Example 9.4
A lightbulb filament at 2500 K emits light having a maximum intensity at
what wavelength?Solution
Using the Wien displacement law, one determines(^) max2500 K 2898 mK
(^) max1.1592 m or 11,592 Å
This wavelength of light is in the infrared region, very close to the visible light
region. This does not imply that no visible light is emitted, only that the
wavelength maximumof the emitted light lies in the infrared region of the
spectrum.
There were several attempts to model blackbody radiation behavior to ex-
plain these relationships, but they were only partially successful. The most suc-
cessful of these started with an assumption by the English baron Lord John
W. S. Rayleigh that light waves come from tiny oscillators within the blackbody.
Rayleigh assumed too that the energy of the light wave is proportional to its
wavelength, so that the smaller wavelengths would be emitted more easily by
these tiny oscillators. Using the equipartition principle from the kinetic theory
of gases (see Chapter 19), Rayleigh proposed and later James Hopwood Jeans
corrected a simple formula for the infinitesimal amount of energy per unit vol-
ume d(also known as an energy density) in a blackbody in a wavelength in-
terval d as
d^8
k
4
T
d^ (9.20)
In this expression,kis Boltzmann’s constant, is the wavelength, and Tis the
absolute temperature. The total energy per unit volume at a particular tem-
perature is given by the integral of the above expression. Equation 9.20 is
known as the Rayleigh-Jeans law.
Though it is an important first step in trying to model the behavior of light,
the Rayleigh-Jeans law has its limitations. It fits the experimentally observed
blackbody intensity curves such as those shown in Figure 9.12, but only at high
temperatures and only in long-wavelength regions of the spectrum. Most
problematic is the short-wavelength intensity predicted by the Rayleigh-Jeans
law: it indicates that as the wavelength gets smaller, the energy density dgiven
off in the wavelength interval d goes up by a factor of the fourth power. (This
is a consequence of the 4 term in the denominator of equation 9.20.) The fi-
nal result is shown in Figure 9.13, which compares the Rayleigh-Jeans equation
with the known blackbody behavior: the intensity predicted by the Rayleigh-
Jeans law approaches infinity as the wavelength of the light approaches zero.
In terms of Rayleigh’s assumption, it suggests that the smaller the wavelength
of the light, the less the energy of the light, and so the easier it should be for a
blackbody to radiate that light. Infinite intensities, however, are impossible to
obtain! It was obvious from experiments of the time that the intensity of light
at shorter wavelengths does not approach infinity. Instead, the intensities ta-
pered off to zero as the wavelength shortened. The Rayleigh-Jeans law predicts
an ultraviolet catastrophethat does not occur.
Other attempts were made to explain the nature of light in terms of black-256 CHAPTER 9 Pre-Quantum Mechanics