Step into sunlight and you are standing in a river. Every second, roughly 1360\ \text{joules} of energy pour through each square metre facing the Sun — the solar constant — after an eight-minute crossing of empty space. Nothing solid made that trip: it was carried by the electric and magnetic fields of a light wave, streaming outward from the Sun at c. The fields do not merely store energy; they move it, and they carry momentum too — enough that a large, thin, mirrored sheet in space feels a steady push and can sail on sunlight with no fuel at all.

This whole page rests on one vector, named for John Henry Poynting (and the pun is deserved): the Poynting vector

It points in the direction electromagnetic energy is flowing, and its magnitude is the power crossing unit area perpendicular to that flow, in \text{W/m}^2. Get comfortable with \mathbf{S} and you can answer three questions at once: where is field energy going, how much is going, and how hard does it push when it gets there.

First: fields hold energy

Before energy can flow it must be somewhere. Both fields store energy, spread through space with an energy density (joules per cubic metre):

The first term is the energy in the electric field (you can dig it out of a charged capacitor), the second is the energy in the magnetic field (stored in an inductor's coil). These are not bookkeeping tricks — pull the charges apart or run down the current and that energy comes back out. In a light wave the two terms turn out to be exactly equal, because B = E/c makes B^2/\mu_0 = \varepsilon_0 E^2: the electric and magnetic halves of a wave carry the same energy, always.

The Poynting vector: energy on the move

Now the flow. The direction is fixed by a cross product: curl the fingers of your right hand from \mathbf{E} towards \mathbf{B} and your thumb points along \mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B} — the way the energy travels. For a wave heading towards you with its electric field pointing up and its magnetic field pointing sideways, \mathbf{S} points straight at you. Reveal the triad below to watch \mathbf{E}, \mathbf{B} and their product \mathbf{S} click into a right-handed set of three mutually perpendicular arrows.

Two features are worth pinning down. First, the units: [\,E\,]\cdot[\,B\,]/\mu_0 works out to \text{W/m}^2, a power per area — an intensity. Second, \mathbf{S} is perpendicular to both fields, so in a light wave the energy flows in the one direction the fields are not oscillating: sideways-shaking fields, forward-moving energy.

Poynting's theorem: energy conservation, written locally

Why should \tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B} be the energy flow? Because it makes the energy books balance at every point in space. Start from Maxwell's equations, dot \mathbf{E} into Ampère's law and \mathbf{B} into Faraday's law, subtract, and a vector identity (\nabla\cdot(\mathbf{E}\times\mathbf{B}) = \mathbf{B}\cdot(\nabla\times\mathbf{E}) - \mathbf{E}\cdot(\nabla\times\mathbf{B})) collapses the mess into a single clean statement:

Read it as an accountant would. The left side is the rate at which field energy is disappearing from a tiny box. It goes to exactly two places: \nabla\cdot\mathbf{S}, energy flowing out through the walls of the box, and \mathbf{J}\cdot\mathbf{E}, the rate at which the field does work on charges inside (the power that heats a resistor). Nothing vanishes and nothing appears from nowhere — this is the continuity equation idea applied to energy, with \mathbf{S} playing the role of the energy current.

Intensity of a plane wave

Put a real light wave into the formula. With B = E/c and c = 1/\sqrt{\mu_0\varepsilon_0}, the instantaneous Poynting magnitude for a plane wave simplifies beautifully:

The energy flow is just the speed of light times the energy density — energy sitting at density u and sweeping past at speed c, exactly as you would guess. But E oscillates, so the useful quantity is the time-averaged value, the intensity I. Averaging \cos^2 over a cycle gives a factor of one half:

where E_0 is the peak field. The graph shows how the intensity climbs as the square of the field amplitude — double the field and you quadruple the power delivered.

Momentum and radiation pressure

Here is the surprise that makes solar sails possible: an electromagnetic wave carries not just energy but momentum. The momentum stored per unit volume — the momentum density — is the energy flow divided by c^2:

When that momentum lands on a surface, it exerts a radiation pressure. If the surface absorbs the light (a black sheet), it soaks up all the incoming momentum, and the pressure is

If the surface reflects the light straight back (a mirror), the light reverses its momentum, so the surface must supply twice the impulse — the pressure doubles:

This is precisely why solar sails are shiny: a mirror gets double the push of a black cloth. Let us put a number on it. Sunlight at I = 1360\ \text{W/m}^2 hitting a perfect mirror gives

Nine millionths of a pascal — feeble by everyday standards (a footstep is a hundred thousand times more). But it never stops, needs no propellant, and scales with area. Spread a sail over a square kilometre (10^{6}\ \text{m}^2) and the total force is F = P\!\cdot\!A \approx 9\ \text{N} — a gentle but ceaseless shove that, over months, can carry a lightweight craft across the Solar System.