On Physical Lines of Force

On Physical Lines of Force: map
Reading the edition Start here Notation and concordance Glossary Index of numbered equations Figures
Reading by level The story: Introduction Plain language: the explainers The modern equations, derived
Maxwell's paper Part I Part II Part III Part IV Sources and editorial method Chronology
Related Hamilton's principle Noether's theorems

A wiki edition of James Clerk Maxwell's paper of 1861–62, with modern derivations and plain-language explainers.

Lines of force about a bar magnet

Between March 1861 and February 1862 Maxwell published four instalments in the Philosophical Magazine under one title. He set out to answer a question nobody had answered: not what the magnetic force does, which was already known with precision, but what is going on in the space where it acts. His answer was a machine — a space packed with spinning cells, separated by little rolling particles. The machine was wrong. On the way to building it he wrote down the displacement current, derived the speed of light from two electrical measurements, and founded field theory.

This wiki reproduces the argument in Maxwell's own order and, so far as possible, in his own voice; derives the modern equations from his results; and explains each major idea for a general reader.

Start here

If you want …	Go to
the story, and why the paper matters	Introduction
a plain-language tour, no mathematics	The explainers
Maxwell's argument, part by part	Part I → II → III → IV
the modern equations, derived	From 1861 to today
help with the symbols	Notation and concordance
what is verbatim and what is not	Sources and editorial method

The four parts

Part I — The Theory of Molecular Vortices applied to Magnetic Phenomena

Phil. Mag., Series 4, Vol. XXI, No. 139 (March 1861), pp. 161–175.

Magnetic force is a state of stress in a medium: a tension along the lines of force and a pressure across them. Maxwell shows that spinning vortices, their axes lying along the lines, produce exactly that stress — and that its variation from point to point reproduces every known magnetic and electromagnetic force at once.

• The introductory discussion — what a line of force is, and why a mechanism is wanted
• Prop. I — the pressure of a vortex does not depend on its size
• Prop. II — the stress resolved on the coordinate planes
• Prop. III — the force on an element of the medium
• Applications — magnetic poles, a straight current, two currents

Part II — The Theory of Molecular Vortices applied to Electric Currents

Phil. Mag., Series 4, Vol. XXI, Nos. 140–141 (April and May 1861), pp. 281–291, 338–348.

Neighbouring vortices turning the same way would grind against each other. Maxwell interposes a layer of particles that roll between them like the idle wheels of a gear train — and identifies those particles with electricity. The current equation is then derived rather than assumed, and induction follows from the inertia of the vortices.

• The mechanism — the difficulty, and the idle wheel
• Prop. IV — the motion of the layer between two vortices
• Prop. V — the transference of particles is the current
• Prop. VI — the energy of the vortices
• Prop. VII — the work done on a vortex by the particles
• Prop. VIII — electromotive force from changing motion
• Props. IX and X — a deformed and rotated element
• Prop. XI — electromotive force in a moving body
• Induced currents — and the anchor-ring experiment

Part III — The Theory of Molecular Vortices applied to Statical Electricity

Phil. Mag., Series 4, Vol. XXIII, No. 151 (January 1862), pp. 12–24.

The substance of the cells is elastic. Force applied to the particles distorts each cell and is resisted; the particles shift a little and stop. That shift is the electric displacement — and its variation is a current. Adding it to the equations of currents gives the displacement current. Then Maxwell computes the speed of transverse waves in his medium, and finds the velocity of light.

• Conduction and induction — two independent qualities
• Prop. XII — the equilibrium of an elastic sphere
• Prop. XIII — electromotive force and displacement
• Prop. XIV — the displacement current
• Prop. XV — the force between two electrified bodies
• Prop. XVI — the velocity of transverse undulations
• Prop. XVII — the capacity of a Leyden jar

Part IV — The Theory of Molecular Vortices applied to the Action of Magnetism on Polarized Light

Phil. Mag., Series 4, Vol. XXIII, No. 152 (February 1862), pp. 85–95.

Faraday had shown that magnetism turns the plane of polarized light. Maxwell argues that this is the decisive evidence that magnetism is a phenomenon of rotation, and works out what his spinning cells predict.

• Why magnetism must be rotatory
• The six results, (A) to (F)
• What became of the vortices

Reference

• Index of numbered equations
• Notation and concordance
• Figures
• Glossary
• Chronology, 1820–1873
• Sources and editorial method

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This wiki was built as a reading edition. It is not a critical edition: see Sources and editorial method for exactly which text is verbatim and which is not.