Theory of thermography
18
where:
W
λb
Blackbody spectral radiant emittance at wavelength λ.
c
Velocity of light = 3 × 10
8
m/s
h
Planck’s constant = 6.6 × 10
-34
Joule sec.
k
Boltzmann’s constant = 1.4 × 10
-23
Joule/K.
T
Absolute temperature (K) of a blackbody.
λ
Wavelength (μm).
Note
The factor 10
-6
is used since spectral emittance in the curves is expressed in
Watt/m
2
, μm.
Planck’s formula, when plotted graphically for various temperatures, produces a family of
curves. Following any particular Planck curve, the spectral emittance is zero at λ = 0,
then increases rapidly to a maximum at a wavelength λ
max
and after passing it ap-
proaches zero again at very long wavelengths. The higher the temperature, the shorter
the wavelength at which maximum occurs.
Figure 18.4
Blackbody spectral radiant emittance according to Planck’s law, plotted for various absolute
temperatures. 1: Spectral radiant emittance (W/cm
2
× 10
3
(μm)); 2: Wavelength (μm)
18.3.2
Wien’s displacement law
By differentiating Planck’s formula with respect to λ, and finding the maximum, we have:
This is Wien’s formula (after
Wilhelm Wien
, 1864–1928), which expresses mathemati-
cally the common observation that colors vary from red to orange or yellow as the tem-
perature of a thermal radiator increases. The wavelength of the color is the same as the
wavelength calculated for λ
max
. A good approximation of the value of λ
max
for a given
blackbody temperature is obtained by applying the rule-of-thumb 3 000/T μm. Thus, a
very hot star such as Sirius (11 000 K), emitting bluish-white light, radiates with the peak
of spectral radiant emittance occurring within the invisible ultraviolet spectrum, at wave-
length 0.27 μm.
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