The introductory discussion

On Physical Lines of Force → Part I → The introductory discussion

Part I opens with several pages of untitled argument before the first Proposition. This is the most-read passage Maxwell ever wrote, and almost all of it can be given here in his own words.

What a line of force is

IN all phenomena involving attractions or repulsions, or any forces depending on the relative position of bodies, we have to determine the magnitude and direction of the force which would act on a given body, if placed in a given position.
In the case of a body acted on by the gravitation of a sphere, this force is inversely as the square of the distance, and in a straight line to the centre of the sphere. In the case of two attracting spheres, or of a body not spherical, the magnitude and direction of the force vary according to more complicated laws. In electric and magnetic phenomena, the magnitude and direction of the resultant force at any point is the main subject of investigation. Suppose that the direction of the force at any point is known, then, if we draw a line so that in every part of its course it coincides in direction with the force at that point, this line may be called a line of force, since it indicates the direction of the force in every part of its course.
By drawing a sufficient number of lines of force, we may indicate the direction of the force in every part of the space in which it acts.

That is a definition, and a purely geometrical one. Nothing physical has been claimed yet. The claim comes next, and it comes from an experiment anyone can do on a table.

The iron filings, and the dissatisfaction

Thus if we strew iron filings on paper near a magnet, each filing will be magnetized by induction, and the consecutive filings will unite by their opposite poles, so as to form fibres, and these fibres will indicate the direction of the lines of force. The beautiful illustration of the presence of magnetic force afforded by this experiment, naturally tends to make us think of the lines of force as something real, and as indicating something more than the mere resultant of two forces, whose seat of action is at a distance, and which do not exist there at all until a magnet is placed in that part of the field. We are dissatisfied with the explanation founded on the hypothesis of attractive and repellent forces directed towards the magnetic poles, even though we may have satisfied ourselves that the phenomenon is in strict accordance with that hypothesis, and we cannot help thinking that in every place where we find these lines of force, some physical state or action must exist in sufficient energy to produce the actual phenomena.

Note what this paragraph does not say. It does not say the action-at-a-distance account is wrong; Maxwell grants that it fits the facts exactly. It says the account leaves him dissatisfied, and that the dissatisfaction is itself worth acting on. That is an unusually honest statement of scientific motive, and it is the hinge of the paper.

The object of the paper

My object in this paper is to clear the way for speculation in this direction, by investigating the mechanical results of certain states of tension and motion in a medium, and comparing these with the observed phenomena of magnetism and electricity. By pointing out the mechanical consequences of such hypotheses, I hope to be of some use to those who consider the phenomena as due to the action of a medium, but are in doubt as to the relation of this hypothesis to the experimental laws already established, which have generally been expressed in the language of other hypotheses.

And on his own earlier work:

I have in a former paper endeavoured to lay before the mind of the geometer a clear conception of the relation of the lines of force to the space in which they are traced. By making use of the conception of currents in a fluid, I showed how to draw lines of force, which should indicate by their number the amount of force, so that each line may be called a unit-line of force (see Faraday's 'Researches,' 3122); and I have investigated the path of the lines where they pass from one medium to another.
In the same paper I have found the geometrical significance of the "Electrotonic State," and have shown how to deduce the mathematical relations between the electrotonic state, magnetism, electric currents, and the electromotive force, using mechanical illustrations to assist the imagination, but not to account for the phenomena.

The last clause is the one to hold on to: to assist the imagination, but not to account for the phenomena. In 1856 the fluid was an aid to thought. In 1861 Maxwell is going to try for an account.

I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed. If, by the same hypothesis, we can connect the phenomena of magnetic attraction with electromagnetic phenomena and with those of induced currents, we shall have found a theory which, if not true, can only be proved to be erroneous by experiments which will greatly enlarge our knowledge of this part of physics.

A theory which, if not true, can only be proved erroneous by experiments that would greatly enlarge our knowledge. He is pre-conceding that the model may be false and arguing that it would be useful anyway.

The survey of rival hypotheses

The mechanical conditions of a medium under magnetic influence have been variously conceived of, as currents, undulations, or states of displacement or strain, or of pressure or stress.

Each is taken in turn, and each is granted what it gets right before it is rejected.

Currents.

Currents, issuing from the north pole and entering the south pole of a magnet, or circulating round an electric current, have the advantage of representing correctly the geometrical arrangement of the lines of force, if we could account on mechanical principles for the phenomena of attraction, or for the currents themselves, or explain their continued existence.

Undulations.

Undulations issuing from a centre would, according to the calculations of Professor Challis, produce an effect similar to attraction in the direction of the centre; but admitting this to be true, we know that two series of undulations traversing the same space do not combine into one resultant as two attractions do, but produce an effect depending on relations of phase as well as intensity, and if allowed to proceed, they diverge from each other without any mutual action. In fact the mathematical laws of attractions are not analogous in any respect to those of undulations, while they have remarkable analogies with those of currents, of the conduction of heat and electricity, and of elastic bodies.

This is the decisive move, and it is made on purely formal grounds: attractions superpose linearly and waves do not. The mathematics chooses the mechanism.

Elastic displacement.

In the Cambridge and Dublin Mathematical Journal for January 1847, Professor William Thomson has given a "Mechanical Representation of Electric, Magnetic, and Galvanic Forces," by means of the displacements of the particles of an elastic solid in a state of strain. In this representation we must make the angular displacement at every point of the solid proportional to the magnetic force at the corresponding point of the magnetic field, the direction of the axis of rotation of the displacement corresponding to the direction of the magnetic force. The absolute displacement of any particle will then correspond in magnitude and direction to that which I have identified with the electrotonic state; and the relative displacement of any particle, considered with reference to the particle in its immediate neighbourhood, will correspond in magnitude and direction to the quantity of electric current passing through the corresponding point of the magneto-electric field. The author of this method of representation does not attempt to explain the origin of the observed forces by the effects due to these strains in the elastic solid, but makes use of the mathematical analogies of the two problems to assist the imagination in the study of both.

Thomson's construction is the closest thing to Maxwell's own, and Maxwell says so — while noting that Thomson too declined to claim it was real.

The turn to stress

We come now to consider the magnetic influence as existing in the form of some kind of pressure or tension, or, more generally, of stress in the medium.
Stress is action and reaction between the consecutive parts of a body, and consists in general of pressures or tensions different in different directions at the same point of the medium.
The necessary relations among these forces have been investigated by mathematicians; and it has been shown that the most general type of a stress consists of a combination of three principal pressures or tensions, in directions at right angles to each other.

The hypothesis of molecular vortices

Here is the sentence the whole paper turns on.

Let us now suppose that the phenomena of magnetism depend on the existence of a tension in the direction of the lines of force, combined with a hydrostatic pressure; or in other words, a pressure greater in the equatorial than in the axial direction: the next question is, what mechanical explanation can we give of this inequality of pressures in a fluid or mobile medium? The explanation which most readily occurs to the mind is that the excess of pressure in the equatorial direction arises from the centrifugal force of vortices or eddies in the medium having their axes in directions parallel to the lines of force.

Not the explanation is, but the explanation which most readily occurs to the mind is. Maxwell introduces the central hypothesis of his paper in the most tentative construction available to him.

There follows the argument for why vortices are the right shape of idea:

This explanation of the cause of the inequality of pressures at once suggests the means of representing the dipolar character of the line of force. Every vortex is essentially dipolar, the two extremities of its axis being distinguished by the direction of its revolution as observed from those points.

A line of force has a north end and a south end. So does a spinning top: look at it from one end and it turns clockwise, from the other, anticlockwise. Nothing else in the mechanical repertoire has that property so naturally.

We also know that when electricity circulates in a conductor, it produces lines of magnetic force passing through the circuit, the direction of the lines depending on the direction of the circulation. Let us suppose that the direction of revolution of our vortices is that in which vitreous electricity must revolve in order to produce lines of force whose direction within the circuit is the same as that of the given lines of force.
We shall suppose at present that all the vortices in any one part of the field are revolving in the same direction about axes nearly parallel, but that in passing from one part of the field to another, the direction of the axes, the velocity of rotation, and the density of the substance of the vortices are subject to change. We shall investigate the resultant mechanical effect upon an element of the medium, and from the mathematical expression of this resultant we shall deduce the physical character of its different component parts.

The pressures in a single vortex

In a circular vortex, revolving with uniform angular velocity, if the pressure at the axis is $p$ , that at the circumference will be $p + ρ v^{2}$ , where $ρ$ is the density and $v$ the velocity at the circumference. The mean pressure parallel to the axis will be $p + \frac{1}{2} ρ v^{2}$ .
If a number of such vortices were placed together side by side with their axes parallel, they would form a medium in which there would be a pressure parallel to the axes, and a pressure in any perpendicular direction. A medium of this kind, filled with molecular vortices having their axes parallel, differs from an ordinary fluid in having different pressures in different directions. If not prevented by properly arranged pressures, it would tend to expand laterally.

The symbols inside this quotation are set in modern type; in the printed paper they appear in Maxwell's own notation. See Sources.

Parallel vortices make an anisotropic medium

The difference between the two pressures — along the axis and across it — is $\frac{1}{2} ρ v^{2}$ for a uniform vortex, and some numerical multiple of $ρ v^{2}$ in general, the multiplier depending on how the angular velocity and density are distributed within the vortex. That difference is the entire magnetic stress. Everything else in Part I is bookkeeping applied to it.

Two further sentences from this stretch of the paper are widely quoted, and both are almost certainly Maxwell's, though this edition could not place them in continuous context:

that the stress along the axis of a line of magnetic force is a tension, like that of a rope;
that the lines of force indicate the direction of minimum pressure at every point of the medium.

The second of these is the compact statement of the whole of Part I, and Maxwell repeats it at the opening of Part II as his settled answer to the first of his questions.

Next: Prop. I · Back to Part I

Plain-language versions: What is a line of force? · Why Maxwell wanted a mechanism