FUNDAMENTAL EQUATION FOR A DIVIDED TOROID 111 as the fundamental equation for a radially divided toroid.' The left-hand member denotes the magnetic difference of potential due to magnetisation measured within the ferromagnetic body between the two faces of the slit; it is approximately equal to the mean value 5, which the magnetic intensity due to magnetisation has within the slit, multiplied by its width. § 76. First Approximation; Limiting Case.-Another expression must now be found for the left-hand member of the fundamental equation (I). To this end, in our first approximation we make the assumption that the magnetisation 3 is constant over the entire cross-section of the toroid, and at right angles to the plane of the section, so that it has, in accordance with the definition of § 43, a uniform peripheral distribution.2 According to the law of saturation' III (§ 57) the actual state of things must approximate more or less closely to this limiting case of our assumption, when we suppose the intensity 5. of the inducing magnetic field to increase without limit, so that finally its value becomes very great in comparison with that of the demagnetising intensity '. In accordance with this assumption there will be no magnetically effective endelements on the convex bounding-surface of the toroid, such elements being confined to the two plane surfaces which bound the slit on either side. These will produce an external field determined by their magnetic strength. But this latter quantity has per unit area of the surface in question the value 3, (§ 49), and since, in the present case the magnetisation is at right angles to the bounding-surfaces of the cleft, = 3. Let us now consider an element of one of the faces of the slit, taken in the form of a plane circular ring of infinitesimal breadth dy and of mean radius y (fig. 16); its area is 2 π y dy, and its magnetic strength is therefore 2 π 3 y dy. Thus at a point P which is on the normal, drawn from the centre of the bounding-face of the slit, at a distance from that face, our elementary plane ring exerts an infinitesimal magnetic intensity 1 du Bois, Wied. Ann. vol. 46, p. 494, equation (7), 1892. 2 It is unnecessary in this case to introduce a mean value I, since the value of I is everywhere the same. d, which in accordance with Coulomb's law is given by the equation (11) d Si = 2 π Jy dy COS a = Here z2x2+ y2 is the distance of each separate point (such as Q) of the elementary plane ring from P. The angle QPO is denoted by a, so that cos a = x/z; and consequently the above equation can be written in the following form: it follows that, for a given position of the point P, that is, for a constant value of x, z dz = y dy And if we substitute this in (12), we obtain This expression has to be integrated over the entire surface which forms one face of the slit, in order to find the resultant magnetic intensity , at P, arising from the action of that face; we shall have, then, The two limits of this definite integral correspond to the centre and the circumference of the surface in question; performing the integration we obtain 1 2 We are now in a position to calculate the change of magnetic potential in passing from the centre of the surface 1 to the point P: denoting this quantity by T11, we have udu = xdx, and if we then take u as a new variable under the last sign of integration, the limits of the definite integral become r, and 2+ r2, and we obtain Let us now return to the consideration of the slit, that is of the region bounded by the pair of surfaces 1 and 2, whose distance apart is d (fig. 16, p. 112); in (15) we must put x = d, and since TT + T12 and the magnetic fields due to the two surfaces are directed in the same sense, we have A = il 12 = 4 π I (d + r1⁄2 √ ď2 + r22) (16) ET, — Thus we have found a second expression for the left-hand member of the fundamental equation (I); inserting it in its proper place we obtain 4π I (d + 12 −√/ û2 + r22) = Ñ ̧ 3 (2 π r1 — d) In the course of the proof of this expression given by the author (Wied. Ann. vol. 46, p. 494, equation (8), 1892), an incorrect term crept in through an error in transcription; this has, however, no influence on the result there obtained. I ∞ The suffix indicates that the value N, strictly speaking, is only attained when the intensity of the magnetising field becomes infinitely great, for it is then alone that our initial assumption holds good. If we now divide the last equation by the factors of N∞, we obtain for this limiting value of the factor of demagnetisation (II) N = ∞ 2 (d + r2 — √ đ2 + r22) ∞ d 2π In accordance with the assumption made at the beginning of this section, the values of N as measured experimentally must approach the value of N given by the expression (II) as the intensity of the magnetising field is made to increase. In practice, however, we cannot reach the limiting case of a uniform peripheral distribution of magnetisation, for the intensity of field with which we can actually work is limited by the overheating of the magnetising coil, and amounts to only a few hundred C.G.S. units. To what extent the approximation is applicable in the most favourable case is a question which must be decided by experiment (compare § 89). § 77. Divergence of the Lines of Induction.-Ordinarily speaking, nearly all the cases with which we have to deal will be those in which we can make use of the assumption of § 59, namely, that to a sufficient degree of approximation I = 2/4π; hence the former quantity may be regarded as having a solenoidal distribution, for under all circumstances the latter possesses this property. Now the distribution of B, can be everywhere represented by means of the lines of induction, and the same lines may therefore be made to furnish a complete representation of the distribution 3, for the constant factor 4 constitutes the only difference between these two quantities. The simplification thus resulting from our assumption makes it far easier to obtain a general notion of the relations which obtain in our problem. We shall also find our work simplified by remarking that, when the intensity of the magnetising field is small, the permeability may always be regarded as a large number. DIVERGENCE OF THE LINES OF INDUCTION 115 Let us consider from this point of view the distribution of I and B, in a divided toroid. The demagnetising effect due to the presence of magnetically effective end-elements at the bounding-faces of the slit will be most perceptible in the neighbourhood of the slit itself. In the foregoing we have generally taken account only of the mean effect over the whole surface of the toroid. The magnetisation, and therefore also the resultant induction, will for this reason have somewhat smaller values in S FIG. 17 the neighbourhood of the slit; and this we may picture by the fact that the tubes of resultant induction, which along the greater part of the circumference of the toroid are almost exactly in the form of co-axial circles, spread out as they approach the slit, so that there is there a divergence among the lines of induction. The lines nearest to the convex surface of the ring meet it at an acute angle, and then proceed into the surrounding medium in a direction nearly coincident |