Question

In the following diagram, the lengths of wires $AB$ and $BC$ are equal, but the radius of wire $AB$ is double that of $BC$. The ratio of potential gradient on wires $AB$ and on $BC$ will be (wires are made of the same material).

• A. $4:1$
• B. $1:4$
• C. $2:1$
• D. $1:1$

Answer 1

The correct option is B $1:4$

Let the length of $AB$ and $BC$ is $P$

And

Radius of $AB$ is $2r$

Radius of $BC$ is $r$

$R_{AB}=\frac{\rho P}{\pi (2r{)}^2}=\frac{\rho P}{4\pi r^2}$

And

$R_{BC}=\frac{\rho P}{\pi r^2}$

Therefore

$R_{AB}=\frac{R_{BC}}{4}$

Potential across $AB$

$V_{AB}=E\left[\frac{R_{AB}}{R_{AB}+R_{BC}}\right]$

$=E\left[\frac{R_{AB}}{5R_{AB}}\right]=\frac{E}{5}$

Potential across $BC$

$V_{BC}=E-\frac{E}{5}=\frac{4E}{5}$

Potential gradient of $AB=\frac{E}{5P}$

Potential gradient across $BC=\frac{4E}{5P}$

Now the ratio of potential gradient $=\frac{E/5P}{4E/5P}=1:4$

Hence, Option $B$ is the correct answer.