Pre-Quantum Electrodynamics
Maxwell's Equationsemd.Me.Me
Full set of equations for the electromagnetic field:
{\bf Maxwell's equations} {\it (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, \hspace{1cm} &\mbox{anonymous} \nonumber \\ (iii) {\boldsymbol \nabla} \times {\bf E} &= -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &\mbox{Faraday}, \nonumber \\ (iv) {\boldsymbol \nabla} \times {\bf B} &= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &\mbox{Ampère + Maxwell}. \label{Gr(7.39)} \end{align}Complement:
{\bf Force law} \[ {\bf F} = q ({\bf E} + {\bf v} \times {\bf B}). \label{Gr(7.40)} \]
These equations summarize the {\bf entire content of classical electrodynamics}.
\paragraph{Note:} even the continuity equation can be derived from Maxwell's equations: take divergence of \((iv)\).
Better way of writing: all fields on left, all sources on right,
\begin{align}
(i) &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
&(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
(ii) &{\boldsymbol \nabla} \cdot {\bf B} = 0,
&(iv) {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = \mu_0 {\bf J},
\label{Gr(7.42)}
\end{align}
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Created: 2022-02-15 Tue 10:14