Internal Energy, Q-Energy, Poynting's Theorem, and the Stress Dyadic in Dispersive Material

Abstract

General expressions are derived for time-domain energy density and the time integral of the Poynting vector that are related to the kinetic, potential, and heat energy densities of the bound charge-polarization carriers and the stored electromagnetic field energy density in passive, nonlinear or linear, lossy or lossless, temporally and/or spatially dispersive polarized media. In the most general linear, lossless, spatially nondispersive media, the energy expressions reveal non-negative quadratic forms for the frequency-domain internal energy densities that are used in expressions for the quality factor (Q) of antennas containing lossless dispersive material. The analysis also reveals useful inequalities that imply that the magnitude of the group velocity in lossless material is always less than or equal to the speed of light. The energy expressions do not, however, predict an internal energy in highly lossy temporally dispersive media that can be used to improve upon the expressions for the Q of antennas in such media. To improve upon the accuracy of the Q of antennas with highly lossy dispersive media, a non-negative Q-energy density is found that maintains the accuracy of the inverse relationship between Q and matched VSWR half-power fractional bandwidth for antennas containing highly lossy dispersive material. Lower bounds for this improved Q are found in terms of previously determined lower bounds. The paper also confirms the result that, for general linear lossy or lossless dispersive material, the steady state time averages of the electromagnetic power density, force density, and stress dyadic with sinusoidal time dependence that turns on at some finite time in the past, unlike the internal energy, does not contain derivatives with respect to frequency.