potential has a definite value for every point of the space surrounding the charged sphere. We may thus define it: The potential at any point of space is the energy required to bring unit positive charge from infinite distance to that point.
We have yet to find a mathematical expression for the potential. To do so we shall assume the approach to the sphere to take place in a straight line. It is quite permissible to restrict the movement to this condition, for if the shape of the path made any difference to the energy expended on approach and recovered on recession, we should be able to construct a perpetual motion machine, as may be easily seen from the following consideration: Imagine that a path of approach could be found which required a smaller expenditure of energy than can be recovered if the pith ball is constrained to follow on its outward journey some other path, then we could by a suitable sequence of the two motions create energy. We know that the creation of energy is impossible, and we must therefore conclude that all paths are equivalent as far as the potential is concerned. We are thus justified to take that shape of path which lends itself most easily to a mathematical investigation, and that is the straight line.
