Abstract
A family of solutions of the Lorentz–Dirac equation is constructed. It consists in the motion of two charges e1 and e2 of masses m1 and m2 in two coplanar and concentric circles of radii a and b. The charges rotate with constant angular velocity, and have an angular separation ψ. The radiation reaction forces and the retarded interactions between the charges are taken into account. The external electromagnetic field that allows the motion consists of a tangential time-independent electric field that takes a fixed value on each orbit, and a homogeneous time-independent magnetic field perpendicular to the plane of the motion. For all the solutions energy conservation is rigorously demonstrated by evaluating the energy radiated, with independence of the equation of motion, through the calculation of the instantaneous energy flux across a sphere of an infinitely large radius.
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