Establish a relation between electric current and drift velocity.
OR
Prove that the current density of a metallic conductor is directly proportional to the drift speed of electrons.

Relation between electric current and drift velocity.
Consider a uniform metallic wire XY of length 1 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.
$E=\frac{V}{l}$........(i)
Due to this electric field, the electrons gain a drift velocity vd opposite to direction of electric field.IF q be the charge through the cross-section of wire in t seconds, then 
Current in wire $I=\frac{q}{t}$..........(i)
The distance traversed by each electron in time t=average velocity x time =vdt
If we consider two planes P and Q at a distance vd 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  x height
$Avdt$
If n is the number of free electrons in the wire per unit volume,then the number of free electrons in the cylinder$$=n(Avd t)
$$
If charge on each electron is $-c(c=1.6\times 10^{-19}C)$,then the total charge flowing through a cross section of the wire.
$q=(n{A}_{v}dt)(-e)=-ne{A}_{v}dt$.........(iii)
$\therefore$ Current flowing in the wire,
$I=\frac{q}{t}=\frac{-v}{t}$
i.e , current$I=-ne{A}_{v}d$.......(iv)
This is the relation between current and drift velocity.Negative sign shows that the direction of current is  opposite to the drift velocity
Numericaly $I=-neA\tau d$...............(v)
$\therefore$ Cureent dendity , $$J=\dfrac{t}P{l}{A-}
$$\therefgfore)$$ Curent density $$J=\dfrac{l}{A}.
$\Rightarrow J\frac{I}{A}=d$
$\Rightarrow J\alpha D$
This is current density of maetallic conductor is directly proportional to the drfyr velocity.