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I understand that an electromagnetic wave contains energy. This is clear given how they will cause a test charge to accelerate. However, I don't understand how this energy gets conveyed from the charge/acceleration source and to the wave, or in other words, how energy being conserved in this case is consistent with what you would get if you were to calculate the dynamics by hand. I've never actually seen this calculation done before, and I've been having trouble doing it myself.

Suppose a charge in empty space is accelerated from rest by a constant force $\vec{F_{0}}$ over a distance $d$. This conveys an amount of energy $W$ to the charge given by $W=F_{0}d$, since the force and displacement are parallel.

If all this energy remains within the charge as kinetic energy, then the electromagnetic wave it emits due to its acceleration should not have any energy, unless energy conservation is violated here. However, we observe that this is not the case. The EM waves contain energy, and the radiating charge loses kinetic energy, such as in a synchrotron.

In other words, by the end of the acceleration, the net energy gained by the charge is not equal to $F_{0}d$; it is less. This seems to suggest that there is some other, potentially time-varying force (call it $\vec{F}(t)$) that has a component opposite the charge's motion, contributing to the total work done on the charge.

Where would this force come from, if it exists? Is it a consequence of Maxwell's Equations requiring the EM wave to produce a component that propagates backwards towards the charge, slowing it down? Or if it doesn't exist, like to me it seems like it should, what is actually going on?

After all, if energy is conserved, it should in theory be possible to verify that the interaction simultaneously obeys Maxwell's Equations and conserves energy, and to show how the dynamics force energy to be conserved. Otherwise, it would seem to me that there is an issue with Maxwell's Equations.

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  • $\begingroup$ > ... an electromagnetic wave contains energy. This is clear given how they will cause a test charge to accelerate. This is a common argument, but it is not valid in general. The wave can simply make the physical field close to and around the particle to change in such a way so that energy in that close field transfers into kinetic energy of the particle. That is, the wave far from the particle need not carry any energy, but it may be an ingredient of a process by which the close field supplies the energy. Like when teacher tells students the lecture is over, students run away. $\endgroup$ Aug 3, 2023 at 2:00
  • $\begingroup$ Wikipedia: Abraham–Lorentz force $\endgroup$
    – Ghoster
    Aug 3, 2023 at 2:40
  • $\begingroup$ Eric Poisson, “The motion of point particles in curved spacetime”: “The particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime.” This paper computes the radiation-reaction self-force due to the particle’s own field. $\endgroup$
    – Ghoster
    Aug 3, 2023 at 2:46

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The issue is not with Maxwell’s equations. The issue is with the concept of a classical point particle. The fields around a classical point particle are infinite, and even worse, so is the total energy. As a result, when you try to calculate the self force on a classical point charge it is mathematically undefined.

So we have to get rid of the concept of a classical point particle. Maxwell’s equations still are used for photons, but the electron is no longer a classical point particle. It is instead a quantum mechanical particle. The specific infinities I mentioned above disappear since the electron no longer has infinite fields or an infinite charge density. Other infinities do arise, but they are renormalizable.

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    $\begingroup$ Thank you all for your replies. I'm going to look at the links sent and see if I understand them. I think I do understand your responses conceptually, at least. $\endgroup$ Aug 5, 2023 at 9:56
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If all this energy remains within the charge as kinetic energy, then the electromagnetic wave it emits due to its acceleration should not have any energy

Correct.

However, we observe that this is not the case. The EM waves contain energy, and the radiating charge loses kinetic energy, such as in a synchrotron.

Here it is important to point out that we observe EM waves due to many charges moving synchronously, and these carry away energy. In synchrotron, charged particles do not move individually, but in closely packed groups, so-called bunches, which can contain billions of charged particles. Synchrotron radiation energy flux is due to acceleration of the whole localized group of particles, it can't be extrapolated that if there was a single particle, the flux pattern would be similar. It is not ruled out experimentally that single electron would not produce such energy flux.

In other words, by the end of the acceleration, the net energy gained by the charge is not equal to $F_{0}d$; it is less. This seems to suggest that there is some other, potentially time-varying force (call it $\vec{F}(t)$) that has a component opposite the charge's motion, contributing to the total work done on the charge.

Where would this force come from, if it exists?

It comes from mutual interaction of different charged particles in the bunch, or other accelerated charged system of particles. This was shown by Abraham and Lorentz, who analyzed EM forces acting on extended charged distribution when in accelerated motion. It is an approximate calculation, but it shows that in addition to external force, the bunch moves as if there is also another force (due to different parts of the body acting on each other), which can be approximately expressed as a series of terms with increasing time derivatives of velocity. The first term is proportional to acceleration, and modifies the effective mass of the charged body. The other term is proportional to rate of change of acceleration, is sometimes called "radiation reaction", and has also earned the name Lorentz-Abraham-Dirac force (LAD force).

The Lorentz and Abraham calculations can't be made and the self-force can't be derived if the particle is a point - there are no interacting parts then (and the Poynting theorem can't be interpreted in terms of energy). Hence in EM theory the bunch as a whole does experience the LAD force (due to mutual interactions of parts), but if the bunch is made of point particles, the point particles themselves should not experience LAD forces, only EM forces due to the other point particles in the bunch. This point of view was analyzed by Tetrode, Fokker and Frenkel, and later by Feynman & Wheeler.

After all, if energy is conserved, it should in theory be possible to verify that the interaction simultaneously obeys Maxwell's Equations and conserves energy

Yes, and all experiments confirm this. However, these experiments all work with immense number of closely distributed charged particles moving in almost the same way. Accounting of energy flux due to single accelerated electron or effect on its trajectory is very hard, the hypothetical radiated energy/trajectory change is extremely small.

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