4.5.2 Introduction of Wien’s Displacement Law
§6. Next to Kirchoff’s theorem of the proportionality of emissive and absorptive
power, the so-called displacement law, discovered by and named after W. Wien,
20
which includes as a special case the Stefan-Boltzmann law of dependence of total
radiation on temperature, provides the most valuable contribution to the firmly
established foundation of the theory of heat radiation. In the form given by
M. Thiesen
21 it reads as follows:
E Á dλ ¼ θ
5
ψ λθ
ð Þ Á dλ
ð4:200Þ
where λ is the wavelength, E Á dλ represents the volume density of the “black-body”
radiation
22 within the spectral region λ to λ + dλ, θ represents temperature, and ψ(x)
represents a certain function of the argument x only.
§7. We now want to examine what Wien’s displacement law states about the
dependence of the entropy S of our resonator on its energy, and its characteristic
period, particularly in the general case where the resonator is situated in an arbitrary
diathermic medium. For this purpose, we next generalize Thiesen’s form of the law
for the radiation in an arbitrary diathermic medium with the velocity of light c. Since
we do not have to consider the total radiation, but only the monochromatic radiation,
it becomes necessary in order to compare different diathermic media to introduce the
frequency n instead of the wavelength λ.
Thus, let us denote by u Á dν the volume density of the radiation energy belonging
to the spectral region ν to ν + dν; then we write: u Á dν instead of E Á dλ; c/ν instead of
λ, and c Á dν/ν
2 instead of dλ. From which we obtain
u ¼ θ
5 c
ν 2 Á ψ
cθ
ν
ð4:201Þ
Now according to the well-known Kirchhoff-Clausius law, the energy emitted per
unit time at the frequency ν and temperature θ from a black surface in a diathermic
medium is inversely proportional to the square of the velocity of propagation c
2 ;
hence the energy density u is inversely proportional to c
3 and we have:
u ¼
θ
5
ν 2 c 3 Á f
θ
ν
ð4:202Þ
20 W. Wien, Proceedings of the Imperial Academy of Science, Berlin, February 9, 1893, p. 55.
21 M. Thiesen, Transactions of the German Physical Society 2 (1900), p. 66.
22 Perhaps one should speak more appropriately of a “white” radiation, to generalize what one
already understands by total white light.
186
4 Unified Mechanics Theory
§6. Next to Kirchoff’s theorem of the proportionality of emissive and absorptive
power, the so-called displacement law, discovered by and named after W. Wien,
20
which includes as a special case the Stefan-Boltzmann law of dependence of total
radiation on temperature, provides the most valuable contribution to the firmly
established foundation of the theory of heat radiation. In the form given by
M. Thiesen
21 it reads as follows:
E Á dλ ¼ θ
5
ψ λθ
ð Þ Á dλ
ð4:200Þ
where λ is the wavelength, E Á dλ represents the volume density of the “black-body”
radiation
22 within the spectral region λ to λ + dλ, θ represents temperature, and ψ(x)
represents a certain function of the argument x only.
§7. We now want to examine what Wien’s displacement law states about the
dependence of the entropy S of our resonator on its energy, and its characteristic
period, particularly in the general case where the resonator is situated in an arbitrary
diathermic medium. For this purpose, we next generalize Thiesen’s form of the law
for the radiation in an arbitrary diathermic medium with the velocity of light c. Since
we do not have to consider the total radiation, but only the monochromatic radiation,
it becomes necessary in order to compare different diathermic media to introduce the
frequency n instead of the wavelength λ.
Thus, let us denote by u Á dν the volume density of the radiation energy belonging
to the spectral region ν to ν + dν; then we write: u Á dν instead of E Á dλ; c/ν instead of
λ, and c Á dν/ν
2 instead of dλ. From which we obtain
u ¼ θ
5 c
ν 2 Á ψ
cθ
ν
ð4:201Þ
Now according to the well-known Kirchhoff-Clausius law, the energy emitted per
unit time at the frequency ν and temperature θ from a black surface in a diathermic
medium is inversely proportional to the square of the velocity of propagation c
2 ;
hence the energy density u is inversely proportional to c
3 and we have:
u ¼
θ
5
ν 2 c 3 Á f
θ
ν
ð4:202Þ
20 W. Wien, Proceedings of the Imperial Academy of Science, Berlin, February 9, 1893, p. 55.
21 M. Thiesen, Transactions of the German Physical Society 2 (1900), p. 66.
22 Perhaps one should speak more appropriately of a “white” radiation, to generalize what one
already understands by total white light.
186
4 Unified Mechanics Theory