Question

Derive an expression for resistivity of a conductor in terms of the number density of charge carriers in the conductor and relaxation time.

Answer 1

The relationship between the relaxation time (τ) and drift velocity `(v_d)` is given by:

`v_d = -e((Eτ)/m)`

∴ `τ = ((v_d m))/e xx E`

Let L = Length of the conductor

A = Area of the conductor

n = free electron density

e = charge of the electron

E = Electric field

m = mass of the electron

τ = Relaxation time

The current flowing through the conductor is

I = `n eAv_d`

I = `n e A((eE)/m)τ`

Also, field E can be expressed as

E = `V/L`

The current flowing through the conductor is:

I = `(n e^2VAτ)/(mL)`

or `V/I = (mL)/(n e^2τA)`

or `R = (mL)/(n e^2τA)` ...`("from Ohm's law" V/I = R)`

or `R = m/(n e^2τ)(L/A)`

Electrical resistivity, `rho = m/(n e^2τ)` `...[∵ R = rho L/A]`