Abstract
An exact time-dependent multipole analysis of macroscopic electromagnetic fields is given for sources which may not have Fourier transforms, e.g., they may spread out indefinitely in the course of time or have nonzero limits in the remote past or future. The multipole fields are determined by charge and energy conservation and the homogeneous Maxwell's equations for vacuum fields, without the use of the inhomogeneous Maxwell's equations or constitutive equations for the source material. The moments at each time are integrals of sources over the region spacelike to the space origin at this time. Since neither spherical Bessel functions nor associated Legendre polynomials are used, only rational functions are needed in this analysis. The problem of relating vector sources and fields is simplified by transforming it to an exactly equivalent scalar one.
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