The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 19 of 36

It is, of course, an immediately obvious corollary, from all that has just been said, that any cutting of a ring or disc which

• Experiments of this kind have been made by M. Borgman. See C(mpte8 JUndus, No. 16, April 21, 1890, p. 849, and also February 3, 1890, VoL OX., p. 233.

318 MUTUAL AND SELF INDUCTION.

hinders the flow of the indnced currents causes the whole of the repulsion effects to vanish. We illustrate this by causing a ring of copper wire to jump off the pole, and then cutting it with pliers, find it has ceased to be capable of giving signs of life. When the metallic masses or circuits which are pre^ sented to the alternating magnetic pole are of very low resistance the electro-magnetic repulsion may become very powerful, many pounds of thrust or push being produced by apparatus of quite moderate size. It is, in fact, quite startling to hold over the pole of a very powerful alternating magnet a very thick plate of high conductivity copper. It would greatdy surprise anyone not acquainted with these principles to be told that a massive copper ring weighing eight or ten pounds could

Fia. 118.— Copper Ring "floatiiig" in air over the pole of an Alternating Current Electro-magnet, when restrained by strings.

be made to float in the air, but it is possible to show this easily. The ring needs to be tethered by light strings (Fig. 118) to prevent it from being thrown off laterally, although these strings in no way support its weight.

One of the most beautiful of Prof. Elihu Thomson's experi- ments exhibits this effect of electro-magnetic repulsion on a closed coil, which is buoyed up in water by a small incandes- cent lamp in circuit therewith. In a glass vase is floated a little glow-lamp like a balloon (Fig. 119). The car con- sists of a coil of insulated wire, and the ends of this coil

MUTUAL AND SELF INDUCTION.

319

are connected with the lamp. The whole arrangement is accurately adjusted to just, or only just, float in water. Placing the vase over an alternating magnetic pole, the magnetic induction creates a current in the coil which lights the lamp, and, moreover, the electro-magnetic repulsion on the coil causes the lamp and coil to rise upward in the water. There is also another class of actions — namely, deflections and rotations — produced by electro-magnetic repulsion on highly conducting discs or rings. If the conducting ring or disc which is presented to the alternating magnet is con- strained by being fixed to an axis around which it can rotate,

Fig. 119. — IncandeBoent Lamp and Secondary Coil floating in water and RepeUed by an Alternating Current Electro-magnet placed beneath.

the action may reduce to a deflective force. On presenting a flat suspended disc to the pole, the disc is prevented by its constraint from being repelled bodily; so it sets its plane parallel to the lines of magnetic induction, and places itself in a position such that the induced currents in it are reduced to a minimum. On this principle, before becoming acquainted with Prof. Elihu Thomson's original work, the author devised

320 MUTUAL AND SELF INDUCTION.

a little copper disc galvanometer for detecting small alter- nating currents.

This deflection by an alternating current of a copper disc suspended within a coil with its plane inclined to the plane of the coils was, in March, 1887, noticed independently by the author, who subsequently described a copper disc galvanoscope for alternating currents based on this fetct (see The Electrician^ May 6, 1887). He did not at the time know how thoroughly Prof. Thomson had explored the phenomena, but the sub- stantial explanation of the facts as above given had already occurred to him.

More interesting than the deflective actions are those which result in the production of continuous rotation in highly

Fio. laa— Alternating Electro-magnet with Shaded Poles, causing a Copper Disc placed between the Jaws to revolve.

conducting bodies placed in an alternating field. We employ lot this purpose an electromagnet having a laminated iron core {see Fig. 120), the ends of the iron circuit being provided with copper bars, which embrace and cover portions of the polar terminations of the magnet. When the magnet is exdted by a periodic current, these secondary circuits become the seat of powerful induced secondary currents. Taking in hand a large copper disc pivoted at the centre ^md held in a fork, we hold this wheel so that part of the disc is inserted between the jaws of the electro-magnet. Immediately, rapid

MUTtTAL AND 8ElF iNDUCTIOik. dSA-

rotation is produoed. The reason is not far to seek. The alternating field creates induced currents, both in the closed coils and in the neighbouring portions of the disc ; and the coA- ductors in which they flow are therefore drawn together. If the polar coils are so placed as to partly shield the poles these attractive actions act unsymmetricaJly on the disc and pull it continuously round. The action is, perhaps, better illustrated by a simpler experiment. If we hold a pivoted copper disc (f*ig. 121) symmetrically over an alternating pole, the action of the pole is one of pure repulsion on the disc, and it causes no rotation in it. When a copper sheet is so placed as to shield or ** shade," as Prof. Thomson calls

Fio. 121.— Revolution of a " Shaded " Copper Plate held over the Pole of an Altemating Current Electromagnet.

it, part of the magnetic pole, currents are induced both in the fixed plate and in the movable one. The fixed disc phields part of the other from the induction of the pole, and J ea '.e causes the induced currents in that plate and disc to be so located that they are in positions to cause continual attrac- tion between the conductors and to continuously pull round the movable disc into fresh positions, so creating regular rotation.

This principle of " shading " a portion of a conductor from the inductive action of the pole, and so causing the eddy currents in it^to be located in a portion of its service and to cause attraction between that conductor and the shading conductor, is capable of being exhibited in various ways*

y

823 MVTVAL AND SELF INDUCTION.

We place on a copper plate a light, hoUow, copper ball {s€€ Fig. 122), and support it in a little depression in a copper plate. Holding the arrangement over the alternating magnet* the ball begins to spin round rapidly when the magnet is excited. This rotation is caused by the continual attraction of the eddy currents induced in the fixed plate and in that part of the ball which is not shielded from the pole by the plate. We may vary the experiment, and exhibit many more or less curious and amusing illustrations of it. If we float these copper balls in water (Fig. 123), and place the glass bowl containing them over the alternating pole, the interposi-

Fio. 122. — Light, Hollow, Coijper Ball BtandiDg in a depression on th« edge of a Copper Plate, and set in rotation when held over the Pole of an Alternating Electro-magnet.

tion of a copper sheet between the pole and the balls causes the latter to begin to spin in a highly energetic manner.'*

Amongst other illustrations of the principles above described Prof. Elihu Thomson invented a novel form of electro- magnetic gyroscope (Fig. 124). Over the altematmg magnet a gyroscope of the usual form is suspended. The wheel of the gyroscope is made of iron, and the tyre of the wheel is a thick copper band. Immediately the iron core is magnetised, the

• For a mathematical discussion of these electro- magnetic rotations, see PhV. Tra/nB, Royal Soc., Vol. CLXXXIIIa., 1892, p. 279, Mr. Q. T. Walker on " Repulsion and Rotation produced by Alternating Electrio Currents."

MUTUAL AND SELF INDUCTION. 828

gyroscope begins to rotate with great rapidity over the pole. In this case theunsymmetrical disposition of the eddy currents in the copper band aroand the wheel is safScient by itself to canse the rotation to occur. The phenomenon, however, which lies at the bottom of all these effects is that the self-induction of the secondary circuit causes the eddy currents to be delayed in phase behind the magnetising field, and hence to persist into the period of reversal of that field, and so produce the repulsion between the primary conducting circuit and that part of the secondary conducting circuit in which the eddy currents are set up.

Fio. 123. — Hollow Copper Ball floatiDg in water over an Alternating Current Electro-niagnet, and caused to revolve by the interpoaition of a " shading " copper plate.

One more experiment in this part of the subject may be referred to. Betuming to the use of the electro-magnet, in which the iron circuit is all but complete, we find that, when a highly-conducting disc is put between the closely approximated half-shielded jaws of this electro-magnet, and an alternating current employed to excite it, the conducting disc is held up in the air gap by reason of the attraction set up between the currents induced in the disc and the shielding polar plates. If, however, the disc has a relatively poor con-

y2

324 MUTUAL AND SELF INDUCTION.

dactmty the attraction is not nearly so marked. A good or bad silver coin can thos be discriminated, because the good silver coin has sufficient conductivity to be the seat of power- ful induced currents, but the bad coin has not.

Closely akin to the foregoing, but more difficult to explain, are the rotations in copper and iron discs which can be caused by the approximation to them of a laminated iron bar alternately magnetised. These actions have been carefully studied by Prof. Elihu Thomson, and applied by him and others in many practical devices. Across the top of an electro-magnet is placed a long bar of laminated iron with the plane of the lamination vertical (Fig. 125). This bar is

FiQ, 124, — ^Electromignetic Gyroscope Revolving over the Pole of an Alternating Current Electro-magnet.

surrounded at intervals by copper bands, which form small closed secondary circuits upon it. If we excite the magnet and hold near the bar an iron disc capable of free rotation, it begins to rotate rapidly. Not only can this be done with a laminated bar throttled by conducting circuits, but oven a solid bar of hard steel will serve the same purpose, and a couple of steel files placed across the poles can cause rapid rotation in pivoted discs of copper or of iron held with their edges dose to the bars so alternately magnetised.

To understand the operations which produce this rotationi we have to return to some elementary principles. Consider

MUTUAL AND SELF INDUCTION.

826

a condaoting ling held in front of the electro-magnet as in Fig. 118. Let a sndden flux of magnetic induction be made through the ring aperture, that is, in common parlance, let ** lines of force " be thrust through the opening of the ring. If these lines proceed out from a north pole, they will create an electromotive force in the ring in such a direction as to make a current flow counter-clockwise round the ring as seen from the pole. This current in the ring itself creates a magnetic field round the ring, and a consideration of the direction of the current will show that in the central aperture of the ring the dire :tion of the inducing field and the field due to the induced current are in opposite directions. The effect of

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Fio. 125. — Alternating Current Magnet with Laminated Iron Bar across its pole causing B^volution of two Iron Discs held near its extremities.

ibis opposition is to retard the formation of the field due to the magnet in the aperture of the ring. In other words, the current induced by the internal field causes the lines of induction due to the external pole to be, as it were, momen- tarily thrust out laterally, and resisted in their endeavour to pass through the aperture of the ring.

If we consider a bar of iron surrounded with a copper band, and imagine that this bar is suddenly acted upon by a magnetising force at one end, the result of the current induced in the ring will be that for a brief time the

326 Mutual and self induction.

magnetio induction in the bar will be causeid to leak out laterally, and go round the copper ring on the out- side, its passage through the ring being resisted. If, then, we throttle a magnetic circuit, such as a laminated iron bar, with copper coils closed upon themselves, and place a mag- netising coil at one end, the closed conducting circuits hinder the rise of magnetic induction in the bar ; or, in other words, give it what may be called magnetic self-induction. If the source of magnetism is a rapidly-reversed pole, the consequences of this delay or << lag " in the induction is that a series of alter- nating magnetic poles are always travelling with retarded speed up the bar, and these may be considered to be represented by tufts of lines of magnetio induction which spring out from and move laterally up the bar. If the bar is not laminated and not throttled, the eddy currents set up in the mass of the bar itself act in the same way, and operate to resist the rise of induction in the bar and to delay the propagation of mag- netism along it. Hence we must think of such a throttled bar, when embraced by a magnetising coil at one end, as sur- rounded by laterally moving bunches of lines of magnetic induction, which move up the bar. Each reversal of current in the magnetising coil calls into existence a fresh mag- netic pole at the one end of the bar, which is, as it were, pushed along the bar to make room for the pole of opposite name, which appears the next instant behind it. When an iron disc is held near such a laminated and throttled bar, these laterally moving lines of force induce poles in the disc which travel after the inducing poles, and hence the disc is continually pulled round. If the disc is a copper disc, the laterally moving lines of magnetic force induce eddy currents in the disc, and these, by the principle already explained, create a repulsion between the pole and that part of the disc in which the eddy currents are set up, causing revolution of the disc.

An interesting application of the above principle has been made in the meter of Messrs. Borel, Wright and Ferranti for measuring alternating currents. It consists of a pair of vertical electro-magnets {see Fig. 126), with laminated iron cores, and each magnet bears at the top a curved horn of laminated iron which is throttled by copper rings. These curved horns, springing from the magnets, embrace and

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328 MUTUAL AND SELF INDUCTION.

nearly touch a light iron-rimmed wheel, free to tarn in the centre. The actions just explained drive the wheel round, when the magnet coils are traversed by an alternating current. The iron wheel carr es on its shaft a set of mica vanes, which retard the wheel by air friction. Under the opposing influ- ences of this retardation and the electro-magnetic rotation forces, the wheel takes a certain speed corresponding to different current strengths in the magnetic coils, and hence the total number of revolutions of the wheel in a given time, as recorded by a counter, serves to determine the total quantity of alternating current which has passed through the meter.

The rotation of iron discs can be shown also by means of a badly-designed transformer. If a closed laminated iron ring

Fia. 127.— Magnetic Leakage acroes a Throttled Iron Ring, cannng rotation of an Iron Disc placed near the Seoondary CoU.

(Fig. 127) is wound with a couple of conducting circuits, such an arrangement constitutes a transformer. If these two circuits are wound on opposite sides of the iron ring, the previous explanations show that the arrangement wiU be productive of great magnetic leakage across the iron circuit. In designing transformers for practical work, one condition amongst others which must be held in view is to so arrange the conductive and magnetic circuits that a great magnetic leakage of lines of force across the air does not take place. If, however, this leakage exists, it indicates that the secondary circuit is not getting the full benefit of the induction created by the primary. To detect it we have merely to hold near the

MUTUAL AND SELF INDUCTION. 329

iron drcnit a little balanced or pivoted iron diso, and if it is set in rapid rotation it indicates that there are laterally- moving lines of magnetic force outside the iron, which have escaped from the iron in consequence of the back-magneto- force of the secondary circuit.

The above described phenomena have been utilized in the <K>nstruction of measuring instruments of various kinds, and the effects due to the magnetic leakage of magnetic fields will be fomid to have applications which will be considered more isarefuUy in discussing the action of transformers.

§ 13. Symmetry of Onrrent and Indaction. — A consideration of the effects described in the present chapter will have dis- closed to the careful reader that there is a complete symmetry between the two fundamental quantities, electric current 4Uid magnetic induction. Let the diagrams in Fig. 128 represent a circuit of iron (magnetic circuit) linked with :a circuit of copper (conductive circuit) and let the iron circuit have wound on it a magnetising coil capable of imposing a magneto-motive force (M.M.F.) on it, whilst the conductive •circuit has a source of electromotive force (E.M.F.), say a battery, introduced into it. Suppose, then, that this E.M.F. is suddenly introduced in the conductive circuit, the linking of this circuit with the iron circuit bestows considerable self- induction on the conductive circuit, and this operates to delay the rise of the current strength in the conductive circuit when the E.M.F. is suddenly applied. If the two circuits were plunged into a good conducting liquid medium, the action of the iron circuit would be to cause a leakage of current across the conductive circuit. Quite similarly, we find that if a magnetomotive force is suddenly applied to the iron circuit, the induced current set up in the conductive circuit opposes the growth of the induction in the magnetic circuit, and, as air is not an insulator for magnetic induction, it causes a leakage of magnetic induction across the circuit. Hence the growth of electric current in the conductive circuit is hindered by linking it with an iron circuit, and the growth of magnetic induction in an iron circuit is hindered by linking it with a copper circuit. This is only one instance of the fact that the laws of current establishment in conductive circuits are similar

3ao

MUTUAL AND SELF INDUCTION,

to the laws of establishment of magnetic induction in magnetic circuits.

The growth of current from surface to centre of con- ductors has been described in sections of this chapter. The gradual soaking in, or growth of the magnetic induction, from the surface to the centre of the iron cores of electro-magnets, when a sudden external magnetising force is applied, has been experimentally examined with great skill by Dr. J. Hopkinson and Mr. E. Wilson, and the reader is referred for a full account

Electric Circuit

Magnetic CvrcwJU

uZ.Jb.^

Copper Iron

Fio. 128. — Diagrams illustrating the Symmetry in relation between

Electromotive Force and £lectric Current, and Magnetomotive Foroe and Magnetic Induction.

of their work to Tl^e Electrician, Vol. XXXIV., 1896, p. 610. Very briefly it may be said that these experiments consisted in showing experimentally that in the case of an electro-magnet with a very large solid iron core the magnetic induction in the iron is not established instantaneously at its full value at all points in the iron when the magnetising foroe ia

MUTUAL AND SELF INDUCTION. 331

applied, but it begins at the surface of the oore and slowly works inwards to the centre. In an entirely similar manner we know that in a conductor of large cross-section the actions involved in the production of a current in the conductor establish the current first at the surface of the conductor, and the central portions of the conductor are only reached by it after a certain finite time. We shall return in the next chapter to the discussion of some of these theoretical questions.

CHAPTER y.

DYNAMICAL THEORY OF INDUCTION.

§ 1. Electromagnetic Theory. — In the matter so far before the reader attention has been directed to the chief facts oi electromagnetic induction without any inquiry into the possible mechanism by which this may be effected. Attention may at this stage be directed to modern views of the subject, which have been the outcome of the work of Faraday and Maxwell and all their illustrious followers in this field of study. The cardinal principle of these methods of viewing the phenomena is the denial of action at a distance. That is to say, if at any point in a £eld we find a force due to a current flowing in some conductor, this force cannot be regarded as appearing there without anything happening in the interspace, but must be the consequence of successive changes in closely contiguous places, and not the result of operations at a distance without intermediate machineiy. Whenever we find an electromagnetic effect taking place at any locality we are directed therefore by these notions to look for its antecedents or consequences at the adjacent places, and the apparent phenomenon is not to be regarded as the whole of it, but to be taken as a portion of the whole of the effects which are produced in every part of the region or medium. The finite velocity of light, and the impossibihty of accounting for its propagation on any other hypothesis than that of actual transmission of something across space, or the propagation of a state of stress and strain or periodic change of some kind through a medium, led to attempts to settle ])etween the rival hypotheses by crucial experiment, with the result that the vast bulk of the accumulated evidence decides

DYNAMICAL THEORY OF INDUCTION. 833

in &Toar of the existenee in space of a medium which has properties not possessed by the ordinary atomic matter, but which may certainly be called a material substance in the sense that it can be the recipient or vehicle of energy. The study of the phenomena of light indicates that along the path of a ray there are certain changes which are periodically repeated, such that at portions of the medium separated by a distance called a wave-length, changes of a similar kind are being coincidently effected. The application of mathematical analysis to optical phenomena has led to the conclusion that we can offer a tolerably satisfactory account of them by making the supposition that there exists such a universally diffused ether or medium in which these changes go on. At this point, however, the profound difficulties of the subject begin. To offer a complete account of the phenomena of light, and to deduce all the observed effects from a fundamental principle, we have to construct a hypothesis as to the structure of this ether and the nature of the periodic changes which constitute the wave motion in it. The periodic changes which in the case of sound and fluid waves are known to exist suggested that in the case of the ether the periodic changes are motions of the parts of the ether relatively to one another, and that these motions are the result of displacements taking place imder certain stresses. We cannot even attempt here a sketch, however brief, of the various hypotheses which have formed as to the sort of motions which may occur. On one assumption the ether has been regarded as capable of having displacements or deformations made in it against internal forces, resisting these changes similar to the shearing strains and stresses in solids. From this point of view, now some- times called the elastic solid theory of light, we may picture this ether to ourselves as a distortable but incompressible jelly-like solid, which exists everywhere and penetrates into the interior of all material bodies. As long as the hypothesis of a universal ether was demanded merely to correlate the observed phenomena of light a Umited order of facts had alone to be considered ; but the conception that electric and magnetic effects also required a similar assumption, increased the difficulties to be dealt with. The mind of Faraday con- tinually turned to the thought that the medium assumed

884 DYNAMICAL THEORY OF INDUCTION.

in both these regions of phenomena might be the same ; and his great disciple, Maxwell, was led more definitely to formu- late a similar conception. If an ether or medium is demanded as a fundamental cause of two or more classes of facts, then it is certainly unscientific to fill space several times over with ethers of different kinds until the attempt has been made to ascertain if one alone cannot be found to fulfil all the required functions. Maxwell was led therefore to the con- clusion that both luminous, electric, and electromagnetic phenomena might be explained by the supposition of one single medium capable of certain internal changes, and possessing certain mechanical properties, and he thus avoided the unscientific process of thought of postulating two different ethers by boldly adopting the hypothesis that the medium on which eleckio effects and optical phenomena depend for their existence is one and the same. We shall see later how this supposition has been supported.

One important element in Maxwell s electric theory is his conception of electric displacement. When an electromotive force acts upon any part of a dielectric which is uniform and non-crystalline it is assumed that at all points along the line of electrostatic induction there is an electric displacement, as Maxwell calls it. The theory does not tell us what is the physical nature of this displacement. We may, in the first place, merely for the purpose of illustration, suppose that the unknown something which we call electricity is moved along a line of induction, and that on the removal of the electric force it returns to its original position, and that a dielectric or insulator is a material in which the electricity, when dis- placed by the application of an electrostatic stress or force* resists this displacement in virtue of an electric elasticity. The apparent charge on conductors, according to this view, is the electricity displaced out of, or into, the dielectric, and positive charge or electrification may be regarded as the possession of an excess which is extruded from the dielectric on to the con- ductor, and negative as a deficit when the conductor gives up some to the dielectric.

Maxwell's next principle is that change in electric displace- ment %$ an electric current whilst the change lasts. He calls this a displacement current, to distinguish it from a current in

DYNAMICAL TBBont OF INDUOTtOtT. 336

conductors called the conduction current. The displacement cttrrent is supposed to have, however, all the properties of an electric current. Conducting bodies must be regarded as those in which there is no elastic force resisting displacement, or, in other words, have no electric elasticity, and in which, therefore, electric displacement can go on continuously. The existence of a current of conduction is recognised by two co-existing effects — ^first, the dissipation of energy into heat ; and second, the existence of magnetic force the direction of which is along closed lines described around the line of the current. The displacement current in dielectrics, which takes place at the instant of applying or changing the electric force, Is also considered to be accompanied by magnetic force. In other words, we must consider the displacement current which takes place in a dielectric when electrostatic force acts on it as a very brief conduction current, and as originating a system of lines of magnetic induction — surrounding it, just as a con- ducting wire is so surrounded, by its loops or closed lines of magnetic induction. Conversely, when lines of magnetic force penetrate through an insulator or dielectric, any change in the density of these lines creates eddy displacement cur- rents in the mass. If the lines penetrate through a conductor they produce, under similar circumstances, eddy currents of conduction, whose energy is ultimately frittered down into heat. Also, if a conductor is moved across a magnetic field so that it '' outs *' lines of induction we have seen that if the conductor is a portion of a closed circuit it has a current of conduction produced in it. Similarly, if a dielectric body is moved in a magnetic field in a like manner it has during the continuance of the motion a displacement current produced in it. Since a dielectric circuit is always a closed circuit, a dis- placement current, or the production of electric displacement in it is always the result of any change in the magnetic field in its interior. For the purpose of obtaining a rough illustrative working model of the actions going on in a dielectric submitted to the action of electric force, it is necessary to fall back on some material hypothesis of elec- tricity— that is, we must conceive of electricity as a something which can be displaced relatively to the molecules of the dielectrici and that it resists this displacementi and that

i» totHTAMtOAL fBMOMt Ot tNDVOTIOK.

when this displacement is made under the action of electric stress the removal of this stress causes a disappearance of the displacement. Dr. Lodge has suggested a form of appazatos serving as a rough working model of this dielectric action, in which buttons sliding along a rod, and held in certain positions by elastic strings, represent the electric particles capable of elastic displacement.*

We may quant tatively define ileetric displacement by saying that in a homogeneous non-crystalline dielectric, if a plane be drawn perpendicular to the line of action of the resultant electric force, then under the operation of this electric force the quantity of electricity displaced normally across a unit of area of this plane is called the eleu^ic displacement. This displace-

Tia, 129.

ment is of the nature of an elastic strain, and is removed when the electric force is removed. Let us fix our ideas by imagin- ing a sphere immersed in a dielectric medium to receive an electric charge of quantity Q. Suppose this sphere to be sur- rounded by a concentric spherical shell (Fig. 129) also immersed in the dielectric. On giving the central sphere a charge + Q we know that on the inside surface of the insulated concentric shell will appear an inductive charge - Q of equal quantity and opposite sign, and a charge + Q on the outside surface.

• See Dr. 0. J. Lodge <<0n a Model Dlustrating MechanicaUj the PMMge of Electricity through Metals and Dieleetrioi," MO. Jtfqf., Korember, 1876b

DYNAMICAL THEORY OF INDUCTION. 337

Let this spherical shell be very thin and be placed at a distance r from the central sphere, supposed to be very small. The electric force due to the central charge Q at the surfiEKse of the

concentric shell is represented by -^, and this force exerts a

displacing action on the electricity of the shell, causing posi- tive electricity to be displaced outwards or in the direction of the force and negative electricity to be displaced inwards or against the force.

The quantity E which appears in the above expression for the magnitude of the electric force is called the dielectric conr gtant, or the specific inductive capacity^ according to Faraday, of the medium. If the dielectric is air or other gas, E is very nearly unity, and the law of the force becomes the ordinary Newtonian law, tiz., force varies as quantity divided by square of distance — that is, the electric force at any point due to a small quantity, Q, collected on a sphere is numerically equal

to -^, where r is the distance of the point from the centre of

the sphere. The quantity E is assumed to have a value of unity in the case of a vacuum, and varies for known dielectrics from a little above unity for dry air up to a value of 6 to 10 for glass. In the C£tse of metals and conducting bodies we may consider E to be infinitely great, and generally E is a number which expresses the ratio of the displacement in the given dielectric to the displacement which would bake place under the same electric force if the dielectric was removed and a vacuum left in its place. The whole quantity of electricity displaced out- wards through the conducting shell is + Q, and since the radius of this spherical shell is r, its surface is 4ir r> and the quantity displaced through unit of area of this shell in the direction of

the force is t-^* ^his quantity, then. Maxwell calls the 4irr«

electric displacement, and denotes by the symbol D.

The electric force or resultant electric intensity at all points

over the spherical shell is ^~ , and this quantity Maxwell calls

the electromotive intensity at that point, and denotes it by E We may also speak of D as the electric strain and E as the eleotiic stress by an extension of osoal meebanieal language*

888 DYNAMICAL THEOBY OF INDUCTION.

The quotient of a itreu by its oorresponding strain is, in mechanics, called the coefficient of elasticity corresponding to that stress. For instance, the quotient of stretching force by longitudinal extension in the case of solid rods subjected to extending forces is called Young's Modulus of Elasticity, or the longitudinal elasticity. By a similar use of language the quotient electric stress by electric strain may be called the ^electrie elasticity y and we have

—^-B -:—. = tIie electiic elasticity.

4fl-r«

Hence the series of numbers obtained by dividing the number 4 x 81416 by the specific inductive capacities give a series of numbers which are the electric elasticities of these substances.

§ % Displacement Ouzrents and Displacement WaTes.—