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Five conducting shells A, B, C, D and E of radii a, 2a, 3a, 4a and 5a are placed concentrically. Shell B is given a charge +q and shell D is given a charge –q. Shell C is earthed. If the potential difference between the shells A and D $(V_{A} - V_{D}) is \frac{P}{48} (\frac{1}{4 {πε}_{0}}) \frac{q}{a}$ Find the value of P.

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answer is 11.

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Detailed Solution

If ‘q’ is the induced charge on shell C,
$\begin{array}{l} V_{C} = 0 \Rightarrow k [\frac{q}{3 a} + \frac{q^{'}}{3 a} - \frac{q}{5 a}] = 0 \\ q^{'} = \frac{- q}{4} \end{array}$
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Now potential of shell $A, V_{A} = k [\frac{q}{2 a} - \frac{q / 4}{3 a} - \frac{q}{4 a}] = \frac{kq}{6 a}$
Potential of shell D, $V_{D} = k [\frac{q - q + q^{'}}{4 a}] = k \frac{- q / 4}{4 a} = \frac{- kq}{16 a}$
Potential difference $V_{A} - V_{D} = \frac{kq}{6 a} - (- \frac{kq}{16 a}) = \frac{11 kq}{48 a} = \frac{11}{48} (\frac{1}{4 {πε}_{0}}) \frac{q}{a}$

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Five conducting shells A, B, C, D and E of radii a, 2a, 3a, 4a and 5a are placed concentrically. Shell B is given a charge +q and shell D is given a charge –q. Shell C is earthed. If the potential difference between the shells A and D VA−VD is P4814πε0qaFind the value of P.