Gustav Kirchhoff

Research

In 1850, Kirchhoff's paper "Über das Gleichgewicht und die Bewegung einer elastischen Scheibe" resolved a specific technical flaw that had dogged plate theory since Germain's original work: the boundary condition problem.

In 1857, Kirchhoff calculated that an electric signal in a resistanceless wire travels along the wire at the speed of light.

In 1859, Kirchhoff proposed a law of thermal radiation, and gave a proof in 1861.

Together, Kirchhoff and Bunsen improved on Joseph von Fraunhofer's 1814 spectroscope, which Kirchhoff used to pioneer the identification of the elements in the Sun, showing in 1859 that the Sun contains sodium. Kirchhoff and Bunsen discovered caesium and rubidium in 1861.

Kirchhoff contributed greatly to the field of spectroscopy by formalizing three laws that describe the spectral composition of light emitted by incandescent objects, building substantially on the discoveries of David Alter and Anders Jonas Ångström. In 1862, he was awarded the Rumford Medal "for his researches on the fixed lines of the solar spectrum, and on the inversion of the bright lines in the spectra of artificial light".

Kirchhoff also contributed to optics, carefully solving the wave equation to provide a solid foundation for Huygens' principle (and correct it in the process).

Kirchhoff's plate theory

The biharmonic plate equation is 4th-order, so mathematically we need two boundary conditions per edge to get a well-posed problem. But naive derivations gave three conditions per edge — an over-determined, inconsistent system. This made Germain's, Poisson's, and others' earlier plate theories internally inconsistent or wrong at the boundary, even though the interior equation was basically right. Kirchhoff provided the first rigorous mathematical grounding for exactly the free-edge boundary conditions that real Chladni plates have.

Kirchhoff showed, using the calculus of variations applied to the plate's strain energy, that the twisting moment along an edge is statically equivalent to a distribution of vertical shear forces (via Kelvin's earlier notion that a twisting couple can be replaced by an equivalent transverse force). It made the plate biharmonic eigenvalue problem well-posed — a solvable, unique 4th-order boundary value problem, matching the order of the governing PDE. It's essentially the origin of using energy/variational methods to derive both the governing equation and its boundary conditions together and consistently — a technique that became standard well beyond plate theory. This is why the standard theory of thin plates today is called Kirchhoff–Love plate theory (Love extended it to shells later, in 1888).

Kirchhoff's circuit laws

Kirchhoff's first law states that at any node in an electrical circuit where current can branch, the sum of the currents leaving the node is equal to the sum of the currents entering the node. The second law states that the algebraic sum of the potential drops along a closed circuit, taken in any direction of flow, is equal to zero.

Kirchhoff's three laws of spectroscopy

1. A solid, liquid, or dense gas excited to emit light will radiate at all wavelengths and thus produce a continuous spectrum.
2. A low-density gas excited to emit light will do so at specific wavelengths, and this produces an emission spectrum.
3. If light composing a continuous spectrum passes through a cool, low-density gas, the result will be an absorption spectrum.

Kirchhoff did not know about the existence of energy levels in atoms. The existence of discrete spectral lines had been known since Fraunhofer discovered them in 1814. That the lines formed a discrete mathematical pattern was described by Johann Balmer in 1885. Joseph Larmor explained the splitting of the spectral lines in a magnetic field known as the Zeeman Effect by the oscillation of electrons. These discrete spectral lines were not explained as electron transitions until the Bohr model of the atom in 1913, which helped lead to quantum mechanics.

Kirchhoff's law of thermal radiation

It was Kirchhoff's law of thermal radiation in which he proposed an unknown universal law for radiation that led Max Planck to the discovery of the quantum of action leading to quantum mechanics.

Kirchhoff's law of thermochemistry

Kirchhoff showed in 1858 that, in thermochemistry, the variation of the heat of a chemical reaction is given by the difference in heat capacity between products and reactants:

${\displaystyle {\left(\frac{\mathrm{\partial }\mathrm{\Delta }H}{\mathrm{\partial }T}\right)}_p=\mathrm{\Delta }{C}_p}$.

Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.

Kirchhoff's theorem in graph theory

Kirchhoff also worked in the mathematical field of graph theory, in which he proved Kirchhoff's matrix tree theorem.

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