Heuristic Derivation of Blackbody Radiation Laws using Principles of Dimensional Analysis

TL;DR

This paper uses dimensional analysis and heuristic principles to derive generalized blackbody radiation laws, including Wien's displacement law and the Rayleigh-Jeans law, providing insights into universal constants and functions that avoid ultraviolet catastrophe.

Contribution

It introduces a heuristic, dimensional analysis-based method to derive and generalize blackbody radiation laws, including the universal constants and functions involved.

Findings

Derived generalized Wien's displacement law.

Identified a universal pi number for Rayleigh-Jeans law.

Established a universal function to prevent ultraviolet catastrophe.

Abstract

A generalized form of Wien's displacement law and the blackbody radiation laws of (a) Rayleigh and Jeans, (b) Rayleigh, (c) Wien and Paschen, (d) Thiesen and (e) Planck are derived using principles of dimensional analysis. This kind of scaling is expressed in a strictly mathematical manner employing dimensional pi-invariants analysis sometimes called Buckingham's pi-theorem. It is shown that in the case of the classical radiation law of Rayleigh and Jeans only one pi number occurs that has to be considered as a non-dimensional universal constant. This pi number may be determined theoretically or/and empirically. It is also shown that dimensional pi-invariants analysis yields a generalized form of Wien's displacement law. In this instance two pi numbers generally occur. Consequently, a universal function is established that is indispensable to avoid the so-called Rayleigh-Jeans catastrophe in the ultraviolet. Unfortunately, such a universal function cannot be inferred from dimensional arguments. It has to be derived theoretically or/and empirically, too. It is shown that such a similarity function can be deduced on the basis of heuristic principles, when criteria like the maximum condition regarding the generalized form of Wien's displacement law, the requirement of the power law of Stefan and Boltzmann, and Ehrenfest's arguments regarding the red and the violet requirements are adopted.