Weber's Theory

The necessity for induced currents may be inferred by general reasoning from the first principles of Weber’s theory.

When a circuit’s moves in the field due to currents, the speed of the vitreous charges in s is, owing to the motion of s, not equal and opposite to that of the resinous charges.

This creates a difference in the forces acting on the vitreous and resinous charges in s. Hence, the charges of opposite sign separate from each other and move in opposite directions.

It is assumed that positive and negative charges move with equal and opposite speeds relative to the matter of the conductor.

• There may be objections to this.
• But it is an integral part of Weber’s theory and cannot be excised from it.

Electric currents would exert forces on electrostatic charges at rest[10]:

• if this condition were not satisfied, and
• if the law of force were Weber’s

This may be seen by the following example.

Let a current flow in a closed circuit formed by arcs of two concentric circles and the portions of the radii connecting their extremities.

If Weber’s law were true, and if only one kind of electricity were in motion, the current would exert an electrostatic force on a charge placed at the centre of the circles.

The assumption [11] of opposite electricities moving with equal and opposite velocities in a circuit is almost inevitable in any theory of the type of Weber’s, so long as the mutual action of two charges is assumed to depend only on their relative (as opposed to their absolute) motion.

Weber’s law is not the only one of its kind.

An alternative was suggested by Bernhard Riemann (b.1826, d. 1866)[12] at Göttingen in 1861. He proposed as the electrokinetic energy of two electrons e(x, y, z) and e′(x′, y′, z′) the expression:

This differs from the corresponding expression given by Weber only in that the relative speed of the 2 electrons is substituted in place of the component of this velocity along the radius vector.

Eventually, the laws of Riemann and Weber were both abandoned in favour of a third alternative.

At the time, however, Weber’s discovery was a great advance. It greatly awakened mathematical physicists to a sense of the possibilities latent in the theory of electricity.

Beyond this, its influence was felt in general dynamics.

Weber’s electrokinetic energy resembled kinetic energy in some respects and potential energy in others. But it could not be precisely classified under either head. Its introduction helped break down the distinction which had hitherto subsisted between the two parts of the kinetic potential.

It prepared the way for the modern transformation-theory of dynamics.[13]

Weber’s work also stimulated the theory of gravitation.

Many generations of physicists had believed that gravitation is propagated by the action of a medium.

The dependence of the force on the distance between the attracting bodies seemed to suggest this idea. An instantaneous propagation would mean that there is a rigid connection between the bodies. This connection would give a force independent of the distance.

If the simple law of Newton is abandoned, there are many rival hypotheses to choose from.

Laplace made the first notable attempt.[14]

He supposed that gravity was produced by the impulsion on the attracted body of a “gravific fluid,” which flows with a definite velocity toward the centre of attraction——say, the sun.

If the attracted body or planet is in motion, the velocity of the fluid relative to it will be compounded of the absolute velocity of the fluid and the reversed velocity of the planet.

The force of gravity will act in the direction thus determined, its magnitude being unaltered by the planet’s motion.

This amounts to supposing that gravity is subject to an aberrational effect similar to that observed in the case of light.

It is easily seen that the modification thus introduced into Newton’s law may be represented by an additional perturbing force, directed along the tangent to the orbit in the opposite sense to the motion, and proportional to the planet’s velocity and to the inverse square of the distance from the sun.

By considering the influence of this force on the secular equation of the moon’s motion, Laplace found that the velocity of the gravific fluid must be at least a hundred million times greater than that of light.

The assumptions made by Laplace are evidently in the highest degree questionable; but the generation immediately succeeding, overawed by his fame, seems to have found no way of improving on them.

Under the influence of Weber’s ideas, however, astronomers began to think of modifying Newton’s law by adding a term involving the velocities of the bodies.

Tisserand[15] in 1872 discussed the motion of the planets round the sun on the supposition that the law of gravitation is the same as Weber’s law of electrodynamic action, so that the force is

where f denotes the constant of gravitation, m the mass of the planet, μ the mass of the sun, r the distance of the planet from the sun, and h the velocity of propagation of gravitation.

The equations of motion may be rigorously integrated by the aid of elliptic functions[16]; but the simplest procedure is

and, regarding F1 as a perturbing function, to find the variation of the constants of elliptic motion.

Tisserand showed that the perturbations of all the elements are zero or periodic, and quite insensible, except that of the longitude of perihelion, which has a secular part. If h be assumed equal to the velocity of light, the effect would be to rotate the major axis of the orbit of Mercury in the direct sense 14" in a century.

A discordance between theory and observation was known to exist in regard to the motion of Mercury’s perihelion; for Le Verrier had found that the attraction of the planets might be expected to turn the perihelion 527" in the direct sense in a century, whereas the motion actually observed was greater than this by 38". It is evident, however, that only

of the excess is explained by Tisserand’s adoption of Weber’s law; and it seemed therefore that this suggestion would prove as unprofitable as Le Verrier’s own hypothesis of an intra-mercurial planet. But it was found later[17] that

of the excess could be explained by substituting Riemann’s electrodynamic law for Weber’s, and that a combination of the laws of Riemann and Weber would give exactly the amount desired.[18]