## Section 4.1: Blackbody Radiation

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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^{−4}*k*_{B}*T* , (4.1)

where *k*_{B} = 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/m^{4} 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 *ultraviolet catastrophe*.

Planck solved this problem by treating energy not as continuously variable, but instead, as if it came in discrete units, *E*_{γ}, and for light, energy was proportional to frequency

*E*_{γ} = *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) = 8π*hc*λ^{−5}/(e^{hc/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,

λ_{max}*T* = 2.898 × 10^{-3} m**·**K, (4.4)

for the 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}

^{1}For 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.

^{2}You may also see the Rayleigh-Jeans and Planck formulas in terms of frequency as

*u*(ν) = 8πν^{2}*c*^{−3}*k*_{B}*T* (4.5)

and

*u*(ν) = 8π*hν*^{3}*c*^{−3}/(e^{hν/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).

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