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Concept of Drift velocity and Relaxation time

Consider a metallic conductor contains a large number of free electrons which are moving randomly in all directions. Let such electrons are marked as 1,2,...n which are moving with random velocities u1,u2,u3....un then taking their average;

uav =(u1 + u2+ ⋯ + un)/n= 0.  Thus, there is no net flow of charge in any direction.

Now when the electric field is applied these electrons are accelerated with the acceleration given by

a =eE/m

These electrons move on for time τ12,τ3,...,τn .  So the average time spent by each electron before they collide with the atom is given by

τ =(τ1 + τ2 + ⋯ + τn )/n

This time is called the relaxation time. The relaxation time is the average time spends by the electrons between two successive collisions. In a good conductor like metals, its value is of the order of 10-14 seconds. The final velocities acquired by these electrons inside the metal in the presence of an electric field is given by,

v1 = u1 + aτ1

v2 = u2 + aτ2

v3 = u3 + aτ3

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vn = un + aτn


Thus the average velocity is given by

vav = 0 + a τ

vav = (eE)/m x τ

This is the drift velocity of electrons. It may be defined as the average velocity gained by the free electrons of a conductor in the opposite direction of the externally applied electric field.