Polarization and the Poynting Vector

Rotate the polarization and watch the energy flow

An electromagnetic wave runs forward. As you rotate the polarization angle with the slider, the direction in which the electric field (cyan) swings changes, and the magnetic field (gold) always follows perpendicular to it. Yet the green arrow, the flow of energy S = E×B, always points forward and keeps the same size no matter how you rotate the polarization. Why?

The energy the wave carries

An electromagnetic wave does not merely 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; since B = E/c the two are exactly equal. Electricity and magnetism share the energy evenly. Together they give u = ε₀E². Sunlight warming your skin and a solar cell making electricity both happen because this energy is carried along.

The Poynting vector, the flow of energy

Where energy flows, and how much, is told by the Poynting vector S = E×B/μ₀ — the energy passing through unit area each second, a power density. Its direction is the direction of travel, perpendicular to both E and B; its magnitude is |S| = EB/μ₀ = E²/μ₀c. Because E swings as a sinusoid, S pulses as cos², and averaged over one period the intensity is I = ⟨S⟩ = ½ε₀cE₀². This is the quantity that measures the brightness of light, the output of an antenna, the intensity of a laser.

Polarization, the direction E swings

Within the plane perpendicular to the direction of travel, which way the electric field swings is the polarization. If it swings along only one direction, it is linearly polarized. Sunlight is unpolarized, a mix of all directions, but reflected off water or a road it lines up along one direction and becomes harsh glare. Polarized sunglasses are a filter that blocks that direction. LCD screens, photographic filters and 3D-movie glasses all use polarization. Polarization is another piece of information light carries, distinct from its frequency or its intensity.

Whatever the polarization, energy goes forward

This is exactly what you saw on the first screen. However you rotate the polarization angle, the magnetic field stays perpendicular to the electric field, so their cross product E×B always points in the direction of travel. Moreover |E×B| is the product of the two magnitudes, independent of the polarization angle. So the energy flow S changes neither direction nor size as you rotate the polarization. Polarization sets how the light vibrates, but it cannot set where the energy goes. The freedom of vibration and the fixedness of flow live together in one wave.

Back to the first screen

As you rotated the polarization angle on the first screen, 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, at the same size. Polarization is the freedom to choose the direction of E within the plane; the Poynting vector is the flow that carries energy in the direction of travel regardless of that freedom. However light vibrates, it sends its energy straight ahead. That intensity is the brightness we feel. I = ½ε₀cE₀².

So far light has run freely through empty space. Next (EM-25) looks at what happens when light meets the boundary of another medium. Part is reflected and part is refracted and transmitted, with the proportions and angles set by the refractive indices of the two media. This is why mirrors reflect and lenses bend.