Larmor's Theory

We have now discussed models in which the magnetic force is represented as the velocity in a liquid, and others in which it is represented as the displacement in an elastic solid.

Some years before the date of Leahy’s memoir, George Francis FitzGerald (b. 1851, d. 1901)[23] had instituted a comparison between magnetic force and the velocity in a quasi-elastic solid of the type first devised by MacCullagh.[24]

An analogy is at once evident when it is noticed that the electromagnetic equation

is satisfied identically by the values

where e denotes, any vector; and that, on substituting these values in the other electromagnetic equation,

we obtain the equation

which is no other than the equation of motion of MacCullagh’s aether,[25] the specific inductive capacity e corresponding to the reciprocal of MacCullagh’s constant of elasticity.

In that analogy:

• electric displacement corresponds to the twist of the elements of volume of the aether
• electric charge is an intrinsic rotational strain.

Mechanical models of the electromagnetic field, based on FitzGerald’s analogy, were afterwards studied by:

• A. Sommerfeld[26]
• R. Reiff[27]
• Sir J. Larmor.[28]

Larmor[29] supposed the electric charge to exist in the form of discrete electrons, for the creation of which he suggested the following ideal process[30]:

A filament of aether, terminating at two nuclei, is supposed to be removed, and circulatory motion is imparted to the walls of the channel so formed, at each point of its length, so as to produce throughout the medium a rotational strain.

When this has been accomplished, the channel is to be filled up again with aether, which is to be made continuous with its walls. When the constraint is removed from the walls of the channel, the circulation imposed on them proceeds to undo itself, until this tendency is balanced by the elastic resistance of the aether with which the channel has been filled up; thus finally the system assumes a state of equilibrium in which the nuclei, which correspond to a positive and a negative electron, are surrounded by intrinsic rotational strain.

Models in which magnetic force is represented by the velocity of an aether are not, however, secure from objection, It is necessary to suppose that the aether is capable of lowing like a perfect fluid in irrotational motion (which would correspond to a steady magnetic field), and that it is at the same time endowed with the power (which is requisite for the explanation of electric phenomena) of resisting the rotation of any element of volume.[31]

But when the aether moves irrotationally in the fashion which corresponds to a steady magnetic field, each element of volume acquires after a finite time a rotatory displacement from its original orientation, in consequence of the motion, and it might therefore be expected that the quasi-elastic power of resisting rotation would be called into play—i.e., that a steady magnetic field would develop electric phenomena.[32]

A further objection to all models in which magnetic force corresponds to velocity is that a strong magnetic field, being in such models represented by a steady drift of the aether, might be expected to influence the velocity of propagation of light, The existence of such an effect appears, however, to be disproved by the experiments of Sir Oliver Lodge[33].

At any rate, unless it is assumed that the aether has an inertia at least of the same order of magnitude as that of ponderable matter, in which case the motion might be too slow to be measurable.

Again, the evidence in favour of the rotatory as opposed to the linear character of magnetic phenomena has perhaps, on the whole, been strengthened since Thomson originally based his conclusion on the magnetic rotation of light. This brings us to the consideration of an experimental discovery.

In 1879 E. H. Hall,[34] at that time a student at Baltimore, repeating an experiment which had been previously suggested by H. A. Rowland, obtained a new action of a magnetic field on electric currents.

A strip of gold leaf mounted on glass, forming part of an electric circuit through which a current was passing, was placed between the poles of an electromagnet, the plane of the strip being perpendicular to the lines of magnetic force. The two poles of a sensitive galvanometer were then placed in connexion with different parts of the strip, until two points at the same potential were found.

When the magnetic field was created or destroyed, a deflection of the galvanometer needle was observed, indicating a change in the relative potential of the two poles. It was thus shown that the magnetic field produces in the strip of gold leaf a new electromotive force, at right angles to the primary electromotive force and to the magnetic force, and proportional to the product of these forces.

From the physical point of view we may therefore regard Hall’s effect as an additional electromotive force generated by the action of the magnetic field on the current; or alternatively we may regard it as a modification of the ohmic resistance of the metal, such as would be produced if the molecules of the metal assumed a helicoidal structure about the lines of magnetic force. From the latter point of view, all that is needed is to modify Ohm’s law

(where S denotes electric current, k specific conductivity, and E electric force) so that it takes the form

where H denotes the imposed magnetic force, and h denotes a constant on which the magnitude of Hall’s phenomenon depends. It is a curious circumstance that the occurrence, in the case of magnetized bodies, of an additional term in Ohm’s law, formed from a vector-product of E, had been expressly suggested in Maxwell’s Treatise[35]: although Maxwell had not indicated the possibility of realizing it by Hall’s experiment.

An interesting application of Hall’s discovery was made in the same year by Boltzmann,[36] who remarked that it offered a prospect of determining the absolute velocity of the electric charges which carry the current the strip. For if it is supposed that only one kind (vitreous or resinous) of electricity is in motion, the force on one of the charges tending to drive it to ono side of the strip will be proportional to the vector-product of its velocity and the magnetic intensity.

Assuming that Hall’s phenomenon is a consequence of this tendency of charges to move to one side of the strip, it is evident that the velocity in question must be proportional to the magnitude of the Hall electromotive force due to a unit magnetic field. On the basis of this reasoning, 1. von Ettingshausen[37] found for the current sent by one or two Daniell’s cells through a gold strip a velocity of the order of 0·1 cm. per second.

If the current consists of both vitreous and resinous charges in motion in opposite directions, Boltzmann’s argument fails; for the two kinds of electricity would give opposite directions to the current in Hall’s phenomenon.

In the year following his discovery, Hall[38] extended his researches in another direction, by investigating whether a magnetic field disturbs the distribution of equipotential lines in a dielectric which is in an electric field; but no effect could be observed.[39]

Such an effect[40] was not to be expected on theoretical grounds. For when, in a material system, all the velocities are reversed, the motion is reversed, it being understood that, in the application of this theorem to electrical theory, an electrostatic state is to be regarded as one of rest, and a current as a phenomenon of motion, and if such a reversal be performed in the present system, the poles of the electromagnet are exchanged, while in the dielectric no change takes place.

We must now consider the bearing of Hall’s effect on the question as to whether magnetism is a rotatory or a linear phenomenon.[41]

If magnetism is linear, electric currents must be rotatory, and if Hall’s phenomenon be supposed to take place in a horizontal strip of metal, the magnetic force being directed vertically upwards, and the primary current flowing horizontally from north to south, the only geometrical entities involved are the vertical direction and a rotation in the east-and-west vertical plane; and these are indifferent with respect to a rotation in the north-and-south vertical plane, so that there is nothing in the physical circumstances of the system to determine in which direction the secondary current shall flow, The hypothesis that magnetism is linear appears therefore to be inconsistent with the existence of Hall’s effect.[42]

There are, however, some considerations which may be urged on the other side. Hall’s effect, like the magnetic rotation of light, takes place only in ponderable bodies, not in free aether; and its direction is sometimes in one sense, sometimes in the other, according to the nature of the substance.

It may therefore be doubted whether these phenomena are not of a secondary character, and the argument based on them invalid. Moreover, as FitzGerald remarked,[43] the magnetic lines of force associated with a system of currents are circuital and have no open ends, making it difficult to imagine how alteration of rotation inside them could be produced.

Various attempts were made to represent electric and magnetic phenomena by the motions and strains of a continuous medium.

• None of those hitherto considered has been found free from objection.[44]