A metal ball of radius R is placed concentric | vedclass

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Question

(A) Let the inner ball be $S_1$ (radius $R$,charge $q_1 = +2Q$).Let the hollow sphere be $S_2$ (inner radius $r_2 = 2R$,outer radius $r_3 = 3R$,total

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$\frac{7Q^2}{24\pi\varepsilon_0 R}$

Vedclass · Answer

(A) Let the inner ball be $S_1$ (radius $R$,charge $q_1 = +2Q$).Let the hollow sphere be $S_2$ (inner radius $r_2 = 2R$,outer radius $r_3 = 3R$,total charge $q_2 = -Q$).Due to electrostatic induction,a charge of $-2Q$ is induced on the inner surface of the hollow sphere $(r=2R)$.Since the total charge on the hollow sphere is $-Q$,the charge on its outer surface $(r=3R)$ must be $q_{out} = -Q - (-2Q) = +Q$.The electrostatic potential energy $U$ of a system of charged conductors is given by $U = \frac{1}{2} \sum q_i V_i$.The potential of the inner ball $(V_1)$ is due to its own charge and the charges on the hollow sphere:$V_1 = \frac{1}{4\pi\varepsilon_0} \left( \frac{2Q}{R} + \frac{-2Q}{2R} + \frac{Q}{3R} \right) = \frac{1}{4\pi\varepsilon_0} \left( \frac{2Q}{R} - \frac{Q}{R} + \frac{Q}{3R} \right) = \frac{1}{4\pi\varepsilon_0} \left( \frac{Q}{R} + \frac{Q}{3R} \right) = \frac{4Q}{12\pi\varepsilon_0 R} = \frac{Q}{3\pi\varepsilon_0 R}$.The potential of the hollow sphere $(V_2)$ is the same at its inner and outer surfaces:$V_2 = \frac{1}{4\pi\varepsilon_0} \left( \frac{2Q}{2R} + \frac{-2Q}{2R} + \frac{Q}{3R} \right) = \frac{1}{4\pi\varepsilon_0} \left( \frac{Q}{R} - \frac{Q}{R} + \frac{Q}{3R} \right) = \frac{Q}{12\pi\varepsilon_0 R}$.Now,$U = \frac{1}{2} (q_1 V_1 + q_2 V_2) = \frac{1}{2} \left( (2Q) \cdot \frac{Q}{3\pi\varepsilon_0 R} + (-Q) \cdot \frac{Q}{12\pi\varepsilon_0 R} \right)$.$U = \frac{1}{2} \left( \frac{2Q^2}{3\pi\varepsilon_0 R} - \frac{Q^2}{12\pi\varepsilon_0 R} \right) = \frac{1}{2} \left( \frac{8Q^2 - Q^2}{12\pi\varepsilon_0 R} \right) = \frac{1}{2} \left( \frac{7Q^2}{12\pi\varepsilon_0 R} \right) = \frac{7Q^2}{24\pi\varepsilon_0 R}$.

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

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