Figure shows a system of three concentric metal shells $A, B$ and $C$ with radii $a, 2 a$ and $3 a$ respectively. Shell $B$ is earthed and shell $C$ is given a charge $Q$ . Now if shell $C$ is connected to shell $A$ , then the final charge on the shell $B is \frac{- Q x}{11}$ , where the value of $x$ is

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Solution

Since $A$ and $C$ are connected, their potentials are equal.
$V_{A} = V_{C}$
$\frac{k q_{a}}{a} + \frac{k q_{b}}{2 a} + \frac{k q_{c}}{3 a} = \frac{k q_{a}}{3 a} + \frac{k q_{b}}{3 a} + \frac{k q_{c}}{3 a}$
Here $q_{a}, q_{b}, q_{c}$ are charges on A, B,C respectively.
$\frac{2}{3} q_{a} = - \frac{q_{b}}{6}$
$\Rightarrow q_{b} = - 4 q_{a}$

Also as B is earthed, $V_{B} = 0$
$\frac{k q_{a}}{2 a} + \frac{k q_{b}}{2 a} + \frac{k q_{c}}{3 a} = 0$
$3 q_{a} + 3 q_{b} + 2 q_{c} = 0$
$3 q_{a} - 12 q_{a} + 2 q_{c} = 0$
$\Rightarrow q_{c} = \frac{9 q_{a}}{2}$

Finally,
$q_{a} + q_{c} = Q$
$\Rightarrow q_{a} + \frac{9 q_{a}}{2} = Q$
$\Rightarrow q_{a} = \frac{2 Q}{11}$
$\Rightarrow q_{b} = - \frac{8 Q}{11}$
$\Rightarrow x = 8$

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Figure shows a system of three concentric metal shells A, B and C with radii a, 2a and 3a respectively. Shell B is earthed and shell C is given a charge Q. Now if shell C is connected to shell A, then the final charge on the shell B is−Qx11, where the value of x is