potential outside an electrostatic charge distribution Consider a discrete charge distribution consisting of point charges with position vectors . We assume the charges to be clustered around the origin, so that for all i: , where has some finite value. The potential , due to the charge distribution, at a point outside the charge distribution, i.e., , can be expanded in powers of . Two ways of making this expansion can be found in the literature: The first is a Taylor series in the Cartesian coordinates , , and , 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 , 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 an ellipsis. Expansion in Cartesian coordinates Let satisfy . Then the Taylor expansion of around the origin can be written with If satisfies the Laplace equation then the expansion can be rewritten in terms of the components of a traceless Cartesian second rank tensor: where is the Kronecker delta and . Removing the trace is common, because it takes the rotationally invariant out of the second rank tensor. Example Consider now the following form of : Then by direct differentiation it follows that Define a monopole, dipole, and (traceless) quadrupole by, respectively, and we obtain finally the first few terms of the multipole expansion of the total potential, which is the sum of the Coulomb potentials of the separate charges: This expansion of the potential of a discrete charge distribution is very similar to the one in real solid harmonics given below. The main difference is that the present one is in terms of linearly dependent quantities, for NOTE: If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance apart, so that , it is easily shown that the only non-vanishing term in the expansion is the electric dipolar potential field. Spherical form The potential at a point outside the charge distribution, i.e. , can be expanded