Moving pointlike charges and electric and magnetic dipoles

Expansions of the volume integrals of strongly localized, continuous, differentiable electric charge and current densities are introduced in terms of the total charge, and variable and moving electric and magnetic dipoles, and conditions under which such expansions are applicable are determined. To enhance an understanding of the effects of external electromagnetic fields acting on very localized densities of electric charges and currents, the resulting Lorentz force, torque, and power are computed by using such expansions. To show how versatile variable and moving dipoles are for building physical models, the following are considered: (a) energy conversions by electric motors, by dielectric and induction heating, and by the magnetic drag force, and (b) relativistic effects in translational movements of steady electric and magnetic dipoles; in particular, the torgue on a magnetic dipole moving parallel to a line carrying a steady current and uniformly distributed electric charges.