Theorem_complex_power.tex

%
% Copyright © 2018 Peeter Joot.  All Rights Reserved.
% Licenced as described in the file LICENSE under the root directory of this GIT repository.
%
\maketheorem{Complex power representation.}{thm:poyntingFComplexPower:300}{
Given a time domain representation of a phasor based field \( F = F(\omega) \)
\begin{equation*}
F(t)
= \Real\lr{ F e^{j \omega t} },
\end{equation*}
the energy momentum tensor multivector \( T(1) \) has the representation
\begin{equation*}
T(1) = \calE + \frac{\BS}{c}
=
\frac{\epsilon}{4} \Real \lr{ F^\conj F^\dagger + F F^\dagger e^{2 j \omega t} }.
\end{equation*}
With the usual definition of the complex Poynting vector
%\label{eqn:poyntingFComplexPower:240}
\begin{equation*}
\calS = \inv{2} \BE \cross \BH^\conj = \inv{2} \lr{ I \BH^\conj } \cdot \BE,
\end{equation*}
the energy and momentum components of \( T(1) \), for real \( \mu, \epsilon \) are
%\label{eqn:poyntingFComplexPower:260}
\begin{equation*}
\begin{aligned}
\calE &=
\inv{4} \lr{
\epsilon \Abs{\BE}^2 + \mu \Abs{\BH}^2 }
+
\inv{4} \Real
\lr{
   \lr{ \epsilon \BE^2 + \mu \BH^2}
   e^{2 j \omega t }
} \\
\BS &= \Real \calS
+
\inv{2} \Real
\lr{
\lr{ \BE \cross \BH }
   e^{2 j \omega t }
}.
\end{aligned}
\end{equation*}
} % theorem