# Historic derivation of Wien's law

Every book I've read, including a lot of websites, Wikipedia, etc, say that Wien derived this:

$$\rho_\nu(T)=\rho(\nu,T)=\nu^3f\left(\frac{\nu}{T}\right)$$

Being $\rho_v(T)$ the spectral enegy density of a black body for a given temperature and electromagnetic wave frequency. And everywhere it's mentioned that he proved this using thermodynamical arguments in a paper from 1893. I haven't been able to find that paper or that thermodynamical argument, which is what I'm interested in. I've been looking for a few days already.

Does anybody know how he did this?

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See webpages.uidaho.edu/~crepeau/ht2009-88060.pdf, which claims that the distribution law was derived by Wien in tandfonline.com/doi/abs/10.1080/14786449708620983 – joshphysics Feb 18 at 22:54
@joshphysics: Those look strong. To avoid link-only answers, could you fill in the argument as an answer? – Emilio Pisanty Feb 18 at 23:44

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(That 1901 picture of Planck is priceless, by the way. A far shout from the usual "statesman of science" pictures.) – Emilio Pisanty Feb 18 at 23:44
@joshphysics Thanks. That article was very interesting. I will try to find that article for free... maybe my university has some copies. Although that article is from year 97, will all sources say that relation was derived in his 93 paper: Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, which is also cited in that article, and that sadly it seems like it has no translated version. – MyUserIsThis Feb 19 at 7:11
– tom May 4 at 15:54
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## Interpretation and assumptions

I expect you want to see the relation $\rho(\nu) = \frac{8\pi h}{c^3}\nu^3$. The derivation requires the assumption of an ideal black body (total absorption and no reflection) of size $L^3\gg\lambda^3$ and the spectral energy density should be homogeneous and isotropic.

## Derivation

I had a lecture in German which explained this topic very well.

1. We calculate the eigenmodes of a box, where the mode index is $j^2 = j_x^2+j_y^2+j_z^2=\left(\frac{2\nu}{c}L\right)^2$ where we used the condition of resonance.

2. We calculate the number of modes $G(\nu)=2\frac{1}{8}\frac{4\pi}{3}j^3$ in the frequency spectrum between 0 and \nu.

3. We calculate the spectral mode density $g(\nu)=\frac{\partial G(\nu)}{\partial\nu}$. The spectral energy density $u(\nu)$ is now the product of $g(\nu)$ and the energy per mode $\epsilon_{Wien}=h\nu e^{-\frac{h\nu}{k_B T}}$ (from classical Boltzmann statistics) per volume $L^3$.

4. The rest (simple math) is up to you or see the german reference. You get the relation from above and $$u(\nu)=\rho(\nu)e^{-\frac{h\nu}{k_B T}}\;.$$

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The claim "the energy per mode is$\epsilon=h\nu\exp(-h\nu/kT)$" is much stronger than anything you've assumed and it is essentially equivalent to Planck's energy quantization condition. As such it would have been unknown to Wien, for whom the equipartition theorem would have governed this. – Emilio Pisanty Feb 18 at 23:40
My derivation is a different than the traditional. I thought you are interested in actual physics and such understanding. It uses the basic knowledge about classical Boltzmann statistics. I am not an expert of physical history. If you want to derive that too, you may do so but I don't do it today. Good night. – strpeter Feb 18 at 23:58
No, your derivation is equivalent to the traditional one. Wien's displacement law is much weaker: it does not give a specific result for $\rho_\nu$ but only constrains the form it may take in terms of some unknown function $f$. Finding $f$ means deriving either Wien's distribution law or the full Planck law, both of which are stronger than the question at hand. – Emilio Pisanty Feb 19 at 0:43
Thanks for your answer, although I was more interested in the historical derivation of the weak version of the function as I wrote it in the question. Just the way Wien derived it from maxwell's electromagnetism and classical thermodynamics. – MyUserIsThis Feb 19 at 7:12
– strpeter Feb 19 at 22:41

## protected by Qmechanic♦May 4 at 16:28

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