THEORY OF CONICAL POLE-PIECES 269 position' (§ 191) at a distance of two metres from the electromagnet; its deflections gave an approximate measure of the absolute amount of leakage. It was found that the latter, as was to be expected, increased with the width of the gap. As the magnetising current was increased, the leakage increased rapidly from zero to a maximum (for gap I about 0.1 C.G.S.), which was attained for a current of about 3 amperes; the leakage then gradually decreased, until at 45 amperes it was only a fifth to a third part-according to the width of the gap-of the above value. This ultimate decrease is evidently still more pronounced with the relative leakage '-that is, indirectly with the coefficient of leakage-and this forms a simple support for Lehmann's result (§ 88, II), which was stated in § 123 to be a 'crucial' experiment, in opposition to the views there put forth.' The two polar coils,' 1 and 2 (fig. 56, p. 262), exert a predominant influence in this direction. When they were switched out of circuit, the difference of magnetic potential, especially with the strongest currents, sank by as much as 20 per cent., while the leakage was increased 4 to 6 times as much. Both these coils, then, in a certain sense, play the part assigned to them (§ 168) of keeping together the flow of induction, and of utilising the stray induction tubes, which would otherwise be lost. The rest of the coils, as special experiments showed, have the object of furnishing ampere-turns-that is, magnetomotive force-and in this respect all of them are sensibly equivalent. The greater part of the stray induction-tubes will be confined to the immediate neighbourhood of the gap. By using 'plate poles' (like P', in fig. 56, p. 262) many more of them will therefore be utilised. § 174. Theory of Conical Pole-pieces. In order to obtain the most intense fields, pointed pole-pieces, sometimes of very peculiar shape, have long been used, since with flat poles, as observed above, the limit of 20,000 C.G.S. can hardly be exceeded. The most suitable form of such pointed poles was first theoreti The disturbing action at a distance will thus be comparatively small, precisely with the strongest currents, and in any case will be considerably less than that of great electromagnets of less simple form. This fact represents an advantage in large laboratories, which must not be under-estimated. 2 This may be the chief reason for the superiority of Ruhmkorff's electromagnets over those with vertical limbs (§ 167). cally investigated (1888) by Stefan, and almost simultaneously by Ewing and Low.' They make a similar assumption to that in § 76-namely, that the distribution of magnetisation is uniform throughout the cross-section up to within the pole-pieces; that is, its direction is FIG. 58 P parallel to the X-axis (fig. 58). This assumption corresponds here also to the ideal limiting case of absolute saturation which in reality is not at all reached (§ 89). The expression obtained represents, therefore, limiting values, which we will again denote by the index ∞. Now consider pole-pieces of the shape repre sented in fig. 58. On the above assumption the end-elements on the conical surfaces will have by § 49 the surface density (36) I, I cos (3,, X) = = I sin a' Now, by the theory of gravitation, it can easily be shown that in the point 0 the self-excited intensity, in so far as it depends on the conical surfaces on each side, is given by the following expression : (37a) (a) Hi 4πsin' a' cos a' log. 12 13 For other points of, along as well as somewhat beside, the X-axis, the intensity may be expressed by spherical harmonics. The differential quotient of the trigonometrical expression in (37a), that is, d (sin2 a' cos a') is obviously zero for = sin a' cos2 a' (2 — tan2 a') a' = tan1√2 = 54° 44′ which expresses that, theoretically, the maximum intensity is obtained when the semi-angle of the cone is 55° in round numbers. This result is quite independent of the distribution of the endelements on the surface of the cone. Moreover, as Ewing and 1 Stefan, Wiener Berichte, vol. 97, p. 176, 1888; Wied. Ann. vol. 38, p. 440, 1889. Ewing and Low, Proc. Roy. Soc. vol. 45, p. 40, 1888; Phil. Trans. vol. 180, pp. 227-232, 1889. See also Czermak and Hausmaninger, Wiener Berichte, vol. 98, p. 1147, 1889. EXPERIMENTS WITH TRUNCATED CONES 271 Low observe (p. 231, loc. cit.), the reluctance of the air-gap will be greater the smaller the angle of the cone. The magnetisation, and indirectly the intensity of the field, are thereby diminished, so that those experimenters consider that in round numbers 50° is the best semi-angle. In case the pole-pieces have either an axial bore, of radius r (fig. 58, right side), or are connected by a cylindrical neck (what is called an 'isthmus,' § 217) of such radius, endelements do not occur except on the surfaces of the cone. But in the case of truncated cones (fig. 58, left), the action of a pair of base-surfaces, of radius 7, at the distance 2 r, cot a' (fig. 16, p. 112), must be added. We obtain their mean intensity if we divide the expression for the magnetic difference of potential [§ 76, equation (16)] by that distance. This portion then The sum of (a) and (b) gives the limiting value of the total self-excited intensity of the field. Ewing and Low showed, further, that if uniformity of the field in axial and in transverse direction is specially desired at the expense of high intensity, it is better to choose a smaller semi-angle than the abovenamely, a' = 39° 14'. For further details as to these instructive theoretical investigations the papers cited must be referred to. Czermak and Hausmaninger treated also the case in which the two summits of the cones do not coincide in the point 0, as has been assumed above. § 175. Experiments with Truncated Cones. With the electromagnet described above the author has investigated how far the above assumptions and results hold good in practice. For this purpose a pair of truncated cones were turned step by step smaller, so that the angle a' became gradually less. The intensity of field was each time determined by the U-tube method (§ 204), and its highest value was, in fact, found for an angle a' = 60°. As it is only a question of a maximum, it does not really matter practically as long as it is between 63° > a' > 57°. No appreciable advantage was found in having the pole-pieces concave. The observed value of the field was always several thousand C.G.S. less than the theoretical. measurements were obtained: The following It follows, from the formulæ of the previous paragraph, and is confirmed by experiment, that high intensity of the field is only attained at the cost of its extent. For many experiments, however, an extent of several sq. mm. is sufficient, or else the methods of investigation must be adapted to satisfy this condition. The bore-holes, which are indispensable in magneto-optical experiments, produce relatively more weakening of the field the wider they are as compared with the distance of the faces. This weakening is, however, less than would follow from the equations in § 174, since there is a kind of internal leakage from the edge of the openings towards the axis. The external leakage and action at a distance when truncated cones are used are similar to that described for plane poles. What has been stated in the previous section may be summed up by saying that a ring electromagnet of manageable size, with truncated conical pole-pieces of 120° aperture, enables us to have fields up to, say, 40,000 C.G.S. over an extent of some square millimetres. To exceed this to any material extent could at present be only accomplished by an undue expenditure of means out of all proportion with the end in view. That follows already from the formula given in which log (r) comes in ; while the weight and cost of an electromagnet are determined rather by the third power of its linear dimesions. D. Inductors and Transformers § 176. Discussion of Mutually Inducing Coils. We have already explained the principal manifestations of self-induction by the example of a uniformly coiled toroid, either closed or divided This field would produce in a piece of thin soft iron between the truncated cones the approximate induction equal to B = 38,000 + 4 × 1750 = 60,000 C.G.S. to which would correspond a total longitudinal stress equal to (§ 103) MUTUAL INDUCTION = : 273 radially (mean radius r1, perimeter 2πr, L, section S). Besides that primary coil 1 (resistance R1, number of turns n1, self-inductance 4,), let there be a secondary one 2 (R2, î2, A2), also wound uniformly on the toroid, as would be the case with the experimental ring described in § 83. Taking the case of such a pair of mutually inducing coils, we will explain as briefly as possible the most important facts in mutual induction, in so far as they are essential for understanding what follows. i2 To a variation of the flux of induction &, produced by the primary current corresponds an electromotive force E, in the secondary, which, from equation (6), § 153, will have the following value: The partial differential quotient of (n, 6) in respect of the primary current I1, which here occurs, is called, in analogy with the definition of § 153, the mutual inductance E12 between the coil 1 and the coil 2. We find, in the present case, 12 If now the parts played by each coil in the phenomenon are interchanged, so that 2 now acts inductively on 1, we have, in an analogous manner, that 12 § 177. Mutual Induction.-Now it is obvious from the above 12 = 21. Hence this quantity may be called, without further definition, the 'mutual inductance' of the two coils.' From (39) and (11) we see, directly, Mutual as well as self inductance has the dimensions of a length, and like this is to be expressed in henries (compare note, p. 237). T |