Gustav Kirchhoff: a birthday

Sad recent event: Donald Glaser, the 1960 Nobel prize winner in physics for his invention of the bubble chamber, died on February 28th.

Gustav Kirchhoff was born on March 12th, 1824, in Königsberg, capital of Prussia (now the island of Kaliningrad, Russia: note that it was easy to correct the country's name) to a lawyer and his wife. He married his math teacher's daughter and got lots of good education by people including Jacobi.

In some sense, we could say that this important 19th century physicist was the ultimate conventional career mainstream scientist. He died at age of 63 in Berlin, Prussia. Much of the 19th century classical physics is encoded in the laws named after Kirchhoff.




First, let me begin with electrical circuits. Gustav was just a 21-year-old student (in 1845) and he already discovered two Kirchhoff's circuit laws that are used all the time even these days:\[

\sum_{k=1}^n I_k = 0, \quad \sum_{k=1}^n V_k = 0

\] The first law is based on the conservation of the electric charge. The currents running through "propagators" (wires) attached to a "vertex" (junction) of a circuit add up to zero (if the signs are all constrained by the same convention, plus for incoming, minus for outgoing, for example).

The second law boils down to the conservation of energy. The voltages are ultimately differences of electrostatic potentials so the sum of the voltages in a loop (counted in the same direction) has to vanish. Both laws may be generalized to oscillating circuits when the currents \(I_k\) and voltages \(V_k\) become complex numbers. The second law on voltages must be fixed in the presence of changing magnetic fields: an electromotive force (emf), an extra contribution expressed in volts, is associated with inductances (coils).

The circuits look like mathematical graphs so Kirchhoff would also invent Kirchhoff's [matrix tree] theorem allowing one to calculate the number of spanning trees in a graph (a subgraph that is a tree and that includes all the vertices of the original graph) from a determinant (Cayley previously quantified the very special case when the original graph is "complete", i.e. has all the edges it can have).




In 1859, he proposed his law of thermal radiation. It relates the emission and absorption by the same body.

The law simplifies for a black body, an idealized object that absorbs all the incoming radiation and emits a specific radiation \(dE=dt\,d\lambda\, dA\,E_{b\lambda}(\lambda,T)\) which is fully determined by a universal function of the temperature and the wavelength, later analytically derived by Max Planck (in 1900).

Note that the ratio of emissivity and absorptivity is equal to \(E_{b\lambda}(\lambda,T)\), Planck's function, for the black body. Now, Kirchhoff stated that the ratio remains the same, \(E_{b\lambda}(\lambda,T)\), for all bodies. The emissivity for a given wavelength drops by the same coefficient as the absorptivity. Because the absorptivity of a non-black body never exceeds the absorptivity of the black body (you can't devour more than 100% of what you're offered), the same must hold for emissivity: a real-world body can't emit more light than the black hole body at the same temperature and wavelength.

He "induced" the law by previously known vague observations of the fact that good emitters were good absorbers. These days, we may prove Kirchhoff's law via the time-reversal invariance of the microscopic laws of physics. Absorption is emission run backwards in time so their likelihoods must be linked and Kirchhoff's law is the right way to relate them. Note that the law doesn't say that the ratio is simply one. It's because the microscopic laws may be time-reversal-symmetric but there are still kinematic factors entering because of the logical arrow of time: the initial state and the final state must always be treated differently. Those naive physicists who don't understand the logical arrow of time (e.g. Sean Carroll) would surely believe – if they could ever think about as advanced questions as this law from 1859 – that the ratio of emissivity and absorptivity is one which is wrong (even dimensionally wrong).

Gustav Kirchhoff also studied fluid dynamics. His "the Kirchhoff equations" govern the motion of rigid bodies through ideal fluids. He also studied stress in rigid bodies so we sometimes talk about the Kirchhoff stress \({\bf \tau}\). In 1888, Augustus Love completed some assumptions by Kirchhoff into the Kirchhoff-Love plate theory that describes how thin plates bend when they're subjected to forces.

Kirchhoff's law of thermochemistry says\[

\frac{d\Delta H}{ dT} = \Delta C_p

\] and was practically used to quantify how the heat of a reaction changes (\(d\Delta h\)) with the temperature (\(dT\)) from the difference \(C_p\) of heat capacities between reactants and products. It looks like a rather derived, non-fundamental law to my eyes today but it was surely helpful when people played with hundreds of reactions and their heat.

Finally, Gustav Kirchhoff formulated his three laws of spectroscopy:

• The emitted spectrum from a hot solid object is continuous.
• The spectrum emitted by a hot tenuous gas has spectral lines (a finite number of colors or wavelengths) and they depend on energy levels in the atoms of the gas.
• When you combine the hot solid object with the tenuous gas around it, you will find gaps in the spectrum at the same (gas') spectral lines (absorption spectrum).

Clearly, some of the explanations involving levels of the atoms were added later; Kirchhoff didn't understand the reason behind these observations in the same modern way we know today. Those concepts started to be appreciated when people began to use the old Bohr model of the atom only.

Gustav Kirchhoff also contributed to wave optics. He solved Maxwell's equations to derive Huygens' principle (he had to correct a minor mistake from Huygens along the way). He also formulated the Hyugens-Fresnel equation in an equivalent way, through Kirchhoff's diffraction formula\[

U(P) = \frac {1}{4 \pi} \int_{S} \left[ U \frac {\partial}{\partial n} \left( \frac {e^{iks}}{s} \right) - \frac {e^{iks}}{s} \frac {\partial U}{\partial n} \right]dS.

\] I won't explain the details here.

At any rate, while we could say that he hasn't made any super shockingly original breakthroughs, he filled lots of holes and clarified tons of inaccuracies in the 19th century physics and formulated some laws in a rather modern way that became natural with the arrival of quantum physics in the 20th century. It's no coincidence that so many laws, equations, expressions, and theorems are named after Gustav Kirchhoff.