Question

1. Define Electrical conductivity. Obtain the expression of electrical conductivity of a conductor in terms of number density and relaxation time of free electrons.
2. Explain qualitative change in resistivity of a conductor with temperature using expression obtained in (a).

Answer 1

a. Electrical conductivity (σ) is the ability of a material to conduct electric current. It is the reciprocal of resistivity and is given by,

σ = `1/rho`

Where ρ is the electrical resistivity of the material.

From Ohm’s law,

J = σE

Where all the alphabet are in their usual meanings.

The current density J is given by,

J = nev_d

Using the drift velocity (v_d) formula now,

v_d = `(e E tau)/m`

Where τ = relaxation time

m = mass of electron

Now,

J = `n e((e E tau)/m)`

= `(n e^2 tau)/m E`

Comparing with J = σE,

ρ = `(n e tau)/m`

b. We can write,

ρ = `1/sigma`

= `3/(n e^2 tau)`

Now the effects of temperature:

• In a conductor, as temperature increases, the number density (n) remains nearly constant.
• The relaxation time τ decreases as collisions between electrons and vibrating ions increase. Since τ decreases, ρ increases.
• Thus, the resistivity of a conductor increases with an increase in temperature.