Spherical Multipole Moments

Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as 1/R. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.

For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density . Through this article, the primed coordinates such as refer to the position of charge(s), whereas the unprimed coordinates such as refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector has coordinates where is the radius, is the colatitude and is the azimuthal angle.