Question

A potential difference V is applied across a conductor of length ‘l’. How is the drift velocity affected when V is doubled and l is halved?

Answer 1

The relation between drift velocity and Potential difference and length of the conductor can be obtained using the relation between Electric field and drift velocity. The variation of electric field with the change in length and potential difference of a conductor is vital in obtaining this relation.

Formula Used: Drift velocity can be expressed using the mathematical expression:
$\left| {{v_d}} \right| = \dfrac{{e\tau }}{m}E$
Here ${v_d}$ is the drift velocity and E is the electric field.

Complete step by step answer:
From the formula for drift velocity we have:
$\left| {{v_d}} \right| = \dfrac{{e\tau }}{m}E$
But we know that $E = \dfrac{V}{l}$where V is the potential difference and l is the length of the conductor.
Now substituting this expression for the value of E in the expression for drift velocity, we have:
$\left| {{v_d}} \right| = \dfrac{{e\tau }}{m}\left( {\dfrac{V}{l}} \right)$
Through this expression it is clear that
-Drift velocity is directly proportional to potential difference applied across the conductor.
-Drift velocity is inversely proportional to length of the conducting wire.
Thus when the potential difference is applied across the conductor is doubled then V becomes 2V. And when the length of the conductor is halved $l$ becomes $l/2$.
Putting these values in the equation for drift velocity mentioned above we have:
$\left| {{v_d}'} \right| = \dfrac{{e\tau }}{m}\left( {\dfrac{{2V}}{{\dfrac{l}{2}}}} \right) \\\ \Rightarrow \left| {{v_d}'} \right| = \dfrac{{e\tau }}{m}\left( {\dfrac{{4V}}{l}} \right) \\\$
When we simplify the equation further we get:
$\left| {{v_d}'} \right| = 4\dfrac{{e\tau }}{m}\left( {\dfrac{V}{l}} \right)$ Now we know that $\dfrac{{e\tau }}{m}\left( {\dfrac{V}{l}} \right) = \left| {{v_d}} \right|$
Substituting this in the above equation we get:
$\left| {{v_d}'} \right| = 4\left| {{v_d}} \right|$
Thus we find that when the value for potential difference of the conductor is doubled and the length of the conductor is halved then the drift velocity increases by four times.

Note: It is important to not mistake drift velocity and mobility. Drift velocity and mobility are two different concepts. Velocity is the rate at which the displacement chances. However, the mobility of an electron is the amount of movability of the electron.