A metal ball of radius R is placed concentrically inside a hollow metal sphere of inner radius $2 R$ and outer radius $3 R$ . The ball is given a charge $+ 2 Q$ and the hallow sphere a total charge $- Q$ . The electrostatic potential energy of this system is:

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$\frac{5 Q^{2}}{16 π ε_{0} R}$

$\frac{7 Q^{2}}{24 π ε_{0} R}$

$\frac{7 Q^{2}}{48 π ε_{0} R}$

$\frac{5 Q^{2}}{8 π ε_{0} R}$

answer is A.

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Detailed Solution

The charge distribution will be +2Q on the outer surface of sphere, -2Q on the inner surface of hollow sphere, a

$T . E = U_{1} + U_{2} + U_{3} + U_{12} + U_{13} + U_{23}$

$= \frac{K {(2 Q)}^{2}}{R} + \frac{K {(- 2 Q)}^{2}}{2 R} + \frac{K {(Q)}^{2}}{3 R} + \frac{K (2 Q \times - 2 Q)}{2 R} + \frac{K (+ 2 Q \times Q)}{3 R} + \frac{K (- 2 Q \times Q)}{3 R}$

$\Rightarrow T . E = \frac{7 {KQ}^{2}}{6 R} = \frac{7 Q^{2}}{24 {πε}_{0} R}$

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