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Electric Field Due to Disk and Infinitely Large Sheet

Two identical...

Question

Two identical thin rings each of radius $R$ are coaxially placed at 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 the other is :

\frac{q (Q_{1} / Q_{2}) (\sqrt{2} + 1)}{4 π ε_{0} \sqrt{2} R}

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\frac{q (Q_{1} - Q_{2}) (\sqrt{2} - 1)}{4 π ε_{0} \sqrt{2} R}

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\frac{q \sqrt{2} (Q_{1} + Q_{2})}{4 π ε_{0} R}

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zero

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Solution

The correct option is A $\frac{q (Q_{1} - Q_{2}) (\sqrt{2} - 1)}{4 π ε_{0} \sqrt{2} R}$
$∵$ work done $W = q (V_{1} - V_{2})$

potential at A is
$V_{1} = \frac{1}{4 π ε_{0}} (\frac{Q_{1}}{R} + \frac{Q_{2}}{\sqrt{2} R})$

potential at B is
$V_{2} = \frac{1}{4 π ε_{0}} (\frac{Q_{2}}{R} + \frac{Q_{1}}{\sqrt{2} R})$

$W = q (V_{1} - V_{2}) W = \frac{q}{4 π ε_{0}} [(\frac{Q_{1}}{R} + \frac{Q_{2}}{\sqrt{2} R}) - (\frac{Q_{2}}{R} + \frac{Q_{1}}{\sqrt{2} R})] W = \frac{q (Q_{1} - Q_{2}) (\sqrt{2} - 1)}{4 π ε_{0} \sqrt{2} R}$

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