EM-24 · Electromagnetic waves

Polarization and the Poynting Vector

An electromagnetic wave carries energy along its direction of travel, and the quantity that measures that flow is the Poynting vector S = E×B/μ₀. The direction in which the electric field swings, meanwhile, is called the polarization. See that however the polarization angle is turned, the magnetic field stays perpendicular to the electric field and energy keeps flowing forward at the same rate.

Rotate the polarization and watch the energy flow

This is an electromagnetic wave running forward. Turning the polarization angle with the slider changes the direction in which the sky-blue electric field swings, and the gold magnetic field turns along with it, always perpendicular. Yet the green arrow, the energy flow S = E×B, always points forward with the same length however the polarization is turned. Why the flow stays put while the swinging direction changes is the question of this lesson.

Polarization angle ψψ = 35°

Drag to orbit the view. The slider changes the polarization angle ψ.

The energy flow and the polarization

S = E×B/μ₀ · energy always flows in the direction of travel

Polarization is the direction E swings; B is always perpendicular to it

Polarization angle ψ 35° · Direction of travel +x · ⟨S⟩ constant (independent of polarization)

The energy the wave carries

An electromagnetic wave does not just ripple through empty space; it carries energy. The energy density stored in the electric field is ½ε₀E² and that in the magnetic field is B²/2μ₀, and since B = E/c in a plane wave, the two are exactly equal. Electricity and magnetism share the energy evenly, and together they give u = ε₀E². Sunlight warming your skin and a solar cell making electricity both come from this energy arriving. Along with energy, an electromagnetic wave carries momentum, so light pushes very gently on any surface it strikes; a solar sail, a broad sheet in space driven forward by this pressure, puts that push to use.

On your ownu = ?

The electric and magnetic fields share the energy evenly.

The Poynting vector, the flow of energy

Where energy flows and how much is given by the Poynting vector S = E×B/μ₀. It is the energy passing through a unit area in 1 second, that is, a power density; its direction is the direction of travel, perpendicular to both E and B, and its magnitude is |S| = EB/μ₀ = E²/μ₀c. Because E swings as a sinusoid, S pulses as cos² at any one point, and averaging over a cycle gives the intensity I = ⟨S⟩ = ½ε₀cE₀². Double the amplitude and the intensity goes up four times. This is the quantity used to measure the brightness of light, the power an antenna sends out, and the intensity of a laser.

ObserveS = (E×B)μ₀

The Poynting vector gives the direction and amount of energy flow.

Choose|S| = ?

Its magnitude is proportional to the square of the electric field.

Fill inI = ?

The time-averaged intensity is proportional to half the square of the amplitude.

Polarization, the direction E swings

The direction in which the electric field swings within the plane perpendicular to travel is the polarization. If it swings along one direction only, the wave is linearly polarized; if the tip of the electric field goes around a circle or an ellipse, it is circularly or elliptically polarized. Sunlight is unpolarized, a mix of all directions, but light reflected from water or a wet road has more of its swing along the horizontal, and polarizing sunglasses cut glare by blocking exactly that horizontal part. LCD screens, photo filters, and 3D movie glasses all use polarization too, and an antenna must match the polarization to receive a signal fully, so an antenna set up for vertical polarization picks up a horizontally polarized signal poorly. Polarization is one more piece of information light carries, separate from its frequency and intensity.

Whatever the polarization, energy goes forward

This is exactly what the first screen showed. However the polarization angle is turned, the magnetic field stays perpendicular to the electric field, so their cross product E×B always points along the direction of travel. On top of that, |E×B| is the product of the two magnitudes, so it does not depend on the polarization angle. That is why the energy flow S changes neither its direction nor its size when the polarization turns. Polarization decides how light vibrates but not where the energy goes: the freedom to choose a direction of vibration and a fixed direction of flow live together in one wave.

Back to the first screen

On the first screen, as you turned the polarization angle, the electric and magnetic fields turned together, but the right angle between them never broke, and the green arrow S = E×B always pointed forward with the same length. Polarization is the freedom to choose the direction of E within the plane, and the Poynting vector is the flow that carries energy along the direction of travel regardless of that freedom. However light vibrates, it sends its energy straight ahead, and that intensity is the brightness we feel. I = ½ε₀cE₀².

The Poynting vector S = E×B/μ₀ is the energy an electromagnetic wave carries through a unit area in 1 second, that is, the flow of energy. Its direction is the direction of travel (perpendicular to both E and B), its magnitude is E²/μ₀c, and its time-averaged intensity is I = ½ε₀cE₀². Polarization is the direction in which the electric field swings within the plane perpendicular to travel. However the polarization is turned, B stays perpendicular to E, so S changes neither direction nor size.

What comes next

So far light has run freely through empty space. The next lesson (EM-25) takes up what happens when light meets the boundary with a material of different properties. Part of it is reflected and part is refracted and transmitted, and the share and the angles are set by the refractive indices of the two materials, the angle of arrival, and the polarization. That is why mirrors reflect and lenses bend light.