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.

Answer:

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

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

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