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

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#### 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.$$E=\dfrac{V}{l}$$........(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 $$I=\dfrac{q}{t}$$..........(i)The distance traversed by each electron in time t=average velocity x time =vdtIf 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$$A vdt$$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.$$q=(nA_v d t)(-e)=-neA_vdt$$.........(iii)$$\therefore$$ Current flowing in the wire,$$I=\dfrac{q}{t}=\dfrac{-v}{t}$$i.e , current$$I=-ne A_vd$$.......(iv)This is the relation between current and drift velocity.Negative sign shows that the direction of current is opposite to the drift velocityNumericaly $$I=-neA \tau d$$...............(v)$$\therefore$$ Cureent dendity , $$J=\dfrac{t}P{l}{A-}$$\therefgfore)$$Curent density$$J=\dfrac{l}{A}.$$\Rightarrow J \dfrac{I}{A}=d$$$$\Rightarrow J \alpha D$$This is current density of maetallic conductor is directly proportional to the drfyr velocity.

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