The Rayleigh-Jeans formula is a classical approximation for the spectral radiance of black-body radiation at different wavelengths. It was derived by Lord Rayleigh and Sir James Jeans in the late 19th and early 20th centuries, respectively. However, this formula fails to accurately describe the behavior of black-body radiation at short wavelengths, which is known as the ultraviolet catastrophe.
In the Rayleigh-Jeans formula, the spectral radiance (Bλ) is proportional to the wavelength (λ) to the power of 4, divided by the temperature (T) and a constant. Mathematically, it can be written as:
Bλ = (2cRλ^4) / (λ^5T),
where c is the speed of light, R is a constant known as the Rayleigh-Jeans constant, λ is the wavelength, and T is the temperature.
At long wavelengths, the Rayleigh-Jeans formula agrees reasonably well with experimental observations and classical physics. This is because at longer wavelengths, the energy of the photons becomes relatively small compared to the thermal energy associated with the temperature of the black-body radiator. Thus, the classical physics assumptions of continuous energy distribution and equipartition of energy among the different modes of vibration are applicable.
However, as the wavelength decreases and approaches the ultraviolet region, the predictions of the Rayleigh-Jeans formula diverge significantly from experimental observations. According to classical physics, the formula predicts an infinite amount of energy radiated at short wavelengths, which is clearly not observed in reality.
This discrepancy was eventually resolved by the development of quantum mechanics and Max Planck's introduction of the concept of energy quantization. Planck's quantum theory successfully explained the observed behavior of black-body radiation, leading to the development of the Planck radiation law, which accurately describes the entire spectrum of black-body radiation. The Planck radiation law incorporates the quantization of energy and provides a more accurate description of black-body radiation at all wavelengths, including the ultraviolet region, where classical physics fails.