The Mathematical Electricians Of The Mid-19th Century: Neumann

Faraday was engaged in discovering the laws of induced currents in his own way through the idea of lines of force.

His contemporary Franz Neumann attacked the problem differently. He used Ampère’s model.

In 1845, he published a memoir,[1] in which the laws of induction of currents were deduced by the help of Ampère’s analysis.

Among the assumptions on which Neumann based his work was a rule which had been formulated, not long after Faraday’s original discovery, by Ennil Lenz,[2] and which may be enunciated as follows: when a conducting circuit is moved in a magnetic field, the induced current flows in such a direction that the ponderomotive forces on it tend to oppose the motion,

Let ds denote an element of the circuit which is in motion, and let Cds denote the component, taken in the direction of motion, of the ponderomotive force exerted by the inducing current on ds, when the latter is carrying unit current; so that the value of C is known from Ampère’s theory. Then Lenz’s rule requires that the product of O into the strength of the induced current should be negative. Neumann assumed that this is because it consists of a negative coefficient multiplying the square of C; that is, he assumed the induced electro- motive force to be proportional to C. He further assumed it to be proportional to the velocity v of the motion, and thus obtained for the electromotive force induced in ds the expression

where ε denotes a constant coefficient. By aid of this formula, ​in the earlier part[3] of the memoir, he calculated the induced currents in various particular cases.

But having arrived at the formulae in this way, Neumann noticed[4] a peculiarity in them which suggested a totally different method of treating the subject. In fact, on examining the expression for the current induced in a circuit which is in motion in the field due to a magnet, it appeared that this induced current depends only on the alteration caused by the motion in the value of a certain function; and, moreover, that this function is no other than the potential of the ponderomotive forces which, according to Ampère’s theory, act between the circuit, supposed traversed by unit current, and the magnet.

Accordingly, Neumann now proposed to reconstruct his theory by taking this potential function as the foundation.

The nature of Neumann’s potential, and its connexion with Faraday’s theory, will be understood from the following considerations:—

The potential energy of a magnetic molecule M in a field of magnetic intensity B is (B.M); and therefore the potential energy of a current i flowing in a circuit’s in this field is

where S denotes a diaphragm bounded by the circuit s; as is seen at once on replacing the circuit by its equivalent magnetic shell S. If the field B be produced by a current i′ flowing in a circuit s′, we have, by the formula of Biot and Savart,

Hence, the mutual potential energy of the two currents is

which by Stokes’s transformation may be written in the form

This expression represents the amount of mechanical work which must be performed against the electro-dynamic ponderomotive forces, in order to separate the two circuits to an infinite distance apart, when the current-strengths are maintained unaltered.

The above potential function has been obtained by considering the ponderomotive forces; but it can now be connected with Faraday’s theory of induction of currents. interpreting the expression

in terms of lines of force, we see that the potential function represents the product of i into the number of unit-lines of magnetic force due to s′, which pass through the gap formed by the circuit s; and since by Faraday’s law the currents induced in s depend entirely on the variation in the number of these lines, it is evident that the potential function supplies all that is needed for the analytical treatment of the induced currents. This was Neumann’s discovery.

The electromotive force induced in a circuit s by the motion of other circuits s′, carrying currents i′, is thus proportional to the time-rate of variation of the potential

so that if we denote by a the vector

which, of course, is a function of the position of the element ds from which r is measured, then the electromotive force induced in any circuit-element ds by any alteration in the currents which give rise to a is

The induction of currents is therefore governed by the vector a; this, which is generally known as the vector-potential, has from Neumann’s time onwards played a great part in electrical theory. It may be readily interpreted in terms of Faraday’s conceptions; for (a.ds) represents the total number of unit lines of magnetic force which have passed across the line-element ds prior to the instant t. The vector-potential may in fact be regarded as the analytical measure of Faraday’s electrotonic state.[5]

While Neumann was endeavouring to comprehend the laws of induced currents in an extended form of Ampère’s theory, another investigator was attempting a still more ambitious project: 110 less than that of uniting electrodynamics into a coherent whole with electrostatics.

Wilhelm Weber (b. 1804, d. 1890) was in the earlier part of his scientific career a friend and colleague of Gauss at Göttingen. In 1837, however, he became involved in political trouble. The union of Hanover with the British Empire, which had subsisted since the accession of the Hanoverian dynasty to the British throne, was in that year dissolved by the operation of the Salic law; the Princess Victoria succeeded to the crown of England, and her uncle Ernest-Augustus to that of Hanover. The new king, who was a pronounced reactionary, revoked the free constitution which the Hanoverians had for some time enjoyed; and Weber, who took a prominent part in opposing this action, was deprived of his professorship. From 1843 to 1849, when his principal theoretical researches in electricity were made, he occupied a chair in the University of Leipzig.

The theory of Weber was in its origin closely connected with the work of another Leipzig Professor, Fechner, who in 1845[6] introduced certain assumptions regarding the nature of ​electric currents. Fechner supposed every current to consist in a streaming of electric charges, the vitreous charges travelling in one direction, and the resinous charges, equal to them in magnitude and number, travelling in the opposite direction with equal velocity. He further supposed that like charges attract each other when they are moving parallel to the same direction, while unlike charges attract when they are moving in opposite directions. On these assumptions he succeeded in bringing Faraday’s induction effects into connexion with Ampère’s laws of electrodynamics.

In 1846 Weber,[7] adopting the same assumptions as Fechner, analysed the phenomena in the following way:—

The formula of Ampère for the ponderomotive force between two elements ds, ds′ of currents i, i′, may be written

Suppose now that λ units of vitreous electricity are contained in unit length of the wire s, and are moving with velocity u; and that an equal quantity of resinous electricity is moving with velocity u in the opposite direction; so that

Let λ, u′, denote the corresponding quantities for the other current; and let the suffix 1, be taken to refer to the action between the positive charges in the two wires, the suffix 2, to the action between the positive charge in s and the negative charge in s′, the suffix 3, to the action between the negative charge in s and the positive charge in s′, and the suffix 4 to the action between the negative charges in the two wires. Then we have

By aid of these and the similar equations with the suffixes 2, 3, 4 the equation for the ponderomotive force may be transformed into the equation

But this is the equation which we should have obtained had we set out from the following assumptions: that the ponderomotive force between two current-elements is the resultant of the force between the positive charge in ds and the positive charge in ds′, of the force between the positive charge in ds and the negative charge in ds′, etc.; and that any two electrified particles of charges e and e′, whose distance apart is r, repel each other with a force of magnitude

Two such charges would, of course, also cxert on each other an electrostatic repulsion, whose magnitude in these units would be ee′c2/r2, where c denotes a constant[8] of the dimensions of a velocity, whose value is approximately 3 x 1010 cm./sec. So that on these assumptions the total repellent force would be

This expression for the force between two electric charges was taken by Weber as the basis of his theory. Weber’s is the first of the electron-theories—a name given to any theory which attributes the phenomena of electrodynamics to the agency of moving electric charges, the forces on which depend not only on the position of the charges (as in electrostatics), but also on their velocity.

The latter feature of Weber’s theory led its earliest critics to deny that his law of force could be reconciled with the principle of conservation of energy. They were, however, mistaken on this point, as may be seen from the following considerations. The above expression for the force between two charges may be written in the form

where U denotes the expression

Consider now two material particles at distance r apart, whose mechanical kinetic energy is T, and whose mechanical potential energy is V, and which carry charges e and e′. The equations of motion of these particles will be exactly the same as the equations of motion of a dynamical system for which the kinetic energy is

and the potential energy is

To such a system the principle of conservation of energy may be applied: the equation of energy is, in fact,

The first objection made to Weber’s theory is thus disposed of; but another and more serious one now presents itself. The occurrence of the negative sign with the term

implies that a charge behaves somewhat as if its mass were negative, so that in certain circumstances its velocity might increase indefinitely under the action of a force opposed to the motion. This is one of the vulnerable points of Weber’s theory, and has been the object of much criticism. In fact,[9] suppose that one charged particle of mass μ is free to move, and that the other charges are spread uniformly over the surface of a hollow spherical insulator in which the particle is enclosed. The equation of conservation of energy is

{\displaystyle {\tfrac {1}{2}}(\mu -ep)v^{2}+V=} constant,

where e denotes the charge of the particle, v its velocity, V its potential energy with respect to the mechanical forces which act on it, and p denotes the quantity

where the integration is taken over the sphere, and where σ denotes the surface-density; p is independent of the position of the particle μ within the sphere. If now the electric charge on the sphere is so great that ep is greater than μ, then v2 and V must increase and diminish together, which is evidently absurd.

Leaving this objection unanswered, we proceed to show how Weber’s law of force between electrons leads to the formulae for the induction of currents.

The mutual energy of two moving charges is

where v and v′ denote the velocities of the charges; so that the ​mutual energy of two current-elements containing charges e, e′ respectively of each kind of electricity, is

{\displaystyle {\frac {ee^{\prime }c^{2}}{2r^{3}}}[-{(\mathbf {r.v^{\prime }} )-(\mathbf {r.v} )}^{2}+{(\mathbf {r.v^{\prime }} )+(\mathbf {r.v} )}^{2}+{-(\mathbf {r.v^{\prime }} )-(\mathbf {r.v} )}^{2}-{(\mathbf {-r.v^{\prime }} )+(\mathbf {r.v} )}^{2}]},

If ds, ds′ denote the lengths of the elements, and i, i′ the currents in them, we have

′{\displaystyle i\mathbf {ds} =2e\mathbf {v} ,i^{\prime }\mathbf {ds} ^{\prime }=2e^{\prime }\mathbf {v} ^{\prime }};

so the mutual energy of two current-elements is

The mutual energy of ids with all the other currents is therefore

where a denotes a vector-potential

By reasoning similar to Neumann’s, it may be shown that the electromotive force induced in ds by any alteration in the rest of the field is

and thus a complete theory of induced currents may be constructed.