The inversion in the sphere or Kelvin transformation, which exchanges the radial coordinate for its
inverse, is used as a guide to relate distinct electrostatic problems with dual features. The exact
solution of some nontrivial problems are obtained through the mapping from simple highly symmetric
systems. In particular, the concept of multipole expansion is revisited from a point of view opposed
to the usual one: the sources are distributed in a region far from the origin while the
electrostatic potential is described at points close to it.