Electromagnetic Theory, Vol. 2 (1899) — part 29 of 31

The commonest knowledge of the transverse motions of a stretched cord is concerning its normal vibrations, the funda- mental mode and its harmonics. But we do not want them in our analogy. For although we may have something similar with short waves, and especially Hertzian waves, it would not be easy to set up such a state of things on a telegraph circuit, which is usually so long as to prevent the assumption of normal modes by reflection at the ends. For the normal vibrations of a oord are complex affairs, depending not only upon propagation ol motion along it^ but also on the constraints to which it ia terminally or intermediately subjected. We desire the propa- gational effect to show itself distinctly. But if we disturb a stationary stretched wire at one end, the pulses produced nm

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to and fro too quioklj for praotioal observation. Take, thai, a long india-rabber oord, suspended firOm a height, as is some- times done by lectuxers. The speed of transmission of poises may be made small enough to allow the disturbances to be conveniently watched. Then, if the oord hang vertioallj at rest to begin with, and its lover end be momentarily jerked, bringing it immediately back to its original position, a hump is generated, which runs along to the fixed end. It is not much changed in transit, and is reflected back, and re-reflected many times. But we only want to consider the passage of a hump one way. Properly speaking, then, we should have so long a cord that reflection is (by frictional loss) insignificant. Per- haps a very long cord laid upon a surface of ice would be more Buitable for the purpose. But if the " fixed end " be movable transversely, and its motion be resisted by a force proportional to its velocity, their constant ratio may have such a value given to it (depending upon the mass and tension of the cord), that the arriving disturbances do not produce any reflected waves, but are absorbed by the terminal resistance. The coid then behaves as if it were infinitely long. This is theoiy, of oourse, but I dare say a sufficiently good practical imitation could be got by means of a terminal block sliding upon a wife with friction. An imperfect cancellation of the reflection is -easily got.

Disrogarding the slow change of lype and the attennatioD, 40 that there is undistorted propsgation of a hump, or of a succession of humps making a train of waves gomg one way only, this case is analogous to the transmission of waves along a uniform diooit of no resistance. The two constants L and 8 in the electrical case are replaced by another two, via., the density (linear, or mass divided by length) and tension of the €ord. Thus, large permittance, that is, small elastanoe, means small tension, or a slack cord ; and large inductance means a massive cord. If I said a heavy " cord, it might be thought that its weight was concerned in the matter in question, but it is not, save as a disturbing factor. The transverse displace- ment of the cord may be compared with the transverse voltage in the electrical circuit. Other comparisons (theoretically better) may be made, but this is perhaps the most convenient if the motion of humps be actually watched.

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Now to imitate the action of the resistance (assumed to be constant) of an electrical circuit, the transverse motion of the cord must be frictionally resisted, and the frictionality (or coefficient of friction) should be constant. That is, the frictional force should be proportional to the transverse velocity. The motion of the cord is frictionally resisted, both internally and externally. It is very unlikely that this should give rise to a constant frictionality, but there is a similarity insofar as the wayes sent along the oord. do attenuate and are distorted as they progress.

But the friction is not anything like enough in the case of a cord in air to imitate the submarine cable ; besides that, the mass is too greats and likewise the tension. Even to oopj tetophomio wayee going along Yery long oopper wires of low misfcanceb we shonld increase the friction in the meohanioal analogae if we desire to have an eqnal amount of attenuation in a wave length. The oord should, therefore, be in a viscous medium, and the viscous resistance to the cord's motion should be so great as to produce a marked attenuation during the transit of a hump^ though it should not be so great as to prcYent its distinct propagation as a hump, without great dis- tortion of shape and spreading out. Under these circumstances we may exemplify low-resistance, long-distance, overland tele- phony, where the inductance is relatively large and the permittance relatively small.

But to imitate a submarine cable we should have a slack cord of small mass in a highly viscous medium. We can then send a hump only a little way along it as a hump, for it becomes greatly distorted and diffused. Consequently, if we compared the displacements at a great distance along the cord with those impressed upon it at the free end, we should find a state of things quite diiferent from what occurs in the case of an india-rubber cord in air. Tiie main features of the originated disturbances might be recognised, unless the attenuation were too great, but they would be out of all proportion. In particular, there is a tenden^ to obliterate small humps, or ripples whilst the big ones persist The propagation is of the sluggish kind characteristic of diffusion, quite unlike that of purely elastic waves. If the slack cord could have no mass at all, we should then imitate a submarine csble wholly without inductance.

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Under these circumstances we could convert the diffusive waves to something like elastic waves by giving mass to the cord. Without changing either its tension or the frictionality, we could, by a continuous increase of density, cause the nature of propagation to pass continuously from the sluggish, diftusive kind to the elastic kind with approximate preservation of form of waves. We might also do the same by increasing the tension and reducing the frictionality, keeping the density small (like, by reducing the permittance and the resistance in an electrical circuit, bringing the inductance into impor- tance). But these are not immediately in question. By inoreasing the mass, or equivalently the inertia of the cord, we impart momentum to the waves, which allows them to be carried forward, in spite of the little tension, against the resistance of the surrounding medium.

So it is in the electrical case. By increasing the inductance we impart momentum to the waves, and that oanries them on. This statement is independent of the particular mechanical analogy that may be employed for illustration. It is probably representative of the actual dynamics of the matter, associating L with inertia, and LC with momentum. The soundness of this dynamical explanation (made by me some years since in asserting and proving the h/ot concenied) was endorsed by Lord Kelvin. ' And I am proud to add, for the benefit of those who may excuse themselves from understanding me by the plea that I am a Schopenhauer (1), that Lord Kelvin expressed his appreciation of the very practical way in which I had reduced my theory to practioe.

Many other dynamical arrangements might be alluded to, exemplifying the power of inertial momentum to overcome frictional resistance, but it is not easy to get a good one exhibiting elastic waves plainly.

To illustrate the nature, and to emphasise the possible magnitude of the effect, suppose we have the power of continuously increasing the inductance of an Atlantic cable from a small quantity up to a very large one, without altering its resistance or permittance, and without introducing any other actions than those to be controlled by resistance, permittance, and inductance. Starting with L = 0, and making, say, 1,000 waves per second at one end, there will be 1,000

THEORY OF PLANE ELECTROMAGNETIC WAVES. 433

waves per second of workable magnitude only in a small part of the (»ble near the sending end ; the amplitude will attenuate to insignificance, not exactly in the shore-end, but before 100 miles is traversed. But let waves of the same size (which will need greater battery power to produce) be kept up at the sending end whilst L is continuously increased all along the cable. The region of practical waves will continuously extend itself right through the cable, by a partial conversion to elastic waves, with little attenuation in transit. With very big induotanoe the waves would reach the distant end nearly in full size. Even then they could be doubled in size by the reflection, thongh there would be no useful purpose served hj that. The amount of inductance required, however, to produce these results would be out of all reason. Nevertheless, the illustration serres its purpose in showing the action of self- induotion, which is, as before stated, wholly beneficial, as it increases the amplitude and lessens the distorUoo. Soma lemarks <m increasing inductance will come later,

Setf-Indnction combined with Leaks. The Bridge System of Mr. A. W. Heavisida^ and suggested DistortionleBS CHrcuit.

§ 216. In the meantime^ some few remarks on self-induction combined with leakage should come in here. Self-induction oame in with the telephone. That was, originally, a mert house-to-house afiair. At any rate, lines were only a few miles long. So condenser action was out of practical question then. On the other hand, the frequency of the currents concerned is very great, and that makes inertia important. So the theory of telephones is a magnetic afliiiir when the lines are short enough, just as if they were lines in a laboratory.

But the telephone soon got beyond that, and it became necessary, with the use of longer and longer lines and under- ground lines, to examine the influence of their permittance in conjunction with their inductance. In hardly any case of telephony can we ignore the self-induction, and take the permittance alone. The resistance would need to be so large that the lines would be wholly unsuitable for telephony, except for mere local lines. So permittance and inductance usually

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go together in considering telephony, and if one of them is ignorable it is the permittance rather than the inductance.

Following the telephone came exchanges. But there was aa exchange before telephones. My brother, Mr. A. W. Heaviside, had an intercommunioatioii exohange at Newcastle, using alphabetical indioaton, whereof the manipulation resembled that of the hurdy-gurdy. These were beautiful instruments mechanically ; but, of course^ the telephone soon stopped their manufacture. The exohange referred to was then turned into a telephone exchange. Now, in certain cases it was needed to have many instruments in one circuit, with local intercommu- nioation; a sort of family arrangement, so to speak; lor example, a number of coal pits and colliery offices belonging to one firm. The custom was to set all the intermediates in sequence (otherwise, but perhaps not so well, called in secies) in the circuit. Why was this donef Simply because they knew no better. It was the custom to do so in the telegraphs generally, being an obvious arrangement — not so much in England, though, as elsewhere. But when this was done with •telephones, or with call-instruments replacing them, it was found that the intermediates had a very deleterious influence, which very soon set a limit to the admissible number of inter- mediates on a circuit and to the length of the circuit, especially when underground wires were included. My brother, who took up the telephone with ardour from the lirst (making the tirst one made in England, I believe), found by experiment that it was quite unnecessary to put the instruments in sequence according to old practice, and that an immense improvement was made by taking them out of the circuit, and putting them across it, as shunts or bridges. Like lamps in parallel, in fact. It seems very easy and obvious now. But it is not so easy to conceive the state of mind of old stagers. Anyhow, the pro- position to put the instruments in bridge was pooh-poolied at first, not being understood. As, however, it was demonstrably a wonderful improvement it was adopted, first at Newcastle^ and then elsewhere, and became known as the Bridge System.

When this system was brought to my knowledge, and I was asked for theoretical explanation, it presented itself in a double aspect. First of all, there was the complete removal from the circuit of intermediate impedance in big lumps. This was

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manifestly the main reason of the increase in working distance possible, and why a far greater number of intermediates could be put on. For they were not put in the circuit, but on it, so to speak.

The other aspect of the matter was this. Considering that the bridge system consists electrically of a oirouit with a number of leaks on it, and bearing in mind my old investigations concern- ing leaks, I inquired whether the intermediates were not ac- tually beneficial to through communication, independently ot the previous effect. Was not the articulation better, not merely than if the intermediates were in series in the oirooit, but better than if the shnnte or bridges weie removed altogether, leaving insulation f My brother's answer was Tes, certainly, in certain but doubtful in others. It appeared, however, to be only a minor efiect, of not much importance compared with the major one. Besides, the bridges caused a weakening of the intensity of the speech received when many bridges were passed. To pre- vent this weakening becoming inconveniently great, the inter- mediate call-instruments in bridge were purposely made to have considerable resistance and inductanoe^ for more than would be needed or desirable for their natural use. This tended to prevent the currents passing along the line from entering the shunts, and especially so as regards the currents of high frequency, and allowed them to be trjuismitted in greater magnitude.

It will be observed here that the expediting effect of leakage considered before (§§ 213, 214) is merely incidental, and is, in fact, partly destroyed by the large impedance of the bridges, in order to mitigate the attenuative evil. The shunts are inductive shunts, intentionally very much inductive. Now, my proposed leaks (§214) for submarine cables were non- inductive, and intentionally so, though it may be remarked that, it would make very little difference with the slow signalling of Atlantic cables if they were inductive. On the other hand, a recent proposal of Prof. Silvanus Thompson is to use induc- tive shunts on cables. The arrangement is electrically as in the bridge system, but the object is different. This point will be returned to.

The investigation of this bridge system of telephony suggested to me the distortionless circuit, as I have before acknowledged

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(" Electrical Papers," Vol. II., p. 402). It may be of interest to state how it came about, since it is not presented to one by the above-described experiences with the bridge system. The ques- tion was, What was the effect of a bridge, or of a succession of bridges, when the self-induction of the line was not negligible t Now the influence of the bridges can be very readily stated for steady currents, of course. There is no difficulty, either, in finding the full formulee required to express the voltage and current and their variations in any part of the system when the inductance and permittance are operative^ as well as a set of leaks, when one knows how to do it. But the general fonnnlse thus obtained ace altogether too complicated to be readily interpretable. I was, therefore, led to examine the effbct produced by a leak on a wave passing it of the elastic type. This implies that the resistance of the line is sufficiently low for waves of the frequency concerned to retain that type approximately. (Im should be greater than B.) Now this was comparatively an easy problem, and the result was to show tliat a leak (or conductance in bridge) had the opposite effect on a wave passing it along the line to that of a resistance inserted in the circuit at the same place, in this sense : — A resistance in the circuit reflects the charge positively and the current negatively, or increases the charge and reduces the current behind the resistance. On the other hand, a leak reflects the current positively and the charge negatively, or in- creases the current and reduces the charge behind the leak. (See also §§ 197, 209.) A resistance and a leak together at the same place therefore tend to counteract one another in producing a reflected wave, leaving only an at- tenuating effect on the transmitted wave. This led to the distortionless circuit. For although the compensation of the leeistanoe in the circuit and the conductance across it is im* perfect when they are finite, it becomes perfect when they are infinitely small. They may then be a part of the circuit itself, namely, the resistance a part of the resistance of the line, and the conductance a part of the leakage. We now have perfect compensation with this one, or any number of successive renst- ances and leaks of the same kind, provided their ratio is pro- perly chosen. This occurs when B and K, the resistance and conductance per unit length of oixouiti of the line and of the

THIOBT OF PLANS KLBOTBOIEAONETIC WAVES. 437

leakage napeotively, are taken in the same ratio as the two other oonatanta^ L and 8. Or, when L/R«-S/K, indicating eqnali^ of the eleotric and maguedc time-conBtants. There is BOW an nndistorted tranamiaaion of any kind of signals. And ■inoe every oironit haa B, L, and S, it follows that it can be made distortionless by the addition of leakage alone of a certain amounti or distortionless to the extent permitted by the ap- proximate constancy of R, L, and S. {See " Electrical Papers," Vol. IL, pp. 119 to 168, for detailed theory of the distortionless circuit.)

In § 213 we saw that even in the electrostatic theory leakage would allow us to signal m fast as we liked provided we could do it with an infinitesimal current. This was not because the circuit was made distortionless, but on account of the infinitely rapid charge and discharge. Now the inclusion of the in- fluence of self-induction shows that we can do the same with a finite leakage conductance, and have finite distortionless sig- nals, due to waves of the purely elastic type. Given R and S then, if L should be so small that the introduction of K of the proper amount to reach the distortionless condition should produce unreasonable attenuation, then one way of curing this would be to increase L. For then a smaller amount of leakage will serve. The bigger L, the smaller need K be, and when it is very big, then veiy little K is needed. This brings us round to the case at the end of § 215, where we had elastic waves with little attenuation on an Atlantic cable got without any leakage^ and since we see now that very little leakage is needed to bring about the distortionless condition, we conclude that it is approximately distortionless without the leakage.

Evidence in Favour of Self-induction. Condition of First-Olaas Telephony. Importance of the Magnetic Beactance.

§ 217. Let us now ask what is the evidence in favour of the

beneficial action of self-induction. There was plenty in 1887; it is overwhelming now. But there are two sorts of evidence, the theoretical and the experiential. The two are not funda- mentally different, however ; only the former is more indirect than the latter. We have certain electrical laws established, mainly by laboratory work followed by mathematical invea*

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tigatioD, and there is no reason whatever to sappose that they do not hold good out of doors as well as inside. The important thing is to oorrectly recognise the prevailing conditions and determine their results. Now, if you can be pretty sure that you have done this, and have confidence in your investigations, then you may consider that yon prove a thing, even though there may be no direct evidence. Discoveries are frequently made in this way. " A foundation of experimental fact there must be ; but upon this a great structure of theoretical de- duction can be based, all rigidly connected together by pure reasoning, and all necessarily as true as the premises, provided no mistake is made. To guard against the possibility of mis- take and oversight, especially oversight, all conclusions must sooner or later be brought to the teat of experiment ; and if disagreeing therewith, the theory itself must be re-examined, and the flaw discovered, or else the theory must be abandoned.** (Oliver Lodge.) But to convince other people is quite another matter. They may not be competent to understand the evidence. Or they may be fully competent, but not have suffi- cient acquaintance with the subject; or not have time to examine it ; or not have the energy ; or have no interest in it. Then, as Elijah said to the priests of Baal, you must Gall him louder ! "

Now, the foundation of fact is the experimental facts of eleo> tridty and the laws deduced therefrom, though these laws are sometimes more comprehensive than the facts upon which they are based. But there was, In 1887, a certain amount of direct evidence as well, quite enough to make the theoretical oon- dusions practically certain, by the confirmation they afibrded* Assuming the electrostatic theory to be true in telephony (for whioli^ however, there was never any sufficient warrant) we can calculate its consequences. They indicate that the product BBP of the total resistance and the total permittance limits the distance I through which telephony is possible. How big it may be is questionable, on account of the very complicated nature of the problem when the human is taken in as a part of the mechanism. We cannot say certainly beforehand how much distortion is permissible before the human will fail to recognise certain sounds as speech. But the electrostatic theory show^s such very considerable distortion even when

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RS/2 is as low as 5,000 or 7,000 ohms x microfarads, that it is scarcely credible that any sort of telephony could be prac- ticable if it were much bigger, as 10,000 for example. Perhaps 5,000 might be taken as the practical limit for rough purposes. But experience showed that good telephony was got up to 10,000 and higher, in this country with circuits including underground cables, as well as in America, where long-distance telephony had already made a good beginning. Manifestly the electrostatic theory was erroneous by direct as well as indirect evidence. Now the inclusion of the influence of self-induction in the theory explained these results (and also the failure of iron wires and the success of copper) and pointed out how to improve upon them, vis., by directing us to lower B aad increase L in order to increase the value of BS/^ possible. So there was sufficient direct evidence then to satisfy anyone com- petent to judge who considered it fairly without being pre- judiced by the law of the squares. Since that time, the amount of this direct evidence has been greatly multiplied, and RS/* has got up to 50,000, which may be about ten times as great as the probable value under purely electrostatic con- ditions. But there is nothing critical about 50,000. It might be 500,000 if the circuit were long enough. We may need to reduce R when we increase but not in proportion to the inverse square of /, or anything like it.

Any long circuit may be made approximately distortionless if the magnetic side of the phenomena can be made important, and without the inconvenience of the leakage needed to remove the distortion, which leakage, however, is essential when the magnetic side cannot be made important. The useful guide in considering telephony is the size of the quantity K/Lii, the ratio of the resistance to the reactance, the two terms appear- ing in (R^ + Ii%l^)^ the impedance of unit length of cirouiti apart from the permittance.

The term " reactance '* was lately proposed in Fiance, and seems to me to be a practical word. It may be genecalised to signify the value of Ln, positive or negativcb of any combina- tion of coils and condensers, L being then the e£feotiv6 induct- ance. When L and the reactance are positive, the magnetio side prevails and the current lags ; and when it is negative^ the electric side prevails and the current leads.

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At present, however, R and L are as before. Now the ratio R/Ln may be very large. If so, the influence of L is small, and we shall have something like the electrostatic theory, provided the range of n be such as to make the ratio always large. But this state of things is quite unsuitable for long- digtanoe telephony. The most suitable state ia when R/Ln is a proper fraction throughout the whole range of n ; that is, when R is less than the smallest value of Ln. We shall then have approximately distortionless propagation of the type

as may be seen by examining the solution for simply periodic waves, and observing that when R/Ln is small we have nearly the same attenuation in a given distance at all frequencies (not too low), so that there is little distortion, and the element of frequency tends to disappear altogether, and we reduce the formula to the above type.

Practice shows, however, that this state of things is unneces- sarily good for commercial telephony, which admits of a considerable amount of distortion. Then R/Lf> ia not less than 1 throughout the whole range of n, but is greater at the low frequencies and less at the higher.'^ For example^ if the lowMt value of n be 625, and L = 20 (centim. per centim., usoal units), the least value of Lfi is 12,500. If also the resistance is 5 ohms per kilometre, then R is 50,000. The value of B/Lti is then 4 at the lowest frequency, and 2, 1, at tiie five SQOcessive octaves, the last being at fi 20,000. Hers we have an excellent state of things from » at 2,500 up to 20,000, and only at the low frequencies does B prevail over Ln. If B be halved, or L be doubled (for it is indifferent which is done), then "RfLn is less than 1 from m« 1,250 upwards. Divide n by 2ir to obtain the frequency. This 2t htm nothing to do with the ridiculous 4ir which the RA. Oommittee want to consecrate. The significant point in calcu- lations of this kind is the position of n in the scale of frequency which produces equality of R and Ln. When it is low we have good quality ; when high, then relatively bad. Perhaps the American electricians may be able from their extended

• 8u my "Electrical Fbpers" for details ; espeoally App. C, p. VoLIL

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ezperienoe to give praotioal infonnatioii on this and other points oonoemed in telephony, and oonstract an empirioal formula of a sound character for determining the limiting dis- tance roughly in terms of the electrical data, with given kinds of instruments for transmission and reception. It should be understood that the above remarks do not apply to short lines, at least in general. We may, indeed, so arrange matters that the above type is still preserved (with R/Ln less than 1) by practical annihilation of reflected waves, but otherwise serious modification may be needed. And with very short lines we may apply a purely magnetic theory, of course.

Varions Ways, good and bad, of increasing the Indnetanos

of Circuits.

§ 218. Supposing, now, that practicians have come to discard the supposed limitations so persistently asserted by official electricians in England, and to open their eyes to the part played by self-induction, they will naturally recognise the wide possibilities that are suggested in future developments, and want to know whether such possibilities can be converted into actualities, and whether the conversion is practicable. For myself, I am not much concerned in this part of the question. It is. for practicians to find out practical ways of doing things that theory proves to be possible^ or not to find them if thej should be impracticable. Kevertheless, since I am respcmBible for these views concerning the parts played by self-induction and leakage, I add a few remarks on the applicational side.

The ideal is the distortionless cirouitt requiring both self- induction and leakage. If we utilise the present existent inductance only, then the needed leakage may be too great hf hi in many cases. On the other hand, if the existent induct- ance is big enough to make R/Ln small, in the sense described in § 217, we do not want the leakage particularly, or not at all practically, it may be. It is clear, then, that we should aim at making L/R, the magnetic time-constant, as big as possible.

Now, here is one way of doing it. If, a few years ago, an old practitioner was asked to put up a long distance telephone circuit, he, being under the belief that R should be high, or at least that it need not be small, and that L should be as smaU as

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possible, in order to prevent the deleterious action of self* induction, would probably carry oat his principles by patting 1^ a circuit, several hundred miles long, oonsisting of a pair of fine wires, put quite dose together. Or, if he was free from the error as regards B, he would use larger wires. But still cht» together, for that makes L the smallest. Also, he would not consider this to be bad as regards S, the permittance, being under the belief that the permittance of the circuit of two wires was just one half that of either with respect to the earth, so that their proximity did no harm in increasing condenser action, whilst it destroyed the deleterious self-induction. Now, such a circuit might do very well for local telephony, but for the purpose required there could not be a worse arrangement. The supposed case is not an extravagant one, since it involves the carrying out of official views thoroughly. The cure would be quite easy. Increase the inductance by separating the wires as widely as possible or convenient. If necessary, reduce R as well. The permittance will also be reduced in about the same ratio as the inductance is increased. By this simple process I«. may be largely multiplied, say from 2 to 20 (cm. per cm.), a tenfold magnification, without separating the wires so much as to require a double line of poles. This way of doing it is the secret of the success of the long-distance telephony which arose in America when iron was discarded. L is made important, and Bis kept low. But although L increases very fast on initial \ separation of wires, it does not continue to increase fust on further separation, so that practical limitatious are soon set to the process. The above example shows how a veiy bad line may be turned into a very good one by increasing the inductance.

Another- way is suggested, but only to be immediately rejected. It is, nevertheless, instructive to study failures. We can increase L to any extent by using finer and finer wires. This would be admirable if it were all. But, unfortunately, this process alters R, and faster still ; so that R/L increases instead of diminishing. Consider, for simplicity, merely a wire with a concentric tube for return. Then R/L is nearly propor- tional to RS. Now Lord Kelvin gave a formula long ago, showing that this quantity was made a minimum when the radii of the tube and wire bore a certain ratio. Then L/R is a maximum, approximately, and on decreasing the radius of

THEORY OF PLANE ELECTROMAGNETIC WAVES. 443

the wire it falls off, ultimately very rapidly, which is the reverse of what is wanted.

A third way must also be immediately rejected. By using iron wires we can increase L largely with steady currents. But then the skin effect, or imperfect penetration, becomes important with telephonic currents in iron wires. It does two tiiiogs, both harmful. It lowers L and increases R. So we do not get the large increase of L suggested at first, but only a oomparatively small one with currents of great frequency, and we do get a lot of extra resistance that should particularly be avoided. Iron wires are failures for long-distanoe telephony, as was found long ago.

The case of copper-coated iron or steel wires is doubtful. Such wires were used in America, and it was said that they were successful for long distance telephony. But here, evidently, the copper is very important in giving conductance. As regards the interior iron core, it must be considered very questionable whether it can be so good as an amount of copper of the same resistance (steady). It is not well placed for increasing the inductance, and the imperfect penetration will operate injuriously.

If we put the iron outside the copper the iron will be better situated for increasing L for steady currents (which is no good), but will act injuriously with rapidly oscillating currents, and partly prevent the utilisation of the conductance of the copper, if it be thick enough to make the increased L important.