Point Charge Potential

The potential (voltage) of a point charge can be evaluated by calculating the work necessary to bring a test charge q in from an infinite distance to some distance r. The zero of potential is chosen at infinity.

Considering a radial path from a to b, the workdone by the Coulomb force is obtained from a line integral which in this case becomes just apolynomial integral since we are moving along a straight radial line: The change in potential is the work done per unit charge against the Coulomb force, so

For a positive charge q, the Coulomb force does positive work in moving it from a to b, so that represents a drop in potential energy.