Supplement 1.4: Energy and Intensity of Electromagnetic Waves
The field energy
The energy density of a wave is defined as its energy per unit volume, given in J/m3 or Ws/m3.
An electromagnetic wave is characterised by an energy density of its electric field and an energy density of its magnetic field :
with the electric permittivity ε and the magnetic permeability μ of the material. On page 3 of supplement 1.2 we found the following relation between the electric and magnetic fied of electromagnetic waves:
where ω and k are the circular frequency and wave number of the wave, and is a unit vector in the direction of the propagating wave. The ratio is the phase velocity c of the wave.
Squaring the equation, and considering Maxwell's relation (supplement 1.3) between the phase velocity and the permittivity and permeability, one obtains:
Hence: ,
the electric and magnetic field energies of electromagnetic waves are identical, and the total energy density is:
The intensity
The energy flux of a wave corresponds to the wave's energy passing a unit area per time interval, given in units of W/m2.
Electromagnetic waves travel with the speed of light c, and the energy flux can thus be calculated from c times the energy density:
We can give this scalar quantity an orientation in the direction of wave propagation by replacing EB with . This is then a vector , denoted as the Poynting vector:
A plane monochromatic wave with
yields:
It is the time-average of the Poynting vector which is seen by the eye or by a photodetector. We use brackets as a symbol for time-averaging.
The sin2 term becomes:
Hence:
Absolute values of the Poynting vector
are given in W/m2. In physical photometry (called radiometry), this corresponds to the irradiance , which is given in the same units.