Spherical Multipole Moments: Electric Potential, Magnetic Potential, Gravitational Potential, Legendre Polynomials, Axial Multipole Moments, Spherical Harmonics, Solid Harmonics

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 rho. hrough this article, the primed coordinates such as mathbf{r^{prime}} refer to the position of charge(s), whereas the unprimed coordinates such as mathbf{r} refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector mathbf{r^{prime}} has coordinates ( r^{prime}, theta^{prime}, phi^{prime}) where r^{prime} is the radius, theta^{prime} is the colatitude and phi^{prime} is the azimuthal angle.