Gustav Kirchhoff

How to use this course

Gustav Kirchhoff’s name sits on more equations than almost any other nineteenth-century physicist’s: circuit laws, a radiation law, laws of spectroscopy, a diffraction formula, a plate theory, a thermochemical equation and a theorem in graph theory. The breadth is real, but the method is unified. Kirchhoff took a physical situation, found the exact mathematical problem behind it, and solved that problem with care. His one great experimental campaign, the spectral analysis he built with the chemist Robert Bunsen, turned the spectroscope into an instrument of chemistry and astronomy and led directly to his theory of black-body radiation.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow the chronology of his work: circuits (1845–1849), elasticity (1850–1859), telegraphy (1857), thermochemistry (1858), spectra and radiation (1859–1862), fluids (1869–1876) and optics (1882).

Timeline

Königsberg, 1824–1847

Gustav Robert Kirchhoff was born in Königsberg, the capital of East Prussia, on 12 March 1824. His father, Friedrich Kirchhoff, was a law councillor with a strong sense of duty to the Prussian state; his mother was Johanna Henriette Wittke. A university professorship was then a form of state service, and his family expected him to pursue one.

In 1842 he entered the Albertus University of Königsberg. It was one of the best places in Europe to learn mathematical physics. Carl Gustav Jacob Jacobi and Franz Ernst Neumann had founded there a mathematical-physical seminar, the first of its kind in Germany, which trained students to turn experiments into exact mathematical problems. Jacobi fell ill in 1843 and soon left for Berlin, so Neumann became Kirchhoff’s main teacher. He learned mathematics also from Friedrich Julius Richelot.

Neumann’s seminar demanded original work, and Kirchhoff produced it early. In 1845, aged 21, he published a paper on the flow of current through a flat conducting plate, a problem set in the seminar. In it he stated the two rules for networks of wires that every engineering student now learns (Lesson 04). He extended the work into his doctoral dissertation and took his degree in 1847.

Berlin and Breslau, 1847–1854

In 1847 Kirchhoff moved to Berlin and qualified as a Privatdozent, an unpaid lecturer living on student fees. These were years of revolution in the German states, but his work in Berlin was quiet and theoretical. He published his paper on the systematic solution of network equations, which contains the matrix-tree theorem of graph theory. In 1849 he showed that the “electroscopic force” in Georg Ohm’s theory of currents is the same quantity as the electrostatic potential of Siméon Denis Poisson, uniting the theories of currents and of static electricity.

In 1850 he became extraordinary professor at Breslau. There he solved a problem in elasticity that had defeated Sophie Germain and Poisson: the correct boundary conditions for a vibrating elastic plate (Lesson 06). His results explained the patterns of Ernst Chladni’s sand figures.

At Breslau he also met the chemist Robert Bunsen, thirteen years older, who spent a short time there in 1851–52 before moving to Heidelberg. The two became close friends. Bunsen was an exceptional experimenter; Kirchhoff was a careful theorist who could also build and use precise instruments. It was Bunsen who drew him to Heidelberg.

Heidelberg and Bunsen, 1854–1875

In 1854 Kirchhoff became professor of physics at Heidelberg. The university was then entering its most brilliant scientific period: Bunsen led chemistry, and from 1858 Hermann von Helmholtz held the chair of physiology. The three formed the centre of a small circle that made Heidelberg a destination for young scientists from across Europe.

In 1857 he married Clara Richelot, daughter of his Königsberg mathematics teacher. They had five children. The same year he published his analysis of signals on telegraph wires, finding that in a wire of low resistance a disturbance travels at a speed equal to that of light (Lesson 05).

From 1859 he and Bunsen developed spectral analysis. Bunsen’s new burner gave a nearly colourless flame in which salts showed their own spectral lines. Kirchhoff built a spectroscope from a prism and telescopes. In October 1859 he passed sunlight through a sodium flame and found that the dark D lines of the solar spectrum grew darker, not brighter. From this he drew both the chemical analysis of the Sun and a general law relating emission and absorption (Lessons 07 and 08). With the new method Bunsen and Kirchhoff found two unknown elements, caesium in 1860 and rubidium in 1861.

Kirchhoff had a disability that often confined him to crutches or a wheelchair, and laboratory work grew harder over time. Clara died in 1869, leaving him to raise the children. In 1872 he married Luise Brömmel, who had been head nurse at a Heidelberg eye clinic.

Circuit laws and spanning trees

For the general reader. In a network of wires, how much current flows in each one? Kirchhoff’s two rules answer this. Charge does not pile up at a junction, so what flows in must flow out. Going around any closed loop, the voltage rises and drops cancel. In 1847 he went further and gave an explicit formula for every current, by counting the ways of connecting the network without loops. This was the first theorem of algebraic graph theory.

The current law is conservation of charge for steady flow. The voltage law states that the electric field in a steady circuit is the gradient of a potential, so that its line integral around a loop vanishes apart from the sources. Kirchhoff derived the voltage law from Ohm’s law; his 1849 paper (Lesson 05) showed why the potential involved is the electrostatic one.

How many equations?

Let the network be a connected graph with \(n\) nodes and \(m\) edges. The current law gives \(n-1\) independent equations (the last is the sum of the others). The voltage law gives one independent equation for each independent cycle, and there are \(\mu=m-n+1\) of them, the cycle rank. Together these make \(m\) equations for the \(m\) unknown currents. Kirchhoff proved that the system always has a unique solution when all resistances are positive.

The matrix-tree theorem

Give edge \(e\) the conductance \(c_e=1/R_e\). The weighted Laplacian of the graph is the \(n\times n\) matrix \(L\) with \(L_{ii}=\sum_{e\ni i}c_e\) and \(L_{ij}=-c_e\) when edge \(e\) joins \(i\ne j\). The node potentials \(\phi\) and injected currents \(J\) satisfy \(L\phi=J\).

Proof sketch. Orient the edges and let \(B\) be the \(n\times m\) incidence matrix, \(C=\operatorname{diag}(c_e)\). Then \(L=BCB^{\mathsf T}\). Delete row \(i\) of \(B\) to get \(B_i\). By the Cauchy–Binet formula, \(\det(B_iCB_i^{\mathsf T})=\sum_{S}\det(B_i[S])^2\prod_{e\in S}c_e\) over sets \(S\) of \(n-1\) edges. A square submatrix \(B_i[S]\) has determinant \(\pm1\) if \(S\) is a spanning tree and 0 otherwise. Kirchhoff stated his result in terms of the currents rather than the Laplacian, but the content is the same.

Potential, conductors and telegraph signals

For the general reader. Ohm’s law of 1827 spoke of an “electroscopic force” driving current through a wire, without saying what it was. Kirchhoff showed in 1849 that it is the electrostatic potential created by charges on the conductor’s surface. Eight years later he worked out how a pulse travels along a telegraph wire. For a wire of low resistance he found that the pulse moves at the speed of light, a striking result he did not interpret as Maxwell later would.

Steady currents

In a homogeneous conductor of conductivity \(\sigma\), Ohm’s law reads \(\mathbf J=\sigma\mathbf E=-\sigma\nabla\phi\). Steady flow requires \(\nabla\cdot\mathbf J=0\), so

Kirchhoff’s point was that \(\phi\) is the same potential as in electrostatics, produced by charges that sit on the conductor’s surface and at interfaces between different materials. His 1845 thesis solved this Laplace problem for a circular plate with point electrodes.

Telegraph equations (1857)

Kirchhoff treated a thin wire with resistance, self-induction and capacitance distributed along it. In modern transmission-line form, with resistance \(R\), inductance \(L\), capacitance \(C\) and leakage conductance \(G\) per unit length, the voltage \(V(x,t)\) and current \(I(x,t)\) satisfy

When \(R\) is small, this is a wave equation with speed \(v=1/\sqrt{LC}\). For a wire in air, \(LC=\mu_0\varepsilon_0\), independently of the wire’s radius and the distance to the return path, so \(v=c\). Kirchhoff expressed the speed through Weber’s electrodynamic constant and found it equal, within the measurement error of Weber and Rudolf Kohlrausch, to the speed of light. When \(R\) is large the inductive term is negligible and the equation becomes a diffusion equation, the regime William Thomson had analysed for the Atlantic cable in 1855. Oliver Heaviside later used the full equations to show how to make lines distortionless.

Elastic plates and rods

For the general reader. When a metal plate is bowed, sand sprinkled on it gathers along lines that do not move, the Chladni figures. Sophie Germain had won a prize in 1816 for a theory of these vibrations, but her boundary conditions at a free edge were wrong. Kirchhoff found the right ones in 1850 by a variational argument. Nine years later he showed that a thin bent rod obeys the same equations as a spinning top, an analogy still used to study DNA and cables.

The plate energy

A thin plate of thickness \(h\), Young’s modulus \(E\) and Poisson ratio \(\nu\) has bending stiffness \(D=Eh^3/12(1-\nu^2)\). Kirchhoff assumed that straight lines normal to the middle surface stay straight and normal after bending. For a small transverse deflection \(w(x,y)\) the strain energy is

Poisson had imposed three conditions at a free edge (moment, shear force and twisting moment), which is one too many for a fourth-order equation. Kirchhoff’s variational derivation shows that the twisting moment and the shear combine into the single effective shear \(V_n\). The second term of \(U\) integrates to a boundary term, which is why it disappears from the field equation but not from the edge conditions. William Thomson and Peter Guthrie Tait later gave a physical explanation of this reduction. The model is now called Kirchhoff–Love plate theory.

The kinetic analogy (1859)

For a thin rod bent and twisted in three dimensions, Kirchhoff showed that the equilibrium equations, written with arc length \(s\) as the independent variable, have the same form as Euler’s equations for a rigid body rotating about a fixed point under gravity, with \(s\) in place of time. A rod bent by end forces corresponds to a heavy top; the shapes of a rod correspond to the motions of the top.

Spectral analysis and the chemistry of the Sun

For the general reader. Every element, heated in a flame, glows with its own set of coloured lines, like a fingerprint. Bunsen and Kirchhoff proved this systematically in 1859–1860 and used it to discover two new elements. Kirchhoff then explained the hundreds of dark lines that Joseph von Fraunhofer had mapped in sunlight: they are the fingerprints of elements in the Sun’s atmosphere, absorbing the light from below. For the first time, the composition of a star could be measured.

The experiment

Fraunhofer had noticed in 1814 that the dark D line of sunlight coincided with a bright yellow line seen in many flames. Others had noticed the coincidence too, but its meaning was unclear, because sodium is so common that its yellow line appears almost everywhere. Bunsen’s burner gave a flame hot and faint enough to show the lines of a salt clearly. Kirchhoff passed sunlight through a sodium flame into his spectroscope, expecting the bright sodium lines to fill in the dark D lines. Instead the D lines became darker. With a strong limelight in place of the Sun, a sodium flame placed in front produced dark D lines where there had been none.

The third law follows from the radiation law of Lesson 08: a gas that emits strongly at a wavelength also absorbs strongly there. If the gas is cooler than the source behind it, it removes more light than it adds. Kirchhoff concluded that the Sun is a hot, dense body surrounded by a cooler atmosphere, and he identified iron, sodium, magnesium, calcium, chromium, nickel and other elements in it by matching hundreds of lines. His map of the solar spectrum appeared in 1861–62.

Spectral analysis was also extremely sensitive. In 1860 Bunsen and Kirchhoff found a pair of blue lines in the spectrum of a residue from Dürkheim mineral water that matched no known element, and named the element caesium, from the Latin for sky blue. In 1861 dark red lines in the mineral lepidolite gave rubidium. Within a few years other workers used the method to find thallium, indium and, in the solar spectrum, helium.

The law of thermal radiation

For the general reader. Why does a good absorber of light also glow brightly when hot? Kirchhoff proved in 1859–60 that, at each wavelength and temperature, the ratio of a body’s emitting power to its absorbing power is the same for every body. The common value is the radiation of a perfectly absorbing body, which he called a black body. Finding that universal function became the central problem of radiation theory, and Max Planck’s solution in 1900 began quantum physics.

Let a body at temperature \(T\) emit power \(e_\lambda\,d\lambda\) per unit area in the wavelength range \(d\lambda\), into a given direction and polarization, and absorb a fraction \(a_\lambda\) of radiation incident on it in that range, direction and polarization.

The argument

Place the body inside an enclosure at uniform temperature \(T\). In equilibrium the body neither heats nor cools, so at each wavelength it must emit what it absorbs: \(e_\lambda=a_\lambda\,I_\lambda\), where \(I_\lambda\) is the radiation in the enclosure. If \(I_\lambda\) depended on the walls, one could connect two enclosures at the same temperature through a filter passing only \(d\lambda\) and get a net flow of energy from one to the other, violating the second law. So \(I_\lambda=E(\lambda,T)\) is universal, and \(e_\lambda/a_\lambda=E(\lambda,T)\). Kirchhoff’s 1860 proof, with idealized plates and mirrors, was later judged incomplete, and modern proofs use detailed balance. Balfour Stewart had reached a related result for heat radiation in 1858, which led to a priority dispute.

Kirchhoff noted that a small hole in the wall of a closed cavity behaves as a black body, since light entering it is almost never reflected back out. This was the source used by Otto Lummer, Ernst Pringsheim and others in Berlin in the 1890s to measure \(E(\lambda,T)\). The answer, found by Planck in 1900, is

Kirchhoff’s diffraction theory

For the general reader. Christiaan Huygens imagined each point of a wavefront as a source of new wavelets, and Augustin Fresnel used the idea to calculate diffraction patterns. But Fresnel had to add ad hoc factors to make it work. In 1882 Kirchhoff derived the principle exactly from the wave equation using Green’s theorem, giving the formula still used to calculate diffraction by apertures.

Proof. Apply Green’s identity \(\int_V(G\nabla^2U-U\nabla^2G)\,dV=\oint_S(G\partial_nU-U\partial_nG)\,dS\) with \(G=e^{iks}/s\), on \(V\) minus a small sphere of radius \(\epsilon\) around \(P_0\). Both functions satisfy the Helmholtz equation there, so the volume integral vanishes. As \(\epsilon\to0\) the small sphere contributes \(-4\pi U(P_0)\).

The diffraction formula

For an aperture \(A\) in an opaque screen, illuminated by a point source at \(P_1\), let \(\mathbf n\) be the unit normal to the aperture pointing toward the source side, \(\mathbf r\) the vector from \(P_1\) to an aperture point and \(\mathbf s\) the vector from \(P_0\) to that point. Kirchhoff assumed that on \(A\) the field and its normal derivative are those of the incident wave, and that on the dark side of the screen both vanish. With \(r\) the distance from \(P_1\) to the aperture point and \(s\) the distance to \(P_0\), for \(kr,ks\gg1\):

The obliquity factor and the phase factor \(-i\) that Fresnel had guessed now follow from the wave equation. The boundary assumptions are mathematically inconsistent: a solution of the Helmholtz equation that vanishes with its normal derivative on part of a surface vanishes everywhere. Arnold Sommerfeld removed the inconsistency with Rayleigh–Sommerfeld formulas using a Green function that vanishes on the screen. For apertures much larger than the wavelength all versions agree to high accuracy.

Heat of reaction and free jets

For the general reader. Two shorter pieces of work show Kirchhoff’s range. In 1858 he showed how the heat released by a chemical reaction changes with temperature, a rule now taught in every physical chemistry course. In 1869 he used complex functions to find the exact shape of a jet of water leaving an opening, the start of the theory of free streamlines used for wakes and cavities.

The proof is one line: \(\Delta H=H_{\text{products}}-H_{\text{reactants}}\) and \((\partial H/\partial T)_p=C_p\) for each substance. Kirchhoff’s paper applied the same reasoning to heats of solution and of vaporization.

Free streamlines (1869)

When liquid leaves a large vessel through a slot in a flat wall, the jet narrows before becoming parallel. On its free surfaces the pressure is atmospheric, so by Bernoulli’s equation the speed is constant there. Helmholtz had introduced such free surfaces in 1868. Kirchhoff showed how to solve these problems exactly by conformal mapping, working in the plane of the logarithm of the complex velocity, where the free boundaries become straight lines.

In his 1876 lectures on mechanics, Kirchhoff also found that an elliptical patch of uniform vorticity \(\omega\) with semi-axes \(a,b\) keeps its shape and rotates rigidly with angular velocity \(\Omega=\omega ab/(a+b)^2\), the Kirchhoff vortex.

Berlin, 1875–1887

Experimental work had become physically difficult for Kirchhoff. In 1875 he accepted a call to the new chair of mathematical physics at the University of Berlin, one of the first chairs of theoretical physics in Germany, together with full membership of the Prussian Academy of Sciences. Helmholtz had moved to Berlin in 1871, and the two again worked side by side.

In Berlin Kirchhoff concentrated on teaching. His lectures were models of clarity and economy, and they defined the course of theoretical physics in German universities for a generation. Max Planck, who heard them as a student in 1877–78, admired their precision but found them dry. The first volume, on mechanics, appeared in 1876. Its opening defined the task of mechanics as describing motions completely and in the simplest manner, without seeking their causes, a programme that influenced Ernst Mach and later Hertz. The other three volumes, on optics, electricity and magnetism, and heat, were edited from his notes after his death.

His doctoral students over his career included Gabriel Lippmann, later a Nobel laureate for colour photography, Arthur Schuster, Loránd Eötvös, Jacob Lüroth, Ernst Schröder, Heinrich Weber and Max Noether. Heinrich Hertz, who studied in Berlin from 1878, attended his lectures. Honours accumulated: the Rumford Medal in 1862, fellowship of the Royal Society of Edinburgh in 1868 and of the Royal Society in 1875, the Prussian order Pour le Mérite for science in 1874, and the Davy Medal, with Bunsen, in 1877.

A story told about the spectral work concerns his banker, who asked what use gold in the Sun could be if it could not be brought down. When Kirchhoff later received prize money from Britain, he deposited it with the banker as gold from the Sun.

Death and legacy

Kirchhoff’s health declined further in the 1880s. He died in Berlin on 17 October 1887, aged 63, and was buried in the Alter St.-Matthäus-Kirchhof in Schöneberg. Planck was appointed to his chair in 1889.

Legacy

• Electrical engineering and graph theory: the circuit laws are the foundation of network analysis; the matrix-tree theorem is the starting point of spectral graph theory and of the theory of electrical networks on graphs.
• Astrophysics and chemistry: spectral analysis made astrophysics possible, from the composition of stars to their temperatures and motions, and it remains a standard tool of analytical chemistry.
• Quantum theory: the universal function \(E(\lambda,T)\) set the problem that Wien, Rayleigh and Planck worked on, and Planck’s solution introduced the quantum of action.
• Mechanics and optics: Kirchhoff–Love plates, Kirchhoff rods, free-streamline theory and Kirchhoff diffraction are standard chapters of their fields.
• Honours: the Kirchhoff Institute for Physics at Heidelberg, the circuit, radiation and diffraction laws, and a lunar crater bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gustav Robert Kirchhoff, MacTutor History of Mathematics.
2. Gustav Kirchhoff, Wikipedia.
3. Kirchhoff Institute for Physics, Heidelberg, 200 Years of Gustav Kirchhoff: his life and work.
4. APS News, This month in physics history: Bunsen and Kirchhoff, May 2018.
5. K. Hentschel, Mapping the Spectrum: Techniques of Visual Representation in Research and Teaching, Oxford University Press, 2002.
6. C. Jungnickel and R. McCormmach, Intellectual Mastery of Nature, vol. 1–2, University of Chicago Press, 1986.

Physics

1. G. Kirchhoff, Über den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisförmige, Ann. Phys. 64 (1845).
2. G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird, Ann. Phys. 72 (1847).
3. G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. reine angew. Math. 40 (1850).
4. G. Kirchhoff, Über die Bewegung der Elektricität in Drähten, Ann. Phys. 100 (1857).
5. G. Kirchhoff, Über das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht, Ann. Phys. 109 (1860) 275–301.
6. G. Kirchhoff and R. Bunsen, Chemische Analyse durch Spectralbeobachtungen, Ann. Phys. 110 (1860).
7. G. Kirchhoff, Zur Theorie freier Flüssigkeitsstrahlen, J. reine angew. Math. 70 (1869).
8. G. Kirchhoff, Zur Theorie der Lichtstrahlen, Ann. Phys. 254 (1883).
9. G. Kirchhoff, Vorlesungen über mathematische Physik, 4 vols., Teubner, 1876–1894.
10. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, chapter 8.
11. M. Planck, The Theory of Heat Radiation, 1914; Dover reprint.