📂Electrodynamics

Ohm's Law, Electrical Conductivity, and the Definition of Resistance

Definition ¹

Let a conductor with area $A$ and length $l$ have a current of $I$, so that the current density is $J = I / A$. It is useful to imagine an electrical circuit.

1. Assume the current density $J$ is proportional to the electric field $E$, so that for some constant $\sigma$ it can be written as follows. $$ J = \sigma E $$ A material that satisfies this relation is said to obey Ohm’s law, and the proportionality constant $\sigma$ is called the electrical conductivity.
2. $R = l / \left( \sigma A \right)$ is called the resistance.
3. The reciprocal of the conductivity $\sigma$, namely $\rho = \sigma^{-1}$, is called the resistivity.

Explanation

Resistance can also be expressed in terms of resistivity as follows. $$ R = \rho {\frac{ l }{ A }} $$

High electrical conductivity simply means that charge flows readily in response to an electric field; such materials are called conductors, while materials with low electrical conductivity are called insulators.