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May 30

Blackbody energy density

In blackbody radiation, how do you go from measuring radiation power per unit area to obtain energy density per unit volume? I am looking for this formula and one of the first publications that uses it, by Wilhelm Wien for example ? Because in the laws of Kirchhoff and Stephan-Boltzmann it is always a question of radiation power.

~~~~ Malypaet (talk) 07:36, 30 May 2023 (UTC)[reply]

You can imagine the radiation crossing a surface at the speed of light into a new volume. In one second it will have traveled c meters. So to get what is in one cubic meter, divide that energy by c to get energy per unit volume.

{\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{3}h^{3}}}

However only half the energy is heading in the correct direction (half is going backwards) and also half is coming at different angles other than perpendicular, so you have to multiply by cos of the angle of incidence and integrate over possible angles. so then you have to multiply the answer by 4. (Another way is to consider the area of a sphere versus the area of the circle 4 πr² versus πr². Graeme Bartlett (talk) 10:04, 30 May 2023 (UTC)[reply]

So when you write the word energy, as you multiple power by one second, you use the system unit W-s, right ?

~~~~ Malypaet (talk) 11:52, 30 May 2023 (UTC)[reply]

Sure, but a joule is identical to a watt-second. --Jayron 32 14:26, 30 May 2023 (UTC)[reply]

Where can I find a reference for this formula? Malypaet (talk) 21:07, 30 May 2023 (UTC)[reply]

see Stefan–Boltzmann law#Energy density for our article. Graeme Bartlett (talk) 01:28, 31 May 2023 (UTC)[reply]

The problem with Wikipedia articles is that they present the Stephan-Boltzmann law with references from the future (Planck, Bose-Einstein...). But at least you gave me a reference on the source. I just have to translate the German! Malypaet (talk) 07:24, 31 May 2023 (UTC)[reply]

I have not tried to discover the first publications as you ask. It appears that people first worked out what was happening in a spherical cavity at a particular wavelength, whereas you are interested in the result over all wavelengths. Though your question is more general than blackbody radiation, and applies to any stuff moving at the speed of light (light, neutrinos, gravitational waves). So there may be an answer for you from before Wien. Graeme Bartlett (talk) 12:22, 31 May 2023 (UTC)[reply]