Poynting vector

The Poynting vector S = (1/μ₀)·E×B is the energy-flux density of the electromagnetic field: the rate at which electromagnetic energy crosses a surface per unit area, with direction given by E×B. Units are watts per square metre. John Henry Poynting introduced it in 1884 in On the Transfer of Energy in the Electromagnetic Field, where he showed that the time derivative of the field energy density satisfies ∂u/∂t = −∇·S − J·E — an exact local conservation statement with S playing the role of the energy flux.

The geometry of S is perpendicular to both E and B, which often produces surprising results. In a simple resistive circuit carrying DC current, the E field inside the conductor is parallel to the current (driving the current, E = J/σ), but the E field outside the conductor (in the surrounding space) points radially inward from the positive-potential end. The B field circles the conductor. Taking E×B gives a Poynting vector that points into the wire from the surrounding space — the energy flows into the wire from the space around it, from the battery through the external fields to the resistor. Energy does not travel "along the wire"; it travels through the space around the wire. This is one of the first deeply counterintuitive results in classical electromagnetism and is due entirely to taking S seriously.

For a plane electromagnetic wave, S points along the direction of propagation and oscillates with the frequency of the wave; its time-average gives the wave's intensity I = (1/2)ε₀cE₀² for amplitude E₀. For sunlight at Earth's surface, I ≈ 1361 W/m² = the solar constant. For radio antenna near-fields, S is complex and reactive (energy cycling in and out of local storage) rather than radiated; for antenna far-fields (beyond a few wavelengths), S becomes purely radiative. The Poynting vector plus Maxwell's four equations plus Poynting's theorem form the complete energy-conservation bookkeeping for electromagnetism — no part of the energy question has been left open since 1884.