Define current density $J$ of a metallic conductor. Deduce the rela...

Question

Define current density $J$ of a metallic conductor. Deduce the relation connecting $J$ and conductivity $\sigma$ when electric field $E$ is applied.

Accepted Answer

Current density: $J = I/A$ (current per unit cross-sectional area). Unit: A m$^{-2}$

Derivation:

Force on electron: $F = eE$

Acceleration: $a = eE/m$

Drift velocity: $v_d = a\tau = eE\tau/m$

Current density:

$$J = nev_d = \frac{ne^2\tau}{m} \cdot E$$

Comparing with $J = \sigma E$:

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

$$\therefore J = \sigma E \quad \text{(Microscopic form of Ohm's law)}$$