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V = IR is the first circuit equation most students learn, and it seems too simple to have caused trouble. But when Georg Ohm published it, much of the German scientific world ignored or attacked it, and he spent years without the university post he wanted. This post covers what Ohm actually found, why it was resisted, a worked example in the form he used, and what a student can take from his way of working.
Who was Georg Ohm?
The Neue Deutsche Biographie, the standard German national biography published by the Bavarian Academy of Sciences, gives his dates: born 16 March 1789 in Erlangen, died 6 July 1854 in Munich. His father, a locksmith, taught him mathematics and physics. He began university in Erlangen in 1806 but left the same year to teach mathematics in Switzerland, returning to take his doctorate in 1811.
In 1817 he became a senior teacher of mathematics and physics at the Jesuit Gymnasium in Cologne. The NDB notes that the school's well-equipped physics collection drew him further into physics. From 1825 he studied the then puzzling behaviour of galvanic currents, looking for a quantitative link between what we now call resistance, voltage and current.
How did he find the law?
The batteries of the time were the problem. According to the NDB, Ohm was at first misled by their current, which was not steady. In 1826 he switched to a thermocouple, where a temperature difference drives current through an all-metal circuit, giving a much steadier source. With it he found the relationship later called Ohm's law: in modern terms, voltage equals current times resistance.
The MacTutor biography from the University of St Andrews adds that in two papers in 1826 he modelled conduction in circuits on Joseph Fourier's theory of heat flow, and that the complete theory appeared in his 1827 book Die galvanische Kette, mathematisch bearbeitet (The galvanic circuit investigated mathematically). The NDB says he also used the law to explain effects such as the influence of measuring instruments on a circuit, and that he stated the rules for branching currents now named after Gustav Kirchhoff.
Why was the law rejected at first?
The NDB is direct: Ohm's work was at first not recognised, sometimes sharply criticised or misunderstood, partly because the link between experimental physics and mathematics was still barely developed in Germany. Few saw its importance. MacTutor notes that the book opened with the mathematics needed to follow it, because even leading German physicists of the time were not used to a mathematical approach.
The cost to Ohm was personal. He took leave in 1826 to work in Berlin, hoping his results would earn him a university post. In 1828 he gave up his teaching post altogether and wrote to many institutions, in 1829 even to King Ludwig I of Bavaria. Munich's assessors praised the attempt to apply mathematics to physics but did not offer him a chair. He got a permanent post only in 1833, at the Polytechnic School in Nuremberg.
Support came slowly. The NDB records that Gustav Fechner confirmed the law in his own experiments in 1831, and that Carl Friedrich Gauss and Wilhelm Weber tested it in Göttingen in 1833. The Royal Society in London awarded Ohm the Copley Medal in 1841, and in 1849 Bavaria finally made him a professor at the University of Munich, five years before his death.
Worked example: Ohm's law with a real source
Ohm's own treatment included the resistance of the source as part of the circuit. In modern form: I = E/(R + r), where E is the emf, R the external resistance and r the internal resistance.
Take a cell with E = 2.0 V and r = 0.50 Ω connected to R = 1.5 Ω.
Total resistance: 1.5 + 0.50 = 2.0 Ω.
Current: I = 2.0/2.0 = 1.0 A.
Terminal voltage: V = IR = 1.0 × 1.5 = 1.5 V. The other 0.50 V is dropped inside the cell.
The instrument effect Ohm noticed. Now put an ammeter with resistance 0.50 Ω in the circuit to measure that current. The total becomes 2.5 Ω and I = 2.0/2.5 = 0.80 A. The meter has changed the very current it was meant to measure. This is why ideal ammeters are taken to have zero resistance and ideal voltmeters infinite resistance.
Changing the wire. Resistance is proportional to length; Ohm's early experiments varied the length of wire in the circuit. Double the external wire to 3.0 Ω and I = 2.0/(3.0 + 0.50) ≈ 0.57 A, not 0.50 A, because r stays the same.
Mistakes to avoid:
Forgetting internal resistance when a question gives r.
Confusing emf with terminal voltage. They are equal only when no current flows.
Assuming every device obeys V = IR. Ohm's law holds for ohmic conductors at constant temperature; diodes and filament lamps do not follow it.
If you keep mixing up which resistances to add, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For multi-loop circuits, see our guide to Kirchhoff's laws and sign conventions.
What can a student take from how he worked?
Fix the source of noise first. His breakthrough came from swapping an unsteady battery for a steady thermocouple.
Borrow ideas across topics. He modelled electric current on heat flow.
Show the working. He opened his book with the mathematics his readers would need.
Keep going when recognition is slow. It took him from 1827 to 1849 to get the post he wanted, but the law outlived its critics.
His branching-current rules were taken further by Gustav Kirchhoff, and the cell he relied on began with Volta's battery.
Try it: with the same cell, find the external resistance that makes the terminal voltage exactly 1.0 V.
