Pre-Quantum Electrodynamics
Maxwell's Equationsemd.Me.Me
- PM 9.3
- Gr 7.3.3
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,
\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}
Except where otherwise noted, all content is licensed under a
Creative Commons Attribution 4.0 International License.
Created: 2024-02-27 Tue 10:31