Poynting’s theorem is the work-energy equation (i.e., the energy conservation law) of electrodynamics. It essentially states that the work done on charges by the electromagnetic force is equal to the decrease in energy remaining in the fields, minus the energy that flowed out of the surface.
In an wave optical setting, we express it as:
where is the electric energy and is the magnetic energy, respectively. The cross product between the electric and magnetic field gives us a power density (in watts per metres squared). Observe that this is a special case of the electromagnetic Poynting theorem without the current density (for Ohmic power).
This cross product is called the Poynting vector, in units of power per area. The direction denotes the direction of {power, energy} transfer. In isotropic media, is in the same direction as .
We define the irradiance (or intensity) are the time average of the Poynting vector.
i.e., for a plane harmonic wave, .