Microscopic form of Ohm's Law Based on the Classical Free Electron Theory

The classical free electron theory is used to verify the Ohm’s law. In order to verify the same, consider that the steady state current density can be written as

eqn (1)

$J=\frac{n e^2 \tau E}{m}$

Similarly the steady state electrical conductivity

eqn (2)

$\sigma=\frac{\mathrm{ne}^2 \tau}{\mathrm{m}}$

On comparing equations (1) and (2)

eqn (3)

$\mathrm{J}=\sigma \mathrm{E}$

We know that the current $\mathrm{J}=\frac{\mathrm{I}}{\mathrm{A}}$ density , the conductivity $\sigma=\frac{1}{\rho}$ and the electric field $\mathrm{E}=\frac{\mathrm{V}}{\ell}$

Therefore equation (3) becomes,

eqn (4)

$\begin{aligned} J & =\sigma \mathrm{E} \\ \frac{\mathrm{I}}{\mathrm{A}} & =\frac{1}{\rho} \frac{\mathrm{V}}{\ell} \\ \mathrm{V} & =\left(\frac{\mathrm{p} \ell}{\mathrm{A}}\right) \mathrm{I} \end{aligned}$