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Optical Sources, Detectors, and Systems by Robert H. Kingston

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14
Chapter
1
Blackbody Radiation^ Image Plane Intensity, and Units
Figure
1.7.
Model
for
calculation
of
background radiation from
a
ptirtially transmitting
medium.
1.3 The Stefan-Boltzmann Law
We
now
calculate
the
total radiance
H of a
surface
of
constant
emissivity
e at
temperature
T; the
result
is
known
as the
Stefan-
Boltzmann law. Starting with
Eq
(1.16),
we
obtain
Inhv^dv Inikjf
"
rj
{rj . f ^rmv av
zmKi
j r ^. ., ^. .
H =
\H,dv
=
el^^^^^^l^^_^^
=
e-^^^\F(x)dx;
F(x)^
(e^'-V
(1.17)
Fortunately,
the
dimensionless integral
has an
explicit value
of n
^115,
and
the
final result may be written
0.18)
and
the
total irradiance striking
the
walls
of a
blackbody chamber
is
/
=
(JV,
where
a,
the
Stefan-Boltzman constant,
is
(7 =
5.67x10-^
W/m^K^
1.3 The Stefan-Boltzmann Law
15
For a unit emissivity surface at 300 K, the radiance beconnes 460 W/m^.
In comparison with this value for room-temperature radiation we may
calculate the radiance of the sun's surface, at 5800 K, to he 6A2 x 1(f
watts/m^. The rapid fourth-power increase in radiance with temperature
is a result of the combination of a linear increase in the frequency
maximum with temperature combined with a cubic increase in total
mode energy, since the mode density goes as the square and the mode
energy directly with frequency.
Next we wish to calculate another more pertinent number with re-
gard to the sun. This is called the
solar
constant, which is the total inten-
sity in
W/nfi
of the solar radiation striking the earth above any absorbing
atmosphere. To obtain the value of this quantity, we return to our
blackbody chamber and assume the walls have unit emissivity and are
heated to 5800 K. In this case, however, as shown in Figure 1.8, we
assume a hemispheric projection from the upper wall which represents
the sun. Just as we removed the whole chamber to determine the power
emitted from a surface, we shall now remove all but the hemisphere and
determine the irradiance /, which is striking a point at the bottom of the
chamber. Since
£2
for the sun is small, all the radiation is effectively
normal to the surface and, using Eq. (1.14), the irradiance becomes
(1.19)
T =
5800K
Figure 1.8 Calculation of solar constant

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