Expansions in Terms of Moments of Time-Dependent, Moving Charges and Currents

Abstract

The time-dependent, co-moving, Cartesian moments of electric charge and current densities localized in a vicinity of a moving reference point are introduced. Using these co-moving moments, the static multipole expansion is generalized to the expansion of volume integrals of time-dependent, continuous, differentiable densities of electric charge and current. Relations implied by the continuity equation are determined between the co-moving moments of the current density and those of the charge density. Lorentz force, torque and power, due to external electromagnetic fields acting on time-dependent, moving, electric charges and currents, are evaluated in terms of their co-moving moments. Multipole expansions of the retarded Lorentz-gauge potentials of time-dependent, moving electric charges and currents are derived. The potentials that are related to the moving dipoles and quadrupoles in the same way as the Liénard–Wiechert potentials are related to a moving pointlike charge are computed explicitly. Relativistic properties of the co-moving moments are studied. The dipole and quadrupole, co-moving moments are computed in terms of the instantaneous-rest-frame, dipole and quadrupole, moments.