The original Note 25) is "Classic Radiation Theory and Planck" introduces
1. Stefan–Boltzmann Law
2. Wien's Displacement law
3. Vien's Radiation Law
4. Rayleigh–Jeans Law
5. Planck's Radiation Law
Planck's Radiation Law or simply Planck's Law is a big issue so it is better to check the originals.
List of references in Wiki " Planck's Law" (08-Aept-2012)
- Planck, M. (1900a). "Über eine Verbesserung der Wienschen Spektralgleichung". Verhandlungen der Deutschen Physikalischen Gesellschaft 2: 202–204. Translated in ter Haar, D. (1967). "On an Improvement of Wien's Equation for the Spectrum". The Old Quantum Theory. Pergamon Press. pp. 79–81. LCCN 66029628.
- Planck, M. (1900b). "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum". Verhandlungen der Deutschen Physikalischen Gesellschaft 2: 237. Translated in ter Haar, D. (1967). "On the Theory of the Energy Distribution Law of the Normal Spectrum". The Old Quantum Theory. Pergamon Press. p. 82. LCCN 66029628.
- Planck, M. (1900c). "Entropie und Temperatur strahlender Wärme". Annalen der Physik 306 (4): 719–737. Bibcode 1900AnP...306..719P. doi:10.1002/andp.19003060410.
- Planck, M. (1900d). "Über irreversible Strahlungsvorgänge". Annalen der Physik 306 (1): 69–122. Bibcode 1900AnP...306...69P. doi:10.1002/andp.19003060105.
- Planck, M. (1901). "Über das Gesetz der Energieverteilung im Normalspektrum". Annalen der Physik 4: 553. Translated in Ando, K.. "On the Law of Distribution of Energy in the Normal Spectrum". Retrieved 13 October 2011.
- Planck, M. (1906). Vorlesungen über die Theorie der Wärmestrahlung. Johann Ambrosius Barth. LCCN 07004527.
- Planck, M. (1914). The Theory of Heat Radiation. Masius, M. (transl.) (2nd ed.). P. Blakiston's Son & Co. OL7154661M.
- Planck, M. (1915). Eight Lectures on Theoretical Physics. Wills, A. P. (transl.). Dover Publications. ISBN 0-486-69730-4.
- Planck, M. (1943). "Zur Geschichte der Auffindung des physikalischen Wirkungsquantums". Naturwissenschaften 31 (14–15): 153–159. Bibcode 1943NW.....31..153P. doi:10.1007/BF01475738.
Very useful articles
1) http://bado-shanai.net/map%20of%20physics/mopPlancksderivBRL.htm
2) Great Experiments in Physics: Firsthand Accounts from Galileo to Einstein. edited by Morris H. Shamos
3) http://publishing.cdlib.org/ucpressebooks/view?docId=ft4t1nb2gv&chunk.id=d0e391&toc.depth=1&toc.id=d0e391&brand=ucpress
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The following part is emphasized by Planck himself (the underlined part) so this part is surely important.
From "On an Improvement of Wien's Equation for the Spectrum" (1900) as well as "On the Law of Distribution of Energy in the Normal Spectrum" (1901).
From "On the Law of Distribution of Energy in the Normal Spectrum".
"
To answer this question we first of all consider the vibrations of the resonators and assign to them arbitrary definite energies, for instance, an energy E to the N resonators v , E' to the N' resonators v', . . . . The sum
E + E' + E'' + . . . = E0
must, of course, be less than Et. The remainder Et −E0 pertains then to the radiation present in the medium. We must now give the distribution of the energy over the separate resonators of each group, first of all the distribution of the energy E over the N resonators of frequency v . If E considered to be continuously divisible quantity, this distribution is possible in infinitely many ways. We consider, however – this is the most essential point of the whole calculation – E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55×10−27 erg · sec. This constant multiplied by the common frequency v of the resonators gives us the energy element " in erg, and dividing E by " we get the number P of energy elements which must be divided over the N resonators. If the ratio is not an integer, we take for P an integer in the neighbourhood.
"
1) Distribution
Planck did not emphasize but he did "grouping" which is actually the process of making "continuously divisible quantity" to "discontinuous divisible quantity". For instance, when you make a distribution chart of the following simple data from the measurement "grouping" is required. Otherwise you cannot find distribution of the data.
The lengths of 10 pencils selected as samples from those shipped from a relatively good pencil factory (specified length 190.50mm +/-2.0mm)
Pencil No. 1 189.73
Pencil No. 2 190.06
Pencil No. 3 191.29
Pencil No. 4 191.14
Pencil No. 5 190.68
Pencil No. 6 190.95
Pencil No. 7 189.87
Pencil No. 8 190.48
Pencil No. 9 190.24
Pencil No. 10 190.55
If you do not do "grouping" the number of each pencil length is 1 so the chance for each number is 1/10. We cannot see "distribution" except "1/10". If you do "grouping", you can find the number for each group.
Group 6) 189.00 - 189.49 - 0pc Chance or distribution 0%
Group 1) 189.50 - 189.99 - 2pc Chance or distribution 20%
Group 2) 190.00 - 190.49 - 3pc Chance or distribution 30%
Group 3) 190.50 - 190.99 - 3pcs Chance or distribution 30%
Group 4) 191.00 - 191.49 - 2pcs Chance or distribution 20%
Group 5) 191.50 - 191.99 - 0pcs Chance or distribution 0%
The "grouping" unit is 0.50mm.
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2) "permutations" and "complexions"
Planck says in "On an Improvement of Wien's Equation for the Spectrum" (1900)
"
From the theory of permutations we get for the number of all possible complexions
N(N + 1) · (N + 2) . . . (N + P − 1) (N + P − 1)!
------------------------------------- = --------------
1 · 2 · 3 . . . P (N − 1)!P!
or to a sufficient approximations,
(N + P)N+P
= -----------
NNPP
"
a) Planck uses "permutations" and "complexions" . But the first part can be gotten by using the following more simple "combination" formula
In the Planck's case n = N + P, r = P
b) No explanation on the approximation here. Planck kindly added the following between the above two formulas in in his "On the Law of Distribution of Energy in the Normal Spectrum". (1901)
"
Now according to Stirling’s theorem, we have in the first approximation:
N! = NN
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3) Planck's Resonators
There is no his explanation on the Resonators he repeated mentioned. So to understand what he says we need some background. The following is a part of the background. From:3) http://publishing.cdlib.org/ucpressebooks/view?docId=ft4t1nb2gv&chunk.id=d0e391&toc.depth=1&toc.id=d0e391&brand=ucpress
"
Planck's Resonators
The
empirical validity of Kirchhoff's law, Stefan's law, and Wien's
displacement law left no doubt about the legitimacy of combining
electrodynamic and thermodynamic laws; and Planck was well aware of
these developments. In 1895 he decided to examine the electrodynamic
mechanisms responsible for the thermalization of radiation, hoping to
find in them the ultimate source of irreversibility and to perhaps
determine the arbitrary function in Wien's displacement law. This grand
program focused on a very simple system, a Hertz resonator: that is, a
small, nonresistive, oscillating electric circuit interacting with
electromagnetic waves, the characteristic wavelength of the oscillator
being much larger than the oscillator. The simplest choice was
considered the most adequate by Planck,
― 30 ―
Figure 6.
Planck's intuition of the source of thermodynamic
irreversibility: the diffusion of a plane electromagnetic
wave by a resonator (at the frequency of the wave).
Planck's intuition of the source of thermodynamic
irreversibility: the diffusion of a plane electromagnetic
wave by a resonator (at the frequency of the wave).
for
in light of Kirchhoff's theorem, the properties of thermal radiation
could not depend on the specific properties of the thermalizing system.[43]
What
first attracted Planck's attention was the apparent irreversibility of
the interaction between radiation and resonator: a plane monochromatic
wave falling upon a resonator forces vibrations of the resonator when
the condition of resonance is approximately met; in turn these
vibrations emit secondary waves over a wide angle (fig. 6). Also, an
excited resonator left to itself emits radiation at its characteristic
frequency and thereby gradually loses its energy. Such processes,
resonant scattering or radiation damping, looked essentially
irreversible, even though the total energy (that of the resonator plus
that of radiation) was strictly conserved (in the absence of the Joule
effect in the circuit). Planck concluded:
The study of conservative damping seems to me highly important, because it opens a new perspective on the possibility of a general explanation of irreversible processes through conservative interactions, a more and more pressing problem in contemporary theoretical physics.[44]
These
results of classical electrodynamics are now very well known; they are
usually obtained through a specific model of the resonator, for instance
an elastically bound electron. In 1895, however, the existence of the
electron had not been proved; and Lorentz's formulation of
electrodynamics, with its detailed analysis of microscopic sources, was
not yet currently known (the famous Versuch was published in the same year).
― 31 ―
Planck
was therefore confined to a different method, first used by Hertz in
1889 for a calculation of the energy radiated by an oscillating dipole.
In this method the detailed structure of the resonator was irrelevant .[45]
As
we shall observe in the following, Planck maintained this generality
throughout his program and even gave it an essential role. It is
therefore useful at this point to explain how the equation of a
resonator in an electromagnetic field can be established with this
method. Another reason for analyzing Planck's reasoning is that it is
typical of a style of theoretical physics, namely, concentrating on the
features of physical systems which can be determined on the basis of
general principles only, without recourse to detailed microscopic
assumptions. Nevertheless, hurried readers may be content with these
general comments and may jump to the equation (62), p. 36, for the
evolution of the dipolar moment f of a resonator.
"
