Question

Two conducting wires X and Y of the same diameter but different materials are joined in series across a battery. If the number density of electrons in X is twice that in Y, find the ratio of the drift velocity of electrons in the two wires.

Answer 1

Step 1: Given Data

For two conducting wires X and Y, have the same diameter(i.e. $A_X=A_Y=A$ where $A$ is the area of cross-section) and the number density of electrons in X is twice that of Y,

i.e. $n_X=2n_Y..........\left(1\right)$

Step 2: Formula used

In a series combination, the current flowing through the components is the same, So

$I_X=I_Y........\left(2\right)$

We know that the current flowing through a conductor due to the drift velocity of electrons can be given by,

$I=neA{V}_d......\left(3\right)$

Where, $n$ is the number density of electrons

$e$ is the charge of electrons

$A$ is the area of cross-section

$V_d$ is the drift velocity

Step 3: To find the ratio between the drift velocity of the electron in the two wires.

Let the drift velocity of the wire X and Y be $V_X$ and $V_Y$ respectively

Now from equations (2) and (3) we can write,

$n_{X}eA{V}_X=n_{Y}eA{V}_Y$

So, applying equation (1) we can write

$\begin{array}{rcl}\frac{V_X}{V_Y}& =& \frac{n_Y}{n_X}\\ & =& \frac{n_Y}{2n_Y}\\ & =& \frac{1}{2}\end{array}$

Hence, the ratio of the drift velocity of electrons in the two wires is $\frac{V_X}{V_Y}=\frac{1}{2}$.