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.