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
These electrons move on for time τ1,τ2,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
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.