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Establish a relation between electric current and drift velocity.
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.
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)
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

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 ,then the total charge flowing through a cross section of the wire.
Current flowing in the wire,

i.e , current.......(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 ...............(v)
Cureent dendity , $$J=\dfrac{t}P{l}{A-}
$$\therefgfore)$$ Curent density $$J=\dfrac{l}{A}.

This is current density of maetallic conductor is directly proportional to the drfyr velocity.


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