Show that electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface.

(N/A) In electrostatic equilibrium,the electric field $\vec{E}$ inside a conductor is zero.
Since the electric field is the negative gradient of the potential $(\vec{E} = -\nabla V)$,if $\vec{E} = 0$,then $\nabla V = 0$,which implies that the potential $V$ is constant throughout the volume of the conductor.
Furthermore,at the surface of the conductor,the electric field must be perpendicular to the surface. If there were a tangential component of the electric field,charges would move along the surface until the tangential component becomes zero.
Since there is no tangential component of the electric field on the surface,no work is done in moving a test charge between any two points on the surface. Therefore,the potential on the surface is the same as the potential inside the conductor.