General problem of mag- netic in- duction. Resolving along the principal magnetic axes at (x, y, z), we get, by the law of induced magnetism, (l lt m lt n^, (Z 2 , m. 2 , n 2 ), (1 3 , m^, ?( 3 ) being the direction cosines of the axes w 1 , .# w 3 , (90), e q ua tions. Multiplying these by l lt 7 2 , 1 3 , adding and so on, we get 30 3/8 + ^7+4*30 c = < 2 o + <!/8 + s 3 -y + 4irC where s l "A 2 + ^g 2 + W 3 Z 3 2 i *1 = w l wl l a l + W 2"V Z 2 + W 3 m i% > &c. ; that is to say, given functions of ce,2/,z. Besides these, we have the conditions of normal continuity for $> viz., da db dc_ ___ _j _ j _ J ax dy dz and at a surface of discontinuity, , /*, v being the direction cosines of the normal at any point from the first medium to the second, (a - a ) + (b - 6> + (c - c> = , dashed letters referring to values on the first side of the surface, undashed letters to values on the second. From these we get finally, for the determination of V. and Here
- r Q ~~r~ T <> r
rfx * dy J 02 . +&c rfV dx &c. dV ts dy~ 3 dy +h dz * dz &C. r + 4*<r = (91). (92). dz Case of homo geneous isotropic medium with no pre existing magnet ization. dx dy <r = ( A - A > + (B - BO )/* + (C - C )v ; i.e., they are Poisson s volume and surface densities for the pre existing magnetization. It may be shown by a method l essentially the same as that used in the article ELECTRICITY, vol. viii. p. 27, that equations (91) and (92), with the condition that V be continuous everywhere and vanish at infinity, lead to a unique determination of V. When V is known, A 1 , B 1 ,C l can be found at once from (90). In what follows we shall confine ourselves to homogeneous iso tropic media, and we shall suppose that in parts of the medium inductively magnetizable there is no pre-existing magnetism,. The equations (91) and (92) then reduce to and + -7 + dy T-.7 dz- -1 dV 4* dx Induced magnet- izatioum solenoi- dal and lamellar. ,d TI , QA , WT- W^-T = O ...... (94), dv dv dv,dv being elements of normals drawn inwards in the two media. Equations (90) reduce to a = wa , 6 = && , c = zr-y ; whence -l dV w-1 dV , . ^ d,j ^~ -iV dy From these last, combined with( 93), we have the im- portant consequence that the induced magnetization is fovfa golenoidal and lamellar. This is true only for homo- g eneous isotropic media in which the pre-existing magnet- ism, if any there be, is solenoidal. In all cases such as we are now considering, the part of ^ e ma gnetic potential due to induced magnetism may be calculated wholly from surface distributions at the surfaces of discontinuity. If o-j be this surface density, we have = (96). dv dv From equations (94) and (96) oTV = /*V 1 ~ KI dv dv where 1 =(w sr r ) j iirzr , KI = (& tr) 1 4-7TZ3- . Let us suppose that a body A of permeability -a is sus- 1 See Thomson, Reprint of Papers on Electrostatics and Magnetism, pp, 548 w. pended in a medium of permeability =/. If the suscepti- Differen bilities be small, the forces arising from the induced tia magnetism will be so small that the direction of the actlon - normal force at the surface of A will be the same as if the field were undisturbed by its presence. First, let the medium be vacuum, for which we suppose w =l, then, if A be paramagnetic (i.e., w>l), K X will be positive, and the surface magnetism will be positive where the lines of force leave the body, and negative where they enter it. If A be diamagnetic (i.e., ra-<l), K X will be negative, and the magnetic polarity of the body as a whole will be opposite to what it was in the former case. Secondly, let the surrounding medium have permeability or , then K I is positive or negative according as > or <w : in the former case A will behave like a paramag netic body in vacuo, in the latter like a diamagnetic body in vacuo. It appears then that, by virtue of differential action, a body may behave paramagnetically or diamagnetically according as it is placed in a less or in a more permeable medium than itself. In practice it is most convenient in general to determine Poisson s Y! instead of V. The above equations can be easily modified equation to admit of this. In fact we get at once, remembering that dV /dv = -dV ldv, since V has no discontinuity at the surface of the media, - JL + f . .^H ^ = (97); ctsc" fty ctz" -T dv - dv (98). These equations, together with the condition that Vj be finite and continuous and vanish at infinity, determine V x completely. Since the induced magnetization is lamellar, we may write A 1 = d<j> 1 jdx &c., we then get by (95) which give the components of moment in terms of the known function V. The number of cases in which the solution of the induc tion problem can be worked out is very small. Besides those already treated synthetically, one or two more, affording examples of the general method, will be men tioned in the historical summary below. Meantime, we J. Neu- must not omit to mention an extremely elegant transf or- ann s mation theorem, due to J. Neumann, 2 which enables us to Jetton 1 " deduce the magnetic moment of any body as a whole under theorem. the action of any forces whatever, when its magnetization in a uniform field is known. Let Aj .B^CV be the components of induced magnetization pro duced in the body A by the uniform field whose components are a o #o >7o ; AjjBj.Cj the magnetization produced in A by any field whatever (a ,& ,y ). Letfdv denote volume integration throughout A ; and consider the function U =
=fdv(ajb.i + &C. ) +/dv(a 1 A 1 + &C. ) -/MlV + &C. ) . Now the last two terms destroy each other ; for they are simply different expressions for the mutual potential energies of the in duced magnetism due to (a , B , y ) and to (a , , y ), regarded as separate rigid systems, although coincident in position. get U whence IT is known, since we suppose a ,/8 ,7 , A/.B/.C/ to be known. But, since A I = K(O O + O I ), &c., A^ = K(O, O + a^), &c., we have i Hence we 2 Crelle s Jour., xxxvii. 44 (1848). The proof given is a modifica-
tior of Kirchhoff s, Crelle, xlviii. 366 (1854).