Maxwell's equations in vacuum predict transverse electromagnetic waves in which oscillating electric and magnetic fields sustain each other, propagating at speed c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s — the speed of light. In an EM wave, E and B are perpendicular to each other and to the direction of propagation, with magnitudes related by E = cB. The Poynting vector S = (1/μ₀) E × B gives the direction and power per unit area (intensity) of the wave. The EM spectrum spans radio waves through gamma rays, all traveling at c in vacuum.
Derive the wave equation from Maxwell's equations by taking the curl of Faraday's law and substituting Ampère's law. Identify c = 1/√(μ₀ε₀) as the wave speed. Then work problems computing intensity, energy density, and radiation pressure.
From Maxwell's equations you know two key coupling laws: Faraday's law (a changing B field induces an E field) and Ampère's law with displacement current (a changing E field produces a B field). Individually these describe how static or slowly varying fields behave near sources. Together, they create a self-sustaining feedback loop that needs no source at all: a changing E drives a changing B, which drives a changing E, which drives a changing B — propagating outward as a wave. Maxwell realized this in 1865 and predicted the existence of electromagnetic radiation before it was observed.
To derive the wave equation, take the curl of Faraday's law: ∇ × (∇ × E) = −∂(∇ × B)/∂t. Using the vector identity ∇ × (∇ × E) = ∇(∇ · E) − ∇²E and noting ∇ · E = 0 in vacuum (no free charges), the left side simplifies to −∇²E. Substituting Ampère's law on the right gives ∂²E/∂t² = (1/μ₀ε₀)∇²E. You recognize this from wave-equation-pde as a wave equation with speed c = 1/√(μ₀ε₀). When Maxwell computed 1/√(μ₀ε₀) from known electromagnetic constants, he got ≈ 3 × 10⁸ m/s — the measured speed of light. This was his stunning prediction: light itself is an electromagnetic wave.
In a plane wave traveling in the +x direction, both E and B lie in the yz-plane (the wave is transverse), are perpendicular to each other, and oscillate exactly in phase. If E oscillates along ŷ, then B oscillates along ẑ with magnitude B = E/c at every point. The Poynting vector S = (1/μ₀)(E × B) then points along +x with magnitude S = E²/(μ₀c) — the intensity, or power per unit area. Because S is proportional to E², doubling the field amplitude quadruples the intensity. This quadratic relationship between amplitude and power is a general feature of waves.
A persistent confusion comes from circuit analysis: in an inductor, voltage leads current by 90°, which might suggest E and B are 90° out of phase in a wave. They are not — that phase relationship belongs to circuits, not traveling waves. In a standing wave (created by two equal and opposite traveling waves), E and B are 90° out of phase in time, but a standing wave is a superposition of traveling waves, not the fundamental solution. For traveling waves, in phase is the correct description.
The electromagnetic spectrum — radio, microwave, infrared, visible light, ultraviolet, X-ray, gamma ray — represents different frequency (and thus wavelength) ranges of exactly the same phenomenon. All travel at c in vacuum; all have E ⊥ B ⊥ propagation direction; all carry energy via the Poynting vector. In a material with refractive index n, the wave slows to c/n because the oscillating fields interact with bound electrons, effectively reducing the coupling speed. The factor n encodes all the material physics in a single number.