A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 9 of 27

467.] Let us now suppose that by processes of this kind, or by the equivalent graphical process of constructing charts of the lines of equal values of the magnetic elements, the values of X and Y, and thence of the potential V, are known over the whole surface of the globe. The next step is to expand V in the form of a series of spherical surface harmonics.

If the earth were magnetized uniformly and in the same direction throughout its interior, V would be an harmonic of the first degree, the magnetic meridians would be great circles passing through two magnetic poles diametrically opposite, the magnetic equator would be a great circle, the horizontal force would be equal at all points of the magnetic equator, and if H0 is this constant value, the value at any other point would be H= //Ocos I', where V is the magnetic latitude. The vertical force at any point would be Z = 2 HQ sin I' , and if Q is the dip, tan 6 = 2 tan I' .

In the case of the earth, the magnetic equator is defined to be the line of no dip. It is not a great circle of the sphere.

The magnetic poles are defined to be the points where there is no horizontal force or where the dip is 90°. There are two such points, one in the northern and one in the southern regions, but they are not diametrically opposite, and the line joining them is not parallel to the magnetic axis of the earth.

468.] The magnetic poles are the points where the value of V on the surface of the earth is a maximum or minimum, or is stationary.

At any point where the potential is a minimum the north end of the dip-needle points vertically downwards, and if a compass- needle be placed anywhere near such a point, the north end will point towards that point.

At points where the potential is a maximum the south end of the dip-needle points downwards, and the south end of the compass- needle points towards the point.

If there are p minima of V on the earth's surface there must be p — \ other points, where the north end of the dip-needle points

124: TERRESTRIAL MAGNETISM. [469.

downwards, but where the compass-needle, when carried in a circle round the point, instead of revolving so that its north end points constantly to the centre, revolves in the opposite direction, so as to turn sometimes its north end and sometimes its south end towards the point.

If we call the points where the potential is a minimum true north poles, then these other points may be called false north poles, because the compass-needle is not true to them. If there are p true north poles, there must be p — I false north poles, and in like manner, if there are q true south poles, there must be y — 1 false south poles. The number of poles of the same name must be odd, so that the opinion at one time prevalent, that there are two north poles and two south poles, is erroneous. According to Gauss there is in fact only one true north pole and one true south pole on the earth's surface, and therefore there are no false poles. The line joining these poles is not a diameter of the earth, and it is not parallel to the earth's magnetic axis.

469.] Most of the early investigators into the nature of the earth's magnetism endeavoured to express it as the result of the action of one or more bar magnets, the position of the poles of which were to be determined. Gauss was the first to express the distribution of the earth's magnetism in a perfectly general way by expanding its potential in a series of solid harmonics, the coefficients of which he determined for the first four degrees. These coeffi cients are 24 in number, 3 for the first degree, 5 for the second, 7 for the third, and 9 for the fourth. All these terms are found necessary in order to give a tolerably accurate representation of the actual state of the earth's magnetism.

To find what Part of the Observed Magnetic Force is due to External

and what to Internal Causes.

470.] Let us now suppose that we have obtained an expansion of the magnetic potential of the earth in spherical harmonics, consistent with the actual direction and magnitude of the hori zontal force at every point on the earth's surface, then Gauss has shewn how to determine, from the observed vertical force, "whether the magnetic forces are due to causes, such as magnetization or electric currents, within the earth's surface, or whether any part is directly due to causes exterior to the earth's surface.

Let V be the actual potential expanded in a double series of spherical harmonics,

472.] SUBTERRANEAN OH CELESTIAL I 125

-2

The first series represents the part of the potential due to causes exterior to the earth,, and the second series represents the part due to causes within the earth.

The observations of horizontal force give us the sum of these series when r — a, the radius of the earth. The term of the order i is

The observations of vertical force give us

Z=* — >

dr '

and the term of the order i in aZ is

Hence the part due to external causes is

and the part due to causes within the earth is _ r-

The expansion of V has hitherto been calculated only for the mean value of V at or near certain epochs. No appreciable part of this mean value appears to be due to causes external to the earth.

471.] We do not yet know enough of the form of the expansion of the solar and lunar parts of the variations of V to determine by tills method whether any part of these variations arises from magnetic force acting from without. It is certain, however, as the calculations of MM. Stoney and Chambers have shewn, that the principal part of these variations cannot arise from any direct magnetic action of the sun or moon, supposing these bodies to be magnetic *.

472.] The principal changes in the magnetic force to which attention has been directed are as follows.

• Professor Hornstein of Prague has discovered a periodic change in the magnetic elements, the period of which is 26.33 days, almost exactly equal to that of the synodic revolution of the sun, as deduced from the observation of sun-spots near his equator. This method of discovering the time of rotation of the unseen solid body of the sun by its effects on the magnetic needle is the first instalment of the repayment by Magnetism of its debt to Astronomy. Akad., Wien, June 1,5, 1871. See Proc. R.8., Nov. 16,1871.

126 TERRESTRIAL MAGNETISM. [473-

I. The more Regular Variations.

(1) The Solar variations, depending on the hour of the day and the time of the year.

(2) The Lunar variations, depending on the moon's hour angle and on her other elements of position.

(3) These variations do not repeat themselves in different years,, but seem to be subject to a variation of longer period of about eleven years.

(4) Besides this, there is a secular alteration in the state of the earth's magnetism, which has been going on ever since magnetic observations have been made, and is producing changes of the magnetic elements of far greater magnitude than any of the varia tions of small period.

II. The Disturbances.

473.] Besides the more regular changes, the magnetic elements are subject to sudden disturbances of greater or less amount. It is found that these disturbances are more powerful and frequent at one time than at another, and that at times of great disturbance the laws of the regular variations are masked, though they are very distinct at times of small disturbance. Hence great attention has been paid to these disturbances, and it has been found that dis turbances of a particular kind are more likely to occur at certain times of the day, and at certain seasons and intervals of time, though each individual disturbance appears quite irregular. Besides these more ordinary disturbances, there are occasionally times of excessive disturbance, in which the magnetism is strongly disturbed for a day or two. These are called Magnetic Storms. Individual disturbances have been sometimes observed at the same instant in stations widely distant.

Mr. Airy has found that a large proportion of the disturbances at Greenwich correspond with the electric currents collected by electrodes placed in the earth in the neighbourhood, and are such as would be directly produced in the magnet if the earth-current, retaining its actual direction, were conducted through a wire placed underneath the magnet.

It has been found that there is an epoch of maximum disturbance every eleven years, and that this appears to coincide with the epoch of maximum number of spots in the sun.

474.] The field of investigation into which we are introduced

474-] VARIATIONS AND DISTURBANCES. 127

by the study of terrestrial magnetism is as profound as it is ex tensive,

We know that the sun and moon act on the earth's magnetism. It has been proved that this action cannot be explained by sup posing these bodies magnets. The action is therefore indirect. In the case of the sun part of it may be thermal action, but in the case of the moon we cannot attribute it to this cause. Is it pos sible that the attraction of these bodies, by causing strains in the interior of the earth, produces (Art. 447) changes in the magnetism already existing in the earth, and so by a kind of tidal action causes the semidiurnal variations ?

But the amount of all these changes is very small compared with the great secular changes of the earth's magnetism.

What cause, whether exterior to the earth or in its inner depth s, produces such enormous changes in the earth's magnetism, that its magnetic poles move slowly from one part of the globe to another ? When we consider that the intensity of the magnetization of the great globe of the earth is quite comparable with that which we produce with much difficulty in our steel magnets, these immense changes in so large a body force us to conclude that we are not yet acquainted with one of the most powerful agents in nature,, the scene of whose activity lies in those inner depths of the earth, to the knowledge of which we have so few means of access.

PART IV.

ELECTROMAGNETISM. CHAPTEK I.

ELECTROMAGNETIC FORCE.

475.] IT had been noticed by many different observers that in certain cases magnetism is produced or destroyed in needles by electric discharges through them or near them, and conjectures of various kinds had been made as to the relation between mag netism and electricity, but the laws of these phenomena, and the form of these relations, remained entirely unknown till Hans Christian Orsted *, at a private lecture to a few advanced students at Copenhagen, observed that a wire connecting the ends of a voltaic battery affected a magnet in its vicinity. This discovery he published in a tract entitled Experiments circa effectum Conflictus Electrici in Acum Magneticam, dated July 21, 1820.

Experiments on the relation of the magnet to bodies charged with electricity had been tried without any result till Orsted endeavoured to ascertain the effect of a wire heated by an electric current. He discovered, however, that the current itself, and not the heat of the wire, was the cause of the action, and that the e electric conflict acts in a revolving manner,' that is, that a magnet placed near a wire transmitting an electric current tends to set itself perpendicular to the wire, and with the same end always pointing forwards as the magnet is moved round the wire.

476.] It appears therefore that in the space surrounding a wire

• See another account of Orsted's discovery in a letter from Professor Hansteen in the Life of Faraday by Dr. Bence Jones, vol. ii. p. 395.

478.]

STRAIGHT CURRENT.

129

transmitting an electric current a magnet is acted on by forces depending on the position of the wire and on the strength of the current. The space in which these forces act may therefore be considered as a magnetic field, and we may study it in the same way as we have already studied the field in the neighbourhood of ordinary magnets, by tracing the course of the lines of magnetic force, and measuring the intensity of the force at every point.

477.] Let us begin with the case of an indefinitely long straight wire carrying an electric current. If a man were to place himself in imagination in the position of the wire, so that the current should flow from his head to his feet, then a magnet suspended freely before him would set itself so that the end which points north would, under the action of the current, point to his right hand.

The lines of magnetic force are everywhere at right angles to planes drawn through the wire, and are there fore circles each in a plane perpendicular to the wire, which passes through its centre. The pole of a magnet which points north, if carried round one of these circles from left to right, would experience a force acting always in the direction of its motion. The other pole of the same magnet would experience a force in the opposite direction.

478.] To compare these forces let the wire be supposed vertical, and the current a de scending one, and let a magnet be placed on an apparatus which is free to rotate about a vertical axis coinciding with the wire. It is found that under these circumstances the current has no effect in causing the rotation of the apparatus as a whole about itself as an axis. Hence the action of the vertical current on the two poles of the magnet is such that the statical moments of the two forces about the current as an axis are equal and opposite. Let % and m2 be the strengths of the two poles, rl and r2 their distances from the axis of the wire, 5\ and T2 the intensities of the magnetic force due to the current at

Fig. 21.

the two poles respectively, then the force on m1 is

and

s

since it is at right angles to the axis its moment l

Similarly that of the force on the other pole is m2T2r2, and since there is no motion observed,

mlT1rl + m2T2r2 = 0.

VOL. II. K

130 ELECTROMAGNETIC FORCE. [479-

But we know that in all magnets

m-L + m^ = 0.

Hence T^ = T2r2,

or the electro magnetic force due to a straight current of infinite length is perpendicular to the current, and varies inversely as the distance from it.

479.] Since the product Tr depends on the strength of the current it may be employed as a measure of the current. This method of measurement is different from that founded upon elec trostatic phenomena, and as it depends on the magnetic phenomena produced by electric currents it is called the Electromagnetic system of measurement. In the electromagnetic system if i is the current,

Tr = 2i.

480.] If the wire be taken for the axis of z} then the rectangular components of T are

Here Xdx+Ydy+Zdz is a complete differential, being that of

Hence the magnetic force in the field can be deduced from a potential function, as in several former instances, but the potential is in this case a function having an infinite series of values whose common difference is 4:iri. The differential coefficients of the potential with respect to the coordinates have, however, definite and single values at every point.

The existence of a potential function in the field near an electric current is not a self-evident result of the principle of the con servation of energy, for in all actual currents there is a continual expenditure of the electric energy of the battery in overcoming the resistance of the wire, so that unless the amount of this expenditure were accurately known, it might be suspected that part of the energy of the battery may be employed in causing work to be done on a magnet moving in a cycle. In fact, if a magnetic pole, m, moves round a closed curve which embraces the wire, work is actually done to the amount of 4 TT m i. It is only for closed paths which do not embrace the wire that the line-integral of the force vanishes. We must therefore for the present consider the law of force and the existence of a potential as resting on the evidence of the experiment already described.

483.] MAGNETIC POTENTIAL. 131

481.] If we consider the space surrounding an infinite straight line we shall see that it is a cyclic space, because it returns into itself. If we now conceive a plane, or any other surface, com mencing at the straight line and extending on one side of it to infinity, this surface may be regarded as a diaphragm which reduces the cyclic space to an acyclic one. If from any fixed point lines be drawn to any other point without cutting the diaphragm, and the potential be defined as the line-integral of the force taken along one of these lines, the potential at any point will then have a single definite value.

The magnetic field is now identical in all respects with that due to a magnetic shell coinciding with this surface, the strength of the shell being i. This shell is bounded on one edge by the infinite straight line. Tho other parts of its boundary are at an infinite distance from the part of the field under consideration.

482.] In all actual experiments the current forms a closed circuit of finite dimensions. We shall therefore compare the magnetic action of a finite circuit with that of a magnetic shell of which the circuit is the bounding edge.

It has been shewn by numerous experiments, of which the earliest are those of Ampere, and the most accurate those of Weber, that the magnetic action of a small plane circuit at distances which are great compared with the dimensions of the circuit is the same as that of a magnet whose axis is normal to the plane of the circuit, and whose magnetic moment is equal to the area of the circuit multiplied by the strength of the current.

If the circuit be supposed to be filled up by a surface bounded by the circuit and thus forming a diaphragm, and if a magnetic shell of strength i coinciding with this surface be substituted for the electric current, then the magnetic action of the shell on all distant points will be identical with that of the current.

483.] Hitherto we have supposed the dimensions of the circuit to be small compared with the distance of any part of it from the part of the field examined. We shall now suppose the circuit to be of any form and size whatever, and examine its action at any point P not in the conducting wire itself. The following method, which has important geometrical applications, was introduced by Ampere for this purpose.

Conceive any surface S bounded by the circuit and not passing through the point P. On this surface draw two series of lines crossing each other so as to divide it into elementary portions, the

K 2

132 ELECTROMAGNETIC FORCE. [484.

dimensions of which are small compared with their distance from P, and with the radii of curvature of the surface.

Round each of these elements conceive a current of strength i to flow, the direction of circulation being the same in all the ele ments as it is in the original circuit.

Along every line forming the division between two contiguous elements two equal currents of strength i flow in opposite direc tions.

The effect of two equal and opposite currents in the same place is absolutely zero, in whatever aspect we consider the currents. Hence their magnetic effect is zero. The only portions of the elementary circuits which are not neutralized in this way are those which coincide with the original circuit. The total effect of the elementary circuits is therefore equivalent to that of the original circuit.

484.] Now since each of the elementary circuits may be con sidered as a small plane circuit whose distance from P is great compared with its dimensions, we may substitute for it an ele mentary magnetic shell of strength i whose bounding edge coincides with the elementary circuit. The magnetic effect of the elementary shell on P is equivalent to that of the elementary circuit. The whole of the elementary shells constitute a magnetic shell of strength i, coinciding with the surface 8 and bounded by the original circuit, and the magnetic action of the whole shell on P is equivalent to that of the circuit.

It is manifest that the action of the circuit is independent of the form of the surface S9 which was drawn in a perfectly arbitrary manner so as to fill it up. We see from this that the action of a magnetic shell depends only on the form of its edge and not on the form of the shell itself. This result we obtained before, at Art. 410, but it is instructive to see how it may be deduced from electromagnetic considerations.

The magnetic force due to the circuit at any point is therefore identical in magnitude and direction with that due to a magnetic shell bounded by the circuit and not passing through the point, the strength of the shell being numerically equal to that of the current. The direction of the current in the circuit is related to the direction of magnetization of the shell, so that if a man were to stand with his feet on that side of the shell which we call the positive side, and which tends to point to the north, the current in front of him would be from right to left.

486.] MAGNETIC POTENTIAL DUE TO A CIRCUIT. 133

485.] The magnetic potential of the circuit, however, differs from that of the magnetic shell for those points which are in the substance of the magnetic shell.

If co is the solid angle subtended at the point P by the magnetic shell, reckoned positive when the positive or austral side of the shell is next to P, then the magnetic potential at any point not in the shell itself is coc/>, where $ is the strength of the shell. At any point in the substance of the shell itself we may suppose the shell divided into two parts whose strengths are ^ and c/>2, where </>! -f c/>2 = c/>, such that the point is on the positive side of c^1 and on the negative side of c/>2 . The potential at this point is

On the negative side of the shell the potential becomes $ (co— • 47r). In this case therefore the potential is continuous, and at every point has a single determinate value. In the case of the electric circuit, on the other hand, the magnetic potential at every point not in the conducting wire itself is equal to ia>, where i is the strength of the current, and co is the solid angle subtended by the circuit at the point, and is reckoned positive when the current, as seen from P, circulates in the direction opposite to that of the hands of a watch.

The quantity ^co is a function having an infinite series of values whose common difference is 4 TT i. The differential coefficients of id) with respect to the coordinates have, however, single and de terminate values for every point of space.

486.] If a long thin flexible solenoidal magnet were placed in the neighbourhood of an electric circuit, the north and south ends of the solenoid would tend to move in opposite directions round the wire, and if they were free to obey the magnetic force the magnet would finally become wound round the wire in a close coil. If it were possible to obtain a magnet having only one pole, or poles of unequal strength, such a magnet would be moved round and round the wire continually in one direction, but since the poles of every magnet are equal and opposite, this result can never occur. Faraday, however, has shewn how to produce the continuous rotation of one pole of a magnet round an electric current by making it possible for one pole to go round and round the current while the other pole does not. That this process may be repeated in definitely, the body of the magnet must be transferred from one side of the current to the other once in each revolution. To do this without interrupting the flow of electricity, the current is split

134 ELECTROMAGNETIC FORCE.

into two branches, so that when one branch is opened to let the magnet pass the current continues to flow through the other. Faraday used for this purpose a circular trough of mercury, as shewn in Fig. 23, Art. 491. The current enters the trough through the wire AB, it is divided at B, and after flowing through the arcs £QP and BRP it unites at P, and leaves the trough through the wire PO, the cup of mercury 0, and a vertical wire beneath 0, down which the current flows.

The magnet (not shewn in the figure) is mounted so as to be capable of revolving about a vertical axis through 0, and the wire OP revolves with it. The body of the magnet passes through the aperture of the trough, one pole, say the north pole, being beneath the plane of the trough, and the other above it. As the magnet and the wire OP revolve about the vertical axis, the current is gradually transferred from the branch of the trough which lies in front of the magnet to that which lies behind it, so that in every complete revolution the magnet passes from one side of the current to the other. The north pole of the magnet revolves about the descending current in the direction N.E.S.W. and if w, o>' are the solid angles (irrespective of sign) subtended by the circular trough at the two poles, the work done by the electromagnetic force in a complete revolution is

mi (ITT — o> — a/),

where m is the strength of either pole, and i the strength of the current.

487.] Let us now endeavour to form a notion of the state of the magnetic field near a linear electric circuit.

Let the value of o>, the solid angle subtended by the circuit, be found for every point of space, and let the surfaces for which co is constant be described. These surfaces will be the equipotential surfaces. Each of these surfaces will be bounded by the circuit, and any two surfaces, o^ and o>2, will meet in the circuit at an angle i(o>1-<i)2).

Figure XVIII, at the end of this volume, represents a section of the equipotential surfaces due to a circular current. The small circle represents a section of the conducting wire, and the hori zontal line at the bottom of the figure is the perpendicular to the plane of the circular current through its centre. The equipotential surfaces, 24 of which are drawn corresponding to a series of values

of CD differing by — > are surfaces of revolution, having this line for

489.] ACTION OF A CIRCUIT ON A MAGNETIC SYSTEM. 135

their common axis. They are evidently oblate figures, being flat tened in the direction of the axis. They meet each other in the line of the circuit at angles of 1 5°.

The force acting on a magnetic pole placed at any point of an equipotential surface is perpendicular to this surface, and varies inversely as the distance between consecutive surfaces. The closed curves surrounding the section of the wire in Fig. XVIII are the lines of force. They are copied from Sir W. Thomson's Paper on 'Vortex Motion*.' See also Art. 702.

Action of an Electric Circuit on any Magnetic System.

488.] We are now able to deduce the action of an electric circuit on any magnetic system in its neighbourhood from the theory of magnetic shells. For if we construct a magnetic shell, whose strength is numerically equal to the strength of the current, and whose edge coincides in position with the circuit, while the shell itself does not pass through any part of the magnetic system, the action of the shell on the magnetic system will be identical with that of the electric circuit.

Reaction of the Magnetic System on the Electric Circuit.

489.] From this, applying the principle that action and reaction are equal and opposite, we conclude that the mechanical action of the magnetic system on the electric circuit is identical with its action on a magnetic shell having the circuit for its edge.

The potential energy of a magnetic shell of strength $ placed in a field of magnetic force of which the potential is T, is, by Art. 410,

T- -J- >

x dy dz'

where I, m, n are the direction-cosines of the normal drawn from the positive side of the element dS of the shell, and the integration is extended over the surface of the shell. Now the surface-integral

where #, I, c are the components of the magnetic induction, re presents the quantity of magnetic induction through the shell, or,

• Trans. R. 8. Edin., vol. xxv. p. 217, (1869).

136 ELECTROMAGNETIC FORCE. [490.

in the language of Faraday, the number of lines of magnetic in duction, reckoned algebraically, which pass through the shell from the negative to the positive side, lines which pass through the shell in the opposite direction being reckoned negative.

Remembering that the shell does not belong to the magnetic system to which the potential V is due, and that the magnetic force is therefore equal to the magnetic induction, we have

dV dV dV

a= -- =-, b= -- =-, c = -- j->

dx dy dz

and we may write the value of M,

M=-<t>N.

If bx1 represents any displacement of the shell, and X1 the force acting on the shell so as to aid the displacement, then by the principle of conservation of energy,

"= 0,

^ or X = 6 --

r §x

We have now determined the nature of the force which cor responds to any given displacement of the shell. It aids or resists that displacement accordingly as the displacement increases or diminishes N, the number of lines of induction which pass through the shell.

The same is true of the equivalent electric circuit. Any dis placement of the circuit will be aided or resisted accordingly as it increases or diminishes the number of lines of induction which pass through the circuit in the positive direction.

We must remember that the positive direction of a line of magnetic induction is the direction in which the pole of a magnet which points north tends to move along the line, and that a line of induction passes through the circuit in the positive direction, when the direction of the line of induction is related to the direction of the current of vitreous electricity in the circuit as the longitudinal to the rotational motion of a right-handed screw. See Art. 23.

490.] It is manifest that the force corresponding to any dis placement of the circuit as a whole may be deduced at once from the theory of the magnetic shell. But this is not all. If a portion of the circuit is flexible, so that it may be displaced independently of the rest, we may make the edge of the shell capable of the same kind of displacement by cutting up the surface of the shell into

49O.] FOKCE ACTING ON A CUKRENT. 137

a sufficient number of portions connected by flexible joints. Hence we conclude that if by the displacement of any portion of the circuit in a given direction the number of lines of induction which pass through the circuit can be increased, this displacement will be aided by the electromagnetic force acting on the circuit.

Every portion of the circuit therefore is acted on by a force urging it across the lines of magnetic induction so as to include a greater number of these lines within the embrace of the circuit, and the work done by the force during this displacement is numerically equal to the number of the additional lines of in duction multiplied by the strength of the current.

Let the element ds of a circuit, in which a current of strength i is flowing, be moved parallel to itself through a space §x, it will sweep out an area in the form of a parallelogram whose sides are parallel and equal to ds and bx respectively.

If the magnetic induction is denoted by 33, and if its direction makes an angle e with the normal to the parallelogram, the value of the increment of N corresponding to the displacement is found by multiplying the area of the parallelogram by 33 cos e. The result of this operation is represented geometrically by the volume of a parallelepiped whose edges represent in magnitude and direction 8ar, ds, and 33, and it is to be reckoned positive if when we point in these three directions in the order here given the pointer moves round the diagonal of the parallelepiped in the direction of the hands of a watch. The volume of this parallelepiped is equal to Xb%.

If 0 is the angle between ds and 33, the area of the parallelogram is ds . 33 sin 6, and if 77 is the angle which the displacement b% makes with the normal to this parallelogram, the volume of the

parallelepiped is

ds . 33 sin 0 . bx cos 77 — 8 N.

Now X bx = i 5 N = i ds . 33 sin 0 fix cos 77,

and X =. i ds . 33 sin 0 cos 77

is the force which urges ds, resolved in the direction 8#.

The direction of this force is therefore perpendicular to the paral lelogram, and is equal to i . ds . 33 sin 0.

This is the area of a parallelogram whose sides represent in mag nitude and direction i ds and 33. The force acting on ds is therefore represented in magnitude by the area of this parallelogram, and in direction by a normal to its plane drawn in the direction of the longitudinal motion of a right-handed screw, the handle of which

138

ELECTROMAGNETIC FORCE.

[491.

South

East

is turned from the direction of the current ids to that of the magnetic induction 33.

We may express in the language of Quaternions, both the direction and

West ^ J^ North the magnitude of this force by saying

that it is the vector part of the result of multiplying the vector ids, the element of the current, by the vector 33, the magnetic induction.

491.] We have thus completely de termined the force which acts on any portion of an electric circuit placed in a magnetic field. If the circuit is moved in any way so that, after assuming various forms and positions, it returns to its original place, the strength of the current remaining constant during the motion, the whole amount of work done by the electromagnetic forces will be zero. Since this is true of any cycle of motions of the circuit, it follows that it is impossible to maintain by electromagnetic forces a motion of continuous rotation in any part of a linear circuit of constant strength against the resistance of friction, &c.

It is possible, however, to produce continuous rotation provided that at some part of the course of the electric current it passes from one conductor to another which slides or glides over it.

When in a circuit there is sliding contact of a conductor over