Theorem_fields_and_potential_wave_equations.tex

%
% Copyright © 2018 Peeter Joot.  All Rights Reserved.
% Licenced as described in the file LICENSE under the root directory of this GIT repository.
%
\maketheorem{The potential wave equations.}{thm:generalPotential:40}{
%%%In terms of the potential components, the electric field vector and the magnetic field bivector are
%%%\begin{equation*}
%%%\begin{aligned}
%%%\BE &=
%%%\gpgrade{\conjstgrad A}{1}
%%%=
%%%   - \spacegrad \phi
%%%   - \PD{t}{\BA}
%%%   - \inv{\epsilon} \spacegrad \cross \BF \\
%%%I \eta \BH &=
%%%\gpgrade{\conjstgrad A}{2}
%%%=
%%%   I \eta
%%%   \lr{
%%%      - \spacegrad \phi_\txtm
%%%      - \PD{t}{\BF}
%%%      + \inv{\mu} \spacegrad \cross \BA
%%%   }
%%%.
%%%\end{aligned}
%%%\end{equation*}
The potentials are related to the sources by
\begin{equation*}
\begin{aligned}
\dAlembertian
\phi &= -\frac{\rho}{\epsilon} - \PD{t}{} \lr{ \spacegrad \cdot \BA + \inv{c^2} \PD{t}{\phi} } \\
\dAlembertian
\BA &= -\mu \BJ + \spacegrad \lr{ \spacegrad \cdot \BA + \inv{c^2} \PD{t}{\phi} } \\
\dAlembertian
\BF &= - \epsilon \BM + \spacegrad \lr{ \spacegrad \cdot \BF + \inv{c^2} \PD{t}{\phi_\txtm} } \\
\dAlembertian
\phi_\txtm &= -\frac{\rho_\txtm}{\mu} - \PD{t}{} \lr{ \spacegrad \cdot \BF + \inv{c^2} \PD{t}{\phi_\txtm} }.
\end{aligned}
\end{equation*}
Reminder: (\(\dAlembertian\): see \cref{dfn:continuity:120})
} % theorem