14.3.3
Stefan-Boltzmann's law
By integrating Planck’s formula from λ = 0 to λ = ∞, we obtain the total radiant
emittance (W
b
) of a blackbody:
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,
W
b
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
λ
max
is only 25 % of the total, which represents
about the amount of the sun’s radiation which lies inside the visible light spectrum.
10399303;a1
Figure 14.7 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 m
2
, 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.
14.3.4
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 wave-
length 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 μm, and
beyond 3 μm 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
Publ. No. 1 557 544 Rev. a121 – ENGLISH (EN) – October 6, 2005
61
14.3 – Blackbody radiation
