Question

Find the relation between drift velocity and relaxation time of charge carries in a conductor. A conductor of length 'L' is connected to a d.c. source of emf 'E'. If the length of the conductor is tripled by stretching it, keeping 'E' constant, explain how its drift velocity would be affected.

Answer 1

All free electrons suffer collisions with the heavy fixed ions inside the conductor. After collisions, these electrons again emerge with the same speed, but in a random direction. So, at a given time, net velocity of the electrons is zero. i.e.,

$\underset{u_{avg}}{\to }=\frac{\overrightarrow{u_1}+\overrightarrow{u_2}+.....+\overrightarrow{u_n}}{n}=0$

If the electric field is established inside the conductor, electrons get accelerated with an acceleration

$a=-\frac{eE}{m}$

Now, the average velocity of all electrons is given by

$\underset{v_d}{\to }=\underset{u_{avg}}{\to }-\frac{eE}{m}\tau$
$\underset{v_d}{\to }=0-\frac{eE}{m}\tau$

where $\tau$ is the average relaxation time.

But $E=\frac{V}{l}$

$\underset{v_d}{\to }=-\frac{eV\tau }{ml}$

$v_d\propto \frac{1}{l}$

As the drift velocity is inversely proportional to its length, so the drift velocity would be reduced by one third if the length of the conductor is tripled.