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Let us suppose that we have a known electromagnetic wave-train of finite size propagating in a certain direction. There is a probe charge on its way. This EMW is an external field for the charge. The EMW has a certain energy-momentum (integral over the whole space). After acting on the probe charge the wave continues its way away. In the end we have the energy of the initial wave (displaced somewhere father), the kinetic energy of the charge (hopefully it starts moving), and the energy of the radiated EMF propagating in other directions. Thus the total energy may become different from the initial one. How to show that the total energy is conserved in this case?

It is not a Compton scattering. Just a regular electrodynamics problem. How EM energy can change appropriately? Via destructive interference? How to show it if the incident field is a known function of space-time?

EDIT: I can emit a half-period long wave from a radio-transmitter:$E(t)=E_0 sin(\Omega t), 0 < t < \pi/\Omega $. Then the final charge velocity will be clearly different from zero:

$ma=F(t), v(t>\pi/\Omega)=\int_{0}^{t}F(t')dt'=\frac{2qE_0}{m\Omega}$.

In addition, the charge itself radiates some new wave during acceleration period. The radiated energy is only a small fraction of $\frac{mv^2}{2}$. What can guarantee that the total energy remains the same?

EDIT 2: OK, let us simplify the task. I wonder if there is a simplest problem in CED where the total energy with a radiating charge is conserved explicitly?

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When you say "integral over the space", is it all space? If the "electromagnetic wave-train of finite size propagating in a certain direction" there must be regions of the 'all space' where EM is null. I think that in the 'physical world' the limits of integrations must be bounded to some value, depending on the lifetime of the EM radiation. –  Helder Velez Apr 26 '11 at 21:52
Yes, I mean something quite ordinary, like a pulse from a radar or so. –  Vladimir Kalitvianski Apr 26 '11 at 22:13
You added a bounty, but it is not clear why you feel the answers are not sufficient. Start with the Lagrangian and apply Noether's theorem using time translation. Griffith's even does this a more elementary way. What exactly are you not understanding that you feel the current answers don't address? –  Edward May 6 '11 at 7:23
@Edward For a given explicit solution the energy conservation should be obtained explicitly. –  Vladimir Kalitvianski May 6 '11 at 8:40
@Edward Griffith proceeds from the Maxwell equations coupled to the Lorentz equations. He implies that the coupled system has physical solutions. But even in case of one point-like charge the system does not have physical solutions. All solutions obtained in the textbooks are approximate. My solutions for the incident wave and for the probe charge are physical but I do not see that the destructive interference can "eat out" exactly the same energy as was transmitted to the charge. –  Vladimir Kalitvianski May 6 '11 at 12:53
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4 Answers

I can't help thinking you are needlessly complicating the physics by focussing on the transient case. In the steady state it is much easier to do the energy balancing. Of course there are several possible versions of this problem in the "steady state": you could have a free charge which is allowed to pick up speed in the direction of the propagating wave (via v X B forces), a free charge which is constrained to move only perpendicular to the propagating wave motion, or a charge on a spring which may oscillate in resonance with the excitation. I am most familiar with the third version: the driven oscillator. In the steady state, there is energy which is scattered everywhere, and there is a "shadow zone" behind the oscillator, roughly paraboloid in shape, where the secondary radiation interferes destructively with the incident wave. The amount of destructive interference is linear in the oscillator strength, and the spherically scattered radiation is of course quadratic. The balance occurs when the linear term is equal to the quadratic term.

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The stress energy tensor $T^{\mu\nu}$ contains all the energy/momentum components of the elctromagnetic field and the conservation of these components is expressed by

$\partial_{\nu}T^{\mu \nu} = 0$

Which states that the change in time of energy/momentum is zero. If the above is non-zero then electromagnetic field energy/momentum is transferred to charged matter and in this case the conservation law becomes.

$\partial_{\nu}T^{\mu \nu} + \eta^{\mu \rho} \, f_{\rho} = 0$

Where $f$ is the force density four vector acting on the charge matter. If we talk specifically about the energy, as in your case, then $f_o$ is given by $\vec{J}\cdot\vec{E}~$ representing charged matter moving up or down a potential field which causes a change in time of the potential energy of the charged matter.

Regards, Hans

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I think you've engaged with Vladimir more than I did, so +1, and Welcome to Physics SE. –  Peter Morgan Apr 25 '11 at 14:55
Welcome to PSE Hans –  Helder Velez Apr 25 '11 at 20:41
@Peter, Thanks for the welcome! –  Hans de Vries Apr 25 '11 at 21:32
@Helder Obrigado pelas boas-vindas e também pelas suas palavras positivas sobre alguns dos meus trabalhos. –  Hans de Vries Apr 25 '11 at 21:36
Fiquei sem palavras. Como é possível? Os elogios são poucos porque o seu trabalho é excelente. –  Helder Velez Apr 26 '11 at 15:11
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The proof that classical electrodynamics conserves energy is found in all sorts of textbooks. I'd start with Griffiths's Introduction to Electrodynamics, and go on to Jackson's Classical Electrodynamics if you want more.

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Although not incorrect, it is a lazy answer that anyone could have given. –  Larry Harson Apr 26 '11 at 16:11
I am afraid the notion of an external field (=solution of Maxwell equations) is incompatible with the energy conservation law (see my edit). –  Vladimir Kalitvianski May 5 '11 at 22:34
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In order for energy to be conserved in a model, one must use a translation invariant formalism, then, with other assumptions about the space of models, one can prove Noether's theorem. As soon as one says that some part of the interaction is "external", one is working in a formalism in which energy will not be conserved. Making the EMW internal to the system is your first step.

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Does that mean the incident field may "create" energy? –  Vladimir Kalitvianski Apr 25 '11 at 13:24
Yes. Short answer--the radiation coming from the particle interferes destructively with the plane wave. –  Jerry Schirmer Apr 25 '11 at 13:24
There is conservation of energy if the model dynamics is time translation invariant (again, assuming various other structures to be able to prove Noether's theorem or something like it). Since the middle 19th Century, it's effectively been said (I think of this as either a philosophical or terminological choice) that something that is not by definition conserved is not energy. In practical calculation this can be fudged if you like. Ad hoc non-translation invariant dynamics are used all the time, but it would usually be said that the details could be filled in if we wanted to spend the time. –  Peter Morgan Apr 25 '11 at 14:03
But the problem is time translation invariant. A transmitter emits a wave-train for a certain time and this wave propagates as a known solution, doesn't it? Then somewhere the wave encounters a probe charge and interacts with it. As a result we will have a new total field and a moving charge. This phenomena is not sensitive to the value of the initial time and this is called time translation invariance. For some reason I cannot figure out how to prove that the total energy is conserved. Should I integrate the equations numerically for that? –  Vladimir Kalitvianski May 3 '11 at 20:51
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