Supplement 1.7: Radiation quantities and radiometry      (5/9)

The irradiance and the radiant exitance

The irradiance $E_{rad}$ corresponds to the radiant power ϕ: incident on a surface A

To distinguish it from photon energy E, the symbol $E_{rad}$ is used for irradiance.

Formula symbol: $E_{rad}$
Unit of measurement: Watts per square metre, $\left[E_{rad}\right]=\frac{\text{W}}{{\text{m}}^{\text{2}}}$

The surface itself can also be an emitter. In this case, the radiant exitance M is equal to the radiant power emitted by the surface:

The relationships given for irradiance also apply mutatis mutandis to radiant exitance.

Examples

• The radiant power of an LED luminaire that consumes 9 W of electrical power is approximately 4 W. For an isotropically emitting luminaire at a distance of 1 m, with $A=4\pi r^2=\mathrm{12<mo>.</mo>57}{\text{m}}^{\text{2}}$, the following applies:
  $E_{rad}=\frac{4}{\mathrm{12<mo>.</mo>57}}\frac{\text{W}}{{\text{m}}^{\text{2}}}=\mathrm{0<mo>.</mo>32}\frac{\text{W}}{{\text{m}}^{\text{2}}}$
• The irradiance of the sun's rays reaching us at midday on a cloudless summer's day is around 1000 W/m².
• The radiant exitance of the sun's surface is calculated using Stefan-Boltzmann's law. It is M=62.9·10⁶ W/m².
• For the Earth, applying Stefan-Boltzmann's law and assuming an average global surface temperature of 16°C, i.e. T=289.25°C, gives a value of M=396.4 W/m. Compare this with the figure given in the graph in the section on the greenhouse effect.

Measurement methods for determining the irradiance $E_{rad}$ are derived from the earlier specifications for measuring radiant power.

The definition of $E_{rad}$ in the left-hand column refers to radiation that strikes a flat surface. This scalar quantity is therefore referred to as plane irradiance.

There is another definition that is explicitly referred to as scalar irradiance $E_o$. It takes into account the radiant power arriving at a point in space from all directions:

Formula symbol: $E_o$
Unit of measurement: Watts per square metre, $\left[E_o\right]=\frac{\text{W}}{{\text{m}}^{\text{2}}}$

A measurement method apparently requires a spherical detector and the determination of the radiant power on its surface; however, this is not feasible in practice. Arrangements that allow measurement will be presented in a subsequent supplement.

Thirdly, there is the vectorial irradiance $\overrightarrow{E}$, which consists of the components of the irradiance in the spatial directions:

The Gershun equation links the vectorial and scalar irradiance with the absorption coefficient a of the medium in which the radiation propagates:

It is intuitively clear that the attenuation of the irradiance, represented by its divergence $\nabla \cdot \overrightarrow{E}$ (the divergence characterises sources and sinks in a vector field), depends on the radiation level $E_o$ and the absorption coefficient. A formal justification of the equation is given in Supplement 1.8 on radiative transfer.