Please note that while I haven't explicitly solved for the electric
field, there must be an electric field because Maxwell's equations
require it when there is a time-varying magnetic field. The wave
equation (5) implicitly involves the electric field
equations as I verified in equations (6)-(7).
Indeed, the electric field can be formed explicitly using Ampere's Law
from Maxwell's equations,

(26) |

(27) |

(28) | |||

(29) |

Note that in the Bessel function has changed from to . The Pointing vector, which is defined as and shows the direction and magnitude of the electromagnetic energy flux, can be computed in a similar fashion. While I will skip the details, the final result is

(30) |

It can be shown that this field configuration is effectively *poloidal* near an antinode in the magnetic field strength. Near such
an antinode, the fields for a single eigenmode can be expanded in
terms of a small vector displacement from the center,
. The results are reasonably straightforward, and
give a Poynting flux of

(31) |

The flux is alternately directed radially *outward* and then *inward*. One might say that that the total energy content in a single
peak is conserved because the influx cancels the outflux, and this is
strictly true! Because an eigenmode extends infinitely throughout
space, and does not decay with time, it must be the case that the
total energy in a peak is conserved over one cycle.

However, I have already shown that a flux loop cannot be expressed as
a single eigenmode, so the above equation is *not* valid for the
flux loop solution. Also, the Poynting flux is not additive, which
means that we cannot simply add the Poynting fluxes due to each
eigenmode separately. Rather, we must first add all the component
fields together as described above, and *then* compute the
Poynting flux.

I have done this, and here are a few examples. All of the plots show the Poynting flux along the radial axis where . An inward-pointing flux is coded as red, and an outward-pointing flux is coded as blue. Shortly after the start, there are an inward- and outward-moving set of fluxes originating from the original toroid.

At a later time the waves have continued to travel. The outward-moving wave proceeds unimpeded, while the inward-moving wave interacts at the origin.

At late times, both waves are moving outward and continue indefinitely beyond the range of computation.

Finally, an animated GIF movie (110 kilobytes) shows the full time evolution of the Poynting flux.