as the relation between permeability and susceptibility. The latter number would probably of course only supposi- 50 quantity. We have therefore =0; from equation (14) it follows that ∞ =1, and in like manner 1/μ∞ ∞ =1 as given above. = As we shall afterwards make no distinction between magnetically indifferent bodies and a vacuum, we shall consequently suppose the permeability of the former unity, whereas it actually differs in the fourth or fifth decimal place. The reluctivity is then also unity, but by equation (14) the susceptibility is zero. § 15. Perfect and Imperfect Magnetic Circuits. Our previous considerations referred to the case of a uniformly wound circular ring, having also a circular section; a body thus shaped we have called a toroid. But we may at once extend them to rings of any given section and any shape. The centroid (§ 6), that is, the curve corresponding to the central circle of the ring (§ 9), may be either a plane or a solid curve, provided only that its radius of curvature be always large compared with the dimensions of the section. With uniform winding there will be no apparent magnetic action at a distance that is, the magnetic condition is restricted to the body of the ferromagnetic ring. Such an arrangement is called a perfect magnetic circuit. We assume that the condition for this perfection is the absence of any action at a distance, and hence an arrangement in which such an action does occur is aptly called an imperfect magnetic circuit.2 In the few places in which action at a distance has been mentioned, it has always been spoken of as apparent; for in the present state of science it may be considered as almost certain, that direct actions at a distance do not take place, but are transmitted through the intervening medium. But since, for our present object, we need not concern ourselves with this deeper insight into the mechanism concerned, the expression 'action at a distance' will, with this reservation, be used without further specification. 2 Compare § 100. CHAPTER II ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS § 16. Action of a Narrow Transverse Cut.-The reason for the comparatively simple behaviour of perfect magnetic circuits in the sense of the previous paragraph, is their geometrical property of having no ends, so that in the literal sense of the words, they are endless. This endlessness entails, as will be seen, the absence of action at a distance, and may therefore just as well be regarded as that property which directly determines the perfection of the magnetic circuit. The correctness of this conception is seen when we provide the perfect circuit with ends by cutting it. Any cut, however fine, then manifests itself by the occurrence of an action at a distance, which is strongest in the immediate neighbourhood; in the part surrounding the cut magnetic conditions are produced, the intensity of which increases with the width of the cut. The occurrence of actions at a distance, conversely, points to the presence of transverse cuts, even when these cannot be seen, as for instance in the case of concealed cracks, or such as have been soldered, or joints in the ferromagnetic substance itself (Chapter IX., § 151). The air-gap which occupies the cut differs in no respect from the indifferent environment, and therefore the action at a distance cannot proceed from it; we must accordingly conclude, as a geometrical necessity, that the action at a distance is due to the ends which the cross cut has produced. In order to acquire an insight into the action of such cuts from a totally different point of view, let us consider the typical case of a uniformly wound toroid. We suppose this to be cut transversely in the direction of the radius say at A (fig. 1, p. 11), in such a manner that the cut is very narrow, compared with the dimensions of the section; let its width be d. Let (4) be the normal curve of magnetisation of the closed ring before it has been cut, as represented in fig. 5 for a particular variety of soft iron. If then the curve is determined for the divided ring, it will be found to run more to the right, like the curve (B) for instance. We thus see that with a given magnetising intensity as abscissa, the ring which has been cut has a smaller magnetisation than the closed one, and that conversely, in order to produce the same to examine them as functions of the ordinates. For narrow transverse cuts we thus obtain, as a first approximation, a straight line OC, which is so inclined that for each given ordinate Absc. (C) = Absc. (B) — Absc. (4). The answer to the question is, therefore, that to obtain a given magnetisation with divided rings, an increase of intensity is necessary, which, as a first approximation, is proportional to the magnetisation to be attained; that is to say, is a certain constant fraction of it. The theory and experiments by which these results are arrived at will be discussed in detail in a special section (Chapter V.); we deal here only with the general qualitative results, in so far as they afford an insight into the action of cuts. SHEARING OF CURVES OF MAGNETISATION 25 § 17. Shearing; Backward Shearing. This is the appropriate place for discussing the graphical operations by means of which, by transforming co-ordinates, normal curves of magnetisation may be transformed into curves for divided rings, or, in general, for imperfect magnetic circuits, and conversely. With this object we draw a straight line OC' through the origin O (fig. 5) into the upper left of the quadrant, which is symmetrical with OC with respect to the axis of ordinates. It is clear from what has been said above that we obtain the new curve (B) from the normal curve (4), when, in conformity with § 13, we effect a shearing from the directrix C O' to the axis of ordinates. By the converse operation, which we may call backward shearing from the axis of ordinates to the line CỠ, we transform again the curve (B) into the normal curve (A).1 The equation of the line O C' we may write generally Δ A5 = N3 in which N is a factor increasing with the width of the cut, to which we shall afterwards refer in more detail. The increase in the intensity of the magnetising field, A5, which has to be applied in consequence of the cut, is, in fact, according to equation (1) proportional to the value 3 of the magnetisation to be induced, as was also mentioned in the previous paragraph. § 18. Action at a Distance of the Ends. As regards the explanation of the facts described, we have already seen that actions at a distance are directly produced by a cut, and the more markedly the wider is the cut. These actions extend over the whole surrounding space, and it is, therefore, natural to suppose that they will also be produced in the space occupied by the ferromagnetic substance of the ring, and will, therefore, act inductively. Now it may be readily shown that such an inductive action must be opposed to that of the field due to the current in the coil (fig. 6, p. 27), so that the latter must be increased, in order to compensate the former and thus to obtain the same total intensity, and, hence, the same magnetisation. It is this increase in intensity which is expressed by the difference of the abscissæ of the curves, and which, therefore, must be equal and 'This construction was first given by Lord Rayleigh, Phil. Mag., 5th series, vol. 22, p. 175, 1886, and generalised by Ewing. |