Demonstrate that the Planck distribution reduces to the Rayleigh-Jeans law at long wavelengths.

The Rayleigh-Jeans Law is a formula used to describe the spectral radiance of electromagnetic radiation at different frequencies from a black body at a given temperature. This law was developed around the early 20th century by Lord Rayleigh and Sir James Jeans.

The law is expressed by the formula:
\[\begin{equation}B_{RJ}(u, T) = \frac{8\:pi kT u^2}{c^3}\end{equation}\]where:

• B_{RJ}(u, T) is the spectral radiance,
• u is the frequency of the radiation,
• T is the absolute temperature of the black body,
• k is Boltzmann's constant, and
• c is the speed of light in a vacuum.

It accurately predicts the radiation at low frequencies (or long wavelengths), but fails at higher frequencies by predicting an infinite amount of radiated energy, known as the 'ultraviolet catastrophe'. This inconsistency led to the development of quantum mechanics and the Planck distribution which corrects this shortfall at short wavelengths.

Blackbody radiation refers to the theoretical spectrum of electromagnetic radiation emitted by an object that absorbs all incident radiation, regardless of frequency or angle of incidence. A perfect black body does not reflect, transmit, or emit radiation due to its temperature.

It is characterized by two key aspects:

• The radiation is solely a function of the blackbody's temperature.
• At thermal equilibrium, the energy emitted at each frequency is equal to the energy absorbed.

As the temperature of a blackbody increases, its radiation becomes more intense and its peak wavelength shifts to shorter wavelengths, which is described by Wien's displacement law. Max Planck's formulation of blackbody radiation led to the quantum theory which marked a major advancement in the understanding of atomic and subatomic processes.

The long wavelengths approximation is applied in the context of blackbody radiation to simplify complex expressions, particularly when the frequency is very low compared to the product of Boltzmann's constant and temperature. When dealing with large wavelengths, or equivalently low frequencies, the energy of the photons becomes very small, and this affects the behavior of the radiation.

In practice, this approximation involves considering the exponent in the exponential of the Planck distribution as very small. As a consequence, the exponential function can be approximated as:\[\begin{equation}e^{\frac{hu}{kT}} \approx 1 + \frac{hu}{kT}\end{equation}\]where:

• h is Planck's constant,
• u is the frequency,
• k is Boltzmann's constant, and
• T is the absolute temperature.

This approximation allows the Planck distribution to reduce to the Rayleigh-Jeans law, thereby bridging classical and quantum physics for the specified conditions.