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derive relation between current density and potential difference across a current carrying conductor of length l,area of cross section A and number density of free electron n

Let us consider a conductor of length l,crossectional area A and having n number of electrons per unit volume drifting with drift velocity vd.While these electrons moving ,they collide with another electrons and in between the two successive collision the time taken is τ called relaxation time.WHen current I flows through the conductor subjected to electric field E, then,I=neAvd----1            where e is electronic charge and m the mass of electron.When electric force acts on this electron, electron gets accelerated, so,m a=eE.       or    m vdτ=eE.      or      vd=eEτm---2putting the value of vd in equation 1,I=neAeEτm.            or       I=ne2AEτm.        OR   J=IA=ne2Eτm----3But  E=POTENTIAL GRADIENT=Vl, putting this value of E in equation 3,J=ne2Vτml.This is the required relation between J and V.

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we know,

I=neAvd ......vd=drift velocity

divide both sides by A,We get

I/A=neAvd/A

J=nevd ......J(current density)=I/A

J=ne(eEt)/m ......vd=eEt/m

where, E=electric field

t= relaxation time

J=ne2(square)Et/m

but, m/ne2(square)t=sigma sigma=conductance

J=sigmaE

hence derived........may god bless you!!!!!

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