A particle with charge q and mass m moves in static external fields E₀(r) and B₀(r). If ϕ₀(r_q) is the electrostatic potential at the position of the particle, and there is negligible radiation, the usual statement of conservation of energy for this system is

Show that this equation agrees with the power balance represented by Poynting’s theorem only if one adds to the latter the power supplied by an external energy source to maintain E0 and B0 against the work done on their sources by the fields produced by the particle. Why is the latter work non-zero?

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1 2 mu nv² +990(ra) = const.