Multipole moments and Maxwell's equations

Maxwell's equations provide the basis for explaining many electromagnetic phenomena, both in vacuum and in matter. While some of these effects may be described within the electric dipole approximation D= in 0E+P and H= mu -10 B, there are others which require the inclusion of higher multipole terms in the fields D and H. Various practices for doing this have been adopted in the past, such as the use of traceless quadrupole and higher moments, and the neglect of certain moments. Some of these practices are shown to lead to Maxwell equations which are not translationally invariant, in that they depend on the choice of origin to which the multiple moments are referred, even though they may also yield expressions for observables which do not possess this defect. It is also shown how to obtain general multipolar forms for D and H which ensure origin-independent Maxwell equations.