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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