9 – Infrared Primer
A6700sc/A6750sc User’s Manual
52
This is the Stefan-Boltzmann formula (after Josef Stefan, 1835–1893, and Ludwig Boltzmann, 1844–
1906), which states that the total emissive power of a blackbody is proportional to the fourth power of
its absolute temperature. Graphically, Wb represents the area below the Planck curve for a particular
temperature. It can be shown that the radiant emittance in the interval
λ = 0 to λmax is only 25 % of
the total, which represents about the amount of the sun’s radiation which lies inside the visible light
spectrum.
Figure 8-11: Josef Stefan (1835–1893), and Ludwig Boltzmann (1844–1906)
Using the Stefan-Boltzmann formula to calculate the power radiated by the human body, at a
temperature of 300 K and an external surface area of approx. 2 m2, we obtain 1 kW. This power loss
could not be sustained if it were not for the compensating absorption of radiation from surrounding
surfaces, at room temperatures which do not vary too drastically from the temperature of the body –
or, of course, the addition of clothing.
Non-Blackbody Emitters
So far, only blackbody radiators and blackbody radiation have been discussed. However, real objects
almost never comply with these laws over an extended wavelength region – although they may
approach the blackbody behavior in certain spectral intervals. For example, a certain type of white
paint may appear perfectly
white
in the visible light spectrum, but becomes distinctly
gray
at about 2
μ
m, and beyond 3
μ
m it is almost
black
.
There are three processes which can occur that prevent a real object from acting like a blackbody: a
fraction of the incident radiation
α
may be absorbed, a fraction
ρ
may be reflected, and a fraction
τ
may be transmitted. Since all of these factors are more or less wavelength dependent, the subscript
λ
is used to imply the spectral dependence of their definitions. Thus:
The spectral absorptance
αλ= the ratio of the spectral radiant power absorbed by an object to that
incident upon it.
The spectral reflectance
ρλ = the ratio of the spectral radiant power reflected by an object to that
incident upon it.
The spectral transmittance
τλ = the ratio of the spectral radiant power transmitted through an object to
that incident upon it.
The sum of these three factors must always add up to the whole at any wavelength, so we have the
relation:
For opaque materials
τ
λ
= 0
and the relation simplifies to:
