Main Idea==		==The Main Idea==

	Electric potential is ~~a fundamental concept in electrostatics that describes~~ the electric potential energy per unit charge at a point in space. It provides ~~a way~~ to ~~analyze~~ electric interactions using ~~energy methods rather than~~ forces, ~~which often makes many problems easier to solve. Instead of tracking vector forces~~ from multiple charges~~, electric potential allows those contributions to~~ be added ~~as scalars~~, ~~making superposition~~ problems ~~much more manageable~~.		Electric potential is the electric potential energy per unit charge at a point in space. It is a scalar quantity and provides an alternative to analyzing electric interactions using forces. Because electric potential is scalar, contributions from multiple charges may be added algebraically through superposition, often simplifying electrostatic problems.

	~~For a point~~ charge, the electric ~~potential at~~ a ~~distance r is given by~~		Electric potential difference is defined as the negative work done per unit charge by the electric field as a test charge moves between two points:

	V=		<math> \Delta V = V_B - V_A = -\int_A^B \vec E \cdot d\vec s </math>
	r
	kq


			If the reference point is chosen to be infinitely far away and

	where k is Coulomb’s constant, q is the source charge, and r is the distance from the charge. This shows that potential decreases with distance and depends on both the magnitude and sign of the source charge.		<math> V_{\infty}=0 </math>

	~~Electric potential is closely related to work and energy. The change in~~ electric potential ~~between two points (voltage) tells how much work is done per unit charge moving between those locations. This relationship is central~~ to ~~understanding charged particle motion, circuits, capacitors, and energy storage~~.		then electric potential at a point can be determined relative to infinity.

	~~Electric potential also connects directly to electric fields through~~ the ~~relationship~~		'''Potential near a Point Charge'''

			Parameters: A point charge with charge <math>q</math>. A point is located a distance <math>r</math> from the charge.

			Using Coulomb’s law,

	E		<math> E=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2} </math>
	=~~−∇V~~

	~~showing that electric fields point in~~ the ~~direction~~ of ~~decreasing~~ potential~~. This connection links energy-based reasoning and field-based reasoning into one framework.~~		and the definition of potential,

	The concept becomes especially powerful when dealing with systems of multiple charges, continuous charge distributions, and conductors, where direct force calculations may be difficult. Because of this, electric potential is one of the most important tools in electromagnetism and serves as a foundation for many later topics in physics and engineering.		<math> V_r=V_r-V_{\infty} =-\int_{\infty}^{r}\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}dr </math>

	~~===A Mathematical Model===~~		which gives

	~~'''The Potential at One Location'''~~		<math> V_r=\frac{1}{4\pi\epsilon_0}\frac{q}{r} </math>

	~~<math>V_{a} = V_{a} - V_{\infty }</math> where '''<math>V_{a}</math>''' is~~ the ~~potential location~~ of ~~interest and '''<math>V_{\infty }</math>''' is~~ the ~~potential at a location infinitely far away~~.		This shows electric potential decreases with increasing distance and depends on the sign of the source charge.

	~~This equation is only valid when~~ <math>~~V_{~~\~~infty }~~</math> ~~is equal to zero.~~		if <math>q>0,\quad V_r>0</math>

	~~'''Potential near a Point Charge'''~~		if <math>q<0,\quad V_r<0</math>

	~~Parameters: A point charge with charge <math>q</math>. We are a distance <math>x</math> from this point charge~~ and ~~we want to find the potential difference between our point charge and the potential at infinity.~~		'''Electric Potential and Electric Field'''

	<math>V_{x} = V_{x} - V_{\infty } = -\int_{\infty }^{x} \frac{1}{4\pi\varepsilon _{0} } \frac{q}{x^2} dx = \frac{1}{4\pi\varepsilon _{0} } (\frac{q}{x}-\frac{q}{\infty}) = \frac{1}{4\pi\varepsilon _{0}} \frac{q}{x} </math>		Electric potential is related to electric field through

	~~The sign of the potential depends on the sign of the charge on the particle:~~		<math> \vec E=-\nabla V </math>

	~~if <math> q < 0, V_{x} < 0 </math>~~		which shows electric fields point in the direction of decreasing potential.

	~~if <math> q > 0~~, ~~V_{x} > 0 </math>~~		For a uniform electric field,

	~~'''Potential Energy at a Single Location'''~~		<math> \Delta V=-Ed </math>

	~~Once the value of the potential at one location~~ is ~~known,~~ the ~~potential energy of a system can be calculated~~.		when displacement is parallel to the field.

	~~If a charged object was placed at the location where the potential is known, then the potential energy~~ of ~~the system would be given the the equation:~~		'''Superposition of Electric Potential'''

	~~<math> U_{A} = qV_{A} </math>~~		For several point charges, electric potential is found by summing contributions from each charge:

	~~Substituting in the derived equation from the previous section for~~ <math>V_{A}</math>~~, we obtain the following:~~		<math> V_{net}=\sum_i \frac{1}{4\pi\epsilon_0}\frac{q_i}{r_i} </math>

	~~<math> U = q_{1} (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{2}}{x}) = (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{1}q_{2}}{x}) </math> where <math> x </math>~~ is ~~the separation between the two particles~~, ~~and <math> q_{1} </math> and <math> q_{2} </math> are the respective charges of the particles~~.		Because potential is scalar, these contributions add algebraically rather than vectorially.

	~~===A Visual Model===~~		'''Potential Energy at a Single Location'''

	~~A visualization~~ of ~~Distance vs. Potential from~~ a ~~point~~ charge is ~~given by Figure 1:~~		The electric potential energy of a charge <math>q</math> placed in potential <math>V</math> is

	~~[[File:Potential_vs_Distance.png]]~~		<math> U=qV </math>

	~~'''Figure 1~~.~~''' Potential vs. Distance for a Point charge with Charge <math> e </math>~~		This relationship connects electric potential directly to energy methods and is fundamental in analyzing charged particle motion, circuits, and capacitors.

	~~The electric field~~ of ~~the point charge can be obtained by differentiation. The negative gradient of the potential with respect to distance would yield the electric field:~~		'''Importance of Electric Potential'''

	~~<math> E_{x} =~~ -~~\frac{\partial V}{\partial x} </math>~~		Electric potential is a foundational idea in electrostatics because it links work, energy, electric fields, voltage, and later topics such as capacitance and circuits. It provides a powerful method for solving problems where force-based analysis may be difficult.

	Here is a video explaining the above concepts with more examples [https://youtu.be/70SsJNE3VFE].		Here is a video explaining the above concepts with more examples [https://youtu.be/70SsJNE3VFE].

Kless32: /* The Main Idea */

2026-04-27T01:13:16Z

The Main Idea

← Older revision		Revision as of 21:13, 26 April 2026
Line 3:		Line 3:
	==The Main Idea==		==The Main Idea==

	~~Potential difference~~ is ~~usually defined as the change~~ in the potential ~~between two distinct points~~. ~~However~~, ~~it is also useful~~ to ~~determine~~ potential ~~at a single location of interest~~.		Electric potential is a fundamental concept in electrostatics that describes the electric potential energy per unit charge at a point in space. It provides a way to analyze electric interactions using energy methods rather than forces, which often makes many problems easier to solve. Instead of tracking vector forces from multiple charges, electric potential allows those contributions to be added as scalars, making superposition problems much more manageable.

	~~In this case~~, ~~we must define~~ the potential at a ~~location (ex. Location A) as the potential difference between a point infinitely far from all other charged particles and the location we defined.~~		For a point charge, the electric potential at a distance r is given by

	As potential ~~has a value at all~~ locations ~~at space~~, ~~it is a scalar~~ field, ~~not~~ a ~~vector field~~.		V=
			r
			kq



			where k is Coulomb’s constant, q is the source charge, and r is the distance from the charge. This shows that potential decreases with distance and depends on both the magnitude and sign of the source charge.

			Electric potential is closely related to work and energy. The change in electric potential between two points (voltage) tells how much work is done per unit charge moving between those locations. This relationship is central to understanding charged particle motion, circuits, capacitors, and energy storage.

			Electric potential also connects directly to electric fields through the relationship

			E
			=−∇V

			showing that electric fields point in the direction of decreasing potential. This connection links energy-based reasoning and field-based reasoning into one framework.

			The concept becomes especially powerful when dealing with systems of multiple charges, continuous charge distributions, and conductors, where direct force calculations may be difficult. Because of this, electric potential is one of the most important tools in electromagnetism and serves as a foundation for many later topics in physics and engineering.

	===A Mathematical Model===		===A Mathematical Model===