Comparing Rayleigh-Jeans and Planck’s Law for Blackbody Radiation

The Rayleigh-Jeans approximation is a classical approach to describing the spectral radiance of blackbody radiation at long wavelengths, including the infrared region. It is derived from the Rayleigh-Jeans law, which expresses the total spectral radiance [math]B_{\lambda }(\lambda ,T)[/math]

[math]B_{\lambda }(\lambda ,T)\approx \frac{2ck_BT}{\lambda ^4}[/math]

The Rayleigh-Jeans approximation works well for long wavelengths, including parts of the infrared spectrum. Since industrial infrared temperature sensors operate in specific infrared bands, this approximation can be useful in estimating radiative behavior in some cases. However, it fails at shorter wavelengths, leading to the well-known “ultraviolet catastrophe,” a problem resolved by Planck’s law. 

The Rayleigh-Jeans approximation is valid for high temperatures, typically in the range of several hundred to thousands of Kelvin. At lower temperatures (<500 K), the approximation significantly deviates from actual blackbody radiation because it does not account for the quantum nature of energy quantization. Figure 1 shows the Rayleigh-Jeans approximation and the overestimation. 

The Rayleigh-Jeans equation and Planck’s law describe blackbody radiation, but they differ fundamentally in their approach to modelling energy distribution across wavelengths. While the Rayleigh-Jeans law provides a simplified understanding of thermal radiation at long wavelengths, it is not the most precise model for infrared thermometry. Instead, the more general Planck’s law should be used for accurate infrared temperature measurement, as it correctly describes blackbody radiation across all wavelengths. Planck’s law, however, correctly models blackbody radiation by incorporating quantum mechanics. It introduces the concept that energy is quantized in discrete packets (photons) with energy proportional to frequency. This prevents excessive radiation at short wavelengths and aligns perfectly with experimental data. Planck’s law is essential for infrared applications because it accurately describes radiation behavior across all wavelengths, including the infrared spectrum where industrial temperature sensors operate. 

The Planck curve in Figure 1 correctly models blackbody radiation, peaking at a specific wavelength and dropping at shorter wavelengths. The Rayleigh-Jeans curve approximates Planck’s curve at long wavelengths but diverges significantly at short wavelengths, increasing unphysically and leading to the ultraviolet catastrophe. 

For most infrared thermometry applications, especially in the short infrared and at low temperatures, Planck’s radiation law should be used for scientific calculations [1,2,3].