the Poynting vector
/ POYN-ting /
If electromagnetic fields carry energy, that energy must be able to move — to stream from a battery down a wire, or from the Sun across empty space. The Poynting vector is the precise bookkeeping of that flow: it points in the direction energy travels and its magnitude is the power crossing unit area.
In vacuum the Poynting vector is S = E x B / mu_0, equivalently S = E x H in media. Its direction is perpendicular to both E and B, given by the right-hand rule, and its units are watts per square metre — an energy flux. For a plane electromagnetic wave, where B = E/c and E is perpendicular to B, the magnitude is S = E^2/(mu_0 c) = epsilon_0 c E^2, pointing along the direction of propagation; its time-average is the intensity of the wave.
The Poynting vector organizes all of electromagnetic energy transport. It shows, surprisingly, that the energy powering a resistor flows in from the surrounding fields through its sides, not down the wire; that a charging capacitor draws energy in radially from the field between its plates; and that a light wave carries a definite intensity. Its divergence enters Poynting's theorem, the local law of energy conservation for fields plus matter. One caveat: only its surface integral (total power through a closed surface) is unambiguous — you may add any curl to S without changing that integral.
Sunlight at Earth's orbit delivers about 1360 watts per square metre — that number is the magnitude of the time-averaged Poynting vector of the Sun's electromagnetic field at our distance. From the wave's electric-field amplitude E, it equals epsilon_0 c E^2 averaged over a cycle.
S = E x B / mu_0 points along energy flow; its magnitude is power per unit area.
Only the flux of S through a closed surface is physically unique; the field S itself is defined only up to the curl of any vector, so 'energy flowing sideways' in a static setup is not directly observable. S = E x B/mu_0 is the standard choice, not the only mathematically possible one.