Where does a light wave's energy go? Poynting answered in 1884: the energy flux is the cross product of the wave's two fields, S = (E × B)/μ₀ — a vector pointing exactly where the light travels, perpendicular to both E and B. Down the center the two fields go in, the engine crosses them, the transported power comes out. The blue team builds and defends it; the red team tries to break it.
source Poynting, J.H., On the Transfer of Energy in the Electromagnetic Field (1884), Phil. Trans. R. Soc. Lond. 175, 343–361 — doi.org/10.1098/rstl.1884.0016. Rendered, not quoted.
The flux is not stipulated; it is constructed from the two fields:
S = (1/μ₀) E × B. A cross product is, by definition, orthogonal to both factors — so S · E = 0 and S · B = 0 identically, for any E and B. For a plane wave the fields are mutually perpendicular and |B| = |E|/c, which fixes the magnitude.
Live components for the current fields (SI units):
| x | y | z |
|---|
The disturbance whose shape obeys uₜₜ = c² uₓₓ is not just a ripple — it carries energy, and Poynting names the rate and direction: S.
Energy density u = ½(ε₀E² + B²/μ₀) streams forward at the wave speed: S = u·c. The shape comes from the-wave-equation; the power in that shape is this sphere. Each sphere is the next one's premise.
The blue team's live check: recompute S from the current fields and confirm S·E = 0, S·B = 0, and |S| = u·c. If red swaps the cross product, this badge is where it shows.
An electromagnetic wave carries two fields at right angles to each other and to the direction of travel:
| field | symbol | along | size |
|---|---|---|---|
| electric | E | y | E₀ |
| magnetic | B | z | E₀/c |
| travel | S | x | derived |
You feed E and B; the panel crosses them. The magnetic field is a factor of c ≈ 3×10⁸ smaller in magnitude — that is why light's "force" feels electric.
Peak: instantaneous flux when the fields are at crest. |S| = E₀²/(μ₀c).
Change any control — E, B, S and the transported power are computed from the cross product on the spot, never looked up.
What the machine proves, live: S is perpendicular to both fields (S·E = S·B = 0); its magnitude is |S| = E₀²/(μ₀c) = ε₀cE₀²; its time average is half the peak, ⟨S⟩ = E₀²/(2μ₀c); and the energy moves at the wave speed, S = u·c. Reverse B and S reverses with it.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
Worse: crossed static fields (a magnet beside a charge) give a nonzero, circulating S with no energy going anywhere — the "hidden momentum" puzzle. So S is exact and indispensable for radiation, yet its pointwise story is a choice, not a fact.
"S = E × B." Cut. In SI the flux is E × B divided by μ₀; without it the units are wrong by 10⁶ and the number is meaningless.
"The Poynting vector is the one true local energy flow." Cut. Unique only up to a curl; the closed-surface integral is what is physical — see the adversary.
"In a DC circuit the energy flows through the wire's electrons." Kept, corrected. The power flows in the field around the wire and enters through its surface, exactly as Poynting says; the electrons carry the current, not the bulk energy.
The red team's move: replace the cross product E × B with the sum E + B. It still returns a vector — but it is no longer orthogonal to the fields. The blue team's witness (window 7) is watching.
Swap the cross product for a sum and S stops being perpendicular — S·E is no longer zero, the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.