Point charge representations of multipoles

Abstract

Abstract We generalize the notion of a dipole made of two point charges placed infinitesimally close together to multipoles of all degrees. The charge distributions are derived first by differentiating the delta function for the monopole in the same way that differentiating the potential 1/r generates the spherical multipoles r − ℓ − 1 P ℓ m ( cos θ ) e i m ϕ . This method surprisingly generates messy distributions for spherical multipoles with m ⩾ 3. So instead we use a ‘bracelet’ of 2m charges for the planar spherical multipoles of order m = ℓ, and stack these to construct multipoles for all m, ℓ. We show that the spherical multipoles of degree ℓ can be made of ℓ + 1 charges for m = 0 and 2m(ℓ − m + 1) charges for 1 ⩽ m ⩽ ℓ. A similar construction is found for the Cartesian multipoles.