EI,ECTROMAGN T ETISM.] E L E C T K1C1 T Y 69 Several other ways of remembering this direction might be given. Although the above may sound arbitrary and look clumsy at first, yet we have found it more convenient in practice than some others we have tried. We may extend what has been said above to the case where part of the magnetic force, it may be the whole of it, is due to the current in the circuit itself ; for we might suppose the magnetic field to be that due to a shell whose boundary coincides infinitely nearly with the circuit. If the circuit is rigid, there will of course be no motion caused by its own action ; but if it be flexible, there may be rela- tive motions; in fact each portion will move until the number of lines of force that pass through the circuit is the greatest possible consistent with the geometrical con ditions. cto-r It is an obvious remark, after what has been said, that
- n- the potential energy of the magnetic shell which repre-
l; * c sents a current depends merely on its boundary, or, in other words, that the magnetic induction or the number of lines of magnetic force which pass through a circuit depends merely on its form. Hence we should expect to find some analytical expression for the surface integral of magnetic induction depending merely on the space relations of the circuit ; in other words, we should expect to find a line integral to represent it. And when the field is that of another circuit, we should expect to find a double line integral for the mutual potential energy of the two repre sentative shells. 1 We shall describe briefly how these ex pectations are realized. In the first place, a vector may be found which has the property that its line integral taken round any circuit is equal to the surface integral of magnetic induction taken over any surface bounded by the circuit. 1 This vector has been called by Maxwell the "vector potential" (18). Let its components be F, G, H. Then applying the definition to small areas dydz, dzdx, dxdy, at the point xyz perpen dicular to the three axes, 3 a, b, c being components of magnetic induction as before, we get dy dz ~dz dx ~~ dx ~dy These equations might be used to determine F, G, H, and would lead to a much more general solution than is here required. The following synthetical solution is simpler. Consider a magnetized particle sn at (fig. 31). Let the positive direction of its axis be OK, and let its moment b m. The resultant force due to sn at any point P is in a plane passing through OK ; hence the vector potential 8 at P must be per pendicular to this plane. Let its direction be taken so as to indicate a rotation round OK, which with translation along OK would give right-handed screw motion. Describe a sphere with as centre and OP ( = D) as radius. Let PQ be a small circle of this sphere whose pole is K. Consider the . line integral round PQ, and the surface integral over the spherical segment PKQ. Since is the same at all points of . PQ by symmetry, the former is 2irDsin0?l, and the latter is 1 It is important to remark here that we say " of the two represen tative shells," not "of the two circuits," or " of the two currents " (see helow, p. 76).
- The mathematical idea concerned here seems to have been origin-
illy started by Prof. Stokes; it is deeply involved in the improve ments effected in the theories of hydrodynamics, elasticity, electricity, &c., by Stokes, Thomson, Helmholtz, and Maxwell. It is to be noted that the rectangular axes here used are drawn thus : ox horizonal, oz vertical (in plane of paper say), and oy fro the reader; thus In this way rotation from y to z and trans- lation along ox give right-handed screw motion, and so on in r-yclical . Equating these we get for vector potential of sn at P its direction being that already indicated. Suppose now the particle sn placed at Q(xyz) so that the direc tion cosines of sn arc A, /u, v. Let the coordinates of P be |, i, ; also let Ql = D= + ^(t-x)- + (n-yF+((-z)*. Then the direction cosines of QP are D s ^, D 2 ^, D -^P, w h cr e p = g; and we get for the component of the vector potential at P and two similar expressions for G and H. The vector potential of a magnetized body may be got by com- Expres- pounding the vector potentials of the different elements ; hence, IA, sion for Ifji, Iv being the components of magnetization at any point of the vector body, we get poten- and two similar expressions for G and II. The first part of our problem is thus solved. Let us, in the second place, apply the above result (10) to the case of the two shells which are equivalent to two currents. In a lamellar distribution of magnetism = . inc.; hence the dz dy volume integral in (10) reduces to a surface integral, and F=/7~ll(>i-mXS .... (11), where I, m, n are the direction cosines of the outward normal to dS . Now the magnetic shell of thickness T and strength i is a lamel- larly magnetized body of constant intensity = t -r-r. It may be looked upon as bounded by two parallel surfaces normal every where to the lines of magnetization, and by an edge generated by lines of magnetization. At every point on either of the parallel surfaces we have therefore Z = A, m = /x, n*=v; and at the edge I = v - u. , and similar! v for m and n. Hence every element of ds rfs the double integral in (11) belonging to either of the parallel surfaces vanishes, and there remain only the parts on the edge which give IT ( f dx dy f dz dx F = 2ir AH M-J--A-T )-*[ * 7~3- D ( ds ds J ds ds dx since j r as (12) gives the vector potential dy dz ds ds due to a magnetic shell S. Let (|TJ) be any point on the boundary of another shell S, of strength i, and let da be the element of arc of the boundary, then 3 + (j 3 + 11 3 }drr .... (13) is the magnetic induction through S due to S with the sign changed, in other words, the mutual potential energy M. Putting for F,G,H their values by (12), we have M -*//l ds dff ds dff
- dsdff
(H), Double line in tegral forM. where e is the angle between ds and da. The result of (14) realizes the second of our expectations. The double integral arrived at is of great importance, not only in the theory of electrodynamics, but also as we shall see in the theory o the induction of electric currents. Hitherto we have spoken only of closed circuits, and con sidered merely the action of a circuit regarded as a whole. When we did speak of the force on an element of a circuit. ^ n) _ we deduced this force directly from the state of the mag- pere *law netic field in its immediate neighbourhood. There is an deduced, order of ideas, however, in which the mutual action of two circuits is considered to be the sum of all the mutual actions of every element in one circuit on every element in the other. Now, we can easily show, by means of (14), that a system of elementary forces of this kind can be found which will lead to the same result for closed circuits as the theory given above. Let the circuit S be supposed rigid and fixed, aid let the circuit.
S be movable in any way with respect to S ; it may even be flexible.