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

• Equation , show that electric potential decreases along the direction of electric field

• Suppose this were not true the potential just inside the surface would be different form that at the surface resulting in a potential gradient. this would mean that outward from the surface.

• these lines cannot at the other end be again on the surface, since the surface is equipotential

• thus lines are possible only if the other ends of the lines are at charge inside, contradicting the premises hence, the entire volume inside must be at the same potential