Draft:Kirchhoff-Clausius's Law

Work in progress... In thermal radiation using geometrical optics, the Kirchhoff-Clausius law was named after Gustav Kirchhoff and Rudolf Clausius, who published their initial findings in 1862 and 1863. The Kirchhoff-Clausius law state 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).' " With the formula:"$I'=n^2 \cdot I$" (where I' = emissive power and n = index of refraction, all in the surrounding medium, and I = emissive power of a perfectly black body in a vacuum).

History
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 noticed that the spectrum of sunlight was amplified in the flame of the Bunsen burner, he found a theoretical explanation in geometric optics with a formula that gave the amplification coefficient with the square of the refractive index ($$I'=n^2 \cdot I$$) 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, but immersed in two different media, such as water and air, and radiating towards each other at different speeds. So, to respect the second law, the mutual radiation between these two black bodies must be equal, and he obtains a different formula

$$I_a \cdot v_a^2=I_c\cdot v_c^2$$.

(where $$I_a$$ and $$I_c$$= emissive power, and $$v_a$$ and  $$v_c$$ = light speed, in each surrounding medium ).

Hence $$I_a \div I_c= v_c^2\div v_a^2$$ and $$I_a \div I_c= n_a^2\div n_c^2$$

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. That gives: $$I_c= n_c^2\cdot I_a$$

Afterwards, this law was mainly used in astrophysics, maybe first by Georges MESLIN in 1872. It was also studied by Marian Smoluchowski de Smolan in 1896. Above all, it became a crucial point in Planck's demonstration of the law of black body radiation in 1901, which earned him his Nobel Prize. In addition, as it also led to the Planck-Einstein relationship, it became a key point in Albert Einstein's demonstration of the photoelectric effect in 1905, earning him his Nobel Prize.

In 1902, R. Straubel extended this law to the plane parallel to the radiation, which is why it is sometimes known as the Kirchhoff-Clausius-Straubel law.

In physics, the Kirchhoff-Clausius Law is defined by:

Temperature
$$B(T)=\frac{1}{c^2}\cdot f(T)$$  (SI units: W⋅m-2)

Wavelength and temperature
$$B(\lambda,T)=\frac{1}{c^2}\cdot f(\lambda T)$$  (SI units: W⋅m-2⋅sr-1⋅nm-1)

Frequency and temperature
$$B(\nu,T)=\frac{1}{c^2}\cdot f(\frac{T}{\nu})$$   (SI units: W⋅m-2⋅sr-1⋅Hz-1)

Max Planck interpreted and used this law for the first time in his 1901 article  > justifying his black body distribution law.

Gustav Kirchhoff's formula in 1862
$$I'=n^2\cdot I$$

So, with n=c/c'

The Clausius form is obtained by:

$$I'c'^2 = Ic^2 $$

and the Planck form:

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

($$I$$ as heat radiation intensity in vacuum and $$I'$$in media, c as speed of light, and n as refractive index)

Rudolf Clausius's statement in 1863
He gives these formulas:

$$e_a v{_a^2} = e_c v{_c^2}$$

$$e_a : e_c = v{_c^2} : v{_a^2}$$

$$e_a : e_c = n{_a^2} : n{_c^2}$$

(Here $$e$$ for heat radiation intensity and $$v$$ for the speed of light, for the medium planes a and c)