Electrodynamics

Electromagnetic waves in vacuum

In source-free vacuum, E and B are mutually perpendicular transverse waves propagating at speed c.

Electromagnetic waves need no material medium: changing electric and magnetic fields sustain one another. Each field is perpendicular to the propagation direction and to the other field.

Interpretation and consequences

In SI, B₀ = E₀/c; electric and magnetic energy densities average equally. The Poynting vector S = E×B/μ₀ gives energy-flux direction and density.

Taking the curl of Faraday's law and using $\nabla\times B=\mu_0\varepsilon_0\partial_tE$ in vacuum yields $\nabla^2E-c^{-2}\partial_t^2E=0$, with the same wave equation for $B$. A plane solution $\cos(kz-\omega t)$ therefore requires $\omega=ck$. Because both fields are transverse, a source-free plane wave has no field component along its direction of propagation.

For a harmonic wave, the time-averaged Poynting vector is $\langle S\rangle=\tfrac12\varepsilon_0cE_0^2\hat k$ for peak amplitude, the power per unit area. The field also carries momentum density $g=S/c^2$; absorption or reflection transfers momentum and produces radiation pressure. These relations show that electromagnetic waves transport energy and momentum even though no material particles travel with them.

In vacuum, the linear relation $\omega=ck$ makes group velocity $d\omega/dk$ equal to $c$, just like phase velocity; a wave packet is nondispersive. At a stationary interface frequency stays fixed, so wavelength changes in proportion to phase speed.

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