(This is being posted as per request from students)
Consider a conductor of length l and area of cross section A. Let the number of free electrons per unit volume be n. When a pd V is applied, let the drift velocity of electrons be v.
The total no. of free electrons in the piece of conductor = nAl (no. of free electrons per unit volume x volume)
Total charge due to free electrons; q = nAle
The time taken by this charge to completely traverse the conductor, t =l/v
Therefore, the current in the conductor, I=q/t = nAle/(l/v) = nAev