Multipole Expansion of the Potentials

Abstract

The integral representations of the scalar and vector potential, Eqs. (3.21) and (3.22), turn out to be unpractical for concrete calculations. Numerically, they can be evaluated for arbitrary source distributions, but such computations do not provide the insights to discuss the physical peculiarities of a particular source distribution. The relevant physics can best be made obvious by expanding a source distribution in a sum of specific contributions. Each of these contributions shall have a clear physical meaning. In this regard, the multipole expansion is a means of abstraction and provides a language to discuss the properties of source distributions. Performing a multipole expansion is in essence the approximation of the potentials or the fields of a source that is characterized by a specific charge or current distribution on a sphere enclosing the entire source, see Fig. 4.1.