Prove that a closed equipotenitial surface with no charge within itself must enclose an equipotential volume.

Explanation

Let the volume inside the closed equipotential surface were not equipotential. As we know that the electric field lines are perpendicular to the equipotential surface and electric potential charges along the direction of the electric field therefore electric potential must change in the direction perpendicular to the closed surface. This requires that electric field lines should originate or terminate somewhere inside the volume of the closed surface and that is possible only when some charges are present inside it. Thus, it must be an equipotential volume inside the equipotential surface if no charge is enclosed by it.