Electric Field and Poynting Flux

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,

$${\bf E}= {\bf E}(t=0) - c^2\int_0^t{\nabla\times{{\bf B}} dt},$$ (26)

and since ${\bf B}$ is now known, the electric field ${\bf E}$ is also determined (except for any constant field terms from ${\bf E}(t=0)$. This term is then computed to be

$${\bf E}= E_r \hat{r} + E_z \hat{z},$$ (27)

where, assuming the initial electric field is zero,

$$E_r = c^2\sum_{ij}{c_{ij}{k_i\over\omega_{ij}}\sin(\omega_{ij} t)\sin(k_i z)J_1(\lambda_j r)}$$ (28)

$$E_z = c^2\sum_{ij}{c_{ij}{\lambda_j\over\omega_{ij}}\sin(\omega_{ij} t)\cos(k_i z)J_0(\lambda_j r)}.$$ (29)

Note that in $E_z$ the Bessel function has changed from $J_1$ to $J_0$. The Pointing vector, which is defined as ${\bf S} = {1\over\mu}{\bf E}\times{\bf B}$ 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

$${\bf S} = {B_\phi\over\mu}\left(E_r \hat{z} - E_z \hat{r}\right).$$ (30)

This expression shows that there is electromagnetic Poynting flux in both the $\hat{r}$ and $\hat{z}$ directions. In the $z=0$ plane, the above equations show that $E_r$ is always zero, so the electric field is always in the $\hat{z}$ direction. Similarly, near a radial peak in the magnetic field, $E_z$ is zero, so the field is purely radial.

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, $\Delta r\ \hat{r} + \Delta z\ \hat{z}$. The results are reasonably straightforward, and give a Poynting flux of

$${\bf S}_{ij} = {c_{ij}\over2\mu\omega_{ij}} \sin(2\omega_{... ...lambda_j^2 \Delta r\ \hat{r} + k_i^2 \Delta z\ \hat{z}\right).$$ (31)

This equation shows several things. First of all, the frequency of oscillations in the Poynting Flux is twice that of the oscillation in either of the fields. This is because the two field components are out of phase by $\pi/2$ in time. The flux is zero when either the magnetic field or electric field is maximum. When either field component is maximum, then the other is zero, which means there cannot be a Poynting flux then.

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 $z=0$. 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.