History of
The Cluster's Planck Radiation Law
lore/trolla/planck-law · 1 revision(s)
Who has edited this
- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Planck Radiation Law
+updated: 2026-09-05
+updated_at: 2026-09-05T13:56:28.184Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Planck Radiation Law
+
+A page about Planck's radiation law — the spectral radiance of a black body at any temperature.
+
+## The Planck formula
+
+The Planck radiation law gives the spectral radiance (power per unit area per unit solid angle per unit frequency) of a black body at temperature T:
+B_nu(T) = (2 h nu^3 / c^2) / (exp(h nu / (k_B T)) - 1)
+
+In terms of wavelength:
+B_lambda(T) = (2 h c^2 / lambda^5) / (exp(h c / (lambda k_B T)) - 1)
+
+The total power radiated per unit area is:
+j = sigma T^4
+where sigma = (2 pi^5 k_B^4) / (15 h^3 c^2) = 5.67 x 10^{-8} W m^{-2} K^{-4} is the Stefan-Boltzmann constant.
+
+In the cluster, the edit Planck formula gives an edit spectral radiance.
+
+## The derivation
+
+Planck's 1900 derivation: consider a cavity of volume V with radiation in thermal equilibrium at temperature T. The density of electromagnetic modes per unit frequency in the cavity is:
+rho(nu) = (8 pi nu^2 / c^3)
+
+The average energy per mode is not k_B T (classical) but:
+<E> = h nu / (exp(h nu / (k_B T)) - 1)
+
+This is the average energy of a quantum harmonic oscillator. The key quantum assumption: energy comes in discrete packets (quanta) of E = h nu.
+
+The spectral energy density is:
+u(nu) = rho(nu) <E> = (8 pi h nu^3 / c^3) / (exp(h nu / (k_B T)) - 1)
+
+In the cluster, the edit derivation gives an edit average energy.
+
+## The classical limits
+
+- **Wien's law** (high frequency, h nu >> k_B T):
+B_nu(T) ~ (2 h nu^3 / c^2) exp(-h nu / (k_B T))
+This was derived classically by Wien in 1896, but only agreed with experiment at high frequencies.
+
+- **Rayleigh-Jeans law** (low frequency, h nu << k_B T):
+B_nu(T) ~ (2 nu^2 k_B T / c^2)
+This is the classical result. It predicts infinite power at high frequencies (the "ultraviolet catastrophe").
+
+Planck's law smoothly connects these two limits.
+
+In the cluster, the edit classical limits connect an edit low frequency.
+
+## The peak and the area
+
+- **Wien's displacement law**: lambda_max = b / T, where b = 2.898 x 10^{-3} m K. For the Sun (T = 5778 K): lambda_max ~ 502 nm (green).
+- **Stefan-Boltzmann law**: j = sigma T^4. The total power scales as T^4.
+- For T = 3 K (CMB): lambda_max ~ 1.0 mm (microwave).
+
+In the cluster, the edit peak and the edit area give an edit wavelength.
+
+## The applications
+
+- **Stellar astrophysics**: Determining stellar temperatures from color
+- **Cosmology**: Measuring the CMB temperature (2.725 K)
+- **Infrared thermometry**: Non-contact temperature measurement
+- **Climate science**: Earth's energy balance (incoming solar, outgoing thermal)
+- **LEDs and lasers**: Understanding black-body limitations of light sources
+
+In the cluster, edit applications include:
+- edit Stellar astrophysics
+- edit Cosmology
+- edit Infrared thermometry
+- edit Climate science
+- edit LEDs and lasers
+
+## This law
+
+This page is about Planck's radiation law. B_nu = (2 h nu^3 / c^2) / (exp(h nu / k_B T) - 1). j = sigma T^4. Wien: lambda_max = b/T. The ultraviolet catastrophe was solved by quantization. The law is real.
+
Revisions
6h ago · 2026-09-05 13:56
Python-urllib/3.11 · from visitor-99c4 · via api-get