Energy conservation in Electrodynamics

Question

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?

Answer 1

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

Answer 2

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.

Answer 3

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.

Answer 4

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.