| Version 22 |
Version 21 |
| Maxwell's equations are a set of four partial differential equations first combined by James Clerk Maxwell. They may also be written as integral equations. Two other important equations, the electromagnetic wave equation and the equation of conservation of charge, may be derived from them. |
Maxwell's equations are a set of four partial differential equations first combined by James Clerk Maxwell. They may also be written as integral equations. Two other important equations, the electromagnetic wave equation and the equation of conservation of charge, may be derived from them. |
| \subsection{Notation} |
\subsection{Notation} |
| As this article considers merely the mathematical aspects of the equations, natural units have been used throughout. For their use in physics see any classical electromagnetism textbook. |
As this article considers merely the mathematical aspects of the equations, natural units have been used throughout. For their use in physics see any classical electromagnetism textbook. |
| \[ |
\[ |
| \mathbf{E} = \mbox{Electric field strength} |
\mathbf{E} = \mbox{Electric field strength} |
| \] |
\] |
| \[ |
\[ |
| \mathbf{B} = \mbox{Magnetic flux density} |
\mathbf{B} = \mbox{Magnetic flux density} |
| \] |
\] |
| \subsection{Gauss' Law of Electrostatics} |
\subsection{Gauss' Law of Electrostatics} |
| Differential form |
Differential form |
| \[ |
\[ |
| \nabla \cdot \mathbf{E} = 0 |
\nabla \cdot \mathbf{E} = 0 |
| \] |
\] |
| Integral form |
Integral form |
| \[ |
\[ |
| \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{S} = 0 |
\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{S} = 0 |
| \] |
\] |
| \subsection{Gauss' Law of Magnetostatics} |
\subsection{Gauss' Law of Magnetostatics} |
| \[ |
\[ |
| \nabla \cdot \mathbf{B} = 0 |
\nabla \cdot \mathbf{B} = 0 |
| \] |
\] |
| \[ |
\[ |
| \oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0 |
\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0 |
| \] |
\] |
| This law can be interpreted as a statement of the non-existence of magnetic monopoles, a fact confirmed by all experiments to date. |
This law can be interpreted as a statement of the non-existence of magnetic monopoles, a fact confirmed by all experiments to date. |
| \subsection{Faraday's Law} |
\subsection{Faraday's Law} |
| Differential form |
Differential form |
| \[ |
\[ |
| \nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}}{\partial t} |
\nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}}{\partial t} |
| \] |
\] |
| Integral form |
|
| \[ |
|
| \oint_{C} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d}}{\mathrm{d} t} \left( \int_{S} \mathbf{B} \cdot \mathrm{d}\mathbf{A} \right) |
|
| \] |
|
| \subsection{Amp\`ere's Law} |
\subsection{Amp\`ere's Law} |
| Differential form |
Differential form |
| \[ |
\[ |
| \nabla \times \mathbf{B} = -\frac{ \partial \mathbf{E}}{\partial t} |
\nabla \times \mathbf{B} = -\frac{ \partial \mathbf{E}}{\partial t} |
| \] |
\] |
| Integral form |
Integral form |
| \[ |
|
| \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \int_S \frac{\partial \mathbf{E}}{\partial t} \cdot \mathrm{d} \mathbf{A} |
|
| \] |
|
| \subsection{Properties of Maxwell's Equations} |
\subsection{Properties of Maxwell's Equations} |
| These four equations together have several interesting properties: |
These four equations together have several interesting properties: |
| \begin{itemize} |
\begin{itemize} |
| \item Lorentz invariance |
\item Lorentz invariance |
| \item Gauge invariance |
\item Gauge invariance |
| \item Derivation from an appropriate Lagrangian |
\item Derivation from an appropriate Lagrangian |
| \item In natural units, the equations are symmetric in $\mathbf{E}$ and $\mathbf{B}$. |
\item In natural units, the equations are symmetric in $\mathbf{E}$ and $\mathbf{B}$. |
| \end{itemize} |
\end{itemize} |
