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 $\frac{(\sqrt{x}-y) q\left(Q_{2}-Q_{1}\right)}{4 \sqrt{2} \pi \varepsilon_{0} R}$. Find $(x-y)$.

Answer 1

Let the charge $q$ be taken from the centre of ring $A$ to the centre of ring $B$.
At $A, U_{A}=\frac{q Q_{1}}{4 \pi \varepsilon_{0} R}+\frac{q Q_{2}}{4 \pi \varepsilon_{0}(\sqrt{2} R)}$
At $B, U_{B}=\frac{q Q_{2}}{4 \pi \varepsilon_{0} R}+\frac{q Q_{1}}{4 \pi \varepsilon_{0}(\sqrt{2} R)}$
Work done, $W=U_{B}-U_{A}$
$=\frac{(\sqrt{2}-1) q\left(Q_{2}-Q_{1}\right)}{4 \sqrt{2} \pi \varepsilon_{0} R}$