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:**

#. :math:`c` (:code:`c`) is :attr:`~symplyphysics.quantities.speed_of_light`.
#. :math:`k_\text{B}` (:code:`k_B`) is :attr:`~symplyphysics.quantities.boltzmann_constant`.

**Conditions:**

#. The black body is isolated from the environment.
#. :math:`h \nu \ll k_\text{B} T`, i.e. photon energy is much smaller than thermal energy.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Jeans_law>`__.

.. py:currentmodule:: symplyphysics.laws.waves.blackbody_radiation.spectral_energy_density_in_low_frequency_limit

.. py:data:: spectral_energy_density

    :attr:`~symplyphysics.symbols.basic.spectral_energy_density`.

Symbol:
    :code:`w_f`

Latex:
    :math:`w_{f}`

Dimension:
    :code:`energy/(frequency*volume)`


.. py:data:: radiation_frequency

    :attr:`~symplyphysics.symbols.classical_mechanics.temporal_frequency` of the radiation.

Symbol:
    :code:`f`

Latex:
    :math:`f`

Dimension:
    :code:`frequency`


.. py:data:: equilibrium_temperature

    Equilibrium :attr:`~symplyphysics.symbols.thermodynamics.temperature` of the ensemble.

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: law

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


    Latex:
        .. math::
            w_{f} = \frac{8 \pi f^{2} k_\text{B} T}{c^{3}}