Define current density and relaxation time. Derive an expression fo...

Question

Define current density and relaxation time. Derive an expression for resistivity of a conductor in terms of number density of charge carriers $n$ and relaxation time $	au$.

Accepted Answer

Current density: $J = I/A$ (current per unit cross-sectional area). Unit: A/m$^2$
Relaxation time $	au$: average time between successive collisions of a free electron.

Derivation:
Acceleration of electron: $a = eE/m$
Drift velocity: $v_d = a	au = eE	au/m$
Current density:
$$J = nev_d = \frac{ne^2	au}{m} \cdot E$$
Comparing with $J = \sigma E$:
$$\sigma = \frac{ne^2	au}{m}$$
Resistivity:
$$ho = \frac{1}{\sigma} = \frac{m}{ne^2	au}$$

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