Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles

An exact calculation of the radiation intensity, angular momentum loss, and the recoil force for the most general type of source, characterized by electric, magnetic, and toroid multipole moments and radii of any multipolarity and an arbitrary time dependence, is presented. The results are expressed in terms of time derivatives of the multipole moments and mean radii of the corresponding distributions. Although quite cumbersome, the formulas found by us represent exact results in the correct multipole analysis of configurations of charges and currents that contain toroidal sources. So the longstanding problem in classical electrodynamics of relating the radiation properties of a system to quantities completely describing its internal electromagnetic structure is thereby exactly solved. By particularizations to the first multipole contributions, corrections to the familiar formulas from books are found, mostly on account of the toroid moments and their interference with the usual electric and magnetic ones.