Spectral energy density at low frequency limit

The Rayleigh-Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wave frequency from a blackbody at a given temperature through classical arguments. The Rayleigh-Jeans law agrees with experimental results at large wavelengths (i.e. at low frequencies) but strongly disagrees at short wavelengths (i.e. at high frequencies). This inconsistency is commonly known as the ultraviolet catastrophe.

Notation:

1. \(c\) (c) is speed_of_light.

2. \(k_\text{B}\) (k_B) is boltzmann_constant.

Conditions:

1. The black body is isolated from the environment.

2. \(h \nu \ll k_\text{B} T\), i.e. photon energy is much smaller than thermal energy.

Links:

1. Wikipedia.

spectral_energy_density

spectral_energy_density.

Symbol:

w_f

Latex:

\(w_{f}\)

Dimension:

energy/(frequency*volume)

radiation_frequency

temporal_frequency of the radiation.

Symbol:

f

Latex:

\(f\)

Dimension:

frequency

equilibrium_temperature

Equilibrium temperature of the ensemble.

Symbol:

T

Latex:

\(T\)

Dimension:

temperature

law

w_f = 8 * pi * f^2 * k_B * T / c^3

Latex:

\[w_{f} = \frac{8 \pi f^{2} k_\text{B} T}{c^{3}}\]