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Spherical multipole moments are the coefficients in a series expansionof 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 ρ {\displaystyle \rho }. Through this article, the primed coordinates such as r ′ {\displaystyle \mathbf {r} '} refer to the position of charge, whereas the unprimed coordinates such as r {\displaystyle \mathbf {r} } refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector r ′ {\displaystyle \mathbf {r} '} has coordinates {\displaystyle } where r ′ {\displaystyle r'} is the radius, θ ′ {\displaystyle \theta '} is the colatitude and ϕ ′ {\displaystyle \phi '} is the azimuthal angle.