Question

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 1

Relaxation time (τ), it is the short time for which a free electron accelerates before it undergoes a collision with the positive ion in the conductor. Or, we can say it is the average time elapsed between two successive collisions. It is of the order 10⁻¹⁴ s. It decreases with increase of temperature and is given as

`vecV_d = vecatau`

`or vecV_d = (-eE)/m  tau    [because veca =-(evecE)/m]`

Where `vecV_d` is the drift velocity E is the applied electric field. e and m are the charge and mass of electron respectively.

Again consider the conductor with length l and A as area of cross-section. Let n be the number of electrons per unit volume in the conductor.

 `V_d = -(eE)/m tau`(Magnitude of drift velocity)

The current flowing through the conductor due to drift

I = nAvd^e

Substituting value of ν_d

`I = nA ((eEtau)/m)e`

`I = (nAe^2Etau)/m`

If V is potential difference applied across the two ends then

`E = V/l`put in above equation

`So I = (nAe^2Vtau)/(ml)`

`V/I = (ml)/("ne"^2tauA)`

Now, According to ohm’s law `V/1 = R`(Resistance of conductor)

Thus,

`R = m/("ne"^2tau) l/A`

Compare this with formula of resistance `R =rho*l/A`

Where ρ is the resistivity of the material we get

`rho = m/("ne"^2tau)`

Thus electrical resistivity depends inversely on the relaxation time τ.