Question

Two identical thin rings each of radius R metre are coaxially placed at a distance R metre apart Q1 and Q2 are respective charges uniformly spread on the two rings, calculate the work done in moving a charge q from the centre of one ring to that of other.

Answer 1

Potential at A,
$V_A=\frac{1}{4\pi {ϵ}_0}[\frac{Q_1}{R}+\frac{Q_2}{\sqrt{2}R}]$
Potential at B,
$V_B=\frac{1}{4\pi {ϵ}_0}[\frac{Q_1}{R}+\frac{Q_2}{\sqrt{2}R}]$
Work done in moving a charge q from the centre of one ring to that of other is given by,
$W=q(V_A-V_B)$
$=\frac{q}{R4\pi {ϵ}_0}[Q_1+\frac{Q_2}{V_2}-Q_2-\frac{Q_1}{V_2}]$
$=\frac{q(Q_1-Q_2)(\sqrt{2}-1)}{\sqrt{2}4\pi R{ϵ}_0}$.