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Potential Energy of a Point Charge Placed at a Point with Non Zero Potential

Two identical...

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.

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Solution

Potential at A,
$V_{A} = \frac{1}{4 π ϵ_{0}} [\frac{Q_{1}}{R} + \frac{Q_{2}}{\sqrt{2} R}]$
Potential at B,
$V_{B} = \frac{1}{4 π ϵ_{0}} [\frac{Q_{1}}{R} + \frac{Q_{2}}{\sqrt{2} R}]$
Work done in moving a charge q from the centre of one ring to that of other is given by,
$W = q (V_{A} - V_{B})$
$= \frac{q}{R 4 π ϵ_{0}} [Q_{1} + \frac{Q_{2}}{V_{2}} - Q_{2} - \frac{Q_{1}}{V_{2}}]$
$= \frac{q (Q_{1} - Q_{2}) (\sqrt{2} - 1)}{\sqrt{2} 4 π R ϵ_{0}}$ .

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