Spherical multipole moments

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. 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 ( r ′ ) {displaystyle ho (mathbf {r^{prime }} )} . Through this article, the primed coordinates such as r ′ {displaystyle mathbf {r^{prime }} } refer to the position of charge(s), 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^{prime }} } has coordinates ( r ′ , θ ′ , ϕ ′ ) {displaystyle (r^{prime }, heta ^{prime },phi ^{prime })} where r ′ {displaystyle r^{prime }} is the radius, θ ′ {displaystyle heta ^{prime }} is the colatitude and ϕ ′ {displaystyle phi ^{prime }} is the azimuthal angle. The electric potential due to a point charge located at r ′ {displaystyle mathbf {r^{prime }} } is given by where R   = d e f   | r − r ′ | {displaystyle R {stackrel {mathrm {def} }{=}} left|mathbf {r} -mathbf {r^{prime }} ight|} is the distance between the charge position and the observation pointand γ {displaystyle gamma } is the angle between the vectors r {displaystyle mathbf {r} } and r ′ {displaystyle mathbf {r^{prime }} } .If the radius r {displaystyle r} of the observation point is greater than the radius r ′ {displaystyle r^{prime }} of the charge, we may factor out 1/r and expand the square root in powers of ( r ′ / r ) < 1 {displaystyle (r^{prime }/r)<1} using Legendre polynomials This is exactly analogous to the axialmultipole expansion. We may express cos ⁡ γ {displaystyle cos gamma } in terms of the coordinatesof the observation point and charge position using the spherical law of cosines (Fig. 2) Substituting this equation for cos ⁡ γ {displaystyle cos gamma } intothe Legendre polynomials and factoring the primed and unprimedcoordinates yields the important formula known as the spherical harmonic addition theorem where the Y l m {displaystyle Y_{lm}} functions are the spherical harmonics.Substitution of this formula into the potential yields