# Derivation of Planck's law and the shape of the cavity

1. Sep 10, 2014

### center o bass

2. Sep 10, 2014

### Avodyne

That's a good question!

First of all, we can easily do a rectangular box with side lengths $L_1$, $L_2$, and $L_3$. If you work this through, you will find that the factor of $L^3$ in the formula for $dN(p)$ on page 5 gets replaced with $L_1 L_2 L_3$, which is just the volume of the rectangular box. So that's a hint that only the volume matters.

To show that this is true for more general shapes requires some higher-level math; the general result is known as Weyl's Law.

3. Sep 10, 2014

### dextercioby

Weyl's Law? Do you have a reference for that? Perhaps Wien's Law.

4. Sep 10, 2014

### Avodyne

5. Sep 12, 2014

### center o bass

Thats interessting! Do you have a reference where Weyl's law is utilized in deriving Planck's law?

6. Sep 12, 2014

### Avodyne

Weyl was motivated by the blackbody problem:

Arendt, W., Nittka, R., Peter, W. and Steiner, F. (2009) Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics, in Mathematical Analysis of Evolution, Information, and Complexity (eds W. Arendt and W. P. Schleich), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. doi: 10.1002/9783527628025.ch1

http://media.johnwiley.com.au/product_data/excerpt/04/35274083/3527408304.pdf

Much more about the history in this paper.

7. Sep 15, 2014

### Jano L.

The cavity is mentioned only for historical reasons; it is the way blackbody was prepared and measured originally. But it does not really play any explicit role in the Rayleigh-Jeans type of derivation of spectrum (treating EM modes like harmonic oscillators in thermal equilibrium). This is because one can assume any cuboid of vacuum with imaginary walls inside the real cavity, expand the field inside into Fourier series and arrive at the result.

The really important assumptions behind the derivation are:

- energy interpretation of the Poynting theorem, where $\frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2$ is density of energy;
- some rule for assigning average energy to one Fourier mode (equipartition rule or the energy of quantum harmonic oscillator etc.).