11.3 Resistance, Resistivity, and Ohm’s Law

Ohm’s law relates current, resistance, and potential difference for ohmic materials:

An ohmic material has constant resistance over the range being considered, so a graph of ${\Delta V}$ versus I is linear. The material property behind resistance is resistivity, written ${\rho}$, and for a uniform wire of length L and cross-sectional area A,

This means longer wires resist current more, while thicker wires resist current less.

A useful derivation comes from combining electric field ideas with current density. If ${J = nqv_d}$ and ${I = JA}$, then for ohmic conduction the current density is proportional to electric field: ${J = \sigma E, where \sigma = 1/\rho}$. Since ${E = \Delta V/L}$, then

So

which gives the resistance formula. This derivation is helpful because it ties together microscopic motion, material properties, and the macroscopic circuit equation.