Maxwell's Equations

Four equations govern all electrical, magnetic, and optical phenomena, revealing that light is electromagnetism.

Equation	Name & Meaning
$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}$	Gauss's Law: Charges create electric fields.
$\nabla\cdot\vec{B} = 0$	No Monopoles: Magnetic field lines form closed loops.
$\nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}$	Faraday's Law: Changing magnetic fields induce electric fields.
$\nabla\times\vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$	Ampere-Maxwell: Currents and changing electric fields create magnetic fields.

Displacement current ( $\varepsilon_0\frac{\partial\vec{E}}{\\partial t}$ ) was added by Maxwell to Ampere's law, making wave propagation possible in vacuum.

A wave that needs no charge, no current, and no loop

Electromagnetic Waves

Maxwell's equations predict self-sustaining waves travelling at $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00\times 10^8\;\text{m/s}$ .

Property	Expression
Speed	$c = f\lambda$
$E/B$ ratio	$E = cB$
Energy density	$u = \varepsilon_0 E^{2} = B^{2}/\mu_0$
Poynting vector	$\vec{S} = \frac{1}{\mu_0}\vec{E}\times\vec{B}$

Common pitfall: An electromagnetic wave needs no medium. The $\vec{E}$ and $\vec{B}$ fields are in phase, mutually perpendicular, and both perpendicular to travel direction. Do not draw them out of phase.

Electromagnetic Waves

Practise this lesson

Principles of Electromagnetism and Optics