Question

Two identical thin rings, each of radius $R$, are coaxially placed a distance $R$ apart . If $Q_1$ and $Q_2$ are respectively the charges uniformly spread on the two rings, the work done in moving a charge $q$ from the centre of one ring to that of the other is equal to

• A. Zero
• B. $\frac{q(Q_1-Q_2)(\sqrt{2}-1)}{\sqrt{2}\left(4\pi {\epsilon }_{0}R\right)}$
• C. $\frac{q(Q_1-Q_2)(\sqrt{2}+1)}{\sqrt{2}\left(4\pi {\epsilon }_{0}R\right)}$
• D. $\frac{q(Q_1+Q_2)(\sqrt{2}-1)}{\sqrt{2}\left(4\pi {\epsilon }_{0}R\right)}$

Answer 1

The correct option is B $\frac{q(Q_1-Q_2)(\sqrt{2}-1)}{\sqrt{2}\left(4\pi {\epsilon }_{0}R\right)}$

Electric potential at point ${\text{C}}_1$,
$V_{C_1}=V_{Q_1}+V_{Q_2}....\left(1\right)$


Electric potential at ${\text{C}}_1$ due to charge $Q_1$,
$V_{Q_1}=\frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{Q_1}{R}$

Electric potential at ${\text{C}}_1$ due to charge $Q_2$,
$V_{Q_2}=\frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{Q_2}{R\sqrt{2}}$

Substituting the values in $\left(1\right)$,
$V_{C_1}=\frac{1}{\left(4\pi {\epsilon }_0\right)}\frac{Q_1}{R}$ $+\frac{1}{\left(4\pi {\epsilon }_0\right)}$ $\frac{Q_2}{R\sqrt{2}}$

$\Rightarrow V_{C_1}=\frac{1}{4\pi {\epsilon }_0}$ $[Q_1+\frac{Q_2}{\sqrt{2}}]$

Similarly, Electric potential at point ${\text{C}}_2$,
$V_{C_2}=\frac{1}{4\pi {\epsilon }_0}[Q_2+\frac{Q_1}{\sqrt{2}}]$

Now, potential difference between points ${\text{C}}_1$ and ${\text{C}}_2$ is,
$\Delta V=V_{C_1}-V_{C_2}$

$\Rightarrow \Delta V=\frac{1}{4\pi {\epsilon }_{0}R}\left[\right(Q_1-Q_2)-\frac{1}{\sqrt{2}}(Q_1-Q_2\left)\right]$

$\Rightarrow \Delta V=\frac{Q_1-Q_2}{\sqrt{2}\left(4\pi {\epsilon }_{0}R\right)}(\sqrt{2}-1)$

Work done in moving charge $q$ from point ${\text{C}}_1$ to ${\text{C}}_2$ is,

$W=q\Delta V=\frac{q(Q_1-Q_2)}{\sqrt{2}\left(4\pi {\epsilon }_{0}R\right)}(\sqrt{2}-1)$

$\therefore$ The correct option is (b).

Why this question ? Tip: Here radius of both rings are same, but due to presence of unequal charges potential at their centres differ. This will cause a difference in potential energy when a charges are moved from centre of one ring to other, $$W_{elec}=-(Q_2{V}_2-Q_1{V}_1)$$