The Wave Equation and the Plane Wave
Watch the plane wave travel
An electromagnetic wave running through empty space. The electric field swinging vertically (cyan) and the magnetic field swinging horizontally (gold) are perpendicular, and they peak together and vanish together, in the same beat. Change the wavelength and the wave still advances at the same speed c. See why the two fields are in phase.
The two curls in empty space
Consider empty space with no charge and no current: ρ = 0, J = 0. Then only the two curl equations of Maxwell remain: Faraday ∇×E = -∂B/∂t and Ampère–Maxwell ∇×B = μ₀ε₀∂E/∂t. A changing electric field makes a magnetic field, and that changing magnetic field makes an electric field again. The two feed each other. What comes out when we solve this interlocking?
The wave equation emerges
Take the curl of one curl equation and substitute the other, and you get an equation the electric field satisfies on its own: ∇²E = μ₀ε₀ ∂²E/∂t². The magnetic field obeys the same form. This is the wave equation. It says the amount a field bends in space is proportional to how fast it changes in time — the very form every wave follows when a disturbance does not stay put but spreads sideways. Ripples and sound obey it too.
The speed is the speed of light
In every wave equation ∂²/∂x² = (1/v²)∂²/∂t², the v is the speed of that wave. Matching it to the electromagnetic equation gives 1/v² = μ₀ε₀, so v = 1/√(μ₀ε₀). Put in the measured values of the two constants and out comes about 3×10⁸ m/s. When Maxwell computed this, it was exactly the already-known speed of light. The speed of light had leapt out of equations built from electricity and magnetism alone. Light was an electromagnetic wave.
The plane wave — transverse and in phase
The simplest solution of the wave equation is the plane wave: E = E₀ cos(kx - ωt), a sinusoid advancing in one direction. The magnetic field has the same form, with magnitude B = E/c. Three things matter. First, E and B are transverse — perpendicular to the direction of travel. Second, E and B are perpendicular to each other. Third, the two are in phase, peaking together and vanishing together, because Faraday's equation ties the spatial slope of one to the time change of the other. E × B always points in the direction of travel.
Back to the first screen
On the first screen the electric field (vertical) and the magnetic field (horizontal) swung perpendicular to each other, and in the same beat. Stretch or shrink the wavelength and the wave always advanced at the same speed. This is what the wave equation says. The two curl equations mesh into ∇²E = μ₀ε₀∂²E/∂t², and its speed, independent of wavelength, is c = 1/√(μ₀ε₀). E and B are in phase because Faraday's equation ties the spatial slope of one to the time change of the other. That light is entering your eyes right now.
In this plane wave, E × B not only points along the direction of travel but is also the flow of energy. Next (EM-24) looks at the Poynting vector S = E × B / μ₀ that measures this energy flow, and at polarization, set by the direction in which the electric field swings. We move into what light carries and how it vibrates.