A time domain energy theorem for scattering of plane electromagnetic waves

A time domain energy theorem for the scattering of plane electromagnetic waves by an obstacle of bounded extent is derived. It is the counterpart in the time domain of the "optical theorem" or the "extinction cross section theorem" in the frequency domain. No assumptions as to the electromagnetic behavior of the obstacle need to be made; so, the obstacle may be electromagnetically nonlinear and/or time variant (a kind of behavior that is excluded in the frequency domain result). As to the wave motion, three different kinds of time behavior are distinguished: (1) transient, (2) periodic, and (3) perpetuating, but with finite mean power flow density. For all three cases the total energy (case 1) or the time-averaged power (cases 2 and 3) that is both absorbed and scattered by the obstacle is related to a certain time interaction integral of the incident plane wave and the spherical-wave amplitude of the scattered wave in the far-field region, when observed in the direction of propagation of the incident wave. The practical implications of the energy theorem are briefly indicated.