The Rayleigh-Einstein-Jeans law

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss blackbody radiation. Max Planck’s blackbody radiation formula is given in Eq. 14.33

where λ is the wavelength and T is the absolute temperature. This equation, derived in December 1900, is the first formula that contained Planck’s constant, h.

Often you can recover a classical (non-quantum) result by taking the limit as Planck’s constant goes to zero. Here’s a new homework problem to find the classical limit of the blackbody radiation formula.

Section 14.8

Problem 26 ½. Take the limit of Planck’s blackbody radiation formula, Eq. 14.33, when Planck’s constant goes to zero. Your result should should be the classical Rayleigh-Jeans formula. Discuss how it behaves as λ goes to zero. Small wavelengths correspond to the ultraviolet and x-ray part of the electromagnetic spectrum. Why do you think this behavior is known as the “ultraviolet catastrophe”?

I always thought that the Rayleigh-Jeans formula was a Victorian result that Planck knew about when he derived Eq. 14.33. However, when thumbing through Subtle is the Lord: The Science and the Life of Albert Einstein, by Abraham Pais, I learned that the Rayleigh-Jeans formula is not much older than Planck’s formula. Lord Rayleigh derived a preliminary version of it in June 1900, just months before Planck derived Eq. 14.33. He published a more complete version in 1905, except he was off by a factor of eight. James Jeans caught Rayleigh’s mistake, corrected it, and thereby got his name attached to the Rayleigh-Jeans formula. I am amazed that Planck’s blackbody formula predates the definitive version of the Rayleigh-Jeans formula.

Einstein rederived Rayleigh’s formula from basic thermodynamics principles in, you guessed it, his annus mirabilis, 1905. Pais concludes “it follows from this chronology (not that it matters much) that the Rayleigh-Jeans law ought properly to be called the Rayleigh-Einstein-Jeans law.”