Example 9.6
Radiation from the Big Bang has been doppler-shifted to longer wavelengths by the expansion
of the universe and today has a spectrum corresponding to that of a blackbody at 2.7 K. Find
the wavelength at which the energy density of this radiation is a maximum. In what region of
the spectrum is this radiation?
Solution
From Eq. (9.40) we have(^) max1.1 10 ^3 m1.1 mm
This wavelength is in the microwave region (see Fig. 2.2). The radiation was first detected in a
microwave survey of the sky in 1964.
Stefan-Boltzmann Law
Another result we can obtain from Eq. (9.38) is the total energy density uof the radiation
in a cavity. This is the integral of the energy density over all frequencies,
u
0
u( ) d T^4 aT^4
where ais a universal constant. The total energy density is proportional to the fourth
power of the absolute temperature of the cavity walls. We therefore expect that the
energy Rradiated by an object per second per unit area is also proportional to T^4 , a
conclusion embodied in the Stefan-Boltzmann law:
Re T^4 (9.41)
The value of Stefan’s constant is
5.670 10 ^8 W/m^2 K^4
The emissivity edepends on the nature of the radiating surface and ranges from 0, for
a perfect reflector which does not radiate at all, to 1, for a blackbody. Some typical values
of eare 0.07 for polished steel, 0.6 for oxidized copper and brass, and 0.97 for matte
black paint.
Example 9.7
Sunlight arrives at the earth at the rate of about 1.4 kW/m^2 when the sun is directly overhead.
The average radius of the earth’s orbit is 1.5 1011 m and the radius of the sun is 7.0 108 m.
From these figures find the surface temperature of the sun on the assumption that it radiates
like a blackbody, which is approximately true.
ac
4
Stefan’s constant
Stefan-Boltzmann
law
8 ^5 k^4
15 c^3 h^3
2.898 10 ^3 mK
2.7K
2.898 10 ^3 mK
T
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