THEORY OF MULTIPOLE RADIATIONS

We develop a systematic analysis of the radiation from a given oscillating system of charges and currents, without any approximations. Using a simple vector identity, the vector potential is separated into irrotational and solenoidal parts. The field may be expressed in terms of the latter alone. A similar vector identity involving the operator L = r × grad (the rotation operator) permits the separation of the field into parts in which the radial components of the electric and magnetic field, respectively, vanish. The energy flux, energy density, and angular momentum density may in each case be expressed in terms of the angular operators L, L2. Expansion in the eigenfunctions of these operators, the spherical harmonics, corresponds to the separation into electric and magnetic multipoles of all orders. Introduction of "tensor spherical harmonics" enables us to exhibit these radiations in terms of natural multipoles (derivatives of 1/r). All calculations are carried out without restriction as to size of radiating system relative to wave length, in the induction as well as the radiation region.