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Finally, rearranging and simplifying the terms in (2.71) leads to

(2.72a)

      (2.72b)

      where the superscript corresponds to the transpose conjugate operation. The final expression of the time-average bianisotropic Poynting theorem for time-harmonic fields is then given by

(2.73)

where , and , are, respectively, provided by (2.69) and (2.72).

Integrating (2.73) over a volume defined by the surface , and applying the divergence theorem to the resulting left-hand side transforms this relation into

(2.74)

      which indicates that the amount of loss or gain that an electromagnetic wave experiences in a given volume surrounding a medium is related to the amount of energy passing across the surface delimiting this volume.

If the medium is perfectly lossless and gainless, the amounts of electromagnetic energy entering and exiting the medium are equal, so that in (2.74), and therefore in (2.73). In the case of a lossy medium, there is less energy leaving than entering the volume, corresponding to , and vice versa for a gain medium, i.e. . Note that the fact that is a necessary but not sufficient condition for the medium to be gainless and lossless, since gain–loss compensation may occur between the terms on the right-hand side of (2.73).

      Substituting (2.69) and (2.72) into (2.73) provides the following alternative form of the Poynting theorem:

(2.75)

In the absence of impressed surface currents, i.e. , then (2.75) reduces to

(2.76)

2.6 Energy Conservation in Lossless–Gainless Systems

From a practical perspective, it is often desirable to implement gainless and lossless systems. This specification requires certain conditions to be satisfied, which may be

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