Stefan's Law states that emissive power ($E$) of a black body is proportional to $T^4$. But how did Stefan arrive at the conclusion? I mean, it is not possible currently to get a perfectly black body, also $0\,\mathrm{K}$ is not possible. Then how did Stefan prove the proportionality? Was it some kind of a thought experiment?


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You can find a couple of biographies of Josef Stefan online and they make it clear that he was an experimentalist at heart with a deep respect for more-cerebral minds like Maxwell; in particular one of his early inventions was a cool thermometer to measure the thermal conductivity of air, just because James Clerk Maxwell argued it'd be nearly impossible.

Anyway, the first link that I provided above contains this excerpt:

Stefan showed empirically, in 1879, that total radiation from a blackbody is proportional to the fourth power of its absolute temperature. This is the result for which he is best known and it was the work which we have described above which set him up to undertake this next piece of research. In fact he was led to the result by data produced by Tyndall in an 1865 book. Tyndall measured the radiation from a platinum wire heated by an electric current. Stefan, using Tyndall's data, wrote in Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (1879):-

From weak red heat (about 525° C) to complete white heat (about 1200° C) the intensity of radiation increases from 10.4 to 122, thus nearly twelvefold (more precisely 11.7). This observation caused me to take the heat radiation as proportional to the fourth power of the absolute temperature. The ratio of the absolute temperature 273 + 1200 and 273 + 525 raised to the fourth power gives 11.6.

Stefan then applied it to determine the approximate temperature of the surface of the Sun. Boltzmann, who was one of Stefan's students, showed in 1884 that this Stefan-Boltzmann law could be demonstrated mathematically.

In other words, it was an empirical observation first observed in someone else's data that when you calculate the ratios between two numbers you seem to get a $T^4$ relationship to within 1% error; and from there it was a straightforward summation: shiny new thermometer invention which corrects for radiative losses, plus apparent $T^4$ radiative loss law in someone else's data, equals experiments to use the shiny new thermometer to test whether radiative losses really always go like $T^4.$

As for the mention about Ludwig Boltzmann, it must be noted that all of this stuff was a bit after his time; Boltzmann was his student in the late 1860s; he wrote an awesome recommendation which got Boltzmann a professorship in Graz in 1869 (Boltzmann was only 25), and then Boltzmann bounced around a bit among a lot of other universities before bouncing back to Graz in 1876. During this time Stefan was pretty firmly in Vienna, as far as I can tell. Then Boltzmann's seminal work on thermodynamics took place at this university in the late 70s at around the same time as Stefan's groundbreaking work on radiation; they were not at that point in a teacher-student relationship but much more distant colleagues with similar interests.


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