MOMENTUM AND THE MAXWELL STRESS TENSOR

In §07.4 the Poynting vector carried energy across empty space. Sunlight warms a thermometer; a radio wave heats a resistor. Energy has a direction of flow and a density per cubic metre — real, locatable stuff.

Momentum is conserved too. When a flashlight beam lands on a sheet of paper and the paper absorbs it, the energy becomes heat — and Newton's second law insists the paper must recoil. A laser pointer on your palm deposits roughly 6 nN. Small, but real. The photon is momentum in a red envelope. Which means, between being emitted and absorbed, that momentum was stored in the field. Energy and momentum live in the same place.

This is the last move §07 makes: we write down the bookkeeping for how electromagnetic fields carry momentum, and every clean result from mechanics — Newton's third law, conservation of momentum, the idea of stress — extends out of matter into pure fields. Maxwell, in his Treatise (1873), wrote the tensor down before anyone had measured the force. He called it "tension along lines of force, pressure across them" — Faraday's mechanical intuition, now algebraically exact.

A stress tensor is a 3×3 matrix T_ij that books-keeps the flow of momentum through surfaces. The number T_ij reads as: "the rate at which the j-component of momentum crosses a surface whose outward normal points along î, per unit area." Diagonal entries T_ii are normal stresses (pressure if they push inward, tension if they pull outward). Off-diagonal entries T_ij for i ≠ j are shears.

For the electromagnetic field in vacuum, Maxwell's expression is:

Read it in two parts. The electric part ε₀(E_i E_j − ½ δ_ij E²) is everything the E field contributes; the magnetic part is the same formula with E → B and ε₀ → 1/μ₀. The Kronecker delta δ_ij is 1 when the indices match and 0 otherwise — it's the bookkeeping symbol that subtracts the "isotropic half" from the diagonal.

The force per unit volume on matter in this field is the divergence of T_ij (plus a small −∂g/∂t correction when things vary in time). Stress in, force out — the same arithmetic as elastic continua. The field has become a medium with its own elastic properties, and all we did was write a 3×3 matrix of quadratic combinations of E and B.

Alongside the stress tensor lives a second quantity: the field momentum density, g. This is the momentum stored per unit volume in the field itself. The expression is:

It points in the direction the field is carrying energy (because S = (1/μ₀)·E × B does, and g is the same cross product scaled down by ε₀μ₀ = 1/c²). Its magnitude is tiny — sunlight's energy flux of 1361 W/m² corresponds to a momentum density of about 1.5 × 10⁻¹⁴ kg/(m²·s) — but over time and volume it adds up to real, measurable effects.

Two checks the scene makes visual. First, E × B points along the beam — right-hand rule with E vertical and B into the page. Second, g is not separate from S; they are two names for the same stuff related by a factor of c². You've met S first for historical reasons: energy flow was discovered before momentum flow, and the c² is large enough that momentum effects were hard to see with Victorian instruments.

Fold the stress tensor and the momentum density into a single sentence: a wave that delivers energy flux I to a surface also delivers a momentum flux of I/c. Conservation of momentum then requires the surface to push back with the same flux — a pressure.

That's the pressure on a perfectly black (perfectly absorbing) surface. If the surface is instead a perfect mirror, the photons bounce back with their momentum reversed, so the total momentum change is doubled:

This is the quantity spaceflight engineers call radiation pressure. At Earth-orbit sunlight (I ≈ 1361 W/m²), the reflector formula gives roughly 9.1 µPa. That's a millionth the pressure of a light breeze — ignore it in a bridge truss, budget for it in a space telescope's pointing system.

The 1/c is why we discovered radiation pressure late. Divide a respectable sunlight flux by the speed of light and you get micropascals; divide by the speed of sound and you'd get reasonable pressures — but light goes a million times faster than sound, so the conversion factor is brutal. Kepler guessed radiation pressure was why comet tails point away from the Sun (1619) — he was right for the dust tails, centuries before the physics was in place.

Radiation pressure became a lab fact on Maxwell's terms in 1900. Pyotr Lebedev, in Moscow, built a torsion balance with a pair of thin mica vanes — one side blackened, the other silvered — suspended in a high vacuum. Carbon-arc light focused on the silvered vane deflected the balance by microradians. He read the deflection with a mirror and a scale; the numbers agreed with P = 2I/c to within experimental error. Fields push on mirrors. Confirmed.

The engineering version arrived a century later. In May 2010, JAXA launched IKAROS — Interplanetary Kite-craft Accelerated by Radiation Of the Sun — a 14-metre square aluminized polyimide sail massing 310 kg total. On the way to Venus, with no propellant burned, sunlight alone changed its orbital velocity by roughly 100 m/s. The acceleration at 1 AU was about 6 × 10⁻⁵ m/s² — feeble, but free, and cumulative. The Planetary Society's LightSail-2 (2019) followed with a similar demonstration in Earth orbit.

The math: a sail of area A, mass m, reflectivity ρ, at distance r from the Sun, accelerates at a = (1 + ρ) · I(r) · A / (m c) with I(r) = L☉ / (4πr²). Slide the sail toward 0.3 AU (Mercury's orbit) and the number climbs by 1/0.3² ≈ 11×. Solar sails want to be close to the Sun.

Here is the subtlest case. Place a stationary charge q next to a stationary magnetic dipole m. Nothing moves, nothing emits. No Poynting flux crosses a large sphere around the pair. So where's the momentum?

Compute g = ε₀·E × B everywhere around the two. The charge's E-field, crossed with the dipole's B-field, gives a nonzero density at nearly every point. Integrate over all space — the total is not zero.

Static momentum stored in the field between static objects. If you suddenly extinguish the dipole's current, that stored momentum has to go somewhere — and it shows up as mechanical momentum in the wire, kicked into motion by an induced E-field that is exactly the back-reaction of the vanishing field momentum. What looks like a free kick is paid for by the field's bookkeeping. This classical "hidden momentum" effect is one of the clean tests that convinced people the stress tensor is not a formal device — it is the actual accountant for Newton's third law when pieces of the system live in empty space.

Five topics in §07 and electromagnetism is now a complete, self-consistent field theory. §07.1 found the displacement current that patches Ampère's law. §07.2 consolidated the result into four equations that fit on a postcard. §07.3 showed the potentials (φ, A) are not unique but the physics is — gauge freedom. §07.4 discovered that energy flows locally through space as S = (1/μ₀)·E × B. And this page closes the loop: momentum flows too, at S/c², and the matter-field system obeys Newton's third law once T_ij is in the accounting.

What we get next for free: light. Take Maxwell's equations in a source-free region, take a curl, and the wave equation falls out with speed c = 1/√(ε₀μ₀) = 299,792,458 m/s — the speed of light, from electricity and magnetism alone. §08 turns that crank.