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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