Math 2.0

Classical thermodynamics, heat transfer Thermal radiation Proposition

Rayleigh–Jeans law

Builds on Black bodyThermodynamic temperaturePlanck's law

Statement

Let a black body be in thermal equilibrium at absolute temperature T>0T > 0, with spectral radiance per unit frequency Bν(ν,T)B_\nu(\nu, T). Let hh be the Planck constant, cc the speed of light and kBk_B the Boltzmann constant. Then, in the limit of low frequency, Bν(ν,T)=2ν2kBTc2(1+O(hν/kBT))as hνkBT→0.B_\nu(\nu, T) = \frac{2 \nu^2 k_B T}{c^2} \big(1 + O(h\nu / k_B T)\big) \quad \text{as } \frac{h\nu}{k_B T} \to 0. So Bν≈2ν2kBT/c2B_\nu \approx 2 \nu^2 k_B T / c^2 when hν≪kBTh\nu \ll k_B T.

Remarks

From Planck's law, with x=hν/(kBT)x = h\nu/(k_B T), one has Bν=2ν2kBTc2⋅xex−1B_\nu = \frac{2\nu^2 k_B T}{c^2} \cdot \frac{x}{e^x - 1}, and x/(ex−1)=1−x/2+O(x2)x/(e^x - 1) = 1 - x/2 + O(x^2) as x→0x \to 0. Classically, the law follows from equipartition: each of the 8πν2/c38\pi\nu^2/c^3 cavity modes per unit volume and frequency gets the mean energy kBTk_B T.

Without the low-frequency hypothesis the law fails badly. It grows like ν2\nu^2 without bound, so its integral over all frequencies diverges; this is the ultraviolet catastrophe, which showed that classical physics cannot describe black-body radiation. The law is named after Lord Rayleigh and James Jeans.

Going deeper

For a student

In radio astronomy and microwave remote sensing, hν≪kBTh\nu \ll k_B T holds, and the law makes radiance proportional to temperature. This is why radio brightness is often reported as a brightness temperature.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Black body, concept, no evidence levelBlack bodyconceptEmissivity, concept, no evidence levelEmissivityconceptThermal radiation, concept, no evidence levelThermal radiationconceptThermodynamic temperature, concept, no evidence levelThermodynamic tempera…conceptPlanck's law, statement, no evidence levelPlanck's lawstatementRayleigh–Jeans law, statement, no evidence levelRayleigh–Jeans lawstatement
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  1. Black body concept
  2. Emissivity concept
  3. Thermal radiation concept
  4. Thermodynamic temperature concept
  5. Planck's law statement
  6. Rayleigh–Jeans law statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Jeans_law
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Kindstatement, proposition
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Changed3 November 2026

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