Why electric field is perpendicular to …

Consider an equipotential surface with electric field lines that are not perpendicular to the surface. These field lines could then be resolved into two components, one perpendicular to the surface and one along the surface. The field line along the surface means that the charges would move along the surface in the direction of the field lines. This means that there would be potential difference along the surface as the charges only move when there is a difference in potential. But that is not possible since the surface was said to be equipotential with equal potential across the surface. Thus there cannot be any field component along the surface which leaves the field lines no option other than to be perpendicular to the surface.