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Figure shows a spherical capacitor consisting of two shells of radii $R$ and $2 R$ . The shells carry charges $Q$ and $- Q$ as shown. For what value of $r$ , the electrostatic energy stored within radius $r$ and outside the radius $r$ is same?

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$1.6 R$

$\frac{3}{2} R$

$\sqrt{2} R$

$\frac{4}{3} R$

answer is B.

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Detailed Solution

It is like two spherical capacitors in series. If energy is same then

$\frac{Q^{2}}{2 C_{1}} = \frac{Q^{2}}{2 C_{2}} \Rightarrow C_{1} = C_{2} \Rightarrow \frac{4 π ε_{0} R r}{r - R} = \frac{4 π ε_{0} r .2 R}{2 R - r} \Rightarrow 2 R - r = 2 (r - R) \Rightarrow r = \frac{4 R}{3}$

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