Let l is the length of the conductor and A uniforms area of cross-section.

Therefore, the volume of the conductor = Al

If n is the number of free electrons per unit volume of the conductor, then the total number of free electrons in the conductor=A/n.

If e is the charge on each electron then total charge on all the free electrons in the conductor

Q=A/ne                                     (1)

Let a constant potential differences V is applied across the ends of the conductor with the help of a battery

The electric field set up across the conductor is given by

E=V/l

Due to this field, the free electrons present in the conductor will begin to move with a drift velocity v_d towards the left hand side of the conductor.

Thus the time taken by the free electrons to cross the conductor is

t=l/v_d (2)

As current I=q/t                                                     (3)

By substituting equation (1) and (2) in equation (3),

We get

I=Alne/l/v_d

Or                                      I=Anev_d (4)

Since A,n and e are constant

I∞v_d

Hence the current flowing through a conductor is directly proportional to the drift velocity.

Explanation and Example. In a conductor, there are large number of free electrons. When we close the circuit, the electric field is established in the circuit with the speed of electromagnetic wave which causes electron drift at every portion of the circuit. Due to which the current is set up in the entire circuit instantly. The current thus set up does not wait for the electrons to flow from one end of the conductor to another end. It is due to this reason, the electric bulb glows immediately when the switch is on.