Poynting vector

The Poynting vector (named after its inventor, John Henry Poynting) is the cross product of the electric field and the magnetic field. It points in the direction of energy flow when an electromagnetic wave is propagating and it's magnitude is the power per unit area crossing a surface whose normal is parallel to it. It is given the symbol S (in bold because it is a vector).

$\mathbf {S} =\mathbf {E} \times \mathbf {H} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B}$

Since the electric and magnetic field of an electromagnetic wave are oscillating, the magnitude of the poynting vector is changing with time. Averaged the magnitude over long times T (longer than the period $\tau$ of the wave), it gives the irradiance :

$I=\left\langle S\right\rangle _{T}$