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Electromagnetic energy-momentum circulation around the dipole-axis of a "classical" electron can be substantiated by determination of the amount of passive power [W] assignable to such energy-circulation.\

The task simply is to integrate the Poynting-vector field $\vec{S}(r, \varphi, \Theta) = \vec{E} \times \vec{H}$ surrounding an electron over an adequate reference plane like the x-z-plane (for $x>0$) where $\vec{S}$ is $\perp$ to that plane at any point as shown in http://vixra.org/abs/1808.0179

In result, the passive power circulating around a "classical" electron of radius $r_{e}$ would be \begin{equation*} P = \frac{c^{2}\hbar}{12 \pi r_{e}^{2}} \approx 31,6 \hspace{5mm} [GW] \end{equation*}

Note that this relatonship reflects a power-singularity for $r_{e} \rightarrow0$.

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closed as off-topic by John Rennie, PM 2Ring, ACuriousMind May 7 at 19:36

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  • $\begingroup$ I think it is clear that, according to classical electromagnetism, field energy flows around the electron (if you treat its magnetic field as classical, which it isn’t). I have not checked the value stated for the power when you use the classical electron radius as the cutoff. But since you would get infinity for the power without a cutoff, we shouldn’t be surprised if the power is large. $\endgroup$ – G. Smith May 7 at 18:10
  • $\begingroup$ This is just Jackson-style EM. It could be a nice homework problem, but I don’t remember it actually being in Jackson. $\endgroup$ – G. Smith May 7 at 18:11
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    $\begingroup$ 1. Please note that physics.SE has a policy that you must disclose if you are the author of any materials you cite. 2. It is not clear what you're really asking for here. The only question is in the title, and it is not clear to me how you arrived from the calculation involving the Poynting vector at "electromagnetic vortex". In so far as the question is supposed to ask us to review the potentially more elaborate content of the paper you link, peer-review is off-topic here. $\endgroup$ – ACuriousMind May 7 at 19:31
  • $\begingroup$ The question has been closed as "non-mainstream". I don't see anything non-mainstream about it. It is standard classical EM. Physicists have investigated what classical EM has to say about the electron ever since the electron was discovered. I believe the calculation is correct (within the assumptions made) so I am voting to reopen it. $\endgroup$ – G. Smith May 7 at 19:59
  • $\begingroup$ @G.Smith : I agree. See this: nature.com/articles/252674a0 . It's a letter published in December 1974 on Relativistic angular momentum of power circulation by U. Kayser Herold. $\endgroup$ – John Duffield May 7 at 20:03
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Technically, you are right: we can assign a circulating energy to the electromagnetic field around a rotating body and, of course, given that energy, and its rate of circulation, define the radial power that represents, in the way you suggest.

Namely, the Poynting vector

$$\mathbf{S} := \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})$$

defines the direction and intensity (i.e. power per unit area) of the energy passing through a point and, the weird result of this is that you can have situations where that the electromagnetic field does not appear to be changing in any way, yet somehow, in some subtle sense, there is energy on the move. The vector more obviously corresponds to moving energy when you consider an electromagnetic wave: then $\mathbf{S}$ points in the direction of wave movement and waves obviously transports energy from one point to another point. And if you do this for the case of a spinning sphere, with a half-plane formed by the axis of rotation, and integrate this to get the energy flux or power, you will get that circulatory flux is present.

And of course, thanks to conservation of energy, if we have an electromagnetic field in a bounded region of space, then we must have (and we will, if it satisfies Maxwell's equations, which it must to be physically sensible, which mathematically can be proven to conserve energy by using Noether's theorem) that the total integrated energy in and out will have to be zero, so the energy can be considered, in some sense, to circulate, and thus an energy "vortex", if one will. Moreover, you can also define, through a volume integral, the total energy in the whole electromagnetic field.

So yes, for a classical electron, this is correct. The trick is, of course, that the real electron is not classical. It is quantum mechanical, and you must treat with quantum electrodynamics, and even there this question - which is just a bit of rephrasing of the notion of the self-energy of the electron - is a difficult problem which I've heard is not fully understood.

That said, the calculation is valid also for a real charged, macroscopic sphere that we are rotating, e.g. a piece of metal to which we have conducted a net electric charge, as in electrostatics experiments with, say, a van de Graaf generator, and the energy in the "vortex" represents energy that has to be implemented in order to cause the sphere to rotate, above and beyond the kinetic energy of rotation. More specifically, it is the energy investment to create the magnetic field associated with the sphere's rotation.

Regarding the linked paper, while the vortex-idea is sound for the classical model, the conclusion it draws at the very end is over-drawn:

"The large amount of passive energy flux ≈ 31, 6 GW can be interpreted as a theoretical upper limit of electromagnetic energy flux."

This is wrong. You can push well more than 31.6 GW through an electromagnetic field: in fact, the Sun alone puts about 383 YW, which is (to 3 sig figs) 383 000 000 000 000 000 GW, through it. If no more than this could travel, Earth would be a cryogenic ball of inert ice, because as of right now, it receives about 174 PW, or 174 000 000 GW, from the Sun. The mere fact that right now, we're warm enough to survive, is observational disproof of the paper's conclusion. And even if the electron really were a classical object with classical flux, that would not affect the energy carrying capability of EM fields as a whole, since the EM fields around an electron are just one kind of EM field configuration that can exist. It is just the energy flux in one particular circumstance. Indeed, if you put two classical electrons very near each other, with aligned spins, then the combined "vortex" of the two will have even more circulating energy flux, as can be derived from the very same principles as were used here.

Instead of drawing grandiose (and errant!) conclusions, it would be better to simply describe the calculation as a pure remark on classical EM theory.

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  • $\begingroup$ You may like to look at this answer I gave to a question about the electron a few years back. $\endgroup$ – John Duffield May 7 at 20:54
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If we follow classical electromagnetism in its present formulation, there is indeed a flow of energy around an electron. Whether it has this value is debatable. In fact when a charged condensor is placed in some static magnetic field, the Poynting vector also gives an energy flow or momentum density circulating as it were from between its plates around the back. This argument avoids the pitfalls of discussions about QM that the example of an electron brings with it. This is counterintuitive, hence a paradox. Panofsky&Phillips therefore warn against the interpretation of the Poynting vector as the field momentum. As Panofsky was once director of the SLAC he must have been part of many discussions on this topic and must have reached his verdict after ample consideration. Indeed, the Poynting vector is not the Noether current for translation of any lagrangian. It is part of the Belinfante-Rosenfeld tensor, which is a modification of the Noether energy-momentum expression. This modification is necessary if gauge invariance of the energy-momentum density is required. It leads to a class of paradoxes involving the EM conservation laws. An alternative formulation of ectromagnetism was published by myself and can be found here: https://arxiv.org/abs/physics/0106078. It resolves these paradoxes at the expense of gauge invariance of the conservation laws, while maintaining the same agreement with experiment as the standard approach.

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