30: Resistance from Microscopic Ohm's Law

INTRO:
Your task is to calculate the resistance of a simple cylindrical resistor with wires connected to the ends, such as the carbon composition resistors that are used on electronic circuit boards. Imagine that the resistor is made by squirting material whose conductivity is σ into a cylindrical mold with length L and cross-sectional area A as shown in (Figure 1) . Assume that this material satisfies Ohm's law. (It should if the resistor is operated within its power dissipation limits.)

Figure 1

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PART A:
What is the resistance R of this resistor?
Express the resistance in terms of variables L, A and σ given in the introduction. Do not use V, I, E or J in your answer.

SOLUTION:
There are two equations that are necessary to complete this problem. 

Eq. (30.18) states that resistivity ρ = 1/σ

and

Eq. (30.22) states that resistance R = ρL/A

plugging (30.18) into (30.22) gives... 

R = L/σA

NOTE

Real resistors vary tremendously in overall size. The larger the size, the more power the resistor can dissipate without heating to the point that it is dangerous to nearby components or that the material of which it is constructed begins to change its conductivity (i.e., so that the resistance would no longer be constant). The amount of resistance is determined by the conductivity of the material of the resistor, which can vary over more than 20 orders of magnitude. Commercially available resistors vary from 0.1 ohm or less to more than 10⁷ ohm.