A hollow sphere of radius 2R is charged to V volts and another smaller sphere of radius R is charged to V/2 volts. Now the smaller sphere is placed inside the bigger sphere without changing the net charge on each sphere. The potential difference between the two spheres would be :

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$\frac{3 V}{2}$

$\frac{V}{4}$

$\frac{V}{2}$

answer is B.

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Detailed Solution

The given point is inside the larger sphere, so the V at his point is same on the surface.
Case I : On Bigger sphere $V = \frac{1}{4 {πε}_{0}} \frac{q_{1}}{2 R}$
On Smaller sphere $\frac{1}{4 {πε}_{0}} \frac{q_{2}}{R} = \frac{V}{2}$
Case II : V on outer sphere
$V_{1} = \frac{1}{4 {πε}_{0}} [\frac{q_{1}}{2 R} + \frac{q_{2}}{2 R}]$
V on inner sphere $V_{2} = \frac{1}{4 {πε}_{0}} [\frac{q_{1}}{2 R} + \frac{q_{2}}{R}]$
$V_{2} - V_{1} = \frac{1}{4 {πε}_{0}} \frac{q_{2}}{2 R} = \frac{V}{4}$