A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 10 of 27

the surface of a smooth solid or a fluid, the circuit can no longer be considered as a single linear circuit of constant strength, but must be regarded as a system of two or of some greater number of circuits of variable strength, the current being so distributed among them that those for which N is increasing have currents in the positive direc tion, while those for which N is diminishing have currents in the negative direction.

Thus, in the apparatus represented in Fig. 23, OP is a moveable conductor, one end of which rests in a cup of mercury 0, while the other dips into a circular trough of mercury concentric with 0.

Fig. 23.

492.] CONTINUOUS KOTATION. 139

The current i enters along AB, and divides in the circular trough into two parts, one of which, #, flows along the arc BQP, while the other, y, flows along BRP. These currents, uniting at P, flow along the moveable conductor PO and the electrode OZ to the zinc end of the battery. The strength of the current along OP and OZ is x + y or i.

Here we have two circuits, ABQPOZ, the strength of the current in which is x, flowing in the positive direction, and ABRPOZ, the strength of the current in which is y> flowing in the negative direction.

Let 23 be the magnetic induction, and let it be in an upward direction, normal to the plane of the circle.

While OP moves through an angle 9 in the direction opposite to that of the hands of a watch, the area of the first circuit increases by i#P2. 0, and that of the second diminishes by the same quantity. Since the strength of the current in the first circuit is #, the work done by it is J x. OP2. 0.33, and since the strength of the second is — y, the work done by it is \y.OP2. 6 33. The whole work done is therefore

i(tf + 3/)OP2.033 or ii.OP2.0«B,

depending only on the strength of the current in PO. Hence, if i is maintained constant, the arm OP will be carried round and round the circle with a uniform force whose moment is \i .OP2 53. If, as in northern latitudes, 33 acts downwards, and if the current is inwards, the rotation will be in the negative direction, that is, in the direction PQBR.

492.] We are now able to pass from the mutual action of magnets and currents to the action of one current on another. For we know that the magnetic properties of an electric circuit C± , with respect to any magnetic system M2, are identical with those of a magnetic shell S19 whose edge coincides with the circuit, and whose strength is numerically equal to that of the electric current. Let the magnetic system M2 be a magnetic shell S2, then the mutual action between ^ and 82 is identical with that between ^ and a circuit C2, coinciding with the edge of S2 and equal in numerical strength, and this latter action is identical with that between Ct and C2.

Hence the mutual action between two circuits, Cl and C2) is identical with that between the corresponding magnetic shells Sl and S2.

We have already investigated, in Art. 423, the mutual action

140 ELECTROMAGNETIC FORCE. [493-

of two magnetic shells whose edges are the closed curves s1 and s2 .

/2 /! COS 6

If we make M= I - &,<&«,

J0 ^0 ?

where e is the angle between the directions of the elements ds1 and ds2, and r is the distance between them, the integration being extended once round s.2 and once round slf and if we call M the potential of the two closed curves ^ and <s2, then the potential energy due to the mutual action of two magnetic shells whose strengths are ^ and a'2 bounded by the two circuits is

and the force X, which aids any displacement 8#, is

The whole theory of the force acting on any portion of an electric circuit due to the action of another electric circuit may be deduced from this result.

493.] The method which we have followed in this chapter is that of Faraday. Instead of beginning, as we shall do, following Ampere, in the next chapter, with the direct action of a portion of one circuit on a portion of another, we shew, first, that a circuit produces the same effect on a magnet as a magnetic shell, or, in other words, we determine the nature of the magnetic field due to the circuit. We shew, secondly, that a circuit when placed in any magnetic field experiences the same force as a magnetic shell. We thus determine the force acting on the circuit placed in any magnetic field. Lastly, by supposing the magnetic field to be due to a second electric circuit we determine the action of one circuit on the whole or any portion of the other.

494.] Let us apply this method to the case of a straight current of infinite length acting on a portion of a parallel straight con ductor.

Let us suppose that a current i in the first conductor is flowing vertically downwards. In this case the end of a magnet which points north will point to the right-hand of a man looking at it from the axis of the current.

The lines of magnetic induction are therefore horizontal circles, having their centres in the axis of the current, and their positive direction is north, east, south, west.

Let another descending vertical current be placed due west of the first. The lines of magnetic induction clue to the first current

496.] ELECTROMAGNETIC MEASURE OF A CURRENT. 141

are here directed towards the north. The direction of the force acting on the second current is to be determined by turning the handle of a right-handed screw from the nadir, the direction of the current, to the north, the direction of the magnetic induction. The screw will then move towards the east, that is, the force acting on the second current is directed towards the first current, or, in general, since the phenomenon depends only on the relative position of the currents, two parallel currents in the same direction attract each other.

In the same way we may shew that two parallel currents in opposite directions repel one another.

495.] The intensity of the magnetic induction at a distance r from a straight current of strength i is, as we have shewn in

Art. 479, i

2-. r

Hence, a portion of a second conductor parallel to the first, and carrying a current i' in the same direction, will be attracted towards the first with a force

where a is the length of the portion considered, and r is its distance from the first conductor.

Since the ratio of a to r is a numerical quantity independent of the absolute value of either of these lines, the product of two currents measured in the electromagnetic system must be of the dimensions of a force, hence the dimensions of the unit current are [i] = [F*] = _M* L* T-*].

496.] Another method of determining the direction of the force which acts on a current is to consider the relation of the magnetic action of the current to that of other currents and magnets.

If on one side of the wire which carries the current the magnetic action due to the current is in the same or nearly the same direction as that due to other currents, then, on the other side of the wire, these forces will be in opposite or nearly opposite directions, and the force acting on the wire will be from the side on which the forces strengthen each other to the side on which they oppose each other.

Thus, if a descending current is placed in a field of magnetic force directed towards the north, its magnetic action will be to the north on the west side, and to the south on the east side. Hence the forces strengthen each other on the west side and oppose each

142 ELECTROMAGNETIC FORCE. [497-

other on the east side, and the current will therefore be acted on by a force from west to east. See Fig. 22, p. 138.

In Fig. XVII at the end of this volume the small circle represents a section of the wire carrying a descending current, and placed in a uniform field of magnetic force acting towards the left-hand of the figure. The magnetic force is greater below the wire than above it. It will therefore be urged from the bottom towards the top of the figure.

497.] If two currents are in the same plane but not parallel, we may apply this principle. Let one of the conductors be an infinite straight wire in the plane of the paper, supposed horizontal. On the right side of the current the magnetic force acts downward, and on the left side it acts upwards. The same is true of the mag netic force due to any short portion of a second current in the same plane. If the second current is on the right side of the first, the magnetic forces will strengthen each other on its right side and oppose each other on its left side. Hence the second current will be acted on by a force urging it from its right side to its left side. The magnitude of this force depends only on the position of the second current and not on its direction. If the second current is on the left side of the first it will be urged from left to right.

Hence, if the second current is in the same direction as the first it is attracted, if in the opposite direction it is repelled, if it flows at right angles to the first and away from it, it is urged in the direction of the first current, and if it flows toward the first current, it is urged in the direction opposite to that in which the first current flows.

In considering the mutual action of two currents it is not neces sary to bear in mind the relations between electricity and magnetism which we have endeavoured to illustrate by means of a right-handed screw. Even if we have forgotten these relations we shall arrive at correct results, provided we adhere consistently to one of the two possible forms of the relation.

498.] Let us now bring together the magnetic phenomena of the electric circuit so far as we have investigated them.

We may conceive the electric circuit to consist of a voltaic battery, and a wire connecting its extremities, or of a thermoelectric arrangement, or of a charged Leyden jar with a wire connecting its positive and negative coatings, or of any other arrangement for producing an electric current along a definite path.

The current produces magnetic phenomena in its neighbourhood.

499-] RECAPITULATION. 143

If any closed curve be drawn, and the line-integral of the magnetic force taken completely round it, then, if the closed curve is not linked with the circuit, the line-integral is zero, but if it is linked with the circuit, so that the current i flows through the closed curve, the line-integral is 4 IT i, and is positive if the direction of integration round the closed curve would coincide with that of the hands of a watch as seen by a person passing through it in the direction in which the electric current flows. To a person moving along the closed curve in the direction of integration, and passing through the electric circuit, the direction of the current would appear to be that of the hands of a watch. We may express this in another way by saying that the relation between the direc tions of the two closed curves may be expressed by describing a right-handed screw round the electric circuit and a right-handed screw round the closed curve. If the direction of rotation of the thread of either, as we pass along it, coincides with the positive direction in the other, then the line-integral will be positive, and in the opposite case it will be negative.

Fig. 24.

Relation between the electric current and the lines of magnetic induction indicated by a right-handed screw.

499.] Note. — The line-integral 4 TT i depends solely on the quan tity of the current, and not on any other thing whatever. It does not depend on the nature of the conductor through which the current is passing, as, for instance, whether it be a metal or an electrolyte, or an imperfect conductor. We have reason for believing that even when there is no proper conduction, but

144 ELECTROMAGNETIC FORCE. [5OO.

merely a variation of electric displacement, as in the glass of a Leyden jar during charge or discharge, the magnetic effect of the electric movement is precisely the same.

Again , the value of the line-integral 4 TT i does not depend on the nature of the medium in which the closed curve is drawn. It is the same whether the closed curve is drawn entirely through air, or passes through a magnet, or soft iron, or any other sub stance, whether paramagnetic or diamagnetic.

500.] When a circuit is placed in a magnetic field the mutual action between the current and the other constituents of the field depends on the surface-integral of the magnetic induction through any surface bounded by that circuit. If by any given motion of the circuit, or of part of it, this surface-integral can be increased, there will be a mechanical force tending to move the conductor or the portion of the conductor in the given manner.

The kind of motion of the conductor which increases the surface- integral is motion of the conductor perpendicular to the direction of the current and across the lines of induction.

If a parallelogram be drawn, whose sides are parallel and pro portional to the strength of the current at any point, and to the magnetic induction at the same point, then the force on unit of length of the conductor is numerically equal to the area of this parallelogram, and is perpendicular to its plane, and acts in the direction in which the motion of turning the handle of a right- handed screw from the direction of the current to the direction of the magnetic induction would cause the screw to move.

Hence we have a new electromagnetic definition of a line of magnetic induction. It is that line to which the force on the conductor is always perpendicular.

It may also be defined as a line along which, if an electric current be transmitted, the conductor carrying it will experience no force.

501.] It must be carefully remembered, that the mechanical force which urges a conductor carrying a current across the lines of magnetic force, acts, not on the electric current, but on the con ductor which carries it. If the conductor be a rotating disk or a fluid it will move in obedience to this force, and this motion may or may not be accompanied with a change of position of the electric current which it carries. But if the current itself be free to choose any path through a fixed solid conductor or a network of wires, then, when a constant magnetic force is made to act on the system, the path of the current through the conductors is not permanently

501-]

RECAPITULATION.

145

altered, but after certain transient phenomena, called induction currents, have subsided, the distribution of the current will be found to be the same as if no magnetic force were in action.

The only force which acts on electric currents is electromotive force, which must be distinguished from the mechanical force which is the subject of this chapter.

Fig. 25.

Relations between the positive directions of motion and of rotation indicated by three right-handed screws.

VOL. II.

CHAPTER II.

AMPERE'S INVESTIGATION OF THE MUTUAL ACTION OF

ELECTRIC CURRENTS.

502.] WE have considered in the last chapter the nature of the magnetic field produced by an electric current; and the mechanical action on a conductor carrying an electric current placed in a mag netic field. From this we went on to consider the action of one electric circuit upon another, by determining the action on the first due to the magnetic field produced by the second. But the action of one circuit upon another was originally investigated in a direct manner by Ampere almost immediately after the publication of Orsted's discovery. We shall therefore give an outline of Ampere's method, resuming the method of this treatise in the next chapter.

The ideas which guided Ampere belong to the system which admits direct action at a distance, and we shall find that a remark able course of speculation and investigation founded on these ideas has been carried on by Gauss, Weber, J. Neumann, Riemann, Betti, C. Neumann, Lorenz, and others, with very remarkable results both in the discovery of new facts and in the formation of a theory of electricity. See Arts. 846-866.

The ideas which I have attempted to follow out are those of action through a medium from one portion to the contiguous portion. These ideas were much employed by Faraday, and the development of them in a mathematical form, and the comparison of the results with known facts, have been my aim in several published papers. The comparison, from a philosophical point of view, of the results of two methods so completely opposed in their first prin ciples must lead to valuable data for the study of the conditions of scientific speculation.

503.] Ampere's theory of the mutual action of electric currents is founded on four experimental facts and one assumption.

505.] AMPERE'S SCIENTIFIC METHOD. 147

Ampere's fundamental experiments are all of them examples of what has been called the null method of comparing' forces. See Art. 214. Instead of measuring the force by the dynamical effect of communicating1 motion to a body, or the statical method of placing it in equilibrium with the weight of a body or the elasticity of a fibre, in the null method two forces, due to the same source, are made to act simultaneously on a body already in equilibrium, and no effect is produced, which shews that these forces are them selves in equilibrium. This method is peculiarly valuable for comparing the effects of the electric current when it passes through circuits of different forms. By connecting all the conductors in one continuous series, we ensure that the strength of the current is the same at every point of its course, and since the current begins everywhere throughout its course almost at the same instant, we may prove that the forces due to its action on a suspended body are in equilibrium by observing that the body is not at all affected by the starting or the stopping of the current.

504.] Ampere's balance consists of a light frame capable of revolving1 about a vertical axis, and carrying1 a wire which forms two circuits of equal area, in the same plane or in parallel planes, in which the current flows in opposite directions. The object of this arrangement is to get rid of the effects of terrestrial magnetism on the conducting wire. When an electric circuit is free to move it tends to place itself so as to embrace the largest possible number of the lines of induction. If these lines are due to terrestrial magnetism, this position, for a circuit in a vertical plane, will be when the plane of the circuit is east and west, and when the direction of the current is opposed to the apparent course of the sun.

By rigidly connecting two circuits of equal area in parallel planes, in which equal currents run in opposite directions, a combination is formed which is unaffected by terrestrial magnetism, and is therefore called an Astatic Combination, see Fig. 26. It is acted on, however, by forces arising from currents or magnets which are so near it that they act differently on the two circuits.

505.] Ampere's first experiment is on the effect of two equal currents close together in opposite directions. A wire covered with insulating material is doubled on itself, and placed near one of the circuits of the astatic balance. When a current is made to pass through the wire and the balance, the equilibrium of the balance remains undisturbed, shewing that two equal currents close together

L 2

148

AMPERES THEORY.

[506.

in opposite directions neutralize each other. If, instead of two wires side by side, a wire be insulated in the middle of a metal

Fig. 26.

tube, and if the current pass through the wire and back by the tube, the action outside the tube is not only approximately but accurately null. This principle is of great importance in the con struction of electric apparatus, as it affords the means of conveying the current to and from any galvanometer or other instrument in such a way that no electromagnetic effect is produced by the current on its passage to and from the instrument. In practice it is gene rally sufficient to bind the wires together, care being taken that they are kept perfectly insulated from each other, but where they must pass near any sensitive part of the apparatus it is better to make one of the conductors a tube and the other a wire inside it. See Art. 683.

506.] In Ampere's second experiment one of the wires is bent and crooked with a number of small sinuosities, but so that in every part of its course it remains very near the straight wire. A current, flowing through the crooked wire and back again through the straight wire, is found to be without influence on the astatic balance. This proves that the effect of the current running through any crooked part of the wire is equivalent to the same current running in the straight line joining its extremities, pro vided the crooked line is in no part of its course far from the straight one. Hence any small element of a circuit is equivalent to two or more component elements, the relation between the component elements and the resultant element being the same as that between component and resultant displacements or velocities.

507.] In the third experiment a conductor capable of moving

508.]

FOUK EXPERIMENTS.

149

only in the direction of its length is substituted for the astatic balance, the current enters the conductor and leaves it at fixed points of space, and it is found that no closed circuit placed in the neighbourhood is able to move the conductor.

Fig. 27.

The conductor in this experiment is a wire in the form of a circular arc suspended on a frame which is capable of rotation about a vertical axis. The circular arc is horizontal, and its centre coincides with the vertical axis. Two small troughs are filled with mercury till the convex surface of the mercury rises above the level of the troughs. The troughs are placed under the circular arc and adjusted till the mercury touches the wire, which is of copper well amalgamated. The current is made to enter one of these troughs, to traverse the part of the circular arc between the troughs, and to escape by the other trough. Thus part of the circular arc is traversed by the current, and the arc is at the same time capable of moving with considerable freedom in the direc tion of its length. Any closed currents or magnets may now be made to approach the moveable conductor without producing the slightest tendency to move it in the direction of its length.

508.] In the fourth experiment with the astatic balance two circuits are employed, each similar to one of those in the balance, but one of them, C, having dimensions n times greater, and the other, A, n times less. These are placed on opposite sides of the circuit of the balance, which we shall call B, so that they are similarly placed with respect to it, the distance of C from B being n times greater than the distance of B from A. The direction and

150

AMPERES THEORY.

[5o8.

strength of the current is the same in A and C. Its direction in B may be the same or opposite. Under these circumstances it is found that B is in equilibrium under the action of A and C, whatever be the forms and distances of the three circuits, provided they have the relations given above.

Since the actions between the complete circuits may be considered to be due to actions between the elements of the circuits, we may use the following method of determining the law of these actions.

Let Alt BI} Cv Fig. 28, be corresponding elements of the three circuits, and let A2, B2, C2 be also corresponding elements in an other part of the circuits. Then the situation of B± with respect to A2 is similar to the situation of C^ with respect to B.2) but the

0

u

distance and dimensions of Cl and B2 are n times the distance and dimensions of Bl and A2i respectively. If the law of electromag netic action is a function of the distance, then the action, what ever be its form or quality, between Bl and A.2, may be written

and that between C1 and B2

'

where #, b, c are the strengths of the currents in A, B, C. But

A — CB and a = c. Hence

^ = Clt

= B

and this is equal to F by experiment, so that we have

or, the force varies inversely as the square of the distance.

511.] FOKCE- BETWEEN TWO ELEMENTS. 151

509.] It may be observed with reference to these experiments that every electric current forms a closed circuit. The currents used by Ampere, being produced by the voltaic battery, were of course in closed circuits. It might be supposed that in the case of the current of discharge of a conductor by a spark we might have a current forming an open finite line, but according to the views of this book even this case is that of a closed circuit. No experiments on the mutual action of unclosed currents have been made. Hence no statement about the mutual action of two ele ments of circuits can be said to rest on purely experimental grounds. It is true we may render a portion of a circuit moveable, so as to ascertain the action of the other currents upon it, but these cur rents, together with that in the moveable portion, necessarily form closed circuits, so that the ultimate result of the experiment is the action of one or more closed currents upon the whole or a part of a closed current.

510.] In the analysis of the phenomena, however, we may re gard the action of a closed circuit on an element of itself or of another circuit as the resultant of a number of separate forces, depending on the separate parts into which the first circuit may be conceived, for mathematical purposes, to be divided.

This is a merely mathematical analysis of the action, and is therefore perfectly legitimate, whether these forces can really act separately or not.

511.] We shall begin by considering the purely geometrical relations between two lines in space representing the circuits, and between elementary portions of these lines.

Let there be two curves in space in each of which a fixed point is taken from which the arcs are measured in a defined direction along the curve. Let A, A' be these points. Let PQ and P Q' be elements of the two curves.

Let AP=s, A'P'=s

and let the distance PPf be de- Fig< 29'

noted by r. Let the angle P*PQ be denoted by 0, and PP'(g by Qf, and let the angle between the planes of these angles be denoted by rj.

The relative position of the two elements is sufficiently defined by their distance r and the three angles 0, 6' , and r/, for if these be

152

AMPERES THEORY.

given their relative position is as completely determined as if they formed part of the same rigid body.

512.] If we use rectangular coordinates and make #, y, z the coordinates of P, and of, y', z' those of P', and if we denote by I, m, n and by I', m', n' the direction-cosines of PQ, and of P'Q' re spectively, then

dx j dy dz -»

-J-—1) -f- = m, = n,

as as as

dx' ,, dy' , dz' ,

(2)

and I \x' — x) + m (y' — y} + n (z' — z) = rcos0,

I' (xf — x] -f- m' (y' — y) -f n (zf — z) = — rcos6\ (3)

II' -f mm' -f nn' = cos e,

where e is the angle between the directions of the elements them selves, and

cos e = — cos 6 cos 6' + sin 0 sin (f cos rj. (4)

r* = (af—x)* + (tf—y)* + (af-z)2, (5)

Again

. whence

•  -*, 
  

dr . , . dx , , N dy , , . dz

• = -(* -*) _(y _,) -(, -z)

= — rcosO.

dr

Similarly r= (^-

. i . <

-^) +(/-*)

\ (6)

= — r cos 6 ; and differentiating r -=- with respect to /,

dr dr dx dx' dy dy dz dz'

CvS CtS CvS CvS CvS CvS CtS dS

(7)

— — (II' -j- mm' + n n'}

= — cos e. j

We can therefore express the three angles 0, 6', and r;, and the auxiliary angle e in terms of the differential coefficients of r with respect to s and s' as follows,

dr

cos 0 =

dr

cose = — r

dr dr

d2r sin 6 sin 6' cos 77 = — r -

(8)

513-] GEOMETRICAL RELATIONS OF TWO ELEMENTS. 153

513.] We shall next consider in what way it is mathematically conceivable that the elements PQ and PQ' might act on each other, and in doing so we shall not at first assume that their mutual action is necessarily in the line joining them.

We have seen that we may suppose each element resolved into other elements, provided that these components, when combined according to the rule of addition of vectors, produce the original element as their resultant.

We shall therefore consider ds as resolved into cos 6 ds — a in the direction of r, and sin 6 ds = /3 fl ^

in a direction perpendicular to \ / *^/

T in the plane P'PQ. p « «'>"'

We shall also consider ds'

as resolved into cos Q' els' = a in the direction of r reversed, mntfoO8ri(tf=P in a direction parallel to that in which /3 was measured, and sin 0' sin 17 els' = y in a direction perpendicular to a and /3'.

Let us consider the action between the components a and j3 on the one hand, and a, /3', / on the other.

(1) a and a are in the same straight line. The force between them must therefore be in this line. We shall suppose it to be an attraction = Aa<xii't

where A is a function of r, and i} i' are the intensities of the currents in ds and els' respectively. This expression satisfies the condition of changing sign with i and with i'm

(2) /3 and (3' are parallel to each other and perpendicular to the line joining them. The action between them may be written

This force is evidently in the line joining (3 and /3', for it must be in the plane in which they both lie, and if we were to measure (3 and ft in the reversed direction, the value of this expression would remain the same, which shews that, if it represents a force, that force has no component in the direction of f3, and must there fore be directed along r. Let us assume that this expression, when positive, represents an attraction.

(3) /3 and y are perpendicular to each other and to the line joining them. The only action possible between elements so related is a couple whose axis is parallel to T. We are at present engaged with forces, so we shall leave this out of account.

(4) The action of a and /3', if they act on each other, must be expressed by

154 AMPERE'S THEORY.

The sign of this expression is reversed if we reverse the direction in which we measure j3'. It must therefore represent either a force in the direction of ft', or a couple in the plane of a and /3'. As we are not investigating couples, we shall take it as a force acting on a in the direction of ft'.

There is of course an equal force acting on /3' in the opposite direction.

We have for the same reason a force

Cay'ii' acting on a in the direction of y', and a force

acting on /3 in the opposite direction.

514.] Collecting our results, we find that the action on ds is compounded of the following forces,

X = (Aaa' + B (3fi')ii' in the direction of r, Y— C(a(B' — aj3)ii' in the direction of (3, (9)

and Z — C ay ii' in the direction of y'.

Let us suppose that this action on ds is the resultant of three forces, Rii'dsds' acting in the direction of r, Sii'dsds' acting in the direction of ds, and S'ii'dsds' acting in the direction of ds' , then in terms of 6, d', and 77,

R = A cos 0 cos 0' + J9sin0sin0'cosr7,

In terms of the differential coefficients of

.r o, r

^ + G-yyJ & = — G--1

ds ds J

In terms of I, m, n, and I', m', n'9

R =-

where f, ??, fare written for af—x, y' — y, and / — z respectively.

515.] We have next to calculate the force with which the finite current / acts on the finite current s. The current s extends from A, where s = 0, to P, where it has the value s. The current / extends from A', where s'= 0, to P', where it has the value /.

5 1 6.] ACTION OF A CLOSED CIRCUIT ON AN ELEMENT. 155

The coordinates of points on either current are functions of s or of /.

If F is any function of the position of a point, then We shall use the subscript (s o) to denote the excess of its value at P over that at A, thus jr(SiQ} = FP-FA,

Such functions necessarily disappear when the circuit is closed.

Let the components of the total force with which A' P* acts on A A be iif Xj ii'Y, and ii'Z. Then the component parallel to X of

the force with which da' acts on ds will be ii' - — 7-7 da ds'.

dsds

Hence -T = R+8l+8'l'. (13)

r

Substituting the values of R, S, and S' from (12), remembering

(14)

and arranging the terms with respect to lt m, n, we find

ds

Since A, B, and C are functions of r, we may write

P = f (A + £)~dr, Q=[ Cdr, (16)

jr r jr

the integration being taken between r and oo because A, JB, C vanish when r = oo.

Hence (A + £)-L = -~, and <? = -^. (17)

516.] Now we know, by Ampere's third case of equilibrium, that when / is a closed circuit, the force acting on ds is perpendicular to the direction of ds, or, in other words, the component of the force in the direction of ds itself is zero. Let us therefore assume the direction of the axis of x so as to be parallel to ds by making I = 1 , m — 0, n == 0. Equation (15) then becomes

To find — , the force on ds referred to unit of length, we must ds

156 AMPERE'S THEORY. [5J7-

integrate this expression with respect to /. Integrating the first term by parts, we find

*X=(Pp-Q)Va-£(2Pr-3-O?l-<U'. (19)

When / is a closed circuit this expression must be zero. The first term will disappear of itself. The second term, however, will not in general disappear in the case of a closed circuit unless the quantity under the sign of integration is always zero. Hence, to satisfy Ampere's condition,

(20)

517.] We can now eliminate P, and find the general value of

When / is a closed circuit the first term of this expression vanishes, and if we make

(22)

& r

/=r

JQ Z T

where the integration is extended round the closed circuit /, we may write c^

Similarly =na'_iyt (23)

u/s

dZ

-j-=lp

ds