%$Id: notes.tex,v 1.105 2023/02/27 07:55:35 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
\usepackage{graphicx,natbib,bm,url,color}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input macros
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{}
\author{}
\date{\today,~ $ $Revision: 1.105 $ $}
\begin{document}
%\maketitle

\section{cgs}

\begin{equation}
\nab\times\BB=\frac{4\pi}{c}\left(\JJ_{\rm ohm}+\JJ_\Theta\right)
\end{equation}
\begin{equation}
\JJ_{\rm ohm}=\sigma\left(\EE+\frac{1}{c}\uu\times\BB\right)
\end{equation}
\begin{equation}
\JJ_\Theta=c\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\frac{4\pi}{c}\JJ_\Theta=4\pi\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\label{4picJ}
\end{equation}
\begin{equation}
\nab\times\BB=\frac{4\pi\sigma}{c^2}\left(c\EE+\uu\times\BB\right)
+4\pi\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\nab\times\BB=\frac{4\pi\sigma}{c^2}\left(c\EE+\uu\times\BB\right)
+\frac{4\alpha}{\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\nab\times\BB=\frac{4\pi\sigma}{c^2}\left(-\frac{\partial\AAA}{\partial t}+\uu\times\BB\right)
+\frac{4\alpha}{\hbar c}\tilde{\mu}_5\BB
\end{equation}
Define
\begin{equation}
\mu_5=\frac{4\alpha}{\hbar c}\tilde{\mu}_5
\end{equation}
\begin{equation}
\frac{\partial\AAA}{\partial t}=\uu\times\BB
+\frac{c^2}{4\pi\sigma}\left(\mu_5\BB-\nab\times\BB\right)
\end{equation}

\newpage

\section{LH}

\begin{equation}
\nab\times\BB=\frac{1}{c}\left(\JJ_{\rm ohm}+\JJ_\Theta\right)
\end{equation}
\begin{equation}
\JJ_{\rm ohm}=\sigma\left(\EE+\frac{1}{c}\uu\times\BB\right)
\end{equation}
\begin{equation}
\JJ_\Theta=c\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\frac{1}{c}\JJ_\Theta=\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\nab\times\BB=\frac{\sigma}{c^2}\left(c\EE+\uu\times\BB\right)
+\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\nab\times\BB=\frac{\sigma}{c^2}\left(c\EE+\uu\times\BB\right)
+\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
\begin{equation}
\nab\times\BB=\frac{\sigma}{c^2}\left(-\frac{\partial\AAA}{\partial t}+\uu\times\BB\right)
+\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
Define
\begin{equation}
\mu_5=\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5
\end{equation}
\begin{equation}
\frac{\partial\AAA}{\partial t}=\uu\times\BB
+\frac{c^2}{\sigma}\left(\mu_5\BB-\nab\times\BB\right)
\end{equation}

\section{Comments}

Eq.~(24) of Alekseev+98 corresponds to
\begin{equation}
\frac{4\pi}{c}\JJ_{k}=c\frac{\alpha}{\pi\hbar c}\tilde{\mu}_5\BB
\end{equation}
which suggests cgs units, but doesn't agree with \Eq{4picJ}.

%\begin{figure}[h!]\begin{center}
%\includegraphics[width=\columnwidth]{X}
%\end{center}\caption[]{
%\url{X}
%}\label{X}\end{figure}

%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%$R$ & $T$ & $\kf$ & $\omf$ & $\EEGW$ & $\hrms$ & $k_{\rm peak}$ \\
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

%r e f
%\begin{thebibliography}{}

%\bibitem[Biskamp \& M\"uller(1999)]{BM99}
%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

%\end{thebibliography}

%\vfill\bigskip\noindent\tiny\begin{verbatim}
%$Header: /var/cvs/brandenb/tex/etc/notes.tex,v 1.105 2023/02/27 07:55:35 brandenb Exp $
%\end{verbatim}

\end{document}
