Question

Two identical thin rings, each of radius $R$, are coaxially placed a distance $R$ apart. $Q_1$ 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 $\frac{(\sqrt{x}-y)q\left(Q_2-Q_1\right)}{4\sqrt{2}\pi {\epsilon }_{0}R}$. Find $(x-y)$.

Answer 1

Answer: 1

Let the charge $q$ be taken from the centre of ring $A$ to the centre of ring $B$.
At $A,U_A=\frac{q{Q}_1}{4\pi {\epsilon }_{0}R}+\frac{q{Q}_2}{4\pi {\epsilon }_0(\sqrt{2}R)}$
At $B,U_B=\frac{q{Q}_2}{4\pi {\epsilon }_{0}R}+\frac{q{Q}_1}{4\pi {\epsilon }_0(\sqrt{2}R)}$
Work done, $W=U_B-U_A$
$=\frac{(\sqrt{2}-1)q\left(Q_2-Q_1\right)}{4\sqrt{2}\pi {\epsilon }_{0}R}$