obtained by adding the abscissæ; it is now represented by the dotted curve (C'). The construction formerly given (§ 17), the analogy of which with the Hopkinson method was thoroughly discussed, undergoes quite similar modifications by taking leakage into account (fig. 21, p. 131), so that the agreement of the two methods is the same as before. We can therefore regard the confirmation of the author's theory (Chapter V.) by the experiments of Lehmann, made expressly with this view, also as a confirmation of the Hopkinson method-at any rate in so far as it deals with the simple case of a toroid with one radial slit. Of the two methods, which at first appear totally different, sometimes the one has the advantage, and sometimes the other, from the point of view of practical application. § 100. Generalisation of the Method. The Hopkinson method thus has the advantage that it can be directly applied to imperfect magnetic circuits of a more general kind than the typical example previously considered. We have already treated theoretically the case of a toroid with several radial slits, taking leakage into account (§ 81). Introducing the generalisation of § 15, according to which the curved axis of the ring may be any arbitrary plane or tortuous curve, provided only its radius of curvature is always great compared with the greatest diameter of the normal cross-section, this latter may have any arbitrary though invariable form. If, finally, we remove these latter restrictions also, so that the curve may now be sharply bent, and the shape, as well as the area of the cross-section, may be variable, and if we further assume that the ferromagnetic parts consist of different materials, we have obviously an imperfect magnetic circuit of the most general kind possible. 0 We now start from the mean flux of induction & in one of the gaps, the length and sectional area of which are L, and S respectively. In the other parts, 1, 2, 3, &c., into which we can separate the magnetic circuit, let the values of the flux of induction be G o, v2 Go, v3 Go, and so on. In like manner, let the particular functions f, which are characteristic of the various ferromagnetic substances forming those parts, be ƒ1, ƒ2, ƒ3, &c. [§ 96, equation (3a)]. If we call the corresponding lengths of path L1, L2, L., &c., and the sectional areas S1, S2, S3, &c., we V ELECTROMAGNETIC STRESS 155 ultimately obtain from equation (II), p. 153, the more general In so far as, besides the parts of the magnetic circuit represented by the first term, there are others of indifferent material for instance, the nth part-to each of these parts there will correspond a linear term since in this case the function f is simply equal to its independent variable. This most general equation has, so far, been scarcely tested by purely magnetic experiments. Any attempt at an accurate test would, moreover, be useless, as the method is intended for practical requirements, and, of course, only allows of mean values and rough approximations. We shall revert to this in Chapter VIII., in discussing the chief applications in connection with dynamos, and show how the Drs. Hopkinson succeeded in applying their determination of the electromotive force of the machine as a function of the current in the field-magnets to give a test of the theory. By their measurements the approximate correctness of the synthetical method could, on the whole, be confirmed (compare especially §§ 128, 129). C. Electromagnetic Stress § 101. Specification of the state of Stress. We will now consider more closely the state of stress which occurs in the ferromagnetic parts of magnetic circuits, or which prevails in the gaps between them. The latter explains the well-known apparent action at a distance, which appears, in general, as an attraction or repulsion between the ferromagnetic parts. To these considerations we will preface a few elementary definitions. By stress is understood, in general, a system of forces which tend, not to move a body, but to strain it. The stress produces, in general, a strain. The closer investigation of the latter, or of its relations to the stress, constitutes a problem of geometry, or of the theory of elasticity respectively. 1 Every stress is to be expressed as a force per unit area, and has therefore, in absolute measure, the dimensions [L-1MT-2]. There are various elementary forms of stress; the most important are shearing stress, pull or tension, and thrust or pressure. That being premised, we may mention that in the theoretical part (§ 65) we have already expressed in a few equations the most general state of stress in a magnetic body, as deduced mathematically by Maxwell. Making the simplification that the induction B and the magnetic intensity' have the same direction, which, on the assumption always made of isotropy and absence of hysteresis (§ 54), does actually hold, those equations assume an elementary form; and, in accordance therewith, the stress may be completely specified in terms of the two following elementary forms: I. A (hydrostatic) pressure, the same in all directions, and equal in absolute measure to 12/8π. II. A simple tension in the direction of the lines of induction, whose value in absolute measure is B''/4π. The investigation of the strains 2 due to the electromagnetic stress concerns the theory of elasticity. The test whether such strain is admissible, from the point of view of construction, is a question of strength of materials. From neither of these points of view need we follow the question in this book; but its importance is apparent from what has been said, for, as we shall see, the stress may in some circumstances be very great. § 102. Resultant Tension in the Gap.-In the preceding paragraph the stress in a ferromagnetic body is fully specified. In the sequel we shall consider only that one of its manifestations which is of greatest experimental and practical interest; we 1 Since, in the sequel we shall only deal with the resultant intensity, and the resultant induction, we shall drop the prefix 'resultant' as well as the corresponding index t from the corresponding symbols (§ 53). 2 We have already alluded to stresses which occur in magnetisation, and to the small charges of size and shape of ferromagnetic bodies due (at any rate partly) to them. RESULTANT TENSION IN THE GAP 157 shall confine ourselves to an investigation of the resultant tension in the direction of the lines of induction. This vector (in the ferromagnetic substance) we shall call 3'. We obtain its value if, from that of the simple tension in the direction of the lines of induction (2), we subtract the pressure mentioned in (1), which of course acts in the direction in question, as well as in all others. We accordingly obtain the equation Let us now consider an infinitely narrow slit in the ferromagnetic substance at right angles to the direction of the lines. of induction and intensity, which does not produce either demagnetising actions or leakage. As this will in practice be always more or less the case, we shall speak of such a break in the continuity of the ferromagnetic substance as an ideal slit. Hand B are there identical, as in an indifferent substance, and are equal to B' in the ferromagnetic substance, as follows from the principle of the normal continuity of induction (§ 58); hence the resultant tension 3 in the ideal slit itself is This value for 3 in the narrow gap is considerably greater than that for 3' in the ferromagnetic substance. We now take the difference of the two quantities and introduce the magnetisation, remembering the fundamental equation By subtracting the above equation (10) from (IV) we obtain This difference 27 32 between the values of the resultant tension in the ferromagnetic substance and in the ideal slit, can only be due to the two terminal faces of the latter. On these, magnetic end-elements are present, the strength of which per unit surface is + 3 or 3. As we have already proved (§ 21), an apparent attraction is exerted by the magnetic end-elements of one surface on those of the other, the latter being of opposite sign. In the language of the old theory, the two faces, being charged with imaginary fluids of opposite signs and uniform density, attract one another, just like the two plates of a plane air-condenser charged to the electric surface-density ± 3. This attraction may be simply calculated,' and has the value, 232 per unit area, which we found in equation (11). § 103. Theoretical Lifting Force of a Diametrically-divided Toroid. Let us first consider for the sake of simplicity a toroid FIG. 26 uniformly and closely wound, and having a uniform peripheral distribution of magnetisation, while at the same time it is cut through radially at two diametrically opposite places (fig. 26). Let us assume the faces quite smooth, and polished, so that the two halves of the toroid fit closely to each other. The width of the cut is then as small as possible, so that it differs as little as possible from the ideal cut postulated in the last paragraph. Its demagnetising action as well as the leakage we may assume to be infinitely small; how far this is actually the case will be subsequently discussed. Let us now investigate the attraction which the two halves exert on each other—that is, supposing we determine it by hanging weights to the lower half, 1 See, for instance, Mascart and Joubert, Electricity and Magnetism, vol. 1, § 81. 2 Compare the theory of toroids with more than one radial slit, § 81. |