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Kirchhoff's Law and Its Derivation

It was a familiar observation that heated object gave off radiation. Metal objects in forges glowed red, or if they got hot enough they glowed white. It was perceived in the 19th century that even heated objects that were not glowing visibly were giving off radiation, a radiation that was called infrared. Still later spectroscopy revealed that in sunlight there was radiation beyond violet in its spectrum. That radiation was called ultraviolet.

These discoveries prompted a search for the formula for the spectrum of radiation from a heated object as a function of its temperature. The search for that formula went on during the entire 19th century and was not completed until Max Planck found it in the early 20th century. Along the way an outstanding bit of analysis was achieved by Gustav Kirchhoff of Prussia.

Gustav Kirchhoff

Gustav Kirchhoff was born March 12th, 1824 in Königsberg in East Prussia, which is now Kaliningrad in the German territory that was transferred to Russia after World War II. He attended Albertus University in Königsberg and graduated in 1847. Even before graduation he was doing first rate research. He formulated in 1845 laws for electrical circuits which are still a fundamental part of electrical engineering and he established that electricity travels at the speed of light in metal wires.

After graduation Kirchhoff became a Privatdozent (unsalaried lecturer) at the University of Berlin. After three years as a Privatdozent he was appointed extraordinary professor of physics at the University of Breslau.

In 1854 he became a professor at the University of Heidelberg where he began collaboration with Robert Bunsen in spectroscopy.

Gustav Kirchhoff (left)
Robert Bunsen (right)

Bunsen is best known for inventing the Bunsen burner in which air and a flammable gas are mixed in a tube and can be ignited at their exit from the tube to give a constant, controllable flame. This was used to exhibit the spectra of elements. A platinum wire is touched in the powder of a compound, such as strontium chloride, and then inserted into the flame of the Bunsen burner. The flame turns to a color associated with the elements of the compound, a beautiful scarlet in the case of a strontium compound. The color of the flame may serve as a test for the presence of a particular element in a material. Kirchhoff and Bunsen were able to discover two elements in that way, caesium and rubidium.

Kirchhoff was able to succinctly summarize what was known in spectroscopy at the time. These were called laws.

These applied not just to visible light but to radiation in general. Kirchhoff understanding that a gas absorbs light at the same wavelengths that it emits enabled him to determine the composition of the Sun from the hitherto mysterious Fraunhofer dark lines of the solar spectrum. Kirchhoff's work is an example of the dictum

Research is to see what everyone has seen and think what no one has thought.

Kirchhoff went on to articulate a surprising general relationship between the absorption and emission of light by materials. That relationship, which has come to be known as Kirchhoff's Law, is as he stated it in a presentation before the Berlin Academy in 1859 that

For rays of the same wavelength at the same temperature the ratio of the emissive power to the absorptivity is the same for all bodies.

Kirchhoff created a thought experiment involving a radiative equilibrium between two emitting and absorbing infinite parallel plates of different materials facing each other.

The analysis is carried out for a particular wavelength, say λ. The emissivity of the plate on the left is denoted at E. It is the amount of energy emitted per unit area per unit time as radiation of wavelength λ. The absorptivity at wavelength λ is the proportion of that radiation which is absorbed. This is designated as A for the left plate. The reflectivity is the proportion reflected. For the left plate this is R and R is equal to 1−A. For the right plate the corresponding quantities are denoted as e, a and r.

Now consider the outflow and inflow of radiation (of wavelength λ) for a unit area of the plate on the right. The outflow is the emissivity e. The inflow comes from two sources. One is that which is absorbed from the left plate either directly or indirectly after reflection and re-reflection. The other is the radiation from the right plate itself after it is reflected from the left.

An amount of radiation E comes from the left plate and a proportion a is absorbed at the right plate. This is the first pass of the radiation emitted by the left plate. In the first pass an amount Er is reflect from the right plate and then goes to the left plate where an amount of (Er)R is reflected back to the right plate. At the right plate an amount of (Er)Ra=Ea(rR) is absorbed. This is the second pass. The third pass results in the absorption of Ea(rR)². The amount absorbed in each pass is rR times the amount absorbed in the previous pass. The total amount of the radiation originating from the left plate that is absorbed by the right plate is then

Ea + Ea(rR) + Ea(rR)² + Ea(rR)³ + …
which simplifies to
Ea[1 + rR + (rR)² + (rR)³ + …]
which reduces to

The first pass of the radiation originating at the right plate results from eR being reflected from the left plate and then eRa being absorbed at the right. A proportion r is reflected to the left plate where a proportion R is reflected back to the right plate. The same infinite repetitions of reflections and re-reflections occur as in the previous case. Thus the total radiation from the right plate that is absorbed by the right plate is

eRa + eRa(rR) + eRa(rR)² + eRa(rR)³ + …
which reduces to

Therefore a balance of outflow and inflow of radiation to the right plate requires that

e = Ea/(1−rR) + eRa/(1−rR)

The same accounting at the left plate determines that

E = eA/(1−rR) + ErA/(1−rR)

These can be put into the forms

[(1−rR) − Ra](e/a) − A(E/A) = 0
− a(e/a) + [(1−rR) − rA](E/A) = 0

This is a set of two linear homogeneous equations in two unknowns, (e/a) and (E/A). It is not a trivial matter to show that

(e/a) = (E/A)

is the solution but that is the case as will be shown below. (There is another matter of the determinant of the coefficients being zero bu that will be left aside for now.)

If (e/a)=(E/A)=γ then the two equations above are equivalent to

[(1−rR) − Ra − A]γ = 0
[(1−rR) − rA − a]γ = 0

These require that

[(1−rR) − Ra − A] = 0
[(1−rR) − rA − a] = 0
and thus
[(1−rR) − Ra − A] = [(1−rR) − rA − a]

Is it true that [(1−rR) − Ra − A] is equal to [(1−rR) − rA − a]?

For proof of this matter consider the following sequence of transformations:

aA = aA
(1-r)A = a(1-R)
A−rA = a−aR
−a − rA = −A −aR
and adding (1−rR) to both sides
(1−rR) −a − rA = (1−rR) −A −aR

Thus [(1−rR) − Ra − A] does equal [(1−rR) − rA − a] and (e/a)=(E/A) is the solution to the equations.

The Significance of (e/a)=(E/A)
Being the Solution

The plates were of arbitrary materials and hence the ratio of emissivity to absorptivity is the same for all materials. This means that for all materials

e/a = f(λ, T)

where f(λ, T ) is a universal function. If this universal function can be found then the emissivity of any material can be determined from its absorption or reflectivity ratio; i.e.,

e(λ, T) = a(λ, T)f(λ, T) = (1-r(λ, T))f(λ, T)

Kirchhoff envisioned a material with zero reflectivity and coined the term black body to label it. According to the analysis determining the emissivity and absorption as a function of wavelength and temperature for a black body would establish the universal function.

Kirchhoff published a rigorous proof of this result in 1861.

The relation Kirchhoff found between emission and absorption was a surprise at his time. Later, with the development of the quantum model of atoms, such a relationship seemed obvious.

In 1875 Kirchhoff was appointed professor of mathematical physics at the University of Berlin. He remained there until he died in 1887. His work increased the interest in the search for a formula for the radiation spectrum of heated bodies. The next step was the discovery of the Stefan-Boltzmann law for emissivity as a function of absolute temperature.

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