W N NW In the NE feeble magnetisations, for which the leakage is most considerable, the greater part of the leakage takes place within the short length which lies between NW and NE, and which contains the air-gap. case of the strongest magnetisation employed (the leakage-coefficient being considerably smaller), the property in question may be expressed by saying that up to the point NE or NW no considerable change occurs in the flux E of induction through the cross-section of the toroid. The distribution of the induction is therefore sensibly uniformperipheral over more than three-fourths of the circumference, and the uniformity of distribution will be the greater the higher the value which is reached by the induction. § 89. Comparison of Theory and Experiment. We are now in a position to compare the results of the experiments described above with the conclusions of the theory previously developed. In equation (III) (§ 80) ᏚᎳ S FIG. 23 SE COMPARISON OF THEORY WITH EXPERIMENT 135 we have a relation between the mean factor of demagnetisation. N and the function n, which is the reciprocal of the leakagecoefficient v. This last quantity, moreover, is represented graphically in fig. 22, p. 133, as a function of the magnetisation for the four widths of the gap (2, 3, 4, 5) which were employed. Again from (III) we easily obtain for the lines of demagnetisation the following equation : which now enables us to construct the lines in question from the curves v = funct. (3) of fig. 22. On the left-hand portion of fig. 21, p. 131, the lines of demagnetisation constructed in this manner are shown. The lines (2) and (3) are continued as far as the ordinate I = 1500, because for higher values of 3 the assumed reciprocity of n and v ceases to hold good with sufficient approximation (compare § 82). On the other hand (4) and (5) are only drawn for the range covered by the directly observed points.' As will be seen, these points lie approximately on the lines of demagnetisation. Thus, the theory leads to a satisfactory coincidence of the lines of demagnetisation plotted from measurements of the leakage with the curves of magnetisation which were determined by an entirely independent method. Fig. 21 also gives for the three narrowest air-gaps, (1), (2), (3), between the values = 1000 and 3 = 1750 C.G.S., the straight lines of demagnetisation whose equation is (29) $ = where N denotes that factor of demagnetisation which is to be found from equation II (§ 76), and which, in accordance with the assumption there made, is, strictly speaking, only applicable for infinitely high values of He. From fig. 21, p. 131, it will now be observed how the values of N, calculated from the measurements of leakage by means of 1 ν 1 Fig. 22 does not give the function = funct. (3) for the gap (1), but, as we shall see in the next section, this can be found by interpolation. The line of demagnetisation (1) in fig. 21 was obtained in this manner. equation III, and, as we have seen, according well with observation, tend also to the limit N as the saturation value 3m 1750 C.G.S. is approached. The lines of magnetisation may, in fact, be produced as in the dotted curves, so as to pass through the points A1, A2, A3. In Table V the somewhat complicated relations under discussion are collected in a form convenient for reference, so far as they correspond to the range of magnetisation 3 = 0 to 3 = 875, in which both the leakage-coefficient v and the factor of demagnetisation N may be considered as constant. The meaning of each column will be sufficiently clear from its heading. TABLE V The agreement between the calculated and the observed values of N is as good as could be expected, when we remember that, on the one hand, the theory deals only with mean values and approximations, and that, on the other hand, the sources of experimental error, especially in relation to the exact form of the gap, may easily introduce an uncertainty of several per cents. Finally, then, we may consider the theory here developed as confirmed by experiment with sufficient accuracy for most purposes; while the experimental data furnish us with the means of determining the function n or its reciprocal v, the To express the condition that magnetisation of the body is to be near the point of saturation, so that Kirchhoff's law of saturation becomes approximately applicable, it was supposed in § 57 that was small in comparison with e (compare Culmann, Wied. Ann. vol. 48, p. 380, 1893). From fig. 21 it will be seen that in reality for J 1500 C.G.S., He was of an order of magni 2 Value found by interpolation; compare the following section. EMPIRICAL FORMULA FOR THE LEAKAGE 137 leakage-coefficient. These two quantities had to le provisionally introduced into our theory as unknown (§§ 78, 80). § 90. Empirical Formula for the Leakage.—On introducing the function n, we denoted it (§ 80) symbolically by which expresses the fact that it depends only on the ratio r2/d determined by the shape of the gap, and not on the radius. of the entire toroid. In the following discussion we shall confine our attention to the range of magnetisation 30 to 3875, which for practical applications is the most important; we can then put n = 1/v, and consider both these quantities to be independent of the magnetisation. The question then arises how the leakage-coefficient v (or the function n) depends on the shape of the air-gap, as determined by the ratio d/r2 (or r/d). In order to obtain the experimental answer to this question, the ordinates v in fig. 22, p. 133, are also plotted as a function of d/2; the second (upper) scale for abscissæ is introduced so as to allow this relation to be read off. We thus arrive at the empirical rule that, within the limits of experimental error, the four observed points lie on a straight line. This line cuts the axis of ordinates for the value v=1, the corresponding law being that, when the width of the gap is reduced to zero, the leakage vanishes. The equation to the straight line is empirically found to be This formula for n = funct. (2/) is represented graphically by an hyperbola. In the square brackets following equations (30) and (31) is given the range of the independent variable within which the corresponding formulæ hold good. Too much weight must not be attached to such purely empirical relations as these. For our physical insight into the phenomena they are quite useless; but, on the other hand, in practical applications it is useful to have at least some method for roughly estimating leakage-coefficients. We shall return to the consideration of this empirical formula for the leakage in § 173. In the present case, the formula can be applied to find by interpolation the value of the leakage-coefficient v 1.31 for the gap (1), for which it was not directly measured. This has already been done in Table V. From the researches here described, it is clearly established that in a radially divided toroid the leakage for moderate values of the magnetisation (0 << 875) remains nearly constant, while beyond this range the leakage decreases with increasing magnetisation. It also follows from our theory that this must be the case. To show this, let us consider once more figs. 13, p. 87, and 17, p. 115. The acute angle (90 a'), which the lines of induction within the toroid make with its bounding-surface, will become still more acute as the magnetisation increases beyond a certain value, since the peripherally directed distribution of the magnetisation (§ 76), and therefore also of the induction, tends to become more completely established in accordance with Kirchhoff's law of saturation. Let us further consider the tangent law of refraction for lines of induction, in accordance with which We have just seen that, as the magnetisation increases, a' becomes greater, while, on the other hand, the permeability becomes smaller (§ 14); that is, 1/μ becomes greater. These two causes conspire to increase the value of a, so that the lines of induction in the external medium will deviate further and further from the normal to the bounding-surface, the leakage thus becoming smaller, as was found by experiment to be actually the case. |