Page:EB1911 - Volume 09.djvu/231

of one conductor conveying a current ${\text{I}}$ upon another element $d{\text{S}}'$ of, another circuit conveying another current ${\text{I}}'$ the elements being at a distance apart equal to $r.$

If $\theta$ and $\theta '$ are the angles the elements make with the line joining them, and $\phi$ the angle they make with one another, then Ampère’s expression for the mechanical force $f$ the elements exert on one another is

$f{=}2{\text{II}}'r^{\_2}\{\cos \phi -{\tfrac {3}{2}}\cos \theta \cos \theta '\}d{\text{S}}d{\text{S}}'.$

This law, together with that of Laplace already mentioned, viz. that the magnetic force due to an element of length $d{\text{S}}$ of a current ${\text{I}}$ at a distance $r$, the element making an angle $\theta$ with the radius vector $o$ is ${\text{I}}d{\text{S}}\sin \theta /r^{2}$, constitute the fundamental laws of electrokinetics.

Ampère applied these with great mathematical skill to elucidate the mechanical actions of currents on each other, and experimentally confirmed the following deductions: (1) Currents in parallel circuits flowing in the same direction attract each other, but if in opposite directions repel each other. (2) Currents in wires meeting at an angle attract each other more into parallelism if both flow either to or from the angle, but repel each other more widely apart if they are in opposite directions. (3) A current in a small circular conductor exerts a magnetic force in its centre perpendicular to its plane and is in all respects equivalent to a magnetic shell or a thin circular disk of steel so magnetized that one face is a north pole and the other a south pole, the product of the area of the circuit and the current flowing in it determining the magnetic moment of the element. (4) A closely wound spiral current is equivalent as regards external magnetic force to a polar magnet, such a circuit being called a finite solenoid. (5) Two finite solenoid circuits act on each other like two polar magnets, exhibiting actions of attraction or repulsion between their ends.

Ampère’s theory was wholly built up on the assumption of action at a distance between elements of conductors conveying the electric currents. Faraday’s researches and the discovery of the fact that the insulating medium is the real seat of the operations necessitates a change in the point of view from which we regard the facts discovered by Ampère. Maxwell showed that in any field of magnetic force there is a tension along the lines of force and a pressure at right angles to them; in other words, lines of magnetic force are like stretched elastic threads which tend to contract.^[1] If, therefore, two conductors lie parallel and have currents in them in the same direction they are impressed by a certain number of lines of magnetic force which pass round the two conductors, and it is the tendency of these to contract which draws the circuits together. If, however, the currents are in opposite directions then the lateral pressure of the similarly contracted lines of force between them pushes the conductors apart. Practical application of Ampère’s discoveries was made by W. E. Weber in inventing the electrodynamometer, and later Lord Kelvin devised ampere balances for the measurement of electric currents based on the attraction between coils conveying electric currents.

Induction of Electric Currents.—Faraday^[2] in 1831 made the important discovery of the induction of electric currents (see Electricity). If two conductors are placed parallel to each other, and a current in one of them, called the primary, started or stopped or changed in strength, every such alteration causes a transitory current to appear in the other circuit, called the secondary. This is due to the fact that as the primary current increases or decreases, its own embracing magnetic field alters, and lines of magnetic force are added to or subtracted from its fields. These lines do not appear instantly in their place at a distance, but are propagated out from the wire with a velocity equal to that of light; hence in their outward progress they cut through the secondary circuit, just as ripples made on the surface of water in a lake by throwing a stone on to it expand and cut through a stick held vertically in the water at a distance from the place of origin of the ripples. Faraday confirmed this view of the phenomena by proving that the mere motion of a wire transversely to the lines of magnetic force of a permanent magnet gave rise to an induced electromotive force in the wire. He embraced all the facts in the single statement that if there be any circuit which by movement in a magnetic field, or by the creation or change in magnetic fields round it, experiences a change in the number of lines of force linked with it, then an electromotive force is set up in that circuit which is proportional at any instant to the rate at which the total magnetic flux linked with it is changing. Hence if ${\text{Z}}$ represents the total number of lines of magnetic force linked with a circuit of ${\text{N}}$ turns, then $-{\text{N}}(dZ/dt)$ represents the electromotive force set up in that circuit. The operation of the induction coil (q.v.) and the transformer (q.v.) are based on this discovery. Faraday also found that if a copper disk A (fig. 6) is rotated between the poles
Fig. 6. of a magnet ${\text{NO}}$ so that the disk moves with its plane perpendicular to the lines of magnetic force of the field, it has created in it an electromotive force directed from the centre to the edge or vice versa. The action of the dynamo (q.v.) depends on similar processes, viz. the cutting of the lines of magnetic force of a constant field produced by certain magnets by certain moving conductors called armature bars or coils in which an electromotive force is thereby created.

In 1834 H. F. E. Lenz enunciated a law which connects together the mechanical actions between electric circuits discovered by Ampère and the induction of electric currents discovered by Faraday. It is as follows: If a constant current flows in a primary circuit ${\text{P}}$, and if by motion of ${\text{P}}$ a secondary current is created in a neighbouring circuit ${\text{S}}$, the direction of the secondary current will be such as to oppose the relative motion of the circuits. Starting from this, F. E. Neumann founded a mathematical theory of induced currents, discovering a quantity ${\text{M}}$, called the “potential of one circuit on another,” or generally their “coefficient of mutual inductance.” Mathematically ${\text{M}}$ is obtained by taking the sum of all such quantities as ƒƒ$d{\text{S}}d{\text{S}}'\cos \phi /r$, where $d{\text{S}}$ and $d{\text{S}}'$ are the elements of length of the two circuits, $r$ is their distance, and $\phi$ is the angle which they make with one another; the summation or integration must be extended over every possible pair of elements. If we take pairs of elements in the same circuit, then Neumann’s formula gives us the coefficient of self-induction of the circuit or the potential of the circuit on itself. For the results of such calculations on various forms of circuit the reader must be referred to special treatises.

H. von Helmholtz, and later on Lord Kelvin, showed that the facts of induction of electric currents discovered by Faraday could have been predicted from the electrodynamic actions discovered by Ampère assuming the principle of the conservation of energy. Helmholtz takes the case of a circuit of resistance ${\text{R}}$ in which acts an electromotive force due to a battery or thermopile. Let a magnet be in the neighbourhood, and the potential of the magnet on the circuit be ${\text{V}}$, so that if a current ${\text{I}}$ existed in the circuit the work done on the magnet in the time $dt$ is ${\text{I}}(d{\text{V}}/dt)dt.$ The source of electromotive force supplies in the time $dt$ work equal to ${\text{EI}}dt$, and according to Joule’s law energy is dissipated equal to ${\text{RI}}^{2}dt$. Hence, by the conservation of energy,

${\text{EI}}dt{=}{\text{RI}}^{2}dt+{\text{I}}(d{\text{V}}/dt)dt.$

If then ${\text{E}}{=}{\text{O}}$ we have ${\text{I}}{=}-(d{\text{V}}/dt)/{\text{R}}$, or there will be a current due to an induced electromotive force expressed by $-d{\text{V}}/dt$. Hence if the magnet moves, it will create a current in the wire provided that such motion changes the potential of the magnet with respect to the circuit. This is the effect discovered by Faraday.^[3]

Oscillatory Currents.—In considering the motion of electricity in conductors we find interesting phenomena connected with the discharge of a condenser or Leyden jar (q.v.). This problem was first mathematically treated by Lord Kelvin in 1853 (Phil. Mag., 1853, 5, p. 292).

If a conductor of capacity ${\text{C}}$ has its terminals connected by a wire of resistance ${\text{R}}$ and inductance ${\text{L}}$, it becomes important to consider

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1. See Maxwell, Electricity and Magnetism, vol. ii. 642.
2. Experimental Researches, vol. i. ser. 1.
3. See Maxwell, Electricity and Magnetism, vol. ii. § 542, p. 178.