Physics 1827

The Galvanic Circuit Investigated Mathematically

Georg Simon Ohm

Push divided by resistance gives flow — the one ratio that made electric circuits calculable.

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In depth · the introduction

Push harder and more electricity flows; make the path harder and less does. Ohm pinned that down as a single, exact ratio.

The big idea

Electricity in a wire behaves a lot like water in a pipe. There is a push that drives it — the voltage. There is the flow itself — the current. And there is how much the wire fights the flow — the resistance. Ohm showed that these three lock together in one tidy rule: the current equals the voltage divided by the resistance.

That single sentence is enormously useful. Double the push and you double the flow. Double the resistance and you halve the flow. Knowing any two of the three lets you work out the third — so for the first time you could predict, before you switched anything on, exactly how much current a circuit would carry.

How it came about

In the 1820s Georg Ohm was a schoolteacher in Cologne with a modest home laboratory. He had read Fourier's brand-new theory of how heat flows through a metal bar, and he suspected electricity flowed by the same kind of rule. To test it he needed a steady source — batteries of the day drifted — so he used a thermocouple, two joined metals kept at a fixed temperature difference, and measured how the current fell as he swapped in longer and longer wires.

The numbers were beautifully regular, and in 1827 he published the theory behind them. Then almost nothing happened. German physicists ignored or scorned the work; Ohm, disappointed, gave up his teaching post and waited. Recognition finally arrived from Britain — the Royal Society's Copley Medal in 1841 — and long after, his name was given to the very unit of electrical resistance.

Why it mattered

Before Ohm, electricity was a qualitative wonder — sparks, shocks, twitching frog legs. After Ohm, it was something you could compute and therefore design. His ratio is the seed of electrical engineering: every circuit you own, from a phone charger that won't fry your phone to the heating coil in a kettle, is built on V = I R.

A way to picture it

Think of water in a hose. Turn the tap higher and the pressure rises — that is voltage. More water rushes out — that is current. Now kink the hose, or swap in a longer, thinner one: the same pressure pushes far less water through. That extra struggle is resistance. Ohm's law is just the exact bookkeeping of this trade-off: flow equals pressure divided by struggle.

Where it sits

Volta's battery (1800) had just given the world a steady current, and Ørsted, Ampère and Faraday were busy mapping the magnetism it produced. Ohm supplied the arithmetic underneath all of it. His law slots between Faraday's discovery of induction (see faraday-1831) and Maxwell's grand unification of electricity and magnetism (see maxwell-1865) — the quiet, quantitative bones on which the electrical age was built.

The original document

Original source text

G. S. Ohm · Die galvanische Kette, mathematisch bearbeitet · Berlin, 1827 (English: Scientific Memoirs, vol. 2, 1841)

Preface

I herewith present to the public a theory of galvanic electricity, as a special part of electrical science in general.

Ohm announces his aim: to deduce the behaviour of the galvanic circuit not from accumulated observation alone but theoretically, from a few principles secured by experiment — and to do so by the same mathematics Fourier had just used for the flow of heat in a conducting bar.

The three quantities

The theory rests on three measurable things defined at every cross-section of the circuit: the electroscopic force — the tension, what we now call electric potential; the magnitude of the current; and the resistance of the conductor, which Ohm represents as a "reduced length" growing with length and falling with cross-section and conductivity.

The law

From these he derives that in an unbranched circuit the current is the same at every cross-section and equals the total driving force divided by the total resistance. His own measurements — a thermoelectric source and test wires of varying length — had already given the empirical form X = a / (b + x), with b the fixed resistance of the apparatus and x the wire; the theory explained why. In modern symbols this is V = I R.

[ … ]

The complete memoir — the heat-flow analogy, the differential equations for the current, and the experimental tables that confirm them — is available in full at the source below.

Berlin · 1827