What is drift velocity? Establish a relation between current and drift velocity?

Explanation

Relation between electric current and drift velocity: Consider a uniform metallic wire XY of length l and cross-sectional area A. A potential difference V is applied across the ends X and Y of the wire. This causes an electric field at each point of the wire of strength `E=V/I`..........(i) .
Due to this electric field, the electrons gain a drift velocity opposite to direction of electric field. If q be the charge passing through the cross-section of wire in t seconds, then
Current in wire `I=q/t`.........(ii)
The distance traversed by each electron in time t = average velocity × time `vd_t`
If we consider two planes P and Q at a distance `v_t` in a conductor, then the total charge flowing in time t will be equal to the total charge on the electrons present within the cylinder PQ. The volume of this cylinder = cross sectional area × height `Av_d t`
If n is the number of free electrons in the wire per unit volume, then the number of free electrons in the cylinder =`n(v_dt)`
If charge on each electron is `-e(e=1.6 times 10^-19C)` then the total charge flowing through a cross-section of the wire
`q=(nAv_Dt)(-e)=-neAv_dt….(iii)` CUrrent flowing in air = `I=q/t=(-nev_dt)/t`
Current =-`I=-nev_d....(iv)`
This is the relation between electric current and drift velocity. Negative sign shows that the direction of current is opposite to the drift velocity.
Numerically `I=nev_d` ...(v) Current density. `therefore J=I/A=nev_d implies j prop v_d`
That is, current density of a metallic conductor is directly proportional to the drift velocity.