Group displayed equations with braces

Question

I would like to display a list of equations with some braces on the left to group some of them. Something that looks like the example below, with text on the left of each brace (aligned vertically), saying for instance "source-free" and "with sources", and both braces aligned with each other. Also, semantically, it would be better if this is done with a single environment instead of two successive ones.

I would prefer a solution using packages/features as standard as possible, and the least ad-hoc as possible, even if the alignments are not perfect.

(Please, no comment on the typesetting of div, rot, etc. The application I have in mind has nothing to do with Maxwell anyway!)

Thanks.

\documentclass{minimal}
\usepackage{empheq}
\begin{document}
Maxwell's equations:
\begin{empheq}[left=\empheqlbrace]{gather}
div B = 0 \tag{Thomson}\\
rot E + \frac{\partial B}{\partial t} = 0 \tag{Faraday}
\end{empheq}
\begin{empheq}[left=\empheqlbrace]{gather}
div E = \frac{\rho}{\epsilon_0} \tag{Gauss}\\
rot B - \frac1c^2 \frac{\partial E}{\partial t} = \mu_0 J \tag{Amp\`ere}
\end{empheq}
\end{document}

Update (2021-01-31): I accepted Bernard's answer which is satisfying. I would have hoped that an environment exists so that one can write:

\begin{gather}
\begin{group}[\text{source-free }\left\{]
a = 0\\
b = 0  
\end{group}
\begin{group}[\text{with sources }\left\{]
c = 0\\
d = 0  
\end{group}
\end{gather}

(since the two groups are connected (the two left braces should be aligned), some options should be passed at the level of the "gather" environment, probably).

Answer 1

You can use eqparbox. The label to use should be unique to the group of boxes to typeset with the same width.

I'd prefer left alignment, change to \eqmath[c]{A}{...} if you want centering.

\documentclass{article}
\usepackage{amsmath}
\usepackage{empheq}
\usepackage{eqparbox}

\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\Rot}{rot}

\newcommand{\eqmath}[3][l]{%
  % #1 = alignment, default l, #2 = label, #2 = math material
  \eqmakebox[#2][#1]{$\displaystyle#3$}%
}

\begin{document}

Maxwell's equations:
\begin{empheq}[left=\empheqlbrace]{align}
&\eqmath{A}{\Div B = 0\vphantom{\Big|}} \tag{Thomson}\\
&\eqmath{A}{\Rot E + \frac{\partial B}{\partial t} = 0} \tag{Faraday}
\end{empheq}
\begin{empheq}[left=\empheqlbrace]{align}
&\eqmath{A}{\Div E = \frac{\rho}{\epsilon_0}} \tag{Gauss}\\
&\eqmath{A}{\Rot B - \frac{1}{c^2} \frac{\partial E}{\partial t} = \mu_0 J} \tag{Amp\`ere}
\end{empheq}
\end{document}

Please, note that it should be

\frac{1}{c^2}

True, \frac12 works, but…

Actually, I'd prefer not to abuse \tag.

\documentclass{article}
\usepackage{amsmath}
\usepackage{eqparbox}

\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\Rot}{rot}

\newcommand{\eqmath}[3][l]{%
  % #1 = alignment, default l, #2 = label, #2 = math material
  \eqmakebox[#2][#1]{$\displaystyle#3$}%
}
\newcommand{\eqtext}[3][l]{%
  % #1 = alignment, default l, #2 = label, #2 = text material
  \eqmakebox[#2][#1]{#3}%
}

\begin{document}

Maxwell's equations:
\begin{gather*}
\left\{\begin{alignedat}{2}
&\eqmath{B}{\Div B = 0\vphantom{\Big|}} &\quad&\eqtext{C}{(Thomson)}\\
&\eqmath{B}{\Rot E + \frac{\partial B}{\partial t} = 0} &&\eqtext{C}{(Faraday)}
\end{alignedat}\right.
\\[1ex]
\left\{\begin{alignedat}{2}
&\eqmath{B}{\Div E = \frac{\rho}{\epsilon_0}} &\quad&\eqtext{C}{(Gauss)}\\
&\eqmath{B}{\Rot B - \frac{1}{c^2} \frac{\partial E}{\partial t} = \mu_0 J} &&\eqtext{C}{(Amp\`ere)}
\end{alignedat}\right.
\end{gather*}

\end{document}

With labels before the braces, both realizations:

\documentclass{article}
\usepackage{amsmath}
\usepackage{empheq}
\usepackage{eqparbox}

\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\Rot}{rot}

\newcommand{\eqmath}[3][l]{%
  % #1 = alignment, default l, #2 = label, #2 = math material
  \eqmakebox[#2][#1]{$\displaystyle#3$}%
}
\newcommand{\eqtext}[3][l]{%
  % #1 = alignment, default l, #2 = label, #2 = text material
  \eqmakebox[#2][#1]{#3}%
}

\begin{document}

\section{With empheq}

Maxwell's equations:
\begin{empheq}[left={\makebox[4em][r]{Source-free }\empheqlbrace}]{align}
&\eqmath{A}{\Div B = 0\vphantom{\Big|}} \tag{Thomson}\\
&\eqmath{A}{\Rot E + \frac{\partial B}{\partial t} = 0} \tag{Faraday}
\end{empheq}
\begin{empheq}[left={\makebox[4em][r]{With sources }\empheqlbrace}]{align}
&\eqmath{A}{\Div E = \frac{\rho}{\epsilon_0}} \tag{Gauss}\\
&\eqmath{A}{\Rot B - \frac{1}{c^2} \frac{\partial E}{\partial t} = \mu_0 J} \tag{Amp\`ere}
\end{empheq}

\section{No empheq}

Maxwell's equations:
\begin{gather*}
\eqtext[r]{1}{Source-free }
\left\{\begin{alignedat}{2}
&\eqmath{B}{\Div B = 0\vphantom{\Big|}} &\quad&\eqtext{C}{(Thomson)}\\
&\eqmath{B}{\Rot E + \frac{\partial B}{\partial t} = 0} &&\eqtext{C}{(Faraday)}
\end{alignedat}\right.
\\[1ex]
\eqtext[r]{1}{With sources }
\left\{\begin{alignedat}{2}
&\eqmath{B}{\Div E = \frac{\rho}{\epsilon_0}} &\quad&\eqtext{C}{(Gauss)}\\
&\eqmath{B}{\Rot B - \frac{1}{c^2} \frac{\partial E}{\partial t} = \mu_0 J} &&\eqtext{C}{(Amp\`ere)}
\end{alignedat}\right.
\end{gather*}

\end{document}

Answer 2

If you don't need to cross-reference the equations, you have a solution with the bigdelim package and tabularx. I added the esdiff package for an easy typing of partial derivatives, and cellspace to add some vertical padding to the table rows:

\documentclass{article}
\usepackage{esdiff, empheq}
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\rot}{rot}
\usepackage{tabularx, bigdelim}
\usepackage[column=O]{cellspace}
\setlength{\cellspacetoplimit}{4pt}
\setlength{\cellspacebottomlimit}{4pt}

\begin{document}

Maxwell's equations:

\noindent\begin{tabularx}{\textwidth}{@{}Xr @{}>{$\displaystyle}Ol<{$}>{\raggedleft\arraybackslash}X @{}}
 &\ldelim\{{3}{*}[Source-free~] & \Div B = 0 & \text{(Thomson)}\\
 & & \rot E + \diffp{B}{t} = 0 &\text{(Faraday)} \\ \strut \\
 &\ldelim\{{4}{*}[With sources~] & \Div E = \frac{\rho}{\epsilon} & \text{(Gauss)}\\
 & & \rot B - \frac1{c^2} \diffp{E}{t} = \mu_0 J &\text{(Ampère)}
\end{tabularx}

\end{document} 

Answer 3

This is a solution I came up with, it's not perfect:

\documentclass{minimal}
\usepackage{amsmath}
\usepackage{cases}

\begin{document}
Maxwell's equations:


\begin{numcases}{source-free = }
    div B &= 0 \tag{Thompson} \\
    rot E + \frac{\partial B}{\partial t} &= 0 \tag{Faraday}
\end{numcases} 
\begin{numcases}{With sources  = }
    div E &= $\frac{\rho}{\epsilon}$ \tag{Gauss}\\
    rot B - \frac1c^2 \frac{\partial E}{\partial t} &= $\mu_0$ J \tag{Amp\`ere} 
\end{numcases}

\end{document}

Answer 4

With the lastest version of nicematrix (v. 5.10 of 2021-02-05), you can use the environment {NiceTabular*} and add the braces with the built-in command \SubMatrix. All the equations are aligned within the same environment.

\documentclass{article}
\usepackage{nicematrix}
\usepackage{esdiff}
\usepackage{lipsum}
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\rot}{rot}

\NiceMatrixOptions{cell-space-limits=4pt}

\begin{document}

\lipsum[1]

Maxwell's equations:

\begingroup
\everymath{\displaystyle}
\noindent
\begin{NiceTabular*}{\textwidth}{r@{\qquad}l@{\extracolsep{\fill}}r@{}}
\Block{2-1}{\hspace{2cm}Source-free~}  
              & $\Div B = 0$                                  & (Thomson) \\
              & $\rot E + \diffp{B}{t} = 0$                   & (Faraday) \\ 
\strut \\
\Block{2-1}{With sources~} 
             & $\Div E = \frac{\rho}{\epsilon}$              & (Gauss) \\
             & $\rot B - \frac1{c^2} \diffp{E}{t} = \mu_0 J$ & (Ampère)
\CodeAfter
\SubMatrix\{{1-2}{2-2}.
\SubMatrix\{{4-2}{5-2}.
\end{NiceTabular*}
\endgroup

\end{document} 

You need several compilations (because nicematrix uses PGF/Tikz nodes).