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Short Version

How exaclty can the Poynting vector be physically interpreted for static electric and magnetic fields?

I know that it describes the magnitude and direction of energy flow in EM-waves. But how does this interpretation hold up when only static fields are present? Does that interpretation only make sense w.r.t. wave packets (i.e. photons)?

Long Version

The question originally comes from a thought experiment, where a cylindrical capacitor is placed into a homogeneous magnetic field that is parallel to the axis of the capacitor. In that case we should have a static electrical field inside the capacitor that is rotation symmetric as well as a static magnetic field that is perpendicular to it. Thus, if we calculate the Poynting vector at each point inside the capacitor, we would get a rotating field that circles the inside of the capacitor.

Down the Rabbit hole

When thinking about this problem, I got even more confused on how to interpret the Poynting vector. If we simplify the original problem to a planar capacitor in a homogeneous magnetic field, all Poynting vectors between the planes would point sideways. Would that mean that somehow there is an energyflow from the vacuum between the planes of the capacitor going to one side?

The Questions

  • Does the interpretation of the Poynting vector as energy flow only make sense when introducing dynamic fields?
  • Can this circular energy flow be interpreted in any physically meaningful way? (If so, how?)
  • What happens in the planar case? How can this be interpreted?
  • If we modulate the fields accordingly in any of the mentioned configurations, do we generate photons (circulating around the cylindrical capacitor or equivalently emitted by the planar capacitor in both directions)?

Thanks for the help.

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    $\begingroup$ The Poynting vector with static fields is described in its Wikipedia article: en.wikipedia.org/wiki/Poynting_vector#Static_fields It even considers the thought example you mentioned, so that should be a good starting point. Maybe you can even answer the question yourself and share your knowledge here! $\endgroup$
    – Cream
    Aug 17, 2021 at 13:53

2 Answers 2

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As Cream stated in his comment, the Wikipedia article was a good starting point. It led me to the corresponding Feynman Lectures that contained much of the answers.

In short for everyone interested:

  • The Poynting vector does represent energy flow density in the static case as well. However, energy flow density and momentum density are linked with one another.
  • Thus, a circulating Poynting vector field can be interpreted as circulating momentum, i.e. angular momentum. If we build up the fields, we add angular momentum to the field (via Lorentz force). If we want angular momentum to be conserved in the whole world, we have to accept that some angular momentum is stored in the field. We get the angular momentum out again if we turn off the fields again.
  • In the planar case without external magnetic fields (or any similar case where we don't have circulating Poynting vector fields) we can interpret the energy as not flowing along the rods to the plates of the capacitor, but from the field towards its center. If we add an external magnetic field, the field lines change as also the charges flowing towards the plates are pulled by the Lorentz force.
  • With the interpretation of a circulating Poynting vector field as angular momentum, we don't need to consider strange things like potential circulating photons or the like as I wrote in the question.

Additionally, considering energy conservation, only the divergence of the Poynting vector field appears in the corresponging equations. Thus, as circulating fields have no sources we do not violate energy conservation in the example.

I hope that I got everything right. Feel free to correct me if I made a mistake. The Feynman Lectures I linked should give a good starting point for further questions.

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    $\begingroup$ You might also be interested in this paper, Poynting vectors are a lot easier to understand in geometric algebra $\endgroup$ Aug 18, 2021 at 10:39
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This is a very good question touching on the paradoxical nature of the electromagnetic conservation laws, as remarked by for example Panovsky&Phillips and discussed in unusual terms in Feynman Lectures 2, ch. 28.

The Poynting vector ascribes momentum density to crossed static electric and magnetic fields. This is counterintuitive. However the total energy-momentum, of which the Poynting vector is a part, is conserved unless we ignore this. This is a paradox of the gauge invariant theory. There are several paradoxes associated with conservation laws in this theory.

In a paper, Eur. Phys. J. D, vol. 8, p 9-12 (2000), accessible as https://arxiv.org/abs/physics/0106078, I describe an alternative theory which makes the same predictions as the standard one but is free of this and other paradoxes.

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