A small conducting sphere of radius r is lying concentrically inside a bigger hollow sphere of radius R as shown in fig. The bigger and the smaller spheres are charged with charge Q and q(Q>q) respectively and are insulated from each other. The potential difference between spheres will be

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$\frac{1}{4 {πε}_{0}} (\frac{q}{R} + \frac{Q}{R})$

$\frac{1}{4 {πε}_{0}} (\frac{q}{r} + \frac{Q}{R})$

$\frac{1}{4 {πε}_{0}} (\frac{Q}{R} - \frac{q}{r})$

$\frac{1}{4 {πε}_{0}} (\frac{q}{r} - \frac{q}{R})$

answer is A.

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Detailed Solution

fig. The potential V₁ of smaller sphere is given by
$V_{1} = \frac{1}{4 {πε}_{0}} \cdot \frac{q}{r} + \frac{1}{4 {πε}_{0}} \cdot \frac{Q}{R} . . . (1)$
The potential V₂ of bigger sphere is
$V_{2} = \frac{1}{4 {πε}_{0}} \frac{Q}{R} + \frac{1}{4 {πε}_{0}} \cdot \frac{q}{R}$
So, the potential difference (V₁ - V₂) is given by
$V_{1} - V_{2} = \frac{1}{4 {πε}_{0}} [\frac{q}{r} - \frac{q}{R}]$