Supplement 1.4: Energy Density and Irradiance 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 $U_{e l}$ of its electric field $\vec{E}$ and an energy density $U_{m a g}$ of its magnetic field $\vec{B}$ :

U_{e l} = \frac{ε}{2} E^{2}

U_{m a g} = \frac{1}{2 μ} 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:

\vec{B} = \frac{k}{ω} \vec{a} \times \vec{E}

where ω and k are the circular frequency and wave number of the wave, and $\vec{a}$ 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 $c = 1 / \sqrt{ε μ}$ (supplement 1.3) between the phase velocity and the permittivity and permeability, one obtains:

B^{2} = \frac{k^{2}}{ω^{2}} E^{2} = \frac{1}{c^{2}} E^{2} = ε μ E^{2}

Hence: $U_{e l} = U_{m a g}$ ,

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

U = U_{e l} + U_{m a g} = ε E^{2} = \frac{1}{μ} B^{2} = \sqrt{\frac{ε}{μ}} E B

Equations ↓

The irradiance

The irradiance 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_{e l} + U_{m a g}) = c \sqrt{\frac{ε}{μ}} E B = \frac{1}{μ} E B

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

\vec{S} = \frac{1}{μ} \vec{E} \times \vec{B}

A plane monochromatic wave with

\vec{E} = {\vec{E}}_{o} sin (\vec{k} \cdot \vec{r} - ω t)

\vec{B} = {\vec{B}}_{o} sin (\vec{k} \cdot \vec{r} - ω t)

yields: $\vec{S} = \frac{1}{μ} {\vec{E}}_{o} \times {\vec{B}}_{o} {sin}^{2} (\vec{k} \cdot \vec{r} - ω t)$

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: $⟨ {sin}^{2} (\vec{k} \cdot \vec{r} - ω t) ⟩ = \frac{1}{2}$

Hence:

⟨ \vec{S} ⟩ = \frac{1}{μ} ⟨ \vec{E} \times \vec{B} ⟩ = \frac{1}{2 μ} ({\vec{E}}_{o} \times {\vec{B}}_{o}) = \frac{c ε}{2} E_{o}^{2} \vec{a} = \frac{c}{2 μ} B_{o}^{2} \vec{a}

$\vec{a}$ is again the unit vector in the direction of wave propagation. Absolute values of the Poynting vector

⟨ S ⟩ = \frac{c ε}{2} E_{o}^{2} = \frac{c}{2 μ} B_{o}^{2}

are given in W/m². In physical photometry (called radiometry), this corresponds to the irradiance $E_{r a d}$ , which is given in the same units.

Question 1: Electric and magnetic field of solar radiation ↓

Understanding Spectra from the Earth

Supplement 1.4: Energy Density and Irradiance of Electromagnetic Waves

The field energy

The irradiance