Consider a point charge in a uniform electric field. Since it is easy to calculate the force, of magnitude qE, it is easy to calculate the work done by the field. We can then define a potential energy as usual, to increase by the work WE would have to do to move the charge q around in the uniform field.

As usual we can define a total energy E = KE + PE, which will be conserved since the electrical force is conservative. Thus if the PE of a charged particle decreases, its kinetic energy must increase by the same amount. It is convenient to define an electric potential V by V = PE/q = U/q. In other words, the potential is the potential energy per unit charge. It is thus a scalar field V(r) defined at every point in space. Using calculus, we can construct a function U(r) or V(r) for all space around a point charge.

It is easy to see that surfaces of equal potential are always perpendicular to E. Since the electric field is everywhere perpendicular to the surface of a conductor, it follows that every point on the surface of a conductor is at the same potential! Since the electric field is zero everywhere inside a conductor in equilibrium, it then follows that whatever V was on the surface, it is the same everywhere inside the conductor.