Question
Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.
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Answer
Let us assume that in a closed equipotential surface with no charge the potential is changing from position to position. Let the potential just inside the surface is different to that of the surface causing in a potential gradient $\Big(\frac{\text{dV}}{\text{dr}}\Big)$ It means $\text{E}\neq0$ electric field comes into existence, which is given by as $\text{E}=-\frac{\text{dV}}{\text{dr}}$ It means there will be field lines pointing inwards or outwards from the surface. These lines cannot be again on the surface, as the surface is equipotential. It is possible only when the other end of the field lines are originated from the charges inside. This contradicts the original assumption. Hence, the entire volume inside must be equipotential.
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