Question

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'.

Answer 1

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 '$-Ee$' on each free electron in the conductor
The acceleration of each electron is given by
$\overline{a}=-\frac{e\overrightarrow{E}}{m}$
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.
${\overrightarrow{v}}_d=\frac{c\overrightarrow{E}}{m}\tau$...............(i)
$E=\frac{-V}{L}$...........(ii)
Where $\tau$=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,
$t=\frac{L}{v_d}$
Current $I=\frac{q}{t}$
$=\frac{nALe}{L}\times v_d$
$=nAe{v}_d$
Using equation (i) and (ii), we get that
$I=nAe\times (-\frac{e(-V)}{mL}\tau )$
$=\left(\frac{n{e}^{2}A}{mL}\tau \right)V$