Consider three concentric metallic spheres $A, B$ and $C$ of radii $a , b, c$, respectively where $a < b < c$. $A$ and $B$ are connected, whereas $C$ is grounded. The potential of the middle sphere $B$ is raised to $V$, then the charge on the sphere $C$ is
KVPY 2012, Diffcult
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(a)
Let potential source (cell) of $V$ volts is connected across $B$ and $C$. Then potential on surface of $B=V_B=\frac{k q}{b}+\frac{k(-q)}{c}$
As given, $\quad V_B=V$
So, $\quad V=k q\left(\frac{1}{b}-\frac{1}{c}\right)$
$=k q\left(\frac{c-b}{b c}\right)$
$\Rightarrow \quad q=\frac{V b c}{k(c-b)}$
So, charge on $C$ is $-q=\frac{-V b c}{k(c-b)}$ where, $k=\frac{1}{4 \pi \varepsilon_0}=$ constant.
Hence, the charge on the sphere is
$-4 \pi \varepsilon_0 \frac{V b c}{(c-b)}$
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