Energy Conservation in a Charged Retarded Field Engine

Energy conservation, rooted in the time invariance of physical laws and formalized by Noether’s theorem, requires that systems with space-time translational symmetry conserve momentum and energy. This work examines how this principle applies to a charged retarded field engine, where the rate of change of total energy—mechanical plus field energy—is balanced by the energy flux through the system’s boundary. Using electric and magnetic field expressions from a Taylor expansion to incorporate retardation effects, we analyze the energy equation order by order for two arbitrary charged bodies. Our results show that total energy is conserved up to the fourth order, with mechanical and field energy changes exactly offset by boundary energy flux. Consequently, the work done by the internal electromagnetic field precisely equals the engine’s gained mechanical kinetic energy, addressing the central focus of this study.