Question

Define the term 'drift velocity' of charge carriers in a conductor. Obtain the expression for the current density in terms of relaxation time ?

Answer 1

 Drift velocity: It is the velocity with which free electrons get drifted towards the positive terminal under the effect of the applied electric field.

Free electrons are in continuous random motion. They undergo change in direction at each collision and the thermal velocities are randomly distributed in all directions.

∴ Average thermal velocity,`u =(u_1+u_2+ ...+u_n)/n = 0 ....... (1)` 

The electric field E exerts an electrostatic force −Ee.

Acceleration of each electron,`veca = (-evecE)/m      ....... (2)` 

where,

m = mass of an electron

e = charge on an electron

`(vecv_1+vecv_2+ ...+vecv_n)/n `

`vecv_d = (vecu_1+vecatau_1)+(vecu_2+vecatau_2)+ ........+(vecu_n+vecatau_n)`

where,

`vecu_1 ,vecu_2 ->`thermal velocities of the electrons

`vecatau_1 ,vecatau_2 ->` velocity acquired by electrons

`tau_1 ,tau_2 ->`  time elapsed after the collision

`vecv = (vecu_1 +vecu_2 +....vecu_n)/n + veca(tau_1+tau_2+.....+tau_n)/n`

Since `vecv = (vecu_1 +vecu_2 +....vecu_n)/n` = 0, we get:

∴ v_d = a τ, where, `veca(tau_1+tau_2+tau_3+.....+tau_n)/n`

is the average time elapsed.

Substituting the value of a from equation (2), we have:

`vecv_d = (-evecE)/m tau  ....... (4)`