Question

Two identical rings $P$ and $Q$ of radius $0.1\text{m}$ are mounted coaxially at a distance $0.5\text{m}$ apart. The charges on the two rings are $2\mu C$ and $4\mu C$ respectively. The work done in transferring a charge of $5\mu C$ from the centre of $P$ to that of $Q$ is

• A. $0.144\text{J}$
• B. $1.28\text{J}$
• C. $2.24\text{J}$
• D. $0.72\text{J}$

Answer 1

The correct option is D $0.72\text{J}$



The net electric potential at centre of any of the ring will be sum of potential at centre due to itself and potential at axial point due to another ring.
$V=V_{centre}+V_{axis}$

$V=\frac{K{Q}_1}{R}+\frac{K{Q}_2}{r}$

$V=\frac{K{Q}_1}{0.1}+\frac{K{Q}_2}{\sqrt{(0.1{)}^2+(0.5{)}^2}}$

At point $P$
$\Rightarrow V_P=\frac{K\times 2\times {10}^{-6}}{0.1}+\frac{K\times 4\times {10}^{-6}}{\sqrt{(0.1{)}^2+(0.5{)}^2}}$
At point $Q$
$V_Q=\frac{K\times 4\times {10}^{-6}}{0.1}+\frac{K\times 2\times {10}^{-6}}{\sqrt{(0.1{)}^2+(0.5{)}^2}}$
$(\therefore \sqrt{(0.1{)}^2+(0.5{)}^2}\approx 0.50)$

Work done in transferring charge from centre of $P$ and $Q$ is,
$W_{ext}=q(V_Q-V_P$) = $q(V_Q-V_P)$
$Wext=(5\times {10}^{-6})[\frac{4K\times {10}^{-6}}{0.1}+\frac{2K\times {10}^{-6}}{0.5}-\frac{2K\times {10}^{-6}}{0.1}-\frac{4K\times {10}^{-6}}{0.5}]$

$\Rightarrow W_{ext}\approx 72\times {10}^{-2}\text{J}$

$\therefore Wext\approx 0.72\text{J}$