SELF-COMPENSATING EFFECT OF LEAKAGE

147

The self-compensating effect of leakage in magnetic circuits affords a certain analogy with the demagnetising tendency of the faces which bound the interspace. Both actions may, as follows, be brought under one point of view. We have seen (§ 50) that any apparent action at a distance is due to local variations in the strength' of the magnetisation. Now it may easily be shown that the magnetic action at a distance arising from these variations, is opposite in direction to the magnetisation at places where the strength is greater, while at places of less strength it is in the same direction; the local variations will therefore indirectly tend to partially neutralise themselves.

In the experiments which have been described the field was so weak as to satisfy the conditions of § 11 (equation 14), and therefore the magnetisation was proportional to the induction, and, like this vector, it had a solenoidal distribution. It is accordingly sufficient to assume magnetic end-elements on the lateral surface as the direct result of leakage. The theoretical formulation last mentioned includes also, such variations in the strength, as take place in the interior of the ferromagnetic substance when the distribution of the magnetisation is no longer solenoidal, so that its convergence is finite, as may readily occur with higher intensities of field (compare §§ 11, 59).

The leakage in the latter case cannot, in general, be determined, nor can we fix the limiting conditions corresponding to a magnetising field whose intensity increases without limit. In any case, however, Kirchhoff's law of saturation will then hold. The question, then, is how the lines of magnetic intensity of the external field would run in reference to the geometrical configuration of the magnetic circuit; for in the limiting case, in accordance with the principle in question, these alone will completely determine and direct the other vectors concerned. The answer to this question depends on the nature of the special case we consider, and will usually present considerable difficulties. Even in the simple case of a toroid magnetised by a local coil covering one special part, it depends on the relation of the dimensions of the coil to the diameter of the toroid; so that any general enun

Compare in this connection the graphical representation of the lines of intensity of a single circular conductor, Maxwell, Treatise, vol. 2, Plate XVIII.

ciation, not to speak of a solution, of the problem, applicable to all cases, seems out of the question.

Finally, since it has been experimentally established and theoretically explained how the value of the vector H, is sensibly constant throughout a ring magnetised by a local coil, the question arises how the mean value of that vector can be calculated. To this end, we observe that the line-integral of H along a path of integration lying within the toroid is, as before, 4πn I, in which n is the number of turns of the local coil, I the current in deca-ampères (i.e. in absolute C.G.S. measure). The line-integral of 5 is, however, zero in this as in all other cases. The mean value is found by dividing the mean circumference of the toroid (2π1) into the sum of the two lineintegrals just mentioned, which sum, in this case, evidently also amounts to 4 Tn I. Accordingly, π

the same expression which we previously found for the case of a uniform winding [§ 72, equation (1)], that is for a uniform peripheral distribution of the magnetising field. We shall consequently drop, in the sequel, the tacit or express assumption that the winding is uniform.

B. Hopkinson's Synthetic Method

§ 96. Principles of the Method.—We now turn to a mode of treating the magnetic circuit, as ingenious as it is fruitful, which was published in 1886 by Drs. J. and E. Hopkinson.' It depends on two fundamental ideas, each of which, again, is based on a mathematical theorem capable of rigorous proof.

The starting-point is the consideration of the total flux of induction, the conservation' of which is the first of the principles referred to. This we have already fully discussedor, what amounts to the same thing, we have established the general solenoidal property of the resultant induction, as

1 J. and E. Hopkinson, Phil. Trans. vol. 177, I. p. 331, 1886. Reprinted in J. Hopkinson, Original Papers on Dynamo Machinery and Allied Subjects, p. 79, New York 1893.

HOPKINSON'S SYNTHETIC METHOD

149

expressed by the equation of continuity, and have shown how this principle governs a series of phenomena (§§ 60-65).

In the second place, the fundamental principle is applied that the line-integral of the resultant magnetic intensity H, along any closed curve is 4πn I, where n is the number of turns of the conductor which embrace the curve, I the current which passes through all of them (§ 56).

The magnetic circuit is separated into its natural parts, through which the curve of integration successively passes. To each such part of the circuit corresponds a portion of the lineintegral in question, the integral being calculated by multiplying the mean value, for each part by the corresponding length of the path of integration. An essential assumption here tacitly made is the tendency of the resultant induction F1, already discussed, to become distributed as uniformly as possible, so that its variations along any one such portion of the curve are but small. From B, we find H, by equation (3a) p. 147.

t

П

The several portions of the integral are finally added, and their sum must amount to 4πn I. We can thus ascertain synthetically the value of the line-integral corresponding to any given or prescribed value of the total flux of induction &1, which we will call M. The relation between these two quantities M and 6, we can represent graphically; and, according to the practice of the Drs. Hopkinson, it is usual to choose the values of the former as abscissæ and those of the latter as ordinates. The curve thus obtained, which represents the 'Hopkinson's function,' (2) M = FH (6) or &1 = ÞH (M) may be called the magnetic characteristic of the corresponding magnetic circuit.

ΦΗ

In order that the method, here represented in its general features, may be available, the relation between the vectors P, and 5, which, as we know, are coincident in direction (§ 54) at each point, must be given for the ferromagnetic substance in question. We can represent it by the equation

H

H

The functions ƒ and ø, as well as F and , are ‘inverse functions.' The former are assumed to be given empirically by the normal curve of induction for the material in question.

§ 97. Application to Radially-divided Toroids.As it may at first prove somewhat difficult to understand Hopkinson's synthetical method, we will once more explain the mode of applying it in the typical case of a toroid with a radial slit. We shall then ultimately see that it leads to the same results as the method which we have developed in Chapter V., and which, at first sight, appears to be totally different.

Using the same notation as before (§ 75, fig. 15), we assume as a first approximation that the width of the slit d is small, so that the leakage may be neglected. Or, as the Drs. Hopkinson say, we suppose that, by some miracle, the tubes of induction are prevented from emerging from the surface of the toroid, so that they can only pass from one face to the other through the Let S [22] be the cross-section of the toroid, as well gap. Let S as of the slit; then, on the above assumption,2

=

Let us now evaluate the separate portions of the line-integral mentioned above. For this we first consider the slit, where B.; consequently (see fig. 15, p. 109),

(4)

=

Next, in the remaining ferromagnetic part of the toroid, by introducing the mean value of the resultant intensity Hi,

(5) Sid L = $ (2 πη

d) = (2 = r; − d) ƒ (;)

E

in which ƒ is the function defined by equation (3a). The sum of the two portions (4) and (5) of the integral must, from the

=

1 In Hopkinson's paper is put f-1 in conformity with the usual notation for inverse functions.

2 It should be remembered that when a symbol is accented, it denotes the value of the corresponding quantity within the ferromagnetic substance.

GRAPHICAL REPRESENTATION

151

preceding paragraph, amount to M = 4T In. Hence we finally

This equation represents Hopkinson's solution of the problem of the radially-divided toroid.

§ 98. Graphical Representation. Transformation of Curves.— In fig. 25 this solution is graphically represented for a concrete case. The toroid is assumed to be of the specimen of iron whose normal curve of magnetisation is represented in fig. 5,

The two terms of equation (I) corresponding to the two portions of the integral are now represented by curves (A) and (B), the first of which is obviously a straight line through the origin. If then we add the abscissæ of these curves (A) and (B), we obtain a third one (C), which represents the relation sought for between , and M, and which is therefore the magnetic characteristic of the divided toroid.

1