Wikipedia:Reference desk/Archives/Science/2024 January 28

Welcome to the Wikipedia Science Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

January 28

1/r² law with area for source and reception

For radio waves, the radiated power responds to the law 1/r$^{2}$ for the power received at the distance r into a receiving antenna. This rule applies to transmission and reception as a theoretical point. But if the power is measured for a surface near the emission and a surface near the reception in watt/m2, wouldn't we have a rule in 1/r$^{4}$ or something else? Malypaet (talk) 10:04, 28 January 2024 (UTC)

no you would not. r is the distance between the transmitter and the receiver, not the side of a square at both receiver and transmitter. You could measure the strength of the wave at the receiver in watts per square meter, but the power transmitted is measured in watts. Graeme Bartlett (talk) 11:09, 28 January 2024 (UTC)

Also, when you are measuring the power, the surface used for the measuring assumes the role of the receiver, so then r is the distance between the transmitter and that surface. The law can easily be understood as a steady amount of power being emitted in concentric spherical shells centred on the emitter, all equally (infinitesimally) thin. The surface of these shells is proportional to r², so the amount of power per unit of surface area decreases inversely.  --Lambiam 13:19, 28 January 2024 (UTC)

If you were to measure the power per unit area (power density) at a surface near the emission and another surface near the reception, the power density at the receiving surface would still follow the inverse square law. The power received per unit area would decrease with the square of the distance from the source. Harvici (talk) 13:19, 28 January 2024 (UTC)

The 1/r² law assumes a point-source, or at least a source that is small compared with r (the wavelength also needs to be small compared with r). One can indeed consider radiation out to an intermediate surface between the source and receiver, followed by radiation onward from the surface to the receiver. This is the surface equivalence principle, a variant of Huygen's principle. However radiation from the intermediate surface is not bound by the 1/r² law (measured from the surface), since the surface is an extended source, not a point source. catslash (talk) 23:53, 28 January 2024 (UTC)

In acoustics we divide the radiating field up into two parts. In the far field intensity falls as 1/r^2. In the near field there is no such relationship, and with an intensity probe you can trace the energy flux of the waves as it loops back on itself. The reason for this is the radiating body can only emit energy into the general far-field atmosphere in a way that is compatible with the physical properties of air, but the radiating body has a different set of properties, so the complex resulting sound energy pattern resolves the differences between the two. The same will happen with radio waves. Greglocock (talk) 21:31, 29 January 2024 (UTC)

Planck's thoughtlessness

On 1901 Planck article: "Ueber das Gesetz der Energieverteilung im Normalspectrum"[1] One find a conjuring trick:

$u=\theta ^{5}\cdot {\frac {c}{\nu ^{2}}}\cdot \psi ({\frac {c\theta }{\nu }})$

According to Kirchhoff-Clausius law, the energy of a temperature ϑ and the number of vibrations ν, when emitted by a black surface per unit of time into a diathermic medium, is inversely proportional to the square $c^{2}$ of the propagation speed. So the spatial energy density u is inversely proportional to $c^{3}$, and one gets:

$u={\frac {\theta ^{5}}{\nu ^{2}c^{3}}}f({\frac {\theta }{\nu }})$

So, constants of the function f are independent of c. Instead, one can write if f denotes a new single-argument function each time, including in the following:

(7) $u={\frac {\nu ^{3}}{c^{3}}}f({\frac {\theta }{\nu }})$

What is this Kirchhoff-Clausius law ?

Why Planck don't use Wien's Displacement law with $\theta =b{\frac {\nu _{m}}{c}}$ where $\nu =\nu _{m}$? Then in the first equation replace directly $\theta ^{5}$ with $b^{5}{\frac {\nu ^{5}}{c^{5}}}$ to get directly the equation (7) ?

. Malypaet (talk) 18:37, 28 January 2024 (UTC)

Kirchhoff-Clausius law is likely a combination of two important principles in thermal radiation: Kirchhoff's law of thermal radiation and the Stefan-Boltzmann law.

And ,If I am correct, Planck's work on blackbody radiation actually predates the development of Wien's Displacement Law,so that's why Planck didn't use Wien's Displacement law . Harvici (talk) 05:26, 29 January 2024 (UTC)

Can you give me a pointer of this Kirchoff-Clausius law ?

And sorry, but here Panck begin with a Thiesen formula which is the result of applying the Wien's Displacement law to Wien's approximation (${\frac {1}{\lambda ^{5}}}f_{1}(\lambda \theta )=\theta ^{5}f_{2}(\lambda \theta )$).

Also how can you have ${\frac {\theta ^{5}}{\nu ^{2}}}=\nu ^{3}$?

. Malypaet (talk) 10:16, 29 January 2024 (UTC)

The Kirchhoff-Clausius law deals with the energy emitted by a black surface at a specific temperature and frequency.It asserts that the energy emitted per unit of time into a diathermic medium is inversely proportional to the square of the propagation speed (c^2).

Planck formulated an expression for spatial energy density (u) based on this law. He introduced a function, denoted as ψ, to represent the spectral distribution of energy. The ultimate expression (equation 7) involves the frequency (ν) raised to the power of 3, divided by the cube of the propagation speed (c^3), and multiplied by the function f(θ/ν).

About why Planck didn't directly use Wien's Displacement Law, the Planck's objective was to derive a formula accurately describing the energy distribution in blackbody radiation. While Wien's Displacement Law offered a practical empirical relationship, it fell short of explaining the complete blackbody radiation spectrum, particularly at longer wavelengths.

To sum up, Planck's work surpassed the limitations of Wien's Displacement Law. I hope you understand. 😊. Harvici (talk) 11:22, 29 January 2024 (UTC)

The equations above are (variants of) Wien's displacement law. Originally, this law was a sort of scaling law for the entire intensity distribution, which at the time was of course unknown. Additional assumptions were needed to turn this scaling law into the actual distribution. Now of course we know the distribution and have no more use for the scaling law, and "Wien's displacement law" has been reduced, so to speak, to the behaviour of the maximum of the intensity. --Wrongfilter (talk) 11:47, 29 January 2024 (UTC)

Wikipedia is an encyclopedia, so if the Kirchhoff-Clausius law exists, it must be referenced there. I can't find it and no one here has given me the reference. The only Kirchhoff's law referenced in wikipedia is the equality of emitted and absorbed radiation power of a black body at thermal equilibrium. Your statement has no value if it is not referenced and associated with an experiment. Malypaet (talk) 12:32, 29 January 2024 (UTC)

And to:

${\frac {\theta ^{5}}{\nu ^{2}}}=\nu ^{3}$?

Have you a serious explanation ?

. Malypaet (talk) 12:56, 29 January 2024 (UTC)

Wien displacement law => $\theta ^{5}={\frac {\nu ^{5}}{c^{5}}}b^{5}$

So ${\frac {\theta ^{5}}{\nu ^{2}}}={\frac {\nu ^{3}}{c^{5}}}$ not $\nu ^{3}$

. Malypaet (talk) 13:11, 29 January 2024 (UTC)

The Kirchhoff–Clausius Law perhaps ought to be referenced in Wikipedia, but this requires someone actually to write the article. It is not a law of the universe that this would already have happened if the Law exists – Wikipedia is incomplete, always will be, and we can only do our best to make it less so.

In an effort to address your original query, I web-searched "Kirchhoff–Clausius law", and found very few results, none of which gave a satisfactory definition, so perhaps it is not surprising that no-one has written an article about it yet. Are there sufficient RS to do so? Could it instead be made a section of an existing article? {The poster formerly known as 87.81.230.195} 90.205.103.187 (talk) 19:06, 29 January 2024 (UTC)

The law of Kirchhoff-Clausius states that at equilibrium the specific intensities of radiation of a certain frequency in two media are in the direct ratio of the squares of the indices of refraction in those media.^[2]  --Lambiam 10:55, 29 January 2024 (UTC)

Ok:

"The law of Kirchhoff-Clausius states that at equilibrium the specific intensities of radiation of a certain frequency in two media are in the direct ratio of the squares of the indices of refraction in thoses media. See, e.g. Planck 1906..."

Only a one more Planck reference.

Where does $c^{2}$ intervenes here, as the indice of refraction is dimensionless ? Malypaet (talk) 12:50, 29 January 2024 (UTC)

The speed of light in a medium depends on the refractive index: $c_{\mathrm {med} }=c_{\mathrm {vac} }/n$, which is why Planck can turn the law from refractive index to propagation speed. The law doesn't seem to be very well known, at least not under that name. --Wrongfilter (talk) 13:43, 29 January 2024 (UTC)

Yes not under that name.

There is only one law giving specific intensity inversely proportionnal to the speed of light squared:

The planck's law !

Planck knew the result before making the demonstration. In my opinion, he stalled in his research and invented a Kirchhoff-Clausius law, which has since fallen into a quantum well.

It's like replacing

$u={\frac {\theta ^{5}}{\nu ^{2}c^{3}}}f_{1}({\frac {\theta }{\nu }})$ by (7) $u={\frac {\nu ^{3}}{c^{3}}}f_{2}({\frac {\theta }{\nu }})$

whithout justification.

But hey, he gets the desired result.. Malypaet (talk) 21:43, 29 January 2024 (UTC)

May be here [3]Gesammelte abhandlungen; von Kirchhoff (1882) p.594. But in german and poor quality.

This line "$I'=n^{2}I$" ??? Malypaet (talk) 22:37, 29 January 2024 (UTC)

Why do you make your contributions sound as if the physics community is after you? From an article published in 1900, well before Planck's article:

Bereits Kirchhoff hat in seiner 1860 erschienenen berühmten Abhandlung den Satz bewiesen, dass die Strahlung vollkommen schwarzer Körper in verschiedene Medien den Quadraten der Brechungsexponenten dieser letzteren proportional ist und ein Jahr später fand Clausius denselben Satz.^[4]

 --Lambiam 00:22, 30 January 2024 (UTC)

The article in de Gesammelte Abhandlungen is apparently a revised version, published in 1862, of the berühmte Abhandlung from 1860 published in Annalen der Physik; they have identical titles.  --Lambiam 00:36, 30 January 2024 (UTC)

Ok, so who can put this into an equation for me and then derive the equation for the inverse of the speed of light squared? Kirchhoff-Clausius laws: "the radiation of completely black bodies in various media is proportional to the squares of the refraction exponents of the latter." Malypaet (talk) 10:30, 30 January 2024 (UTC)

If you accept Kirchhoff's result, then ${\frac {I_{\mathrm {med} }}{I_{\mathrm {vac} }}}=n^{2}=\left({\frac {c_{\mathrm {vac} }}{c_{\mathrm {med} }}}\right)^{2}$, therefore $I\propto {\frac {1}{c^{2}}}$ (with c the propagation speed in the medium). --Wrongfilter (talk) 12:30, 30 January 2024 (UTC)

So, if I consider the thermal intensity of a black body wall equivalent to a light intensity $I'$ which will be converted into light intensity $I$ in a vacuum, we obtain the

Kirchhoff-Clausius law:

$I={\frac {1}{c^{2}}}(I'C'^{2})$.

Then with $(I'C'^{2})$ as a function with energy related to the temperature of the black body wall.

. Malypaet (talk) 14:01, 30 January 2024 (UTC)

Hey here is a mention of Kirchhoff-Clausius law in BSBM varying-alpha theories, on page 7 Harvici (talk) 13:41, 31 January 2024 (UTC)

Thanks, it is clear for me now . Malypaet (talk) 20:27, 2 February 2024 (UTC)

Extract:

In fact this is implied by the Kirchhoff-Clausius law, which states that "the rate at which a body emits heat radiation is inversely proportional to the square of the speed at which the radiation propagates in the medium in which the body is immersed".
Malypaet (talk) 21:02, 2 February 2024 (UTC)

Science is not a religion. If this law exists with well-demonstrated equations, I will add an article in Wikipedia: "Kirchhoff-Clausius law".

There is a first law: “If the boss is wrong, the boss is always right.” But it's a joke at work. Malypaet (talk) 11:45, 30 January 2024 (UTC)