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One of the first failures of classical theory came about during the analysis of radiation from opaque, or black, objects. Such black bodies radiated and had energy densities, energy per volume per wavelength, u(λ), that depended on their temperature. Restart. The Rayleigh-Jeans formula for blackbody radiation, derived from the classical equipartition of energy theorem, gives the following functional form for such an energy density:
u(λ) = 8πλ−4kBT , (4.1)
where kB = 1.381 × 10−23 J/K is Boltzmann's constant
and T is the temperature in Kelvin. Select the "R - J" button on the animation
and change the temperature to see how this curve varies. Note the units on
the graph (J/m4 vs. microns) and that the graph's scale changes as you change the temperature. The Rayleigh-Jeans
formula agrees well with the experimental results for very long wavelengths (at
low frequencies). As the wavelength of the radiation gets smaller (at high
frequencies), the Rayleigh-Jeans formula states that the energy density of the radiation approaches infinity. This
does not agree with experiment, however, and the failure of this classical result to agree with experiment is called the
Planck solved this problem by treating energy not as continuously variable, but instead, as if it came in discrete units, Eγ, and for light, energy was proportional to frequency
Eγ = hν = hc/λ , (4.2)
where h = 6.626 × 10−34 J·s was a new constant, now called Planck's constant, tuned to fit the blackbody radiation data. When Planck did the u(λ) calculation with this assumption, he found:1
u(λ) = 8πhcλ−5/(ehc/λ kBT−1) , (4.3)
which agrees with the experimental data. Select the "Planck" button on the animation and change the temperature to see how this curve varies with temperature. In addition, Wien's displacement law,
λmaxT = 2.898 × 10-3 m•K, (4.4)
wavelength corresponding to the
maximum energy density per wavelength, can be verified by looking at the Planck curve.
Because of the agreement between the data and the Planck blackbody radiation law, selecting the "Planck and R - J" button shows just how poorly the Rayleigh-Jeans formula does in replicating the true blackbody radiation curve in the small-wavelength limit.2
1For more details on the derivation of Planck's blackbody
radiation law, see P. A. Tipler and R. A. Llewellyn, Modern Physics, W.
H. Freeman and Company (1999) or see Section 15.5.
2You may also see the Rayleigh-Jeans and Planck formulas in terms of frequency as
u(ν) = 8πν2c−3kBT (4.5)
u(ν) = 8πhν3c−3/(ehν/kBT − 1), (4.6)
respectively. Note that the difference in form is not just due to the substitution ν = c/λ. Also note that because of this difference, the graphs of u(ν) vs. ν will look "flipped around" as compared to u(λ) vs. λ. These graphs will also be peaked at different values of λ (c/ν).