The Poynting vector

Maxwell's equations describe fields, not energy transport directly, so a separate quantity is needed to say where electromagnetic energy is going and how fast: the Poynting vector, S = E × B/μ₀. Being a cross product, it is automatically perpendicular to both the electric and magnetic fields — and for a wave, where E and B are themselves perpendicular to the direction of travel, that makes S point exactly along the direction the wave propagates. Energy in an electromagnetic wave flows along neither field individually; it flows along the direction both fields agree is "forward".

This is the rigorous answer to "how does light carry energy from the Sun to a leaf, or a radio wave from a transmitter to an antenna": the fields themselves are the energy-carrying medium, and the Poynting vector's magnitude gives the power crossing a unit area per second, which is why it appears directly in calculations of a laser's intensity or a star's luminosity.

The genuinely surprising application is to something not usually thought of as "radiating" at all: a plain wire carrying steady current, heating up under Joule heating. Ohm's intuition says the energy travels down the metal itself, carried by the moving charges. The Poynting vector says otherwise. Outside the wire, the electric field points along it (driving the current) and the magnetic field circles around it (from the current, via Ampère's law); their cross product points radially inward, converging on the wire's surface from the surrounding space. The energy dissipated as heat is delivered from the field in the space around the wire, not carried through the metal along it — a case where the Poynting vector's answer runs directly against the common-sense picture, and is correct anyway.