The resultant force at any point of the surface is in the direction of the normal to the surface, and the magnitude of the force is such that the work done on an electrical unit in passing from the surface V to the surface V' is V-V'.
No two equipotential surfaces having different potentials can meet one another, because the same point cannot have more than one potential, but one equipotential surface may meet itself, and this takes place at all points and lines of equilibrium.
The surface of a conductor in electrical equilibrium is necessarily an equipotential surface. If the electrification of the conductor is the whole surface, then the potential will diminish as we move away from the surface on every side, and the conductor will be surrounded by a series of surfaces of lower potential.
But if (owing to the action of external electrified bodies) some regions of the conductor are electrified positively and others negatively, the complete equipotential surface will consist of the surface of the conductor itself together with a system of other surfaces, meeting the surface of the conductor in the lines which divide the positive from the negative regions. These lines will be lines of equilibrium, so that an electrified point placed on one of these lines will experience no force in any direction.
When the surface of a conductor is electrified positively in some parts and negatively in others, there most be some other electrified body in the field besides itself. For if we allow a positively electrified point, starting from a positively electrified part of the surface, to move always in the direction of the resultant force upon it, the potential at the point will continually diminish till the point reaches either a negatively electrified surface at a potential less than that of the first conductor, or moves off to an infinite distance. Since the potential at an infinite distance is zero, the latter case can only occur when the potential of the conductor is positive.
In the same way a negatively electrified point, moving off from a negatively electrified part of the surface, must either reach a positively electrified surface, or pass off to infinity, and the latter case can only happen when the potential of the conductor is negative.
Therefore, if both positive and negative electrification exists on a conductor, there must be some other body in the field whose potential has the same sign as that of the conductor but a greater numerical value, and if a conductor of any form is alone in the field the electrification of every part is of the same sign as the potential of the conductor.
