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Most physics students know Kirchhoff from two laws in the current electricity chapter. Fewer know that the same man helped found spectroscopy, the method that tells us what stars are made of. This post covers both sides of his work, a fully worked two-loop circuit, and what a student can take from his way of working.
Who was Gustav Kirchhoff?
The MacTutor biography from the University of St Andrews gives his dates: born 12 March 1824 in Königsberg, Prussia (now Kaliningrad), died 17 October 1887 in Berlin. He studied at the University of Königsberg and from 1843 to 1846 attended the mathematics-physics seminar led by Franz Neumann. He taught in Berlin and Breslau, where he met the chemist Robert Bunsen, and in 1854 moved to Heidelberg as professor of physics, at Bunsen's encouragement. In 1875 he took the chair of mathematical physics in Berlin. MacTutor notes that a disability meant he spent much of his life on crutches or in a wheelchair.
What did he find as a student in 1845?
His first paper, published in Annalen der Physik in 1845, is signed "Studiosus Kirchhoff", a student and member of the physics seminar at Königsberg. It dealt with current flowing through a flat conducting plate. MacTutor records that in 1845 he announced the laws that let you calculate currents, voltages and resistances in circuits with several loops, extending Ohm's work:
Junction law: the total current flowing into a junction equals the total current flowing out.
Loop law: around any closed loop, the sum of the emfs equals the sum of the potential drops across the resistances.
His follow-up paper of 1847 dealt with solving the equations that arise for networks of currents. The junction law is conservation of charge; the loop law is conservation of energy.
Worked example: a two-loop circuit
Two branches join the same pair of junctions, top and bottom. The left branch has a 12 V cell and a 2 Ω resistor. The right branch has a 6 V cell and a 3 Ω resistor. A middle branch has a 6 Ω resistor. Both cells have their positive terminal towards the top junction.
Let I₁ flow up through the left branch, I₂ up through the right branch, and I₃ down through the middle.
Junction law at the top: I₃ = I₁ + I₂.
Left loop (left branch and middle): 12 − 2I₁ − 6I₃ = 0. Substituting I₃ gives 8I₁ + 6I₂ = 12.
Right loop (right branch and middle): 6 − 3I₂ − 6I₃ = 0, which gives 6I₁ + 9I₂ = 6.
Solve. Multiply the first equation by 3: 24I₁ + 18I₂ = 36. Multiply the second by 2: 12I₁ + 18I₂ = 12. Subtract: 12I₁ = 24, so I₁ = 2.0 A. Then 6(2) + 9I₂ = 6 gives I₂ = −0.67 A.
Interpret. I₃ = 2.0 − 0.67 = 1.33 A. The minus sign on I₂ means that current actually flows down through the 6 V cell, so the 12 V cell is charging it.
Check: the voltage across the middle resistor is 6 × 1.33 = 8.0 V. In the left branch, 12 − 2 × 2.0 = 8.0 V. In the right branch, 6 − 3 × (−0.67) = 8.0 V. All three agree.
Students lose marks here by changing their sign convention halfway round a loop, or by "correcting" a negative current instead of reading it as a direction. Our guide to Kirchhoff's laws and sign conventions covers these errors in detail. If you want a step-by-step check of your own circuit work, the Fermi Problem Library has a solver that explains the reasoning rather than printing an answer.
How did Kirchhoff and Bunsen start spectroscopy?
In Heidelberg the two men passed the light from heated substances in a flame through a prism. Their 1860 paper Chemische Analyse durch Spectralbeobachtungen (Chemical analysis by spectral observations) showed that each element gives its own pattern of bright lines, so the lines identify the element even from far away. A BibNum commentary on the paper calls it the founding article of spectral analysis, with uses from chemistry and mineral analysis to astronomy.
The method found new elements at once. The Royal Society of Chemistry credits Kirchhoff and Bunsen with discovering caesium in 1860, named from the Latin for sky blue after its flame colour, and rubidium in 1861, from the Latin for deepest red.
Why are there dark lines in sunlight?
Joseph von Fraunhofer had seen dark lines in the solar spectrum, at positions matching some bright flame lines. Kirchhoff explained them. In his 1860 paper on emission and absorption, he connected how strongly a body emits light of a given wavelength to how strongly it absorbs it. MacTutor summarises the consequence: a gas absorbs at the same wavelengths it emits. So when sunlight passes through cooler gas in the Sun's atmosphere, those elements remove their own wavelengths, leaving dark lines. MacTutor notes that he introduced the term black body in 1862, an idea that became central to the later development of quantum theory.
What can a student take from how he worked?
Start early and write it up. His circuit laws came from a seminar project while he was a student.
Turn rules into equations. The junction and loop laws reduce any network to simultaneous equations.
Work with people whose skills differ from yours. Kirchhoff the theorist and Bunsen the chemist founded a field together.
Look for one principle behind two effects. Bright lines in a flame and dark lines in sunlight are the same physics seen from two sides.
Kirchhoff built on Georg Ohm's law, and his black body led to Max Planck's quantum.
Try it: change the middle resistor in the worked example to 3 Ω and find all three currents again.
