Draft:Kirchhoff-Clausius's Law
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The Kirchhoff-Clausius law, also known as Clausius's law, defines the relationship between the radiance of a black body thermal radiation and the medium through which the emitted light passes. It establishes that the radiance (total power radiated per unit of surface and unit of solid angle, expressed in watts per square meter per steradian) is directly proportional to the square of the refractive index of the medium (Kirchhoff) and, therefore, inversely proportional to the square of the speed of propagation of the radiation in the medium (Clausius):
(where = emissive power as radiance and n = index of refraction, all in the surrounding medium, and = emissive power of a perfectly black body in a vacuum; one assigns the index zero for all quantities relating to the vacuum).
This law is a theoretical discovery based on geometrical optics.[1][2] Its name is in tribute to Gustav Kirchhoff and Rudolf Clausius, who published their initial findings in 1862.[3] and 1864.[4]
The law was proved experimentally by Marian Smoluchowski de Smolan in 1896 [5], comparing the emission of hot, blackened copper in carbon dioxide and hydrogen.
The Kirchhoff-Clausius law states that:
"The emissive power of perfectly black bodies is directly proportional to the square of the index of refraction of the surrounding medium (Kirchoff), and therefore inversely proportional to the squares of the velocities of propagation in the surrounding medium (Clausius)."[6][7][8][9][10][11][12][13][14]
This is one of the 4 classical laws on which Planck's law is based, in order of discovery:
- Kirchhoff's law on thermal radiation.
- Kirchhoff-Clausius law.
- Stefan-Boltzmann law.
- Wien's displacement law.
History
[edit]In 1849, Foucault noticed that bright lines occur where the Fraunhofer double line D of the solar spectrum is found and that this dark line D is produced or made more intense when the rays of the sun, or those from one of the incandescent carbon poles, are passed through the luminous arc.[15] Gustav Kirchhoff discovered the law of thermal radiation in 1859 while collaborating with Robert Bunsen at the University of Heidelberg, where they developed the modern spectroscope. He proved it in 1861 and then, in 1862, defined the perfect black body. The same year, as he had noticed, like Foucault, that the spectrum of sunlight was amplified in the flame of the Bunsen burner,[16][17] he found a theoretical explanation in geometric optics with a formula that gave the amplification coefficient with the square of the refractive index () for a new law, which will become the Kirchhoff-Clausius law.
In 1863, Rudolf Clausius revisited Kirchhoff's study in the spirit of the second law of thermodynamics. For this, he considered two perfect black bodies (a and c) side by side and at the same temperature, immersed in two different adjacent media, such as water and air for example, and radiating towards each other at different speeds. So, to respect the second law, the mutual thermal radiation between these two black bodies must be equal, and he obtains a different form of the formula:
(where and = emissive power, and and = light speed, in each surrounding medium ). Hence and .
As written Clausius, Kirchhoff used only one black body in a vacuum and radiated it in another media, so it had a vacuum refractive index of one. So, to simplify, you obtain the Kirchhoff form:
Marian Smoluchowski de Smolan also studied it and validated it experimentally in 1896 during his stay in Paris,[18][5] by comparing the emission of hot and blackened copper in carbon dioxide and hydrogen.
Above all, it became a crucial point in Planck's demonstration of the law of black body radiation in 1901. With the Kirchhoff-Clausius law, he demonstrated that the energy density emitted by a black body is the same in any media and is a universal function of its temperature and frequency. If you replace with the energy density , the Kirchoff-Clausius law becomes . Then, as ( for wavelength), you obtain . So, the energy of an equilibrium radiation localized in a cube with an edge equal to the wavelength is the same in any media.[11] In addition, as it also led to the Planck-Einstein relationship, it became indirectly a key point in Albert Einstein's demonstration of the photoelectric effect in 1905.
In 1902, Rudolf Straubel extended this law to the plane parallel to the radiation[19], which is why it is sometimes known as the Kirchhoff-Clausius-Straubel law.[20]
The law will be demonstrated again by Wolfgang Pauli,[21] then by D. Sivoukhin under the supervision of V. Ginzburg.[22]
Later, a link will be established with electromagnetism for radiative transfert.[23][24]
Demonstration
[edit]One gives the following definitions:
One names the frequency , the wavelength and the speed of light c.
The quantity “” is the energy intensity (radiance) of isotropic radiation at thermal equilibrium whose one gives the spectral decomposition by the following integral:
It is now a matter of calculating the energy intensity at thermal equilibrium in a homogeneous and isotropic transparent medium of refractive index n. This equilibrium radiation is established in a closed cavity entirely occupied by the medium considered and whose walls are at a constant temperature. The same radiation will exist in the medium, which occupies only part of the cavity. Consider a cavity with a given medium for one part and the vacuum in the remaining. The equilibrium radiation in the medium and vacuum depends neither on the shape nor the properties of the separation surface between the medium and the vacuum. One can assume that this separation surface is flat and perfectly smooth.
Figure 1.
One considers that the energy exchange between the medium and the vacuum will only result from reflections and refractions of the radiation on the separation surface. This exchange of energy applies the principle of detailed balance, and this exchange cannot alter the state of equilibrium between the radiation in the medium and the vacuum. From there, one can establish a relationship between the radiation intensity in a vacuum and the same quantities in the medium.
As one verifies the principle of detailed balance, it is sufficient to consider only a part of the total radiation, including the frequencies between and . The flux per unit time from the vacuum, falling on the unit area of the separation surface, and contained in the solid angle cone where is the angle of incidence (Figure 1), is:
According to the principle of detailed balance, an equal flow of energy must propagate in the opposite direction. That consists of two flows: the first flow results from the reflection of the flow and has the value:
The second results from refracting the flow from the medium ( being the absorbing power coefficient for frequency ). According to Fresnel's formulas, the reflection coefficients on the separation surface of rays propagating in opposite directions are equal; the second flow is, therefore, equal to:
being the angle of refraction and the solid angle in the medium, which, after refraction, becomes equal to . After division by , one writes the detailed balance condition as follows:
It gives:
Let's take for the solid angle (not shown in Figure 1) delimited by the cones whose generators make the angles with the normal to the separation surface and ; it results in:
Likewise:
Equality (2) is now written:
According to the law of refraction, , and subsequently:
Therefore:
Hence, by the same law of refraction:
Kirchhoff found this solution in 1860, and Clausius, who knew it, found it in 1863 by another route and with another form:
It is called the Kirchhoff-Clausius law.
As and is invariant with the medium, you can also write:
This means that in an equilibrium radiation the energy fluxes passing through a rectangular area of side equal to the wavelength (in the medium) are the same for all media. This is valid both for the total flux and for its monochromatic components.
Notable applications
[edit]References
[edit]- ^ Pauli, Wolfgang (1973). "Chapter 1: Geometrical Optics". Pauli Lectures on Physics: Volume 2. Optics and the Theory of Electrons. Translated by Margulies, S.; Lewis, H. R. Cambridge, Massachusetts, and London, England.: Charles P. Henz . The MIT Press. pp. 1–27. ISBN 0262160471.
- ^ Sivoukhine, D. V.; Ginsburg, V. (1984). "$ 114. Formule de Kirchhoff-Clausius". Sivoukhine Cours De Physique Générale - Tome 04 - Optique 2 Ème Partie [D. V. Sivukhin. General course of physics - Volume 04 - Optics Part 2] (in French). Translated by Medvédev, Sergueï. Moscou: Éditions Mir. pp. 296–299. Archived from the original on 2022-10-03.
- ^ Kirchhoff, Gustav; Boltzmann, Ludwig (1882). "KIRCHHOFF, GESAMMELT ABHANDLUNGEN". Kirchhoff, Collected Treatises. By Ludwig Boltzmann (in German). LEIPZIG: Johann Ambrosius BARTH.: 571.
- ^ Clausius, Rudolf (1867). "The Mechanical Theory of Heat, with its Applications to the Steam-Engine and to the Physical Properties of Bodies". Google Books from T. ARCHER HIRST, F.R.S., 1867. P: 290.
- ^ a b Smoluchowski de Smolan, Marian (1896). "Recherches sur une loi de Clausius au point de vue d'une théorie générale de la radiation". J. Phys. Theor. Appl. (in French): 488-499. Archived from the original on Jan 15, 2024.
- ^ Clausius, Rudolf (1879). "Mechanical Theory Of Heat" (PDF). Internet Archive Tr. By Walter R. Browne 1879. P: 315, 330–331.
- ^ MESLIN, Georges (1872). "Sur le renversement complexe des raies spectrales dans les couches chromosphériques". Le Journal de physique théorique et appliquée (in French): 456.
- ^ Smoluchowski de Smolan, Marian (1896). "Recherches sur une loi de Clausius au point de vue d'une théorie générale de la radiation". J. Phys. Theor. Appl. (in French): 488. Archived from the original on Jan 15, 2024.
- ^ Planck, Max (1914). "The theory of heat radiation" (PDF). Project Gutenberg: 43.
- ^ NORTHRUP, EDWIN F. (1917). "LAWS OF PHYSICAL SCIENCE A REFERENCE BOOK". Philadelphia and London J. B. Lippincott Company (Book): 184.
- ^ a b SIVOUKHINE, D. (1984). "COURS DE PHYSIQUE GENERALE Tome IV OPTIQUE Deuxième partie Chapitre X $ 114. Formule de Kirchhoff-Clausius". Editions MIR (in French): 298.
- ^ HALL, Carl W. (2000). "Laws and models : science, engineering, and technology". Boca Raton CRC Press: 261–262.
- ^ Sharkov, Eugene A. (2003). "Black-body radiation" (PDF). Passive microwave remote sensing of the Earth: physical foundations. Springer-Praxis books in geophysical sciences. Berlin ; New York : Chichester, UK: Springer ; Praxis Pub. p. 210. ISBN 978-3-540-43946-2.
- ^ Barrow, John D.; Magueijo, João (2014). "Redshifting of cosmological black bodies in Bekenstein-Sandvik-Barrow-Magueijo varying-alpha theories". Phys. Rev. D90 (2014) 123506. 90 (12): 6. arXiv:1406.1053. Bibcode:2014PhRvD..90l3506B. doi:10.1103/PhysRevD.90.123506. S2CID 53700017.
- ^ Longair, Malcolm (2006). "The Cosmic Century: A History of Astrophysics and Cosmology". Cambridge University Press: 6.
- ^ Kirchhoff; Bunsen (1860). "Chemical Analysis by Spectrum-observations". London, Edinburgh and Dublin Philosophical Magazine and Journal of Science: 108.
- ^ Genin, Christophe (2008-09-01). "Kirchhoff et Bunsen fondent l'analyse spectrale". Bibnum. Textes fondateurs de la science (in French). doi:10.4000/bibnum.746. ISSN 2554-4470.
- ^ Teske, Andrzej A. "SMOLUCHOWSKI, MARIAN". Encyclopedia.com.
- ^ Straubel, R. (1902). "On a general theorem of geometric optics and some applications" (PDF). Phys. Zeit. 4 (1902-03), 114-117.
- ^ Ilinsky, Roman (2014). "Fluence Rate in UV Photoreactor for Disinfection of Water: Isotropically Radiating Cylinder". International Journal of Chemical Engineering. 2014 (1): 1–13. doi:10.1155/2014/310720.
- ^ Pauli, Wolfgang (1973). "Optics and the Theory of Electrons". Physics. 2: 12. ISBN 0-486-41458-2.
- ^ D. Sivoukhine (1984). Sivoukhine Cours De Physique Générale - Tome 04 - Optique 2 Ème Partie [Sivukhin General Physics Course - Volume 04 - Optics Part 2 - Kirchhoff-Clausius formula] (in French). Moscou: Éditions Mir. pp. 296–299.
- ^ Bal, Guillaume (2006-07-01). "Radiative transfer equations with varying refractive index: a mathematical perspective". JOSA A. 23 (7): 1639–1644. doi:10.1364/JOSAA.23.001639. ISSN 1520-8532.
- ^ Ben Abdallah, P.; Charette, A.; Le Dez, V. (2001-08-01). "Influence of a spatial variation of the thermo-optical constants on the radiative transfer inside an absorbing–emitting semi-transparent sphere". Journal of Quantitative Spectroscopy and Radiative Transfer. 70 (3): 341–365. doi:10.1016/S0022-4073(01)00006-1. ISSN 0022-4073.
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