Theorem_maxwell_statics_solution.tex

%
% Copyright © 2018 Peeter Joot.  All Rights Reserved.
% Licenced as described in the file LICENSE under the root directory of this GIT repository.
%
\maketheorem{Maxwell's statics solution.}{thm:statics:100}{
The solution to the Maxwell statics equation is given by
\begin{equation*}
F(\Bx)
= \inv{4\pi} \int_V dV' \frac{\gpgrade{(\Bx - \Bx') J(\Bx')}{1,2}}{\Norm{\Bx - \Bx'}^3} + F_0,
\end{equation*}
where \( F_0 \) is any function for which \( \spacegrad F_0 = 0 \).
The explicit expansion in electric and magnetic fields and charge and current densities is given by
\begin{equation*}
\begin{aligned}
\BE(\Bx)
&=
\inv{4\pi} \int_V dV' \inv{\Norm{\Bx - \Bx'}^3}
\lr{
{\color{DarkOliveGreen}
   \inv{\epsilon}(\Bx - \Bx') \rho(\Bx')
}
   +
   (\Bx - \Bx') \cross \BM(\Bx')
} \\
\BH(\Bx)
&=
\inv{4\pi} \int_V dV' \inv{\Norm{\Bx - \Bx'}^3}
\lr{
{\color{Maroon}
  \BJ(\Bx') \cross (\Bx - \Bx')
}
+ \inv{\mu} (\Bx - \Bx') \rho_m(\Bx')
}.
\end{aligned}
\end{equation*}
} % theorem