Rayleigh-Jeans Law Calculator
Enter temperature and frequency to compute the Rayleigh-Jeans spectral radiance Bν=2ν²kT/c² and check the ultraviolet catastrophe flag. T=2.725 K (CMB), ν=160 GHz → Bν≈3.8e-18; low-frequency approximation valid.
Input Data
Results
At a glance:The Rayleigh–Jeans law (Rayleigh 1900, Jeans corrected 1905): in the low-frequency limit (ν≪kT/ℏ, i.e. hν≪kT) the blackbody spectral radiance is Bν=2ν²kT/c² (power per unit frequency per steradian). Derivation: classical electrodynamics treats cavity modes as oscillators and, by the equipartition theorem, assigns each mode kT energy (ignoring quantisation), summing to Bν=2ν²kT/c². Per-wavelength form Bλ=2ckT/λ⁵=2ν⁵kT/c⁴. Physical meaning: at low frequency the classical theory matches experiment (hν≪kT, quantum effects negligible); but at high frequency Bν∝ν² diverges (the ultraviolet catastrophe), contradicting measurement. Planck (1900) introduced the energy quantum ε=hν, yielding the correct formula Bν=2hν³/c²/(exp(hν/kT)−1), which reduces to Rayleigh–Jeans in the low-frequency limit. History: Rayleigh missed a factor of 8, Jeans corrected it in 1905; this contradiction directly gave birth to quantum theory. Applications: (1) low-frequency radio astronomy (2.725 K cosmic background); (2) microwave thermal radiation; (3) infrared calibration; (4) antenna noise temperature; (5) judging where the high-frequency failure occurs.
Formula
Per-frequency: Bν = 2ν²kT/c²
Per-wavelength: Bλ = 2ckT/λ⁵ = 2ν⁵kT/c⁴
Thermal frequency: ν_th = kT/ℏ (UV-catastrophe threshold)
Full Planck: Bν = 2hν³/c² / (exp(hν/kT) − 1)
Low-freq limit: hν ≪ kT ⇒ Planck → Rayleigh-Jeans
$$B_\nu = \frac{2\nu^2 k T}{c^2}, \quad B_\lambda = \frac{2c k T}{\lambda^5} = \frac{2\nu^5 k T}{c^4}, \quad \nu_{\text{th}} = \frac{kT}{\hbar}$$How to Use
- Enter temperature T (K; room 300).
- Enter frequency ν (Hz; low like 1e11).
- The tool computes Bν, Bλ and an ultraviolet-catastrophe flag.
Case Studies
Cosmic microwave background
The CMB temperature is 2.725 K, peak frequency ~160 GHz — within the Rayleigh-Jeans range.
Radio telescopes use the RJ approximation to convert brightness temperature: T = Bν·c²/(2kν²). Penzias-Wilson discovered the CMB in 1964 (Nobel).
The CMB is relic radiation from the Big Bang, supporting the expanding-universe model.
Infrared thermometer and thermal camera
At room T=300 K, the Rayleigh-Jeans approximation holds below ~20 THz (far/mid IR).
A thermal camera's detector noise temperature is calibrated via RJ: T_noise = Bν·c²/(2kν²).
Above the threshold Planck must replace RJ, or the radiance is hugely overestimated (UV catastrophe).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.