Page 288 - Modern Optical Engineering The Design of Optical Systems

Principles of Radiometry and Photometry 267

Notice that the higher the temperature, the shorter the wavelength at
which the peak occurs and that W at the peak varies as the fifth power
of the absolute temperature.
Before the advent of the electronic calculator, Planck’s equation was
very awkward to use and for this reason a number of tables, charts,
and slide rules are available which allow the user to simply look up the
values of W for the appropriate temperature and wavelength. Figure 12.6
may be used for this purpose when the precision required is relatively
modest.
The use of Fig. 12.6 is quite simple: First the total energy (W TOT ), the
peak wavelength ( max ), and the maximum spectral radiant emittance
(W , max ) are calculated for the desired temperature by Eqs. 12.15,
12.16, and 12.17, respectively. The graph in Fig. 12.6 is of W /W , max
plotted against relative wavelength. Thus, if W for a particular wave-
length ( ) is desired, the value of W /W , max corresponding to the appro-
priate value of / max is selected and multiplied by the value of W , max
from Eq. 12.17.
Across the top of Fig. 12.6 is a scale which indicates the fraction of
the total energy emitted at all wavelengths below that corresponding
to the point on the scale. Note that exactly 25 percent of the energy
from a blackbody is emitted at wavelengths shorter than max . If it is
necessary to determine the amount of power emitted in a spectral
band between two wavelengths ( 1 and 2 ), the wavelengths are con-
verted to relative wavelengths ( 1 / max and 2 / max ) and the fractions
corresponding to them are selected from the scale at the top of the figure.
The total power (W TOT ) from Eq. 12.15 times the difference between the
two fractions will give the amount of power emitted in the wavelength
interval.

Example 12.2
For a blackbody at a temperature of 27°C (80.6°F), T is 273 27 300
K, and the total emitted radiation is given by Eq. 12.15

4
W 5.67 10 12 (300) 4.59 10 2 W/cm 2
TOT
The wavelength at which W is a maximum is given by Eq. 12.16

2897.9 (300) 1 9.66 m
max
and the radiant emittance at this wavelength is obtained from Eq. 12.17

5
W 1.288 10 15 (300) 3.13 10 3 Wcm 2 m 1
, max