Explain the term -drift velocity- of electrons in a conductor- Hence obtain the expression the current through a conductor in terms of -drift velocity-

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The average velocity of all the free electrons in the conductor is called the drift velocity of free electrons of the conductor. When a conductor is connected to a source of emf an electric field is established in the conductor, such that $E = \frac{V}{L}$
When $V$ = potential difference across the conductor and $L$ =length of the conductor
The electric field exerts an electrostatics force ' $- E e$ ' on each free electron in the conductor
The acceleration of each electron is given by
$¯ a = - \frac{e \to E}{m}$
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.
${\to v}_{d} = \frac{c \to E}{m} τ$ ...............(i)
$E = \frac{- V}{L}$ ...........(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 = $n A L$
Total charge on the conductor , $q = n A L e$
Time taken by this charge to cover the length $L$ of the conductor,
$t = \frac{L}{v_{d}}$
Current $I = \frac{q}{t}$
$= \frac{n A L e}{L} \times v_{d}$
$= n A e v_{d}$
Using equation (i) and (ii), we get that
$I = n A e \times (- \frac{e (- V)}{m L} τ)$
$= (\frac{n e^{2} A}{m L} τ) V$

Explain the term 'drift velocity' of electrons in a conductor. Hence obtain the expression for the current through a conductor in terms of 'drift velocity'.