Question

Why is work done by the electric field in moving a charge on a conducting surface is zero? What is the potential difference between two points along the surface of a charged conductor?

Answer 1

A conductor has all of its charge equally distributed on its surface. If there were a potential difference between any two points on the surface of the conductor, the electrons would rearrange to equalize this distribution. Since the charges would be moving, this would result in a current moving on the surface of the conductor, which we can see does not happen. Since the charges are not moving, electric force integrated along the path of displacement is 0, so there is no transfer of potential energy, which means no work done.