Electromagnetic Waves: How E and B Chase Each Other at the Speed of Light

Maxwell's equations, distilled to one wave

An electromagnetic wave is a self-sustaining oscillation of the electric field E and the magnetic field B, each generating the other as it moves. Maxwell's equations in empty space, with no charges or currents nearby, combine into a single wave equation for each field component:

∂²E/∂t² = c² ∂²E/∂x²
∂²B/∂t² = c² ∂²B/∂x²
c = 1 / √(μ₀ε₀)  ≈ 299,792,458 m/s

That the speed constant falls straight out of two purely electrical and magnetic quantities — the permeability μ₀ and the permittivity ε₀ — and equals the measured speed of light was the moment, in 1865, that light itself was recognised as an electromagnetic phenomenon. A plane wave travelling along x looks like E(x,t) = E₀ sin(kx − ωt), with wavelength λ = 2π/k, angular frequency ω = 2πf, and the two locked together by c = fλ: turn a wave's frequency up and its wavelength shrinks in exact proportion.

Why E and B are perpendicular — and to the direction of travel

Faraday's law says a changing B induces a circulating E; Ampère–Maxwell's law says a changing E (plus any current) induces a circulating B. Wire those two inductions together and you get a wave that only propagates if E, B and the direction of travel k form a mutually perpendicular right-handed triad — this is a transverse wave, unlike sound, which is longitudinal. The two fields stay exactly in phase: they peak together and cross zero together, and their ratio is fixed at every instant, |E|/|B| = c.

The electromagnetic spectrum is one equation, many scales

Radio, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays are the same physical phenomenon at different frequencies — the only thing that changes across fourteen-plus orders of magnitude is λ and f, locked by c = fλ. A radio station broadcasting at 100 MHz emits waves about 3 metres long; visible light sits around 400–700 nanometres, roughly a million times shorter, at frequencies near 500 THz. The boundaries between bands are conventions, not physical thresholds — there is no wall at 400 nm, only a gradual change in how the wave interacts with matter and, eventually, with human eyes.

Energy, momentum and the inverse-square law

An electromagnetic wave carries real energy and momentum, even though nothing material is moving through space. The energy flux — power per unit area — is given by the Poynting vector, S = E × B / μ₀, which points along the direction of propagation with magnitude proportional to E². Because a wave spreading from a point source distributes the same total power over a sphere of growing area 4πr², intensity falls as 1/r² — the familiar inverse-square law that governs everything from starlight to Wi-Fi signal strength.

S = E × B / μ₀          Poynting vector, W/m²
I(r) = P / (4πr²)       intensity vs distance from a point source

Polarisation: the direction E chooses to oscillate in

Because E only needs to stay perpendicular to the direction of travel, it still has an entire plane of directions to oscillate within. If it stays along one fixed line, the wave is linearly polarised; if the tip of E traces a circle as the wave advances, it is circularly polarised, built from two linear components 90° out of phase. Polarising filters, LCD screens, polarised sunglasses and radio antenna orientation all exploit this extra degree of freedom that a longitudinal wave like sound simply does not have.

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