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.
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Updated on : 2022-09-05
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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=lV........(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=tq..........(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=1.6×10−19C),then the total charge flowing through a cross section of the wire. q=(nAvdt)(−e)=−neAvdt.........(iii) ∴ Current flowing in the wire, I=tq=t−v i.e , currentI=−neAvd.......(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=−neAτd...............(v) ∴ Cureent dendity , $$J=\dfrac{t}P{l}{A-} $$\therefgfore)$$ Curent density $$J=\dfrac{l}{A}. ⇒JAI=d ⇒JαD This is current density of maetallic conductor is directly proportional to the drfyr velocity.