Define the term electrical resistivity of a material. Write its SI unit. Derive an expression for resistivity of a metal in terms of number density and mass of free electrons present in it.

Solution

1. Definition

Electrical resistivity (ρ) is the property of a material that opposes the flow of electric current. It is defined as the resistance of a conductor of unit length and unit cross-sectional area.

2. SI Unit

SI Unit of Resistivity: Ω m (ohm meter)

3. Derivation

Let:

n = number density of electrons

e = charge of electron

m = mass of electron

τ = relaxation time

Drift velocity:

v_d = (eEτ) / m

Current density:

J = ne v_d

Substitute:

J = (n e² τ / m) E

From Ohm’s law:

J = σE

So,

σ = (n e² τ) / m

Hence,

ρ = 1/σ = m / (n e² τ)

Final Result (To Remember)

ρ = m / (n e² τ)

SI Unit: Ω m