thousand C.G.S. less than the theoretical. measurements were obtained:

The following

for 13 = 2.5 mm. 36,800 C.G.S.

for r3 = 1.5 mm. 38,000 C.G.S.1

It follows, from the formula of the previous paragraph, and is confirmed by experiment, that high intensity of the field is only attained at the cost of its extent. For many experiments, however, an extent of several sq. mm. is sufficient, or else the methods of investigation must be adapted to satisfy this

condition.

The bore-holes, which are indispensable in magneto-optical experiments, produce relatively more weakening of the field the wider they are as compared with the distance of the faces. This weakening is, however, less than would follow from the equations in § 174, since there is a kind of internal leakage from the edge of the openings towards the axis. The external leakage and action at a distance when truncated cones are used are similar to that described for plane poles.

What has been stated in the previous section may be summed up by saying that a ring electromagnet of manageable size, with truncated conical pole-pieces of 120° aperture, enables us to have fields up to, say, 40,000 C.G.S. over an extent of some square millimetres. To exceed this to any material extent could at present be only accomplished by an undue expenditure of means out of all proportion with the end in view. That follows already from the formula given in which log (7) comes in ; while the weight and cost of an electromagnet are determined rather by the third power of its linear dimesions.

D. Inductors and Transformers

§ 176. Discussion of Mutually Inducing Coils. We have already explained the principal manifestations of self-induction by the example of a uniformly coiled toroid, either closed or divided 1 This field would produce in a piece of thin soft iron between the truncated cones the approximate induction equal to

B = 38,000+ 4 × 1750 = 60,000 C.G.S.

to which would correspond a total longitudinal stress equal to (§ 103)

MUTUAL INDUCTION

273

radially (mean radius r1, perimeter 2πr, = L, section S). Besides that primary coil 1 (resistance R1, number of turns n1, self-inductance 4,), let there be a secondary one 2 (R2, 12, 4,), also wound uniformly on the toroid, as would be the case with the experimental ring described in § 83. Taking the case of such a pair of mutually inducing coils, we will explain as briefly as possible the most important facts in mutual induction, in so far as they are essential for understanding what follows.

To a variation of the flux of induction 6, produced by the primary current corresponds an electromotive force E2 in the secondary, which, from equation (6), § 153, will have the following value:

The partial differential quotient of (n, G1) in respect of the primary current I1, which here occurs, is called, in analogy with the definition of § 153, the mutual inductance E12 between the coil 1 and the coil 2. We find, in the present case,

12

If now the parts played by each coil in the phenomenon are interchanged, so that 2 now acts inductively on 1, we have, in an analogous manner,

12

21

L

d He

=

§ 177. Mutual Induction.-Now it is obvious from the above that E12 = 21. Hence this quantity may be called, without further definition, the 'mutual inductance' of the two coils.' From (39) and (41) we see, directly,

that for n ̧n, we have 4, 4,

Mutual as well as self inductance has the dimensions of a length, and

like this is to be expressed in henries (compare note, p. 237).

T

(42)

Further, for any given value of B we have

These conclusions are based on the assumption that in (39) and (41) S as well as dB / d He are identical; that is, in other words, the common flux of induction threading through the two coils is the same. When a ferromagnetic core is used, this is always the case with sufficient approximation, since the induction tubes outside it produce no appreciable difference (compare, however, § 183).1

As the mutual inductance E differs from the differential quotient d B/d, by a constant factor only, it will be sufficient to refer to § 154 and fig. 36, B, p. 225, where the character of that differential quotient is completely discussed. For closed magnetic circuits there can be no question of a constancy for E, any more than there can for 4; for unclosed magnetic circuits, on the contrary, that assumption is here also the more admissible, the greater the value of the demagnetising factor.

2

3

By means of an arrangement quite similar to that here assumed, in which, however, each of the two coils was severally placed on one half of a ring, Faraday, as is well known, discovered, in 1831, the existence of induction phenomena. Almost simultaneously, and quite independently of him, J. Henry 3 also carried out his 'fundamental researches in this department, more especially on 'extra current,' as it was called; he already introduced the expression self-induction.' The kind of apparatus used by these experimenters had undergone a long course of development, until, after the lapse of half a century, a division into two classes was made, inductoriums, or induction coils, and transformers, of which the first has mainly a scientific interest; in the decade which has elapsed since that separation, the latter class has from its technical importance undergone an

4

1 The circumstances are quite different with two coils without cores, in any given relative position; such cases, however, may be disregarded for our present purposes.

Faraday, Exp. Res. vol. 1, Series I. § 27 (Plate I, fig. 1).

3 J. Henry, Collect. Scient. Writings, vol. 1, p. 73 et seq.; Silliman, Americ. Journ. vol. 22, p. 403, 1832.

G. Wiedemann, loc. cit. vol. 4, §§ 409-430; Fleming, loc. cit. vol. 2, chapter i.

ACTION OF INDUCTION COILS

275

extraordinary development, both in the theoretical relations and in the practical details of construction.

§ 178. Action of Induction Coils.—In an induction apparatus, in the narrower sense of the word, extremely high electromotive forces are induced in a secondary coil, which has a great many turns, by making or breaking the primary current. These are mostly used to produce electric discharges of various kinds between the ends of this coil. The discharge on making-corresponding to the gradual rise of a given primary current, as described in § 155, starts a quantity of electricity in a closed secondary coil, equal to the break discharge which takes place in the opposite direction; because in both cases the same total variation G of the flux of induction and the same resistance are concerned [§ 64, equation (14)]. Although, then, the time-integral of the secondary electromotive force is in both cases the same, its maximum value is far greater on breaking the primary current; for this process, though never instantaneous, is, however, considerably more rapid than the gradual increase of the current up to a large fraction of its steady value, which takes place on making (§ 15). In the case therefore of spark discharges, only the break-spark' passes across wide spaces of air; with the same distance of the electrodes the 'make-spark' cannot traverse the air-path. It follows from this, that it is of paramount importance to break the primary as rapidly as possible; for this purpose the following means are more especially applied.

In the first place, contact-breakers are used, which reduce as much as possible the primary spark due to the self-induction of the primary coil, which prevents an absolutely instantaneous break of the current. Experience shows that in this respect mercury contact-breakers are best, working under insulating liquids, such as alcohol or petroleum. (Note 2, p. 265.)

In the second place, it is usual to adopt Fizeau's plan of interposing a condenser near the break. This diminishes the injurious spark by partially storing up the primary discharge. For any given primary coil the most suitable capacity, that which most effectually extinguishes the primary spark, while it prolongs the secondary one, is to be found by experiment. This probably depends on one of the phenomena of resonance which

by the brilliant discoveries of Hertz have recently claimed prominent attention. The oscillating primary discharge is then, as it were, thrown back by the condenser, and produces a primary current in the opposite direction; in this way it is obvious that longer secondary break-sparks are formed than when the primary current merely vanishes without altering its direction.

§ 179. Magnetic Circuit of Induction Coils. It is not sufficient that the break or the reversal be as rapid as possible; it is also necessary that the corresponding variation of the flux of induction in the core shall directly follow it.

E

=

Let us assume at first that the primary current only falls to zero; for the induction in the secondary coil the corresponding decrease of the flux of induction-that is to say, the evanescent magnetisation-denoted in § 149 by 3-is of paramount importance. Let us imagine, for example, a closed core of good soft wrought iron; with an intensity of H. = 20 C.G.S., let the magnetisation 3 1000 be attained; as the retentivity in this case might easily amount to 90 per cent., 3 would be =900, hence 3 would only equal 100 (see § 149 and fig. 36, A, p. 225). The case is different with an unclosed core, the demagnetising factor of which we suppose to be, say, N = 0·02, corresponding to (m 45, Table I, p. 41); taking the coercive intensity 2 C.G.S., we have 3 = c/ N= Sc 100, and hence, JE = 900. At the same time a demagnetising intensity Hi = N 3r 20 will have to be compensated, so that the field of the primary current must be doubled, that is H, must be increased to 40 C.G.S. From the purely magnetical point of view, there would scarcely be any object in making the dimension-ratio of the core less than 45, and thereby needlessly increasing the demagnetising action.

=

R

e

But if the primary current does change its direction owing to the action of the condenser, a closed core appears again more advantageous, provided that the opposing primary field attains at least the coercive intensity; an inspection of fig. 36, A, p. 225, shows the correctness of this contention. On the other hand, the smaller the demagnetising factor the greater is the value of the differential quotient dB / d H. (§ 154); the selfinduction of the primary coil is hereby increased in a manner

e