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When $V$= potential difference across the conductor and $L$=length of the conductor

The electric field exerts an electrostatics force '$−Ee$' on each free electron in the conductor

The acceleration of each electron is given by

$aˉ=−meE $

Where, $e$=electric charge on the electron and

$m$=mass of electron

Acceleration and electric field are in opposite directions, so the electrons attain a velocity in addition to thermal velocity in the direction opposite to that of electric field.

$v_{d}=mcE τ$...............(i)

$E=L−V $...........(ii)

Where $τ$=relaxation time between two successive collision

Let $n$=number density of electrons in the conductor

N0. of free electrons in the conductor =$nAL$

Total charge on the conductor , $q=nALe$

Time taken by this charge to cover the length $L$ of the conductor,

$t=v_{d}L $

Current $I=tq $

$=LnALe ×v_{d}$

$=nAev_{d}$

Using equation (i) and (ii), we get that

$I=nAe×(−mLe(−V) τ)$

$=(mLne_{2}A τ)V$

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