Compensated Free and Bound Sources and Nonradiating Conditions in Classical Electrodynamics

Abstract

Classical macroscopic electrodynamics allows free and bound sources to cancel as distributions while the auxiliary equations retain nonzero free-source terms. A neutralized uniformly polarized sphere has $\mathbf{E}=0$ and $\mathbf{D}=\mathbf{P}$, and a compensated uniformly magnetized sphere has $\mathbf{B}=0$ and $\mathbf{H}=-\mathbf{M}$. These auxiliary fields are fixed by $\mathbf{P}$ and $\mathbf{M}$ rather than forming independent degrees of freedom. Time dependence separates charge cancellation from complete four-current cancellation. Canceling the charge alone leaves a divergence-free total current whose on-shell transverse transform controls radiation. For a neutralized polarized sphere of radius $R$ maintained by a tangential free-current sheet, the exterior field equals that of a point electric dipole whose effective moment is proportional to the spherical Bessel function $j_2(kR)$, where $k=\omega /c$. Every charge multipole and the ordinary magnetic dipole vanish, yet the source radiates at generic frequencies, and the complete exterior field vanishes at nonzero roots of $j_2$. Complete four-current cancellation removes all retarded source-generated fields. Among the compactly supported separable currents that maintain the same charge, it is the only choice that never radiates. In vacuum the auxiliary fields reduce to $\mathbf{D}={\epsilon }_0\mathbf{E}$ and $\mathbf{H}=\mathbf{B}/{\mu }_0$, so no radiation can be carried by $\mathbf{D}$ and $\mathbf{H}$ alone.

arXiv comment28 pages, 3 figures