# R Circuit

Claimed by Matthew Munns - Fall 2017

Resistors may exist in a circuit and resist electric charge. The resistance of a resistor depends on the properties of the material and the area of the resistor. These two attributes will be investigated later in the page.

Resistors may be ohmic or non-ohmic. An ohmic resistor has a conductivity that is nearly constant, while a non-ohmic resistor will vary depending on the conditions. This page will focus on ohmic resistors in a complex circuit.

The following is an image of an ohmic resistor like the resistor used in lab:

## History

Georg Simon Ohm, born in 1789, discovered Ohm's law. This law relates the current, voltage, and resistance across a resistor. His law was widely accepted by 1850. Ohm's law became the fundamental law in circuit analysis. In 1959, inventor Otis Boykin filed a patent for a resistor that allows manufacturers to designate a value of resistance for a piece of wire in equipment. In 1962, he patented an improved version, which could withstand more extreme conditions. This reduced the cost and increased the reliability of a variety of electronic products. ^{[1]}

## Mathematics

As discussed above, resistance of a material is dependent on two factors: the properties of the material, and the area of the resistor.

From the above factors, we get the following definition of resistance: [math]\displaystyle{ R = \frac{L}{σ A} }[/math]

In this definition [math]\displaystyle{ L }[/math] is the length of the resistor, [math]\displaystyle{ A }[/math] is the cross-sectional area of the resistor, and [math]\displaystyle{ σ }[/math] is the conductivity of the material. [math]\displaystyle{ σ }[/math] can be further broken down to show [math]\displaystyle{ σ = |q|nu }[/math].

## In A Circuit

There is a change in potential energy across a resistor, which is useful when evaluating the loop rule in a circuit with resistors. The change in potential across a resistor is [math]\displaystyle{ I = \frac{ΔV}{R} }[/math] Knowing this equation allows one to solve for [math]\displaystyle{ ΔV }[/math] if given [math]\displaystyle{ I }[/math] and either [math]\displaystyle{ R }[/math] or the components of [math]\displaystyle{ R }[/math] enumerated above.

In the case of a circuit resistors, the loop rule might look something like: [math]\displaystyle{ |emf| = I_{1}R_{1} + I_{2}R_{2} }[/math]

## Equations Derived

From the definition of resistance and the voltage potential formula discussed above, a number of useful equations can be derived and solved. The first equation is [math]\displaystyle{ v = uE }[/math] where [math]\displaystyle{ E }[/math] is electron field , [math]\displaystyle{ u }[/math] as electron mobility, and [math]\displaystyle{ v }[/math] is drift speed.

This equation offers insight into the microscopic view of the resistor.

The second equation that can be derived is [math]\displaystyle{ I = |q|nAv }[/math]. This equation can be used, in addition to the above equation [math]\displaystyle{ I = \frac{ΔV}{R} }[/math] to solve for macroscopic properties of the resistor.

## Example Problems

### Easy Example

**Question:**
Use the loop rule to calculate the voltage drop across each resistor. Then ensure that your answer is correct.

**Answer:**

### Difficult Example

https://www.youtube.com/watch?v=BW7U_5BtH-8

## Applications

**1.How is this topic connected to something that you are interested in?** I am very interested in computers and I know that circuitry is the foundation of computers and other electronic equipment.

**2.How is it connected to your major?** My major is Industrial Engineering, and although I won't study resistors directly in any of my classes, I understand that resistors are crucial in the creation of the tools all engineers and scientists use such as computers and calculators.

**3.Is there an interesting industrial application?** As stated above, resistors are widely used in electric circuits. They are used to reduce current flow, change signal levels, and terminate transmission.