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
equal to

Question

Aicharya · Accepted Answer

From given conditions, $V_{A}=V_{C}$ and $V_{B}=0$ $\Rightarrow V_{B}=\frac{k\left(Q-q_{1}\right)}{3 a}+\frac{k q_{2}}{2 a}+\frac{k q_{1}}{2 a}=0$
$\Rightarrow 2 Q+q_{1}+3 q_{2}=0 \quad \ldots \ldots \ldots$ (i)
Using $V_{A}=V_{C}$
$\frac{k\left(Q-q_{1}\right)}{3 a}+\frac{k q_{2}}{3 a}+\frac{k q_{1}}{3 a}$
$=\frac{k q_{1}}{a}+\frac{k\left(Q-q_{1}\right)}{3 a}+\frac{k q_{2}}{2 a}$
$\Rightarrow q_{1}=-\frac{q_{2}}{4}$
$\ldots \ldots \ldots$ (ii)
Using it in $(1), q_{2}=-\frac{8}{11} Q$

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Figure shows a system of three concentric metal shells $A, B$ and $C$ with radii $a, 2 a$ and $3 a$…

Solution