Supplement 1.4: Energy and Intensity of Electromagnetic Waves

The field energy

The energy density U 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 U el of its electric field E and an energy density U mag of its magnetic field B :

U el = ε 2 E 2                U mag = 1 2μ B 2 ,

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:

B = k ω a × E

where ω and k are the circular frequency and wave number of the wave, and a is a unit vector in the direction of the propagating wave. The ratio ω/k is the phase velocity c of the wave.

Squaring the equation, and considering Maxwell's relation c=1/ εμ (supplement 1.3) between the phase velocity and the permittivity and permeability, one obtains:

B 2 = k 2 ω 2 E 2 = 1 c 2 E 2 =εμ E 2

Hence:                                        U el = U mag ,

the electric and magnetic field energies of electromagnetic waves are identical, and the total energy density is:

U= U el + U mag =ε E 2 = 1 μ B 2 = ε μ EB

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:

c( U el + U mag )=c ε μ EB= 1 μ EB

We can give this scalar quantity an orientation in the direction of wave propagation by replacing EB with E × B . This is then a vector S , denoted as the Poynting vector:

S = 1 μ E × B

A plane monochromatic wave with

E = E o sin( k r ωt )            B = B o sin( k r ωt )

yields:                             S = 1 μ E o × B o sin 2 ( k r ωt )

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:        sin 2 ( k r ωt) = 1 2

Hence:

S = 1 μ E × B = 1 2μ ( E o × B o )= cε 2 E o 2 a = c 2μ B o 2 a

Absolute values of the Poynting vector

S = cε 2 E o 2 = c 2μ B o 2

are given in W/m2. In physical photometry (called radiometry), this corresponds to the irradiance E rad , which is given in the same units.

Question 1: Electric and magnetic field of solar radiation