Models of the Aether

In order to represent the magnetic field by a conservative dynamical system, we shall suppose that it is produced by a number of rings of perfectly conducting material, in which electric currents are circulating.

• The surrounding medium is free aether.

Any perfectly conducting body acts as an impenetrable barrier to lines of magnetic force.

For, as Maxwell showed,[13] when a perfect conductor is placed in a magnetic field, electric currents are induced on its surface in such a way as to make the total magnetic force zero throughout the interior of the conductor.[14]

Lines of force are thus deflected by the body in the same way as the lines of flow of an incompressible fluid would be deflected by an obstacle of the same form, or as the lines of flow of electric current in a uniform conducting mass would be deflected by the introduction of a body of this form and of infinite resistance.

If, then, for simplicity we consider two perfectly conducting rings carrying currents, those lines of force which are initially linked with a ring cannot escape from their entanglement, and new lines cannot become involved in it.

This implies that the total number of lines of magnetic force which pass through the aperture of each ring is invariable.

If the coefficients of self and mutual induction of the rings are denoted by L1, L2, L12, the electrokinetic energy of the system may be represented by

…

where i1, i2, in denote the strengths of the currents.

The condition that the number of lines of force linked with each circuit is to be invariable gives the equations

…

When the system is considered from the point of view of general dynamics, the electric currents must be regarded as generalized velocities, and the quantities

…

as momenta.

The electromagnetic ponderomotive force on the rings tending to increase any coordinate x is ∂T/∂x. In the analogous hydrodynamical system, the fluid velocity corresponds to the magnetic force.

Therefore the circulation through each ring (which is defined to be the integral ∫vds, taken round a path linked once with the ring) corresponds kinematically to the electric current.

The flux of fluid through each ring corresponds to the number of lines of magnetic force which pass through the aperture of the ring.

But in the hydrodynamical problem the circulations play the part of generalized momenta; while the fluxes of fluid through the rings play the part of generalized velocities.

The kinetic energy may indeed be expressed in the form

…

where κ1, κ2, denote the circulations (so that κ1 and κ2 are proportional respectively to i1, and i2), and N1, N12, N2, depend on the positions of the rings.

But this is the Hamiltonian (as opposed to the Lagrangian) form of the energy-function,[15] and the ponderomotive force on the rings tending to increase any coordinate x is - ∂K/∂x.

Since ∂K/∂x is equal to ∂T/∂x, the ponderomotive forces on the rings in any position in the hydrodynamical system are equal, but opposite, to the ponderomotive forces on the rings in the electric system.

The reason for the difference between the 2 cases is that the rings cannot cut through the lines of magnetic force in the one system.

• But they can cut through the stream-lines in the other.

Consequently, the flux of fluid through the rings is not invariable when the rings are moved, the invariants in the hydrodynamical system being the circulations.

If a thin ring, for which the circulation is zero, is introduced into the fluid, it will experience no ponderomotive forces.

But if a ring initially carrying no current introduced into a magnetic field, it will experience ponderomotive forces, owing to the electric currents induced in it by its motion.

Imperfect though the analogy is, it is not without interest.

A bar-magnet, being equivalent to a current circulating in a wire wound round it, may be compared (as W. Thomson remarked) to a straight tube immersed in a perfect fluid, the fluid entering at one end and flowing out by the other, so that the particles of fluid follow the lines of magnetic force.

If 2 such tubes are presented with like ends to each other, they attract; with unlike ends, they repel. The forces are thus diametrically opposite in direction to those of magnets; but in other respects the laws of mutual action between these tubes and between magnets are precisely the same.[16]

Thomson, moreover, investigated[17] the ponderomotive forces which act between two solid bodies immersed in a fluid, when one of the bodies is constrained to perform small oscillations.

If, for example, a small sphere immersed in an incompressible fluid is compelled to oscillate along the line which joins its centre to that of a much larger sphere, which is free, the free sphere will be attracted if it is denser than the fluid;

while if it is less dense than the fluid, it will be repelled or attracted according as the ratio of its distance from the vibrator to its radius is greater or less than a certain quantity depending on the ratio of its density to the density of the fluid.

Systems of this kind were afterwards extensively investigated by C. A. Bjerknes.

Bjerknes showed that 2 pulsating spheres (i.e. changing in volume) immersed in an incompressible fluid exert on each other (by the mediation of the fluid):

• an attraction that follows the inverse square law, if the pulsations are concordant
• a repulsion if the phases of the pulsations differ by half a period

The medium is supposed incompressible so that all pulsations are propagated instantaneously.

Otherwise, attractions would change to repulsions and vice versa at distances greater than a quarter wave-length.

If the spheres, instead of pulsating, oscillate to and fro in straight lines about their mean positions, the forces between them are proportional in magnitude and the same in direction, but

..

opposite in sign, to those which act between two magnets oriented along the directions of oscillation.[20]

The results obtained by Bjerknes were extended by A. H. Leahy to the case of 2 spheres pulsating in an clastic medium.

The wavelength of the disturbance is supposed large compared with the distance between the spheres.

For this system, Bjerknes’ results are reversed. The law is now that of:

• attraction in the case of unlike phases
• repulsion in the case of like phases

The intensity is as before proportional to the inverse square of the distance.

He discussed the oscillations which may be produced in an elastic medium by the displacement, in the direction of the tangent to the crosssection, of the surfaces of tubes of small sectional area.

The tubes either forming closed curves, or extending indefinitely in both directions.

The direction and circumstances of the motion are in general analogous to ordinary vortex-motions in an incompressible fluid.

Leahy showed that if the period of the oscillation be such that the waves produced are long compared with ordinary finite distances, the displacement due to the tangential disturbances is proportional to the velocity due to vortex-rings of the same form as the tubular surfaces.

One of these “oscillatory twists,” as the tubular surfaces may be called, produces a displacement which is analogous to the magnetic force due to a current flowing in a curve coincident with the tube.

The strength of the current being proportional to b2ω sin pt, where b denotes the radius of the twist, and ω sin pt its angular displacement.

If the field of vibration is explored by a rectilineal twist of the same period as that of the vibration, the twist will experience a force at right angles to the plane containing the twist and the direction of the displacement which would exist if the twist were removed.

If the displacement of the medium be represented by F sin pt, and the angular displacement of the twist by ω sin pt, the magnitude of the force is proportional to the vector-product of F (in the direction of the displacement) and ω (in the direction of the axis of the twist).

A model of magnetic action may evidently be constructed on the basis of these results. A bar-magnet must be regarded as vibrating tangentially, the direction of vibration being parallel to the axis of the body.

A cylindrical body carrying a current will have its surface also vibrating tangentially; but in this case the direction of vibration will be perpendicular to the axis of the cylinder.

A statically electrified body, on the other hand, may, as follows from the same author’s earlier work, be regarded as analogous to a body whose surface vibrates in the normal direction.