Gustav Kirchhoff: A Life in Five Chapters

Chapter 1 · Twenty-One

Two rules, published as a student

1824 – 1845 · Königsberg

Gustav Robert Kirchhoff was born at Königsberg in East Prussia in March 1824, the son of a lawyer.

He studied at Königsberg under *Franz Neumann*, who ran a mathematical-physical seminar of a kind that scarcely existed elsewhere — training students to treat physical problems with full mathematical rigour rather than descriptively.

In 1845, while still a student, aged twenty-one, Kirchhoff published two rules for electrical networks.

The *junction rule*: the total current arriving at any junction equals the total current leaving it.

The *loop rule*: the sum of the potential differences around any closed loop in a circuit is zero.

The reason these matter is that Ohm's law alone is not enough. Ohm gives the relationship for a single conductor. A real circuit — a network with branches, junctions, several sources and many components — cannot be solved by proportionality; you need something that handles the *topology*.

Kirchhoff's two rules do. Between them they generate enough simultaneous equations to solve any network of any complexity, and every circuit analysis since — from a school breadboard to the design of a microprocessor — uses them.

What makes them powerful is *where they come from*. The junction rule is *conservation of charge*: charge does not accumulate at a junction or vanish there, so what goes in must come out. The loop rule is *conservation of energy*: a charge taken round a loop and returned to its starting point must have gained and lost exactly the same energy.

A law derived from a conservation principle admits no exceptions, because violating it would mean creating or destroying charge or energy.

Kirchhoff regarded this as a minor early exercise. Posterity has disagreed, and it is what most people know his name for.

“It is a matter of the greatest interest that the composition of the Sun's atmosphere can be determined from the dark lines of its spectrum.”

— Gustav Kirchhoff, on the interpretation of the Fraunhofer lines (1859)

Why this matters

Kirchhoff's rules follow from conservation of charge and energy, which is why they hold for any circuit of any complexity without exception.

You have two rules published by a student. What would you ask him?

Ask Kirchhoff

• “Why is Ohm's law not enough for a real circuit?”
• “What conservation principle is behind each rule?”
• “Why can a rule from a conservation law have no exceptions?”
• “Why did you think this was minor work?”
• “What did Neumann's seminar teach you?”

Chapter 2 · Heidelberg, With Bunsen

Why coloured glass fails and a prism does not

1854 – 1859 · Heidelberg

Kirchhoff moved to Heidelberg in 1854 and began working with the chemist *Robert Bunsen*.

Bunsen was trying to use *flame colour* as a chemical test. Sodium colours a flame yellow, potassium lilac, strontium red, barium green. He was viewing flames through coloured glass filters to try to separate the colours of a mixture, and it did not work reliably — the colours of several elements together blend into something no filter can untangle.

Kirchhoff, a physicist, made the suggestion that solved it: do not look at the *colour*. Pass the light through a *prism* and look at the *spectrum*.

What then appears is that each element produces a set of *bright lines* at particular, fixed positions in the spectrum. Sodium gives a pair of close yellow lines. Each element's set is characteristic and invariant.

And a mixture gives *all the lines of all its components*, superimposed but *separable*, because each line sits at its own place. Nothing blends.

They built the *spectroscope* — burner, slit, prism, viewing telescope — in 1859.

It is astonishingly sensitive. Bunsen calculated that it could detect around a three-millionth of a milligram of sodium. Kirchhoff reportedly noticed sodium lines appearing whenever anybody in Heidelberg shook a tablecloth, from the salt in the dust.

They found two new elements almost immediately: *caesium* in 1860, from two blue lines in the spectrum of mineral water, and *rubidium* in 1861, from two red lines in a lithium mineral.

So the instrument justified itself twice within two years.

Why this matters

Spectral lines from a mixture superimpose without blending, because each line sits at its own fixed position — which is why a prism succeeds where a colour filter cannot.

You have the prism and the fixed bright lines. What is your question?

Ask Kirchhoff

• “Why does a coloured filter fail on a mixture?”
• “What does a spectrum show that a colour does not?”
• “How sensitive is the instrument really?”
• “How do you know a set of lines belongs to a new element?”
• “What did a chemist and a physicist each bring?”

Chapter 3 · The Dark Lines in the Sun

Fraunhofer's puzzle, answered

1859 – 1861 · Heidelberg

In 1814 *Joseph von Fraunhofer* had examined the Sun's spectrum with a good prism and found it crossed by hundreds of *dark lines* — narrow gaps where light is missing. He mapped and labelled them carefully and had no idea what they were.

They had been a puzzle for forty-five years.

Kirchhoff noticed the crucial coincidence. The prominent pair of dark lines Fraunhofer had labelled *D* in the solar spectrum lie at *exactly the wavelengths* where sodium emits its bright yellow pair.

He then did the experiment. He passed sunlight through a sodium flame — expecting, perhaps, that the flame's bright lines would fill in the dark ones.

The dark lines got *darker*.

That gives the answer, and it is beautiful. A substance *absorbs* exactly the wavelengths it *emits*. The Sun's hot interior emits a continuous spectrum, all wavelengths. Passing outward through the cooler gases of the Sun's own atmosphere, sodium in that atmosphere absorbs precisely the wavelengths sodium emits, removing them from the light and leaving dark gaps where they should be.

So each dark line is the *fingerprint of an element in the Sun's atmosphere*.

Kirchhoff and Bunsen identified sodium, iron, magnesium, calcium, chromium, nickel and others in the Sun by matching the dark lines against laboratory spectra.

The chemical composition of a body ninety-three million miles away had been determined without going near it.

The philosopher Auguste Comte had written in 1835 that we would never by any means be able to determine the chemical composition of the stars — offering it as an example of permanently unknowable knowledge. He was wrong within twenty-six years, and it is one of the most cited misjudgements in the history of philosophy.

This is where astronomy becomes *astrophysics*. Everything known about the composition of stars, galaxies and interstellar gas descends from it.

Why this matters

A substance absorbs exactly the wavelengths it emits, so each dark line in the Sun's spectrum is the fingerprint of an element in its atmosphere.

You have the darker dark lines and the composition of the Sun. What would you ask?

Ask Kirchhoff

• “What happened when you passed sunlight through a sodium flame?”
• “Why does a substance absorb what it emits?”
• “How can you know what a star is made of?”
• “What would you say to Comte?”
• “What did Fraunhofer think the lines were?”

Chapter 4 · The Black Body

A problem posed and left for somebody else

1859 – 1862 · Heidelberg

Kirchhoff generalised the emission–absorption relationship into a law of thermal radiation, stated in 1859.

For a body in thermal equilibrium at a given temperature, the ratio of its *emissive power* to its *absorptivity*, at any given wavelength, is the *same for all bodies* — a universal function of wavelength and temperature alone, independent of what the body is made of.

That is a startling claim. It says a universal function exists, governing thermal radiation for every material in the universe, and that finding it is a well-defined problem.

He coined the term *black body* for the idealised case: a body that absorbs all radiation falling on it, and therefore, by the law, is also the best possible emitter. Its emission is the universal function itself.

And he issued what amounts to a challenge: find that function.

It looks approachable. It defeated physics for forty years.

Stefan and Boltzmann found how the *total* emitted power depends on temperature. Wien found where the *peak* lies and got the short-wavelength end right. Rayleigh and Jeans derived from classical physics a formula that works at long wavelengths and predicts *infinite* energy at short ones — the ultraviolet catastrophe, an absurdity that classical physics could not avoid.

On 14 December 1900 *Max Planck* found the answer, and could only get it by assuming that energy is emitted and absorbed in *discrete packets* — quanta — proportional to frequency. He regarded it as a mathematical device and was uncomfortable with it for years.

It was not a device. It is quantum theory, and it is the foundation of twentieth-century physics.

Kirchhoff, by posing a well-defined problem that classical physics could not solve, set the question whose answer broke classical physics. He died thirteen years before it was answered.

Why this matters

Kirchhoff posed a well-defined problem that classical physics could not solve, and its answer forty years later was the beginning of quantum theory.

You have the universal function and the forty-year gap. What is your question?

Ask Kirchhoff

• “What is a black body?”
• “Why should the function be the same for every material?”
• “What is the ultraviolet catastrophe?”
• “What would you have made of energy coming in packets?”
• “Is posing a problem a contribution?”

Chapter 5 · On Crutches

A body that failed, and a mind that did not

1862 – 1887 · Heidelberg · Berlin

Kirchhoff was disabled for much of his working life. An accident in his youth — variously reported as a fall on ice or on a staircase — left him with a permanent injury, and he spent years on *crutches* or in a *wheelchair*.

It curtailed his laboratory work considerably. Much of his later output is theoretical, and colleagues did the bench work for him.

He continued to produce first-rate physics. He gave the first satisfactory mathematical treatment of *diffraction*, putting Huygens's principle on a rigorous footing. He worked on elasticity and on the deformation of plates. And in 1857 he calculated that an electrical signal travels along a perfectly conducting wire at a speed *equal to the speed of light* — a striking result which he obtained from purely electrical quantities, and whose significance he did not press. Maxwell made that connection central to electromagnetic theory a few years later.

His *Lectures on Mathematical Physics*, published from 1876, are famously rigorous and famously demanding. He held that physics should say precisely which principles a result depends on, and he defined mechanics as the complete and simplest *description* of motion — pointedly declining to say anything about causes. That positivist position influenced Hertz, Mach and, through them, Einstein.

His first wife Clara, daughter of his Königsberg mathematics professor, died in 1869, leaving him with children to raise while disabled. He remarried in 1872.

In 1875 his health made experimental work impossible, and he took a chair in *mathematical physics* at Berlin. Max Planck attended his lectures there and later succeeded to his chair.

He died in Berlin in October 1887, aged sixty-three.

Thirteen years later Planck answered his question.

Why this matters

Kirchhoff calculated in 1857 that a signal travels along a conducting wire at the speed of light, obtained purely from electrical quantities, and did not press the point.

You have the crutches, the lectures and the question left open. What would you ask?

Ask Kirchhoff

• “How did the injury change what physics you could do?”
• “You found the speed of light in a wire — why not pursue it?”
• “Why refuse to talk about causes in mechanics?”
• “What was Planck like as a student?”
• “Which of your results do you actually value most?”

What Kirchhoff changed

Kirchhoff's two rules are still how any circuit is solved, from a school breadboard to a microprocessor. And spectrum analysis turned astronomy into astrophysics: everything known about what stars and galaxies are made of descends from his reading of the Sun's dark lines. The black-body problem he posed and named was answered by Planck in 1900, and the answer was quantum theory.

A debate that continues

Kirchhoff regarded his circuit rules as a minor student exercise and posterity has disagreed; and his 1857 calculation that signals travel along a wire at the speed of light was a connection to electromagnetism he did not pursue.

Keep exploring — ask Kirchhoff

• “What else can be read out of light at a distance?”
• “Why can't classical physics give the black-body curve?”
• “Which of your problems would you set a student now?”

Related lives

• Robert Bunsen — Who Gave the Flame a Collar
• James Clerk Maxwell — Who Bound Light to Electromagnetism
• Georg Ohm — Whose Proportion Was Called A Fancy
• Charles-Augustin de Coulomb — Who Weighed A Force With A Twisted Wire

Related themes

Circuits and Kirchhoff's laws · Spectra and the composition of stars · Thermal radiation