A historic failure of classical physics is its description...

A historic failure of classical physics is its description of the electromagnetic radiation from a black body. Consider a simple model for an ideal black body consisting of a cubic cavity of side $L$ with a small hole in one side. (a) Assuming the classical equipartition of energy, derive an expression for the average energy per unit volume and unit frequency range (RayleighJeans' Law). In what way does this result deviate from actual observation?

A historic failure of classical physics is its description of the electromagnetic radiation from a black body. Consider a simple model for an ideal black body consisting of a cubic cavity of side $L$ with a small hole in one side.
(a) Assuming the classical equipartition of energy, derive an expression for the average energy per unit volume and unit frequency range (RayleighJeans' Law). In what way does this result deviate from actual observation?

Key Concepts

Equipartition Theorem

This theorem in classical statistical mechanics states that energy is equally distributed among all degrees of freedom in a system. In the context of blackbody radiation, it implies that each mode of the electromagnetic field should carry an average energy proportional to the temperature (kT), which forms the basis of the classical derivation.

Density of Electromagnetic Modes

This concept refers to the number of allowed standing wave modes per unit volume within a cavity, quantified over a specific frequency interval. It is essential in calculating how energy is distributed among various frequencies in a blackbody, as it enables the integration of contributions from all the modes.

Rayleigh-Jeans Law

Derived using classical physics principles such as the equipartition theorem and the density of electromagnetic modes, the Rayleigh-Jeans Law gives the spectral energy density of a blackbody. It accurately predicts the energy distribution at low frequencies but leads to a divergence at high frequencies.

Ultraviolet Catastrophe

This term describes the failure of the classical Rayleigh-Jeans Law to match experimental observations at high frequencies, where the prediction of infinite energy density (due to the divergence) contradicts the finite energy observed in reality. This discrepancy was a major impetus for the development of quantum mechanics.

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