Derive Expression for Drift Velocity | Define Relaxation time or mean free time.

In a metal Conductor, there are extremely large number of free electrons. These free electrons move randomly with a thermal speed of the order of $10^{5}$ to $10^{6}$ m/s at room temperature.Im any portion of the conductor, the flow of electrons is so oriented that the average thermal velocity of total number of free electrons in a conductor is zero. 

That is 

$U_{av}= \frac{u_1+u_2+u_3+u_4+u_5....u_6}{n}$

$U_{av} = 0$

Where $u_1 , u_2 , u_3 , u_4 ...... u_n$ are thermal velocities of free electrons and n is the total number of free electrons.

When an electric field is applied across the conductor , the free electrons accelerate in a direction opposite to the direction of the applied field. Due to this acceleration, the electrons gain extra velocity but for a short time because the accelerated electrons collide with the ions in the conductor and during this collision , the extra velocity gained is destroyed . Again , the electron is Accelerated and comes to rest after collisions with ions. 

Therefore , the motion of electrons acquire a small velocity called drift velocity (vd) in the direction opposite to that of the applied electric field . The flow of electrons with drift velocity from one end to another end of the conductor constitutes an electric current.

Expression for Drift Velocity:

Drift velocity is defined as the average velocity with which free electrons in a conductor get drifted in a direction opposite to the direction of the applied electric field.

Consider a conductor under the influence of electric field E . The force experienced by a free electrons in the conductor placed in the electric field is given by : 

$F = -eE$

Negative sign shows that the direction of F and E are opposite to each other.

The acceleration produced in the electron is given by .

$a = \frac{F}{m}$

$a = -\frac{eE}{m}$

This acceleration lasts for a short time and is interrupted ( i.e made zero ) , when the accelerated electrons collide with the vibrating ions of the conductor . 

This small interval of time between two successive collisions between electrons and ion in the conductor is called relaxation time or Mean free time (t)

Therefore , the drift velocity is given by 

$v_d = u + at$

$v_d = 0 + (\frac{-eE}{m})t$

$v_d = -\frac{eE}{m}.t$

Note : The Drift Velocity is of the order of $10^{-4}$ m/s which is negligible as compared to the average electrons thermal velocity of $10^6$ m/s at room temperature.