MAGNETISM 245 less for a diamagnetic body, than it ivould be if the sphere were absent. 3. The sphere is in equilibrium ivhen the number of lines of force passing through it is a maximum or a minimum, the equilibrium being stable in the former case, and unstable in the latter. duced Homogeneous Isotropic Sphere in Uniform Field. This iet- case is obtained by putting r^ = r. 2 = r 3 = K in the above itionof formula. The magnetization is parallel to the undisturbed her? 10 ^ e ^ ? n( * ^ ie cou pl e vanishes, so that the sphere is in equilibrium in all positions. If the strength of the field be F, we get for the intensity of magnetization alao In order to familiarize the reader with this important case, we give two figures of the lines of force from Sir W. Thomson s Reprint, pp. 490, 491, one for a paramagnetic. Fiu. 32. Lines of Force for a Paramagnetic Sphere. (fig. 32) having w = 2 8, and another for a diamagnetic (fig. 33) having ^-= 48. The former represents a para magnetic whose susceptibility is something like ^-^th of the maximum observed for the best Norway iron. The latter corresponds to a diamagnetic having a susceptibility some 16,000 times that of bismuth, which is the most powerfully diamagnetic substance known. The reader should observe that, although the field inside the isotropic sphere is uniform, this is not the case outside, FIQ. 33. Lines of Force for a Diamagnetic Sphere. a fact sometimes forgotten by experimenters. Of course the disturbance in the case of a bismuth sphere would be infinitesimal. ?olotro- Homogeneous ^Eolotropic Ellipsoid in a Uniform Field. ic elhp- j n j ae case O f a sphere the tendency to set in a uni- f rm fi e ^ ^ s wholly dependent on the seolotropy of the sphere, and is independent of its form. It is important, in order to get a complete picture of the behaviour of inductively magnetized bodies, to obtain a solution for some case where the form has an effect upon the result. A solid bounded by a surface of the second degree affords such a case. If an ellipsoid be uniformly magnetized so that the components of magnetization parallel to its three principal axes a,b,c be A^B^Cj, this magnetization gives rise to a force a^-AiL, 0^-BjM, -y^-C^N; when L. M, N have the values given above, p. 232. If we now place this ellipsoid in a uniform field (o , , 7 ), the force inside will be given by a = a -A 1 L, ^3 = ^-3^, ^-yo-CjL . . (79). It is obvious, therefore, that the aquations (75) of induction can, as in the case of a sphere, be satisfied by the assumption of uniform magnetization. There is no difficulty in dealing with the general case in which the principal magnetic axes are not parallel to the principal axes of figure ; we shall content ourselves, however, with the case in which the principal magnetic axes r lt r. 2 , r 3 are parallel respectively to a, b, c. Equations (76) then give at once A^^K-AjL), B^r.jOo-BiM), C^r.f-y,,- CN) (80); whence - 1 l l+r a M "I+^N The components of the magnetic moment are of course obtained at once from these by multiplying by the volume. For the components of couple, |T, |tt, |t, tending to turn the ellipsoid about the axes a, b, c, we get From these equations we can draw the following im portant conclusions, first as to the magnetization of the ellipsoid. 1. When r v r 2 , r 3 are so small that their squares may be With neglected, as in fact is the case with all bodies except iron, weakly nickel, and cobalt, the components of magnetization reduce m tf? e to r 1 a , r 2 /3 , r 3 y . A glance at equations (79) will ces the show that what happens is simply that the part of the form does internal inducing force which depends on the squares of llot ^ ffect the susceptibilities is not sensible. In other words, the * he m " form of the body is without influence on the induced magnet- magnetization. Or, what is again equivalent to the same ization. thing, the induced magnetism may be supposed to produce no disturbance in the inducing field. These conclusions are of course not limited to the ellipsoidal form in particular ; but we have the general result that, if the squares of the susceptibilities are negligible, then the form of the body has no effect on the induced magnetism. 2. On the other hand, wlien the susceptibilities (and With consequently the permeabilities) are very great, since b. l = a /(l/r 1 + L), &c., it is clear that the influence of form of the body predominates. The extreme case is that the iaflu- of a body of infinite permeability, in which the induced ence of magnetism is wholly determined by the form. formpre- 3. If, however, the ellipsoid be very elongated in the direction of a, then L will be very small, and r^L may be very small, notwithstanding the largeness of r r In that tj * p case Aj = ? ja . 4. From 1, 2, and 3 we have the following most Best form important results. In experimenting with weakly magnetic for dett-r- bodies in a uniform field in order, say, to determine their susceptibility the form of the body is indifferent. On the other hand, with strongly magnetic bodies an elongated form must be used, because in that case only does the induced magnetism depend mainly on the susceptibility of the material. With bodies approaching the spherical form differences in form produce far more effect on the experimental results than differences in the susceptibility of the material, so that in such cases the experimenter really measures the accuracy of his instrument maker l more than the magnetic susceptibility of his material. 5. For a flat disk (infinitely oblate ellipsoid), having Flat disk. its T*! axis parallel to the lines of force, L=47r, and A 1 -r 1 o v /(l + 4; 1 )-^i-l)/4^. If such a body were diamagnetic, and had r l = - 1/47T, i.e., had zero 1 As a matter of history, Riecke did unwittingly obtain in this way a tolerable approximation to the ratio of the circumference of a circle
to the diameter. See Stoletow, Phil. Mag., 1874, p. 202.