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'''Закон Стефана-Больцмана''' дає- залежністьінтегральний енергіїзакон випромінювання абсолютно чорного тіла, який стверджує, що енергія [[випромінювання]] з одиниці [[площа|площі]] [[поверхня|поверхні]] в одиницю [[час]]у відабсолютно чорного тіла пропорційна четвертій степені [[ефективна температура|ефективної температури]] тіла, що випромінює.
== Загальний вигляд ==
Загальна енергія теплового випромінювання визначається як:
== Доведення закону ==
<!-- The law can be derived by considering a small flat [[black body]] surface radiating out into a half-sphere. This derivation uses [[spherical coordinates]], with ''φ'' as the zenith angle and ''θ'' as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where ''φ'' = <sup>π</sup>/<sub>2</sub>. -->
[[Інтесивність випромінювання]] енергії [[абсолютно чорне тіло|абсолютно чорним тілом]] в залежності від [[частота|частоти]] випромінювання визначається [[закон Планка|законом Планка]] як:
: де <math>\sigma\,</math> є [[стала Стефана-Больцмана|сталою Стефана-Больцмана]]
== Джерела ==
<!--
* [https://sites.google.com/site/osnoviteplotehnikitagidravliki/rozdil-tretij-teoria-teplomasoobminu/-3-3-promenistij-teploobmin/2-osnovni-zakoni-teplovogo-viprominuvanna Основні закони теплового випромінювання - Основи теплотехніки та гідравліки]
The quantity <math>I(\nu,T) ~A ~d\nu ~d\Omega</math> is the [[Power (physics)|power]] radiated by a surface of area A through a [[solid angle]] ''dΩ'' in the frequency range <math>\left(\nu , \nu + d\nu \right) \,</math>.
* [https://studopedia.com.ua/1_379649_zakon-stefana-boltsmana-zakon-vina.html Закон Стефана Больцмана. Закон Віна. - studopedia.com.ua]
* {{книга
The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,
|автор = гол.ред. А. М. Прохоров.
::<math>\frac{P}{A} = \int_0^\infty I(\nu,T) d\nu \int d\Omega \,</math>
|частина =
To derive the Stefan–Boltzmann law, we must integrate ''Ω'' over the half-sphere and integrate ''ν'' from 0 to ∞. Furthermore, because black bodies are ''Lambertian'' (i.e. they obey [[Lambert's cosine law]]), the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle ''φ'', and in spherical coordinates, ''dΩ'' = sin(''φ'') ''dφ dθ''.
|заголовок = [[Фізична енциклопедія|Физическая энциклопедия]]
|оригінал =
:: <math>
|посилання = http://femto.com.ua/
\begin{align}
|видання =
\frac{P}{A} & = \int_0^\infty I(\nu,T) \, d\nu \int_0^{2\pi} \, d\theta \int_0^{\pi/2} \cos \phi \sin \phi \, d\phi \\
|відповідальний =
& = \pi \int_0^\infty I(\nu,T) \, d\nu
|місце = Москва
\end{align}
|видавництво = "Советская энциклопедия"
</math>
|рік = 1988
|том = 4
Then we plug in for ''I'':
|сторінки = 689-690
|сторінок = 704
:: <math>\frac{P}{A} = \frac{2 \pi h}{c^2} \int_0^\infty \frac{\nu^3}{ e^{\frac{h\nu}{kT}}-1} d\nu \,</math>
|isbn = 5-85270-034-7
}}
To do this integral, do a substitution,
* {{Citation |last=Stefan |first=J. |title=Über die Beziehung zwischen der Wärmestrahlung und der Temperatur |journal=Sitzungsberichte der Mathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften |language=de |trans-title=On the relationship between heat radiation and temperature |url=http://www.ing-buero-ebel.de/strahlung/Original/Stefan1879.pdf |volume=79 |year=1879 |pages=391–428}}
* {{Citation |last=Boltzmann |first=L. |title=Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie |language=de |trans-title=Derivation of Stefan's little law concerning the dependence of thermal radiation on the temperature of the electro-magnetic theory of light |journal=Annalen der Physik und Chemie |volume=258 |issue=6 |year=1884 |pages=291–294 |doi=10.1002/andp.18842580616 |bibcode = 1884AnP...258..291B |doi-access=free }}
::<math> u = \frac{h \nu}{k T} \,</math>
::<math> du = \frac{h}{k T} \, d\nu </math>
which gives:
: <math>\frac{P}{A} = \frac{2 \pi h }{c^2} \left(\frac{k T}{h} \right)^4 \int_0^\infty \frac{u^3}{ e^u - 1} \, du.</math>
The integral on the right can be done in a number of ways (one is included in this article's appendix) – its answer is π<sup>4</sup>/15, giving the result that, for a perfect blackbody surface:
: <math>j^\star = \sigma T^4 ~, ~~ \sigma = \frac{2 \pi^5 k^4 }{15 c^2 h^3}. </math>
An alternative form of the Stefan–Boltzmann constant, more fundamental to physics:
:<math>\sigma = \frac{\pi^2 k^4}{60 \hbar^3 c^2}</math>
Finally, this proof started out only considering a small flat surface. However, any [[differentiable]] surface can be approximated by a bunch of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all [[convex set|convex]] blackbodies, too, so long as the surface has the same temperature throughout.
=== Thermodynamic derivation ===
The fact that the energy density of the box containing radiation is proportional to <math>T^{4}</math> can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure <math>P</math> is related to the internal energy density:
:<math>P=\frac{u}{3}</math>
The total internal energy of the box containing radiation can thus be written as:
:<math>U=3PV\,</math>
Inserting this in the [[fundamental thermodynamic relation]]
:<math>dU=T dS - P dV\,</math>
yields the equation:
:<math>dS=4\frac{P}{T}dV + 3\frac{V}{T}dP</math>
This equation can be used to derive a [[Maxwell relations|Maxwell relation]]. From the above equation it can be seen that:
:<math>\left(\frac{\partial S}{\partial V}\right)_{P}\!\!=4\frac{P}{T}</math>
and
:<math>\left(\frac{\partial S}{\partial P}\right)_{V}\!\!=3\frac{V}{T}</math>
The [[symmetry of second derivatives]] of <math>S</math> with regard to <math>P</math> and <math>V</math> then implies:
:<math>4\left(\frac{\partial \left(P/T\right)}{\partial P}\right)_{V}\!\!= 3\left(\frac{\partial \left(V/T\right)}{\partial V}\right)_{P}</math>
Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume. In the derivative on the right hand side. the temperature is thus a constant. Evaluating the derivatives gives the differential equation:
:<math>\frac{1}{P}\frac{dP}{dT}=\frac{4}{T}</math>
This implies that
:<math>u=3P \propto T^{4} </math>
-->
{{Без джерел|дата=листопад 2016}}
[[Категорія:Закони термодинаміки]]
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