PlanetPhysics/Maxwell's Equations

Maxwell's equations are a set of four partial differential equations first combined by James Clerk Maxwell. Together, they completely describe classical electromagnetic phenomena, just as Newton's laws completely describe classical mechanical phenomena. All four are named after persons other than Maxwell, but Maxwell was the first to add the displacement current term to Amp\`ere's Law, which led to the association of electromagnetic waves with light and paved the way for the discovery of special relativity. All four equations can be written in both integral and differential forms, with both forms convenient for specific problems. Note that strictly speaking these are Maxwell's equation in vacuo, with different forms for interaction with matter.

Notation
Throughout this article SI units are adopted for clarity, but the interesting mathematical aspects of the equations are independent of the constants $$\mu_0$$ and $$\epsilon_0$$, and indeed of the physical meaning of the equations. $$ \mathbf{E} = \mbox{Electrical field strength, SI units Volt m}^{-1} $$ $$ \mathbf{B} = \mbox{Magnetic flux density, SI units Tesla} $$ $$ \mathbf{J} = \mbox{Current density, SI units Amp\`ere m}^{-3} $$ $$ \mathbf{\epsilon_0} = \mbox{Permittivity of free space} \approx 8.85 \times 10^{-12} \mbox{m}^{-1} $$ $$ \mathbf{\mu_0} = \mbox{Permeability of free space} = 4\pi \times 10^7 \mbox{Henry m}^{-1} $$

Gauss' Law of Electrostatics
Differential form $$ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} $$ Integral form $$ \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {q}{\epsilon_0} $$ where $$q$$ is the charge enclosed in the volume bounded by the surface $$S$$.

Gauss' Law of Magnetostatics
$$ \nabla \cdot \mathbf{B} = 0 $$ $$ \oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0 $$ This law can be interpreted as a statement of the non-existence of magnetic monopoles, a fact confirmed by all experiments to date.

Faraday's Law
Differential form $$ \nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}}{\partial t} $$

Amp\`ere's Law
Differential form $$ \nabla \times \mathbf{B} = - \mu_0 \epsilon_0 \frac{ \partial \mathbf{E}}{\partial t} $$ Integral form

Properties of Maxwell's Equations
These four equations together have several interesting properties:


 * Lorentz invariance
 * Gauge invariance
 * Invariance under the transformation $$B \rightarrow \frac{E}$$, $$E \rightarrow B c$$