Generally, charged particles like electrons in a conductor move with velocities in random directions such that the average velocity of the electrons is zero. When a conductor is placed in a uniform electric field E, the electrons are accelerated in a particular direction due to the field. If the force acting on a charge q due to the electric field is given by
F = qE
a = F/m = -qE/m
Where a is the acceleration of the electron; -q is the charge of the electron; m is the mass of the electron
Let us consider the case of the nth electron at a given time t. We know that electrons inside a conductor undergo random collisions with each other. Let tn and vn be the time elapsed between successive collisions of the nth electron and velocity of the nth electron after the last collision respectively. The velocity of the nth electron at time t be Vn.
Vn = vn + (-qE/ m). tn
The average of all the Vn values gives us the average velocity of the electrons at time t. As we know that the electrons move in random direction after each collision, the average value of all the vn is zero. Let us denote the average time between successive collision of electrons by t. This is called the relaxation time t , which in other words is the average of tn of all the electrons.
Thus (Vn )average = (vn)average + (-qE/ m). (tn) average
(Vn )average = Vd = 0 + (-qE/ m). t
Vd = -qEt/ m
Drift velocity Vd is the average velocity of electrons in a conductor placed in an electric field. The drift velocity is independent of time and is responsible for transport of electrons across any area perpendicular to the electric field E.
Vd = -qEt/ m
Due to the phenomenon of drift, there will be a net transport of charges across an area perpendicular to the electric field. considering a circular plane of area a inside the conductor such that the normal to the area is parallel to E, in a certain time ∆t, all the electrons to the right ofthe area would have crossed the area.