Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/264

MAGNETISM Mag netic couple, two part 3 of. permeability, the normal magnetization would be infinite for any finite force. Next, we have the following conclusions as to the mag netic couple. Let us suppose tho ellipsoid free to move about its a axis, and let the direction of the field be per pendicular tocr, so that /3 = Fcos0, y = Fsin0; the coupls tending to turn the b axis parallel to the undisturbed direc tion of the field is the sum of two parts : and - M) S1 For small suscep tibilities the effect of seolo- tropy predomi nate*. For large suscepti bilities the effect of form predomi nates. 1. If the susceptibilities are so small that their squares and products are negligible, then ^ reduces to If 1 = %TrabcF*(r 2 - r 3 ) sin 20. In other words, the form of the body has no effect, and it behaves exactly like an seolotropic sphere of the same volume; i.e., it will tend to turn its axis of greatest per meability parallel to the lines of force. 2. If the susceptibilities be very large, then the most important part of |T will be |T 2 . Now a glance at the values of M and N (34) shows that N - M has the same sign as 6 2 c 2 ; hence the ellipsoid will tend to place its longest dimension parallel to the lines of force. 1 This is the general effect of the influence olform in the case of strongly magnetic bodies, or, if we choose to put it so, the effect of the disturbance of the field by the induced magnetism. 3. It is of course possible in the case of strongly magnetic bodies that both parts of JT may be sensible, so that the resultant action would be affected both by form and by the magnetic structure, either predominating according to cir cumstances ; for by properly shaping the ellipsoid we can give N - M any value positive or negative from to 27r. In this way, given an aeolotropic body for which l/? 3 V r 2^ s not greater than 2-n-, we might so shape it that it would turn its longest dimensions parallel to the lines of force, or so that it would turn its shortest dimensions parallel to the lines of force, the shortest axis in the second case being the axis of greatest permeability ; or we might so shape it that the equilibrium would be neutral. And, in general, given a body seolotropic within certain limits, we might shape it in such a manner that the effect of its form would exactly neutralize the effect due to its structure, so that, as far as setting in a uniform field is concerned, 2 it would behave like an isotropic sphere. Homogeneous Isotropic Ellipsoid in a Uniform Field. The formulae for this case are of course at once obtained by putting r l r. 2 = r. A = K in the above formulae. We thus get " Q K 7o raz C i = nr^v (85); (86). The following conclusions are worthy of notice : 1. The resultant magnetization will not, as in the case of an isotropic sphere, be parallel to the resultant magnetic force. Influence 2. The ellipsoid will tend to set its longest dimension of form parallel to the lines of force; and, since K is involved in the the same numera tor in the form K 2 , this conclusion is the same for a magnetic diamagnetic as for a paramagnetic body. Of course we s for a exclude the mathematically possible case of one of the para- factors 1 + M or 1 + *N vanishing or becoming negative. magnetic p or a jj vveakly magnetic bodies, however, K is so small body. J 1 Assuming, as is always the case in nature, that r a and r 3 have the same sign. 2 In other respects it would not in general behave as if it were isotropic. Isotropic ellip soid. l + /cL f ^ 4 7rafo/c 2 (N - M^t/y,) that the tendency of an elongated isotropic body to set in a uniform field is insensible. Ring Electromagnet. A simple case, 3 which has recently Tore. acquired practical importance, is that of an electromagnet having a soft iron core shaped like an anchor ring, whose mean diameter is II, and radius of section a, wound uni formly with n turns of a primary coil in wLich flows a current i. The lines of force and the lines of magnetiza tion will evidently be circles, and, since the Poisson s surface and volume distributions vanish, the whole mag netic force |p will be simply that due to the current. At a distance p from the axis of the ring |) = 2ni/p ; for the whole work done on a unit pole in passing round any coaxial circle of radius p is |) x 2irp 47m; . 4 The inten sity of magnetization is, therefore, 1 = InKi p, and |j= 2ni(4-!rK+ l)/p = 2n-zri/p. Hence it appears that the total induction through a secondary coil of n windings is 2nn i(4arKfdS/p+fd$ lp), where fdSjp is taken over the section of the core, and/cAS /p over the section of the coil. In the case of an anchor ring of circular section, if we neglect the difference between the radius of the primary coil and the radius of the core, the expression for the total induction through the secondary is 47r?mW(R - ^/K 2 - a-). In a non-uniform field the problem of magnetic Small induction becomes very difficult for bodies of finite size. s l liere If, however, we deal with infinitely small bodies we may llon . suppose the field uniform throughout the body, and apply the ne y. results already obtained to find the induced magnetism. Small JEolotropio Sphere in a Non-uniform Field. Let A lf B 1; Cj be the components of tho induced magnetization parallel to the principal magnetic axes of the sphere, a ,/3 ,y () the components of the strength of the undisturbed field at the centre of the sphere in the same direction; then, denoting i/(l + l rr i) by s i and so on, wo have A 1 = s 1 a , B 1 = s. ! ^ , C^^,,. If the mag netization of the small sphere (of volume v) were rigid, its potential energy W would be V = -^AjOq+B^y + C^o). The actual potential energy, W, of the inductively magnetized sphere is different, because its magnetism varies as it passes from one part of the field to another. In any infinitely small displacement, how ever, we may calculate the work on the supposition that the magnetism is temporarily rigid. In other words, we may put oTW = cAV, where the latter is taken on the supposition that Aj, Bj, C : do not vary, while on the other hand a 0) /3 ,y do vary, because the resultant force both alters its magnitude and its direction relative to the principal axes of the sphere. We thus get In integrating we must take account of the fact that variable. Substituting their values, we get whence "W = - -- ( This important formula contains the whole of the theory of the movement of small spherical masses of inductively magnetizable matter in any field of force. We can deduce Farnda from it at once the position of equilibrium of an reolotropic mague- sphere suspended in a uniform magnetic field, with freedom crystal] llV ( 1C- to rotate about a given diameter. duced Let A., ,u, v and I, m, n be the direction cosines of the given diameter from tl and of the direction of the field relative to the principal magnetic princip axes of the sphere, and R the strength of the field; then W= of least - ^vR-^l" + s. 2 m 2 + s%n-). For stable equilibrium W must be a potenti minimum, and for unstable equilibrium a maximum, i.e., there is energy. stable or unstable equilibrium according as sj 2 + s.jn- + s 3 n is a maximum or a minimum under the given kinematical conditions, which will be expressed by a relation between A, /u, v and 1, m, n. It is needless to work out the analytical solution ; for it leads to results easily obtainable from formula} already given. It is im portant, however, to show the identity of this method of treatment 3 See art. ELECTRICITY, vol. viii. p. 68. 4 Kirchhoff, Fogg. Ann., Ergbd. v. 1870. In the same paper he discusses the effect of a rectilinear current in a cylindrical iron wire, and finds that the circular magnetization in a wire of length L gives rise to an apparent increase of the coefficient of self-induction equal

tO 2lTKL.