Multipole Expansion of A Potential Outside An Electrostatic Charge Distribution
Consider a discrete charge distribution consisting of N point charges qi with position vectors ri. We assume the charges to be clustered around the origin, so that for all i: ri < rmax, where rmax has some finite value. The potential V(R), due to the charge distribution, at a point R outside the charge distribution, i.e., |R| > rmax, can be expanded in powers of 1/R. Two ways of making this expansion can be found in the literature. The first is a Taylor series in the Cartesian coordinates x, y and z, while the second is in terms of spherical harmonics which depend on spherical polar coordinates. The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. Its disadvantage is that the derivations are fairly cumbersome (in fact a large part of it is the implicit rederivation of the Legendre expansion of 1/|r-R|, which was done once and for all by Legendre in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion—usually only the first few terms are given followed by some dots.
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