#### Maxwell's Equations
emd.Me.Me

Full set of equations for the electromagnetic field:

Maxwell's equations (in vacuum)

\begin{align} (i)~ {\boldsymbol \nabla} \cdot {\bf E} &= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ (ii)~ {\boldsymbol \nabla} \cdot {\bf B} &= 0, &\mbox{anonymous} \nonumber \\ (iii)~ {\boldsymbol \nabla} \times {\bf E} &= -\frac{\partial {\bf B}}{\partial t}, &\mbox{Faraday}, \nonumber \\ (iv)~ {\boldsymbol \nabla} \times {\bf B} &= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, &\mbox{Ampère + Maxwell}. \tag{Max_vac}\label{Max_vac} \end{align}

Complement:

Force law LorFo \[ {\bf F} = q ({\bf E} + {\bf v} \times {\bf B}). \label{Gr(7.40)} \]

These equations contain the entirety of pre-quantum electrodynamics.

Note: even the continuity equation can be derived from Maxwell's equations: take divergence of \((iv)\) and use \((i)\).

Better way of writing: all fields on left, all sources on right,

- Gr (7.42)

\begin{align} (i)~ {\boldsymbol \nabla} \cdot {\bf E} &= \frac{\rho}{\varepsilon_0}, \nonumber \\ (ii)~{\boldsymbol \nabla} \cdot {\bf B} &= 0, \nonumber \\ (iii)~ {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} &= 0, \nonumber \\ (iv)~ {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} &= \mu_0 {\bf J}, \tag{Max_vac_s}\label{Max_vac_s} \end{align}

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42