The Rayleigh–Jeans law agrees with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe, and its resolution was a foundational aspect of the development of quantum mechanics in the early 20th century.
In 1900, the British physicist Lord Rayleigh derived the λ−4 dependence of the Rayleigh–Jeans law based on classical physical arguments and empirical facts. A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans in 1905. The Rayleigh–Jeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with this prediction.
where h is the Planck constant and kB the Boltzmann constant. The Planck law does not suffer from an ultraviolet catastrophe, and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since,
then in the limit of low temperatures or long wavelengths, the term in the exponential becomes small, and the exponential is well approximated with the Taylor polynomial's first-order term,
This results in Planck's blackbody formula reducing to
which is identical to the classically derived Rayleigh–Jeans expression.
The same argument can be applied to the blackbody radiation expressed in terms of frequency ν = c/λ. In the limit of small frequencies, that is ,
This last expression is the Rayleigh–Jeans law in the limit of small frequencies.
Consistency of frequency and wavelength dependent expressions
When comparing the frequency and wavelength dependent expressions of the Rayleigh–Jeans law it is important to remember that
even after substituting the value , because has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency. To be consistent, we must use the equality
where both sides now have units of power (energy emitted per unit time) per unit area of emitting surface, per unit solid angle.
Starting with the Rayleigh–Jeans law in terms of wavelength we get
Depending on the application, the Planck function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per spectral unit. In this form, the Planck function and associated Rayleigh–Jeans limits are given by
Alternatively, Planck's law can be written as an expression for emitted power integrated over all solid angles. In this form, the Planck function and associated Rayleigh–Jeans limits are given by
In other cases, Planck's law is written as for energy per unit volume (energy density). In this form, the Planck function and associated Rayleigh–Jeans limits are given by