# Geometry Effect on Black Body Radiation with Different Boundary Conditions

## TL;DR

This paper investigates how the geometry and boundary conditions of cavities influence black body radiation, deriving formulas for energy spectral densities and highlighting deviations from Planck's law in small or specific geometries.

## Contribution

It provides new formulas for energy spectral densities in various geometries and boundary conditions, and analyzes deviations from classical black body radiation laws.

## Key findings

- Energy spectral densities deviate from Planck's formula in small or specific geometries.
- Boundary conditions significantly affect the energy spectral densities.
- An asymptotic formula for closed cavities with Dirichlet boundary condition is derived.

## Abstract

We study the black body radiation in cavities of different geometry with different boundary conditions, and the formulas of energy spectral densities in films and rods are obtained, also the approximate energy spectral densities in cubic boxes are calculated. We find that the energy spectral densities deviate from Planck's formula greatly when the length(s) at some directions of the cavity can be compared to the typical wavelengths of black body radiation in infinite volume and the boundary conditions also affect the results. We obtain an asymptotic formula of energy spectral density for a closed cavity with Dirichlet boundary condition and find it still fails when the size of the cavity is too small.