The solution of a problem asking for the electric field of a given charge distribution involves no more than a sum or an integral of the electric fields of all the individual charges [see Eqs. (2.17) and (2.18)]. The solution of this kind of problem is therefore a trivial but possibly tiresome exercise in summation or integration. Unfortunately, many of the problems in electrostatics do not completely specify the charge distribution in advance. Instead, these problems specify some given potentials on some conducting surfaces and then ask for the resulting potentials and electric fields in the space surrounding these surfaces.

In practice, this kind of problem arises whenever we connect conducting bodies to batteries or other sources of electromotive force (emf) that maintain given potential differences between these bodies. For example, we might connect the plates of a capacitor to a battery and then ask: What are the potential and the electric field in the space between the plates? If the plates have some complicated shape, the solution of this problem will be quite difficult. Furthermore, there might be some extra electric charges placed in the space between the plates, adding to the difficulty of the solution.

This kind of problem is a boundary-value problem. It involves the calculation of the potential and electric field within some given region of space, starting from the known values of the potential (or the electric field) at the boundaries of the region and from the known values of the charge within the region.