Define relaxation time of the free electrons drifting in a conductor: How is it related to the drift velocity of free electrons? Use this relation to deduce the expression for the electrical resistivity of the material.

Relaxation time is the time gap between two successive electron collisions in a conductor.

The relationship between the relaxation time (T) and drift velocity (Vd) is given below.

$v_{d} = \left ( e\frac{E}{m} \right )T$

Where

vd = drift velocity

e = charge of electron

E = field

m = mass of electron

T = Relaxation time

So the expression for relaxation time (T) is

$T = \left ( v_{d}\frac{m}{e} \right )E$

Let L = Length of the conductor

A = Area of the conductor

n = current density

then current flowing through the conductor is

$I = -neAv_{d}$ $I = neA\left ( e\frac{E}{m} \right )T$ $I = \frac{ne^{2}EA}{m}T$

Feild E can be expressed as

E = V/L

Then current flowing through the conductor becomes

$I = \frac{ne^{2}VA}{mL}T$ $\frac{V}{I} = \frac{mL}{ne^{2}TA}$

From ohm’s law

V = IR

R = V/I

$R = \left ( \frac{m}{ne^{2}T} \right )\frac{L}{A}$ $R = \rho \frac{L}{A}$