Light Is a Traveling Wave

Light is an electric field and a magnetic field oscillating together, at right angles to each other and to the direction of travel. The wave moves at c, so its wavelength and frequency are locked together: λf = c. The magnetic field is always the electric field divided by c.

How it works

Electromagnetic waves

In one sentence

A linearly polarized plane wave traveling along $z$ in vacuum has $\mathbf{E} = E_0\cos(kz-\omega t)\,\hat{\mathbf{x}}$ and $\mathbf{B} = \frac{E_0}{c}\cos(kz-\omega t)\,\hat{\mathbf{y}}$ with $\omega = ck$ , $c = 1/\sqrt{\mu_0\varepsilon_0}$ , and time-averaged intensity $I = \frac{1}{2}c\varepsilon_0E_0^2$ .

General formula

{\htmlData{sym=0}{\mathbf{E}}} = {\htmlData{sym=2}{E_0}}\cos({\htmlData{sym=3}{k}}z - {\htmlData{sym=4}{\omega}} t)\,\hat{\mathbf{x}}, \qquad {\htmlData{sym=1}{\mathbf{B}}} = \frac{{\htmlData{sym=2}{E_0}}}{\htmlData{sym=5}{{c}}}\cos({\htmlData{sym=3}{k}}z - {\htmlData{sym=4}{\omega}} t)\,\hat{\mathbf{y}}, \qquad {\htmlData{sym=5}{c}} = {\htmlData{sym=6}{\lambda}} {\htmlData{sym=7}{f}} = \frac{1}{\sqrt{{\htmlData{sym=8}{\mu_0}}{\htmlData{sym=9}{\varepsilon_0}}}}

A changing electric field makes a magnetic field and a changing magnetic field makes an electric field. Each keeps regenerating the other, and the pair runs off at the speed of light.

What the symbols mean

$\mathbf{E}$
Electric field
Oscillates along x as the wave passes.
$\mathbf{B}$
Magnetic field
Oscillates along y, in step with E and c times smaller.
$E_0$
Amplitude
The largest electric field in the wave, in V/m.
$k$
Wavenumber
2π divided by the wavelength.
$\omega$
Angular frequency
2π times the frequency.
$c$
Speed of light
3.00 × 10⁸ m/s, the speed of every electromagnetic wave in vacuum.
$\lambda$
Wavelength
The distance between crests.
$f$
Frequency
Crests passing a point per second.
$\mu_0$
Permeability of free space
4π × 10⁻⁷ T·m/A, the magnetic constant.
$\varepsilon_0$
Permittivity of free space
8.85 × 10⁻¹² F/m, the electric constant.

In empty space Maxwell's equations combine into the wave equation $\frac{\partial^2 E}{\partial z^2} = \mu_0\varepsilon_0\frac{\partial^2 E}{\partial t^2}$ , whose waves travel at $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3.00 \times 10^8$ m/s.
A crest moves one wavelength $\lambda$ per period $1/f$ , so $c = \lambda f$ : green light at 500 nm oscillates about $6 \times 10^{14}$ times a second.
Faraday's law ties the fields together: $B = E/c$ , in phase, with $\mathbf{E}$ ⟂ $\mathbf{B}$ ⟂ the direction of travel.
The wave carries intensity $I = \frac12 c\varepsilon_0 E_0^2$ (watts per square meter); the lab shows $E$ (top) and $B$ (bottom, scaled up to be visible) along the direction of travel.

Learn the math

Electromagnetic WavesE&M · Lesson 28, Maxwell's Equations and Light Energy, Intensity and the Poynting VectorE&M · Lesson 29, Maxwell's Equations and Light