Question

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

Answer 1

Step 1. Given data

It is given that the potential difference is given as $\mathrm{V}$ and length is $‘\mathrm{l}’$.

We have to determine how is the drift velocity affected when $\mathrm{V}$ is doubled and $‘\mathrm{l}’$ is halved?

Step 2. Concept and formula to be used

1. The average velocity obtained by the electrons under the influence of an electric field is termed the drift velocity.
2. In the conductor, the free electrons are always moving randomly., when conductor is attached to a battery, an electric field is generated.
3. When these free electrons are exposed to an electric field, they slowly drift in the direction of the electric field applied, while maintaining the random motion, at the point, the net velocity of the electrons. is called its drift velocity.
4. The relation between drift velocity and potential difference and length of the conductor can be attained using the relation between electric field and drift velocity.

Step 3. Formula to be used

The drift velocity is,

$\left|{\mathrm{v}}_{\mathrm{d}}\right|=\frac{\mathrm{e\tau }}{\mathrm{m}}\mathrm{E}$ ----------- $\left(1\right)$

Where, ${\mathrm{v}}_{\mathrm{d}}$ is the drift velocity and $\mathrm{E}$ is the electric filed.

We know that,

$\mathrm{E}=\frac{\mathrm{V}}{\mathrm{l}}$ ------------ $\left(2\right)$

Here, $\mathrm{V}$ is potential difference and $\mathrm{l}$ is the length of the conductor.

Substitute equation $\left(2\right)$ in $\left(1\right)$.

So,

$\left|{\mathrm{v}}_{\mathrm{d}}\right|=\frac{\mathrm{e\tau }}{\mathrm{m}}\left(\frac{\mathrm{V}}{\mathrm{l}}\right)$ ---------- $\left(3\right)$

From the above expression, drift velocity is directly proportional to potential difference applied over the conductor, and it is inversely proportional to length of the conducting wire.

Step 4. Examine how the drift velocity affect when potential difference is double and length is halved.

Since, When a conductor is subjected to a potential difference that is twice its original value, $\mathrm{V}$ becomes $2\mathrm{V}$ and $\mathrm{l}$ becomes $\frac{\mathrm{l}}{2}$.

Substitute these values in equation equation $\left(3\right)$, we get,

$\left|{\mathrm{v}}_{\mathrm{d}}\text{'}\right|=\frac{\mathrm{e\tau }}{\mathrm{m}}\left(\frac{2\mathrm{V}}{{\displaystyle \frac{\mathrm{l}}{2}}}\right)$

$=\frac{\mathrm{e\tau }}{\mathrm{m}}\left(\frac{4\mathrm{V}}{\mathrm{l}}\right)$

$=4\frac{\mathrm{e\tau }}{\mathrm{m}}\left(\frac{\mathrm{V}}{\mathrm{l}}\right)$

Now, we know that, $\left|{\mathrm{v}}_{\mathrm{d}}\right|=\frac{\mathrm{e\tau }}{\mathrm{m}}\mathrm{E}$.

So,

$\left|{\mathrm{v}}_{\mathrm{d}}\text{'}\right|=4\left|{\mathrm{v}}_{\mathrm{d}}\right|$

Hence, 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 $4$ times.