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The electromagnetic field itself contains energy distinct from the energy of charged bodies, the energy in a given volume of empty space can be found by integrating the energy densities $\frac{1}{2}\epsilon E^2$ and $\frac{1}{2} \frac{B^2}{\mu}$ over the region. When the EM fields increase the kinetic energy of charged particles, there is a corresponding decrease in the energy of the EM field in that region, so total energy is unchanged. The general proof that any combination of fields and charges obeying Maxwell's equations will conserve energy is known as Poynting's Theorem, proved for example on pages 346-348 of Introduction to Electrodynamics, Third Edition by David J. Griffiths, or on this page from physicspages.com