The intensity or flux of electromagnetic radiation is equal to the time average of the
Poynting vector over the wave's period. For radiation propagating through a typical medium the energy density of the radiation, u, is related to the Poynting vector \mathbf{S} by -\frac{\partial u}{\partial t}=\nabla\cdot\mathbf{S}, which is derived from
Poynting's theorem. Integrating over a volume of space gives -\iiint\frac{\partial}{\partial t}\frac{dU}{dV}\, dV=\iiint(\nabla\cdot\mathbf{S})\, dV where U is the energy of the electromagnetic radiation. Applying the
divergence theorem, the rate of flow of energy out of the volume is seen to be related to the
surface integral of the Poynting vector over the surface of the volume of space: {{block indent | \frac{dU}{dt} = -}}
Point sources A common example is the intensity or flux of a
point source of given power output P. Considering a spherical volume centered on the source, the formula above becomes {{block indent | P=\left \langle -\frac{dU}{dt} \right \rangle = }} where the angle brackets denote a time average over the period of the waves. Since the surface area of a sphere of radius r is A = 4\pi r^2 this gives P = \langle S \rangle \cdot 4\pi r^2, therefore the intensity from the point source at distance r is I=\frac{P}{4\pi r^2}. This is known as the
inverse-square law.
Electromagnetic waves For a monochromatic propagating electromagnetic wave such as a
plane wave or a
Gaussian beam travelling in a non-magnetic medium, the time-averaged Poynting vector is related to the amplitude of the
electric field, , by \left\langle\mathsf{S}\right\rangle = \frac{cn\epsilon_0}{2} E^2, where is the
speed of light in
vacuum, is the
refractive index of the medium, and \epsilon_0 is the
vacuum permittivity. The relationship to intensity can also be seen by considering the time-averaged
energy density of the wave: \left\langle U \right \rangle = \frac{n^2 \epsilon_0}{2} E^2. The local intensity is just the energy density times the wave velocity {{tmath|\tfrac{c}{n} }}: I = \frac{\mathrm{c} n \epsilon_0}{2} E^2. For non-monochromatic waves, the intensity contributions of different spectral components can simply be added. The treatment above does not hold for arbitrary electromagnetic fields, but it is still often true that the magnitude of the time-averaged Poynting vector is proportional to the time-averaged energy density by a factor c: I = \langle S\rangle\propto c\langle U\rangle An
evanescent wave may have a finite electrical amplitude while not transferring any power. The intensity of an evanescent wave can be defined as the magnitude of the
Poynting vector. ==Electron beams==