Derive an expression for the resistivity of a good conductor, in terms of the relaxation time of electrons.

Solution

Using ohm's law we have,
V = IR $\Rightarrow$ $\frac{v}{I} = R$

Drift velocity is given by, v_d = $\frac{eE}{m} τ$

But, Electric field, E = V/l

$∴ v_{d} = \frac{e V}{m l} τ$

Also, we have I=Anev_dTherefore, I = $Ane (\frac{eV}{ml} τ) = (\frac{{Ane}^{2} τ}{ml}) V$

i.e., $\frac{V}{I} = \frac{m l}{A n e^{2} τ} = R$

On comparing this equation with $R = ρ \frac{l}{A}$

We can deduce an expression for resistivity $ρ$ which is given by,
$ρ = \frac{m}{{ne}^{2} τ}$

where, $τ$ is the relaxation time.

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Current Electricity

Derive an expression for the resistivity of a good conductor, in terms of the relaxation time of electrons.

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