If we have a distribution of electric charge on a conducting surface, the whole surface must be at the same potential, since if some part of the surface has a different potential from some other part, the charges on the surface will move so as to equalise the potential, with electrons moving to lower potentials. As shown in the diagram below, this also means that the interior of a conducting surface is at a constant potential.
It should also be noted
-
the electric charge on a conductor lies entire on the surface. If part of the surface charge penetrates to the interior of the conductor then there will be a non zero electric field in the region of that charge, so the field inside the condcutor is not zero and the surface is not at a constant potential.
-
At any point on the surface, since the surface is at a constant potential, the electric is always perpendicular to the surface.