User Manual

Theory of thermography32
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 approaches
zero again at very long wavelengths. The higher the temperature, the shorter the wave-
length at which maximum occurs.
Figure 32.4 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)
32.3.2 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 mathematically
the common observation that colors vary from red to orange or yellow as the temperature
of a thermal radiator increases. The wavelength of the color is the same as the wave-
length 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 ra-
diant emittance occurring within the invisible ultraviolet spectrum, at wavelength 0.27 μm.
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