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Consider three concentric metallic spheres $$A$$, $$B$$ and $$C$$ of radii of $$a$$, $$b$$, $$c$$ 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 what is the charge on the sphere $$C$$?

Solution:

Three concentric metallic spheres $$A$$, $$B$$ and $$C$$ have radii of $$a$$, $$b$$, $$c$$ respectively where a<b<c . $$A$$ and $$B$$ are connected whereas C is grounded The potential of the middle sphere $$B$$ is raised to $$V$$.

$$V=\frac{Kq}{b}+\frac{KQ}{c}$$ $$\Rightarrow \frac{k(q+Q)}{c}=0$$
$$\Rightarrow q+Q=0$$
$$\Rightarrow q=-Q$$
$$\frac{k(-Q)}{b}+\frac{KQ}{c}=V$$
$$\Rightarrow KQ(\frac{1}{c}-\frac{1}{b})=V$$
$$Q=\frac{bcV}{(b-c)}4\pi\epsilon_0$$