EM-23 · Electromagnetic waves

The Wave Equation and the Plane Wave

In empty space, interlocking the two curl equations, Faraday's law and the Ampère–Maxwell law, gives a wave equation. See that its solution is a plane wave in which the electric and magnetic fields run in the same phase, perpendicular to each other and to the direction of travel, and that its speed is c = 1/√(μ₀ε₀), the speed of light.

Watch the plane wave travel

This is a side view of an electromagnetic wave running through empty space. The sky-blue line is the electric field swinging up and down, and the gold line is the magnetic field swinging front to back; the two are perpendicular and, in step, reach their maximum together and pass through 0 together. Lengthen or shorten the wavelength with the slider and the frequency changes to match, so their product stays the same and the wave moves on at the same speed. Why the two fields move in the same phase is the question of this lesson.

Wavelength λλ = 2.5

Drag to orbit the view. The slider changes the wavelength λ.

The wave's speed and fields

c = λf = 1/√(μ₀ε₀) · the speed is the same for every wavelength

E ⊥ B ⊥ direction of travel · E and B are in phase

Wavelength λ 2.5 · Frequency f 0.44 · Speed c = λ·f 1.10 (constant)

The two curls in empty space

In empty space with no charges and no currents, that is, where ρ = 0 and J = 0, only the two curl equations of Maxwell's equations are left: Faraday's law ∇×E = -∂B/∂t and the Ampère–Maxwell law ∇×B = μ₀ε₀∂E/∂t. A changing electric field makes a magnetic field, and that field, as it changes, makes an electric field again, so the two fields keep each other going. Solving this interlock shows whether fields can carry on by themselves where there are no sources, and if so, in what shape they spread.

The wave equation emerges

Take the curl of one curl equation once more and substitute the other, and you get an equation in the electric field alone: ∇²E = μ₀ε₀ ∂²E/∂t². The magnetic field follows exactly the same form, and this is the wave equation. It says that how sharply the field bends in space is proportional to its acceleration in time: the equation of a wave whose ripple does not stay in place but moves sideways keeping its shape. A vibration running along a taut string and sound in air follow equations of the same form; all that differs is what ripples and how fast.

Observe∇²E = μ₀ε₀ ∂²E/∂t²

Interlocking the two curl equations gives the wave equation.

The speed is the speed of light

In the wave equation ∂²/∂x² = (1/v²)∂²/∂t², v is the speed of the wave. Comparing it with the equation for electromagnetic waves gives 1/v² = μ₀ε₀, so v = 1/√(μ₀ε₀). Putting in the measured values of the two constants gives about 3×10⁸ m/s. When Maxwell worked out this value from the electrical measurements of his day, it agreed, within the measurement error, with the separately measured speed of light. The speed of light had come out of equations built from electricity and magnetism alone, which led to the conclusion that light is an electromagnetic wave. In vacuum this speed is the same for every wavelength, but in glass or water it differs slightly with wavelength, which is how a prism splits white light into colors.

Choosec = ?

Its speed is the speed of light.

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 moving in one direction. The magnetic field has the same form, with magnitude B = E/c. Three things can be read from it. First, E and B swing perpendicular to the direction of travel: the wave is transverse. Second, E and B are perpendicular to each other. Third, the two are in the same phase, reaching their maximum together and passing through 0 together, because Faraday's equation ties the spatial slope of one to the time change of the other. E × B always points along the direction of travel. The picture draws the two fields at the same length, but in SI units B is E divided by c, so as a number it is far smaller; even so, the two fields carry exactly half of the energy each.

Fill inEB = ?

The ratio of the electric field strength to the magnetic field strength is c.

On your ownE = E₀ cos(?)

A plane wave is a sinusoid moving in one direction.

Back to the first screen

On the first screen, the electric field (up and down) and the magnetic field (front to back) swung perpendicular to each other and in step. However you stretched or shrank the wavelength, the wave moved on at the same speed, and that is what the wave equation says. The two curl equations interlock into ∇²E = μ₀ε₀∂²E/∂t², and in vacuum its speed is c = 1/√(μ₀ε₀) whatever the wavelength. E and B share a phase because Faraday's equation ties the spatial slope of one to the time change of the other. The light lighting up this screen right now is this very wave.

The wave equation ∇²E = μ₀ε₀∂²E/∂t² comes from interlocking the two curl equations, Faraday's law and the Ampère–Maxwell law, in empty space. Its solution, the plane wave E = E₀cos(kx - ωt), is a transverse wave in which the electric and magnetic fields are perpendicular to each other and to the direction of travel, and the two swing together in the same phase (B = E/c). In vacuum its speed, whatever the wavelength, is c = 1/√(μ₀ε₀), the speed of light.

What comes next

In this plane wave, E × B not only points along the direction of travel but is also the direction in which energy flows. The next lesson (EM-24) looks at the Poynting vector S = E × B / μ₀, which measures that energy flow, and at polarization, set by the direction in which the electric field swings. It goes into what light carries and how it vibrates.