former may be deduced in the usual way from the difference of abscissæ A of the two curves of magnetisation. For the purpose of this calculation we will consider the bar in its closed yoke (fig. 38, p. 235); in first approximation as a toroid, slit radially, the perimeter of which is 2πr1 = L (the length of the bar). In so narrow a slit the simple equation (VII) § 82 may be used putting this value into the equation AH = NI, we find By means of the last equation Ewing and Low calculated the width of the equivalent air-gap, which for two iron bars was found equal to about 0.03 mm. This value was found to be fairly independent of the magnetisation, as is shown by the almost straight line in fig. 37, which represents a value N0003. It is difficult to decide whether that equivalent air-gap did, in fact, represent the mean distance of the faces, although such a considerable value of the latter is not probable. It can, with certainty, be alleged that a surface produced in the manner described differs materially from a highly polished plane mirror, and still more from a geometrical plane. Ewing and Low have further confirmed that the interposition of a single gold foil had not any further appreciable influence on the magnetic reluctance. The thickness of such a foil amounts, as we know, to only a fraction of the wave-length of sodium light—that is, it is of the same order of magnitude as that by which a good metal mirror differs from an absolute geometrical plane. INFLUENCE OF LONGITUDINAL PRESSURE 235 § 152. Influence of Applied Longitudinal Pressure. -- The same experimenters then investigated the influence of longitudinal pressure on the magnetic behaviour of a joint between two halves of a bar by the apparatus represented in fig. 38. At the same time they examined how far the magnetic properties of the material itself were affected by pressure (§ 107). They found that the demagnetising action-or, in other words, the magnetic reluctance-of a joint arranged as above decreases with increase of pressure, and that for a pressure of over 200 kg.-weight per sq. cm. no difference could be detected between a divided and an undivided bar. In this case, as was to be expected, the interposition of a gold foil had a small, but appreciable, influence. It is here to be observed that the magnetic pull (§ 102) was less than 1 kg.-weight per sq. cm., so that it need scarcely be considered of any influence in comparison with the above value of the external pressure. Ewing and Low further made experiments with rough joints that is to say, such as were bounded by surfaces simply turned and not further fitted. It was found that the width of the equivalent air-gap for such rough joints was about 0.05 mm., and was only reduced to 0.04 mm. by a pressure which would have entirely nullified the magnetic reluctance of a joint between properly prepared faces. Experiments were made with bars which had not one transverse joint only, but three to seven joints. The results given offer considerable practical interest. They lead to the rule that the less a magnetic circuit is to differ from a closed one, the more carefully must the detrimental magnetic reluctance of superfluous joints be avoided; that, moreover, where, for constructive reasons, joints cannot be avoided, the surfaces must be carefully fitted and pressed against each other with great force. In magnetic circuits with a wide air-gap an additional slit of 1 mm. equivalent width is not of much moment, so that in such cases the joints have scarcely any influence.' § 153. Time-variations of the Magnetic Conditions. — We have hitherto limited ourselves to the consideration of invariable lasting magnetic conditions-that is, to the case of stationary magnetisation. We have, however, incidentally (§ 64) discussed the induction of electromotive forces E in consequence of varying magnetisation; it was there mentioned that this may be expressed in absolute measure, as the time-rate of variation of n-times the flux of induction & where it is encircled n-times by the conductor; that is, We have then, further, exclusively considered the timeintegral of E-or, in other words, the total current impulse corresponding to a variation —quite apart from its timevariation, and which affords the most suitable method of measuring it. We shall now investigate more minutely variations of this kind, and we shall again elucidate the somewhat complicated phenomena which occur by the example of a toroid, either closed or divided radially, wound uniformly with n turns (resistance R, radius of the toroid r1, perimeter 2 πr1 = L, crosssection S). At the beginning of the time (T = 0) let a constant external electromotive force E. act suddenly on such an induction coil'; this would correspond to the steady current IER, to the immediate establishment of which there is, however, an impediment in the form of a self-induced counter 6 1 Compare an investigation by Czermak and Hausmaninger, Wiener Berichte, vol. 98, 2 Abth., p. 1142, 1889. T being the time, and I the actual current. d (n ) dl Besides the constants E and R, and the single variable 1, the current flowing in each instant, the derived function or differential coefficient d (n )/d I occurs, which plays an important part in what follows. It is quite generally called the coefficient of self-induction, or the self-inductance, ▲ of the coil, independently of the simple arrangement assumed in the present example. From the well-known equations we obtain, in accordance with the above definition, (8) = 2 n2 S dB 2 The variable coefficient 4 depends then, in the first place, on the geometrical configuration of the coil, and has the dimension S/L that is, a length. It is, further, proportional to the differential coefficient of the induction B in the ferromagnetic core, with respect to the intensity 5. of the field of the coil. We shall, therefore, at once specially consider this differential coefficient. § 154. Discussion of the Function d B/de.-In the first place, we get from the fundamental equation [§ 11, equation (13)] B = 4π I + He 1 We shall here use suffixes similar to those in § 53. Parasitic (Foucault's) eddy currents in the ferromagnetic substance itself will be expressly disregarded in the sequel; for this purpose we may consider it as having infinitely small electrical conductivity, or as being divided so that it has an infinitely fine-grained texture. Further, the electrostatic capacity of the coil will be disregarded, and its resistance R will be considered constant-that is, the current will be assumed constant for the whole length of the conductor, and uniformly distributed throughout its cross section. In the extremely rapid alternations of current, to which the attention of experimenters is now being pre-eminently directed, these assumptions are inadmissible; but in the present case we are considering variations which are comparatively slow. 2 The C.G.S. unit for self-inductance, therefore, is of course the centimetre; the official delegates to the International Congress of Electricians at Chicago in 1893 fixed the henry (109 cm.) as practical unit, a length which hitherto has also been termed the Secohm or Quadrant. In so far as we may neglect the second term on the right of the former equation, in comparison with the first, we can do so with the corresponding unit in the second equation. From the value of [dB/d H.] or of [d I1 / d e], for a given value of the induction B, or the magnetisation 3, with a closed toroid,' we shall deduce that which, ceteris paribus, corresponds to a finite demagnetising factor N. Apart from the algebraical sign [§ 53, equation (1)], we have He = Hi + Hi = [He] + NI By differentiation with respect to I we obtain which also results from the fact that by shearing,' the tangent of the angle of inclination of each element of the curve to the axis of ordinates is increased by an amount proportional to N. If now the latter equation is put in (9), we have The general graph of the function [dB/dH.], or of dB/dHe, which, when there is hysteresis, is no longer a single-valued function, is represented in fig. 36, B, p. 225. These curves exactly correspond to the hysterestic cycle for annealed steel wire represented in A (§ 147), and, therefore, scarcely require any further explanation. The full-line curves for a closed toroid show, in the first place, a characteristic prominence, which is still more strongly marked with soft iron, and is therefore difficult to represent (cf. fig. 41, p. 243); and, in the second place, a discontinuity corresponding to G, the top of the hysteresis loop. The full-line curve The expressions referring to closed toroids, N = 0, are in the sequel put in square brackets. Knott has proposed the names differential susceptibility,' or respectively permeability,' for the differential coefficients [dI / d He] and [d B1 d He], which we, however, shall not adopt here. |