heaviside oliver - electrical papers vol 1

ELECTRICAL PAPERS BY OLIVEK HEAVISIDE IN TWO VOLUMES VOF/ T I f * fe'TSt^C/^'rw,/^ •':*"' -"''V JUN3 0]9gg' MACMILLAN AND CO. AND NEW YORK 1892 [^?/ ri(7/t^'! reserved] ^J\ LIBRARY C / f 'I htwwwbmhc^^^ PKEFACE. This Eeprint of my Electrical Papers comes about by the union ot a variety of reasons and circumstances. First, there was a demand for certain of my papers, especially for a set relating to Electromagnetic Waves. Although I distributed 49 copies in a collected form, I was asked for more, and also received assurances that a republication of my papers in general would be useful. But this demand was too small to lead to an immediate supply. Secondly, however, at the beginning of 1891 it was proposed to me by the publisher of The Electrician that my articles on "Electro- magnetic Theory," then commencing and now continuing in that journal, should be brought out later in book form. This was satis- factory so far as it went, but it brought the question of a reprint of the earlier papers to a crisis. For, as the later work grows out of the earlier, it seemed an absurdity to leave the earlier work behind. Thirdly, the experimental work of Hughes in 1886, furnishing the first evidence (in the sense ordinarily understood, though other evidence was convincing to a logical mind) of the truth of the theory of surface conduction along wires under certain circumstances, first advanced by me a year previously; followed in 1887-8 by the experimental work of Hertz and Lodge on electrical vibrations and electromagnetic waves, still further confirming the above, and also broadly confirming the truth of the theory of the propagation of disturbances along wires I had worked out on the basis of Maxwell's doctrine of the ether in its electromagnetic aspect, and the correctness of Fitzgerald's ideas concerning electrical radiation, and of the nature of the energy-flux developed by Poynting and myself from Maxwell's theory," were the means of stirring up an amount of interest in this theory that was quite wonderful to witness. That electrical disturbances were pro- pagated in time through a medium was raised from a highly probable yl ELECTRICAL PAPERS. speculation to an established fact. A careful study by electrical physicists of Maxwell's development of Faraday's ideas became im- perative, especially on the Continent, where Maxwell's work had hitherto met with a singular want of appreciation, arising, I believe, mainly from misconception of his theory of electrical displacement. This misconception, I think, exists even now, since some writers apply to Maxwell's theory ideas and processes which seem to me to be thoroughly antagonistic to his views. But even in England the theory had licen much neglected. For one thing, much attention was being devoted to the dynamo. Then again, the form in which Maxwell presented his theory did not, I think, display its merits in a manner they deserve, and suited for legitimate development. Moreover, the contrast between the old notions of electricity and Maxwell's was so great that mere natural conservatism stood in the way. A stimulus was wanted in favour of a theory so ill-understood and (apparently) so far removed from actual observation. But the experimental stimulus having come, the result has been a tiood of other experimental work,, mostly tending to confirm the general theory. A work, therefore, like the present, which is, in the main, devoted to the elucidation and extension of Maxwell's theory, and of the mathematical methods suited to it, should have a legitimate place amongst others. Though it was nearly all done before the electrical "boom" began, it may not be out of date, and may perhaps be, in some respects, ahead. Fourthly, it had been represented to me that I should rather boil the matter down to a connected treatise than republish in the form of detached papers. But a careful examination and consideration of the material showed that it already possessed, oh the whole, sufficient continuity of subject-matter and treatment, and even regularity of notation, to justify its presentation in the original form. For, instead of being, like most scientific reprints, a collection of short papers on various subjects, having little coherence from the treatise point of view, my material was all upon one subject (though with many branches), and consisted mostly of long articles, professedly written in a connected manner, with uniformity of ideas and notation. And there was so much comparatively elementary matter (especially in what has made the first volume) that the work might be regarded not merely as a collection of papers for reference purposes, but also as an educational work for students of theoretical electricity. As regards the question, "Will it pay?" httle need be said. For, fifthly, however absurd it may seem, I do in all seriousness hereby PREFACE. vii declare that I am animated mainly by philanthropic motives. I desire to do good to my fellow-creatures, even to the Cui bonos. Having thus justified the existence of this reprint, it remains for me to indicate the general nature of the contents, and, in doing so, I will imagine myself (usually) to be addressing an intelligent and earnest student, who means business. The first twelve articles, pp. 1 to 46, are on matters dealing mainly with telegraphy, and are but loosely connected. But a sort of continuity then begins, for the next eight articles, up to p. 179, deal mainly with the theory of the pro- pagation of variations of current along wires, beginning with applica- tions of the simple electrostatic theory of Sir W. Thomson (1855) to cables under different circumstances (terminal resistances, condensers, etc., intermediate leakage, etc.), and folloAved by extensions to include self-induction, or the influence of the inertia of the magnetic medium, and the mutual influence, both electrostatic and magnetic, of parallel wires. The last of this set. Art. xx., has not been printed before. It is, however, in its right place, having been written in 1882 as a sequel to the papers preceding it. It may be found useful to those who are interested in the subject as an intermediate between the papers of this set and the later series in the second volume, wherein the subject is treated from a more comprehensive point of view, viz., Maxwell's theory of the ether as a dielectric. There is no conflict. The later investigations are generalizations of the earlier, or the earlier are specializations of the later; and I can recommend the earnest student to read the earlier set first, before proceeding to the more advanced treatment in the later set. We next come to a series of papers published in The Electrician between the autumn of 1882 and the autumn of 1887, when under the editorship of Mr. C. H. W. Biggs, to whom I desire to express my obligations for the opportunity he gave me of exercising my philanthropic inclinations, in the face, as I afterwards learnt, of con- siderable opposition. These papers extend over about 500 pages, mostly in this, partly in the second volume, and are usually long articles, with continuity. They relate to electrical theory in general. Beginning with the abstract relations of the electrical quantities, and the mathematics of the subject in vector form (of an elementary kind), including a general theory of potentials and connected quantities expressed in the rational units I introduced, we pass on to the con- sideration of the energy of the electric and magnetic fields, and the transformations concerned in the phenomenon of the electric currents viii ELECTRICAL PAPERS. including an account of Sir W. Thomson's theory of thermo-electricity. Next comes a pretty full study of the theory of the propagation of induction and electric current in round cores, to which I was led by my experiments with induction balances, in the endeavour to explain certain phenomena observed. The analogy with the motion of a viscous liquid is also introduced and developed. Lastly we come (1885) to a more comprehensive treatment of electromagnetism, based upon Maxwell's theory, in "Electromagnetic Induction and its Propagation," of which the first half is in this volume. I here introduce a new method of treating the subject (to which I was led by considering the flux of energy), which may perhaps be appropriately termed the Duplex method, since its main character- istic is the exhibition of the electric, magnetic, and electromagnetic equations in a duplex form, symmetrical with respect to the electric and magnetic sides, introducing a new form of fundamental equation connecting magnetic current with electric force, as a companion to Maxwell's well-known equation connecting magnetic force and electric current. The duplex method is eminently suited for displaying Maxwell's theory, and brings to light many useful relations which were formerly hidden from view by the intervention of the vector- potential and its parasites. There is considerable difficulty in treating electromagnetism by means of Maxwell's equations of propagation in terms of these quantities, as presented in his treatise. The difficulty is greatly increased, if not rendered practically insuperable, when we pass to more advanced cases involving heterogeneity and eolotropy and motion of the medium supporting the fluxes. Here the duplex method furnishes what is wanted in general investi cations, and is the basis of " Electromagnetic Induction " and of the whole of the second volume. The electric and magnetic forces (or fluxes) and their variations are the immediate objects of attention in the duplex method, whilst potentials are treated as auxiliary quantities Avhich do not possess physical significance as regards the actual state of the medium, though they may be useful for calculating purposes. Towards the end of this volume the electric and magnetic stresses are considered. The treatment was interrupted, but a later paper, "On the Forces, Stresses, etc.," in the second volume contains what was to have been its continuation, and developments thereof. The reason of the break was that the interest excited by Professor Hughes's 188G experiments made it desirable that I should at once publish other matter long in hand, namely, developments of the views relating to PREFACE. ix the functions of wires and of the dielectric surrounding them, explained in Section ii. of "Electromagnetic Induction." These developments are contained in the second half of that article (Art. XXXV., vol. ii.) and in the article "On the Self-Induction of Wires " (Art. XL., vol. ii.), published in the Philosophical Magazine in 1886-7. The reader is re- commended to read the former first, as it is much more elementary than the latter, which contains mathematical developments and ex- aminations unsuited to The Electrician. The subject is the diffusion of electrical waves into wires from their boundaries and the propaga- tion of waves along the wires through the insulator surrounding them, supplying the wires themselves with the energy they absorb. Also the self-induction of various arrangements of apparatus, and the theory of induction balances. But in the year 1887 I came, for a time, to a dead stop, exactly when I came to making practical applications in detail of my theory, with novel conclusions of considerable practical significance relating to long-distance telephony (previously partly published), in opposition to the views at that time officially advocated. On the official side the electrostatic theory was upheld, with full application of the re- tardation law of the inverse-squares to telephony ; inertia being regarded as a disturbing factor, assumed to be of a harmful nature, but argued to be quite negligible in long copper-circuits, because telephony through such circuits of low resistance was so successful. On the other side was my theory asserting that owing to the .rapidity of telephonic changes of current inertia was not negligible, that it was often important, and sometimes, as in the case of wires of low resistance, even a dominating factor. Furthermore, that it was not harmful, but was, on the contrary, beneficial in its efi'ects, which was, in fact, the very reason why long-distance telephony was successful. Then, as regards the measure of the inductance, it was asserted on the official side that the inductance per centim. of a copper suspended circuit was (in electromagnetic units) only a minute fraction of unity ; whilst on the other side it was declared to be some hundreds of times as big, say from 10 to 20 per centim. of circuit. Here was the most complete possible antagonism between my views and official views, both in principle and in detail, and a careful consideration and dis- cussion of the matter was desirable. Yet I found it next to impossible to ventilate the matter. First of all, I was prevented by circumstances which need not be mentioned from bringing the matter before the S. T. E. and E. in the spring of 1887 (Art. xll, vol. ii.). Next, a little X ELECTRICAL PAPERS. later, the editor of the Philosophical Magazine could no longer aflPord space for the continuation of my article on "The Self-induction of Wires," Part VIII., dealing with the non-distortional circuit and telephony (p. 307, vol. II.). Thirdly, after a partial exposition in Sections XL. to XLVI. of " Electromagnetic Induction," a change of editor occurred, and the new editor asked me to discontinue. He politely informed me that although he had made particular enquiries amongst students who would be likely to read my papers, to find if anyone did so, he had been unable to discover a single one. Fourthly, he returned a short article (Art. XXXVIII., vol. II.) on the same subject of long-distance telephony, which pointed out official errors in detail, and directed attention to the contrary results indicated by my theory, this paper having been in official hands. And lastly, three other journals declined the same, for reasons best known to themselves. Perhaps it was thought that official views were so much more likely to be right that it was safe to decline the discussion of novel views in such striking opposition thereto. There seemed also to be an idea that official views, in virtue of their official nature, should not be controverted or criticized. But there seems something wrong here, as the above facts, and the later evidence in support of my views, have shown. For what other object have scientific men than to get at the truth, and how is it to be done without free discussion 1 The student is particularly recommended to read the articles referred to, not merely on account of the telephonic application, but because of the simplicity of treatment which the distortionless circuit allows, and as a preliminary to the study of Electromagnetic Waves, to which it supplies a royal road. The action of leakage in promoting quick signalling is treated of in the early set in this volume; now the inductance of the circuit has also a beneficial eff'ect ; and the two together conspire to annihilate the distortion which the resistance of the circuit produces. The same occurs (approximately) without the leakage, by the action of self-induction, if the frequency of alternation be sufficiently rapid, and the wires of not too great resistance. Now in the theory of electromagnetic waves there is a similar pro- perty, which throws considerable light upon the subject of waves in general. I had introduced, in 1885, for purposes of symmetry, the fictitious quality of magnetic conductivity. When its effects upon the propagation of waves in a real conducting dielectric are enquired into, it is found to act contrary to the real conductivity, so that the distortion due to the latter can be entirely removed by having duplex PREFACE. xi conductivit3^ How this strange result comes to pass may be readily understood in detail by studying the theory of the distortionless circuit, in which the leakage conductance and the resistance of the circuit act oppositely in respect to distortion. The remainder of the second volume consists of investigations growing out of "Electromagnetic Induction," viz., the set relating to electromagnetic waves ; the electromagnetic wave-surface ; propaga- tion in a uniform conducting (duplex) dielectric, with the application to plane waves, either free or along straight wires ; the connected theory of convection currents; the theory of resistance and conduct- ance operators ; with a few miscellaneous papers concerning propaga- tion in moving media; finishing with an article discussing the forces and stresses concerned in the electromagnetic field. Acting under advice, I have not carried out my original design to make large additions. Limitations of space prevented this, and T have confined myself to an occasional small addition or footnote. These are put in square brackets, all such signs in the original papers being cancelled. For the rest, I have corrected misprints and obvious slips, and have made verbal improvements and omitted occasional redundant matter. The scientific reader may therefore refer to this work as to the original papers. Their dates, etc., are given at the commencement of the articles. I have introduced uniformity in the notation connected with vectors, though there was little change to be made except to put all vectors into Clarendon black type, as in some of the later of the original papers. The vector-algebra, I should mention, is of a rudimentary kind, and has nothing to do with quaternions ; first, only addition and the scalar product are used, whilst later on the vector product is introduced and freely employed. On the vexed question of vectors, the conclusions to which I have gradually settled down are as follows : —The notorious difficulty of understanding and working Quaternions will always be a bar to their serious practical use by any but mathematical experts. But, on the other hand, a vector algebra and analysis of a simple kind, independent of the quaternion, and readily understandable and workable, can with great advantage take the place of much of the usual cumbrous Cartesian investigations, and be made generally useful in all physical mathematics concerning vectors, and be employed, comparatively speaking, by the multitude. It should obviously be harmonized with the Cartesian mathematics. The quaternionic system is defective in this respect; xii ELECTRICAL PAPERS. in its very nature it cannot be thus harmonized. The system I recommend is fully explained in "Electromagnetic Theory," chapter in. {The Electrician, Nov. 13, 1891, and after). The numerous letter prefixes of the quaternionic system, which greatly contribute to the difficulty of reading quaternionic investigations, are abolished, retaining only the symbol V before a vector product. Another difficulty is in the scalar product of Quaternions being always the negative of the quantity practically concerned. Yet another is the unreal nature of quaternionic formulae. The terms do not stand for physical quantities. Again, in most physical mathematics, the quaternion does not even present itself for consideration, or, at any rate, may be readily dispensed with. Lastly, the establishment of vector-algebra on a quaternionic basis is very hard to understand, as chapter II. of Professor Tait's treatise shows. These troubles are obviated by the method I follow, basing the whole upon the definition of a vector, and of the scalar and the vector product of a pair of vectors. The notation is harmonized with Cartesians and transition is readily made. We may, indeed, regard a vector investigation, from this point of view, as a systemati- cally abbreviated Cartesian investigation, and the latter as the full expansion of the former. And, considering that the bulk of special investigations are necessarily scalar, it seems to me that we should keep in touch with them as far as possible, and not try to abolish the Cartesian method, but make it a useful auxiliary to the vector method. That quaternionic experts may do valuable work is un- doubted, but how can the bulk of mathematicians possibly under- stand it ? Lastly, on the question of units, it is not, I think, generally under- stood that the ordinary electrical units involve an absurdity similar to what would be introduced into the metric system of common units were we to define the unit area to be the area of a circle of unit diameter. A rational system of units founded upon a rational defini- tion of a pole (electric or magnetic), associating the unit pole with one line of the corresponding force or flux instead of with iir, was employed by me in some of the earlier papers (1882-3), but was not carried out further because I believed that a reform of the electrical units was impracticable. Now, I had commenced " Electromagnetic Theory" in January, 1891, with rational units merely to exhibit the theory in a fitting manner, intending to transform later to the common units. But 1 came afterwards to the definite conclusion that a thorough reform of the electrical units is practicable and perhaps indeed inevit- [...]... + 1^ a quantity between and 1, being 1 for the most sensitive balance few numerical examples will serve to show the variation of the Suppose a; = 4,000, r/= 1, 000, /= 90 sensitiveness in different balances ohms Then, in the following balances p is A a 1 c 1 5 10 10 0 1, 000 10 ,000 6 1 2 3 4 7 10 8 10 0 9 1, 000 10 10 1 10 10 0 1, 000 10 ,000 10 10 0 1, 000 10 ,000 1 : : : : : : : : : : : : : : : : : : : 11 10 0... Section 21 Con- Remarks on Normal Electromagnetic Systems - Dielectric Conductors, or of •''36 ELECTRICAL PAPERS ?AGK Section 22 The Mechanical Forces and Stresses The Simple Maxwellian Stress Preliminary - - - 542 - - 545 The Mechanical Action between two Regions - - 548 Summary - - 548 - - 549 - 550 - - 5 51 - - 553 - - 554 - 554 First Electromagnetic Application Section 23 of some Results - of Vector... Energy Magnet of - - their Potential Energy 455 - 457 done by Impressed Forces during Transient States - - 462 - - - - 466 - - - 468 - - - 4 71 - - - 476 - - - 4 81 Circuital Displacement Simple Example 9 and 447 E and 6 Section Force 447 Energy and Magnetic Energy Section - Electric between Differential Equations of r 446 - Rough Sketch Second Connection Magnetic Force The Equation 4 41 of Closed Displacement... : : : : : 11 10 0 10 : : 12 1, 000 10 ,000 10 0 : : 1, 000 : : 13 14 10 0 1 15 16 1, 000 10 ,000 17 300 10 10 0 1, 800 4,000 4,000 4,000 4,000 4,000 : : : : : : : : 4,000, 4,000, we have/5= 4,000, 4,000, /)= -9 0 4,000, P= -5 8 4,000, 4,000, 4,000, 4,000, P =-0 024 4,000, 4,000, /> =-0 029 4,000, 4,000, ,, 4,000, 400 400 400 400 4,000, 40,000 40,000 40,000 40,000 400,000 400,000 400,000 : -0 03 -0 29 -2 4 P= p= p= p=... EXTRA CURRENT - 53 ON THE SPEED OF SIGNALLING THROUGH HETEROGENEOUS TELEGRAPH CIRCUITS 61 71 Art 15 - Art 16 ON THE THEORY OF FAULTS IN CABLES Art 17 ON ELECTROMAGNETS, ETC - - - 95 ELECTRICAL PAPERS xvi Paois Art 18 MAGNETO-ELECTRIC CURRENT GENERATORS Akt ON INDUCTION BETWEEN PARALLEL WIRES 19 Art 20 - - 11 '2 - - lliJ "CONTRIBUTIONS TO THE THEORY OF THE PROPAGATION OF CURRENT IN WIRES Art 21 DIMENSIONS... Coils, with Cores - - - 408 - - - 410 Function of a Linear and Derivation of the Differential Characteristic of Conductors, Equation - - - - A.RT 29 REMARKS ON THE VOLTA FORCE, Abt 30 412 ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION Section 1 ETC - (FiKST Half.) Rough Sketch of Maxwell's Theory - Conductivity, Capacity, and Permeability Sectiox 2 416 On - - - 429 - - - 429 -. - the Transmission of Energy... 2 91 Energy 297 Current Section 7 .-. - The Minimum Heat Property or Continuous Section 8 Thermo-electric Force in Conductors, Linear and Thomson Peltier Section 9a The Section 9b Application of the Second Sectiox 10 The Thermo-electric Diagram and Section 11 The Thermo-electric Theory First On of Thermo-dynamics Law to 309 - 315 Thermo-electricity 318 its Theory - - - tions thereto Section 12 ... AND - - - RESISTANCE - 14 1 17 9 OP 18 1 Art 23 THE EARTH AS A RETURN CONDUCTOR Art 24 THE RELATIONS BETWEEN MAGNETIC FORCE AND - - - 19 0 ELECTRIC CURRENT Section 1 The Universal Relation between a Vector and Section 2 The Section 3 Connected Potentials of Scalars and Vectors General Theorems in - its Curl - Electricity - 4 The : Characteristic Equation of a Potential, and Solution Section 5 6 206 its -. .. Sketch Extensions - 4 41 of 4 41 and Electric Force 443 - Real Transient, and Suggested Dissipative Magnetic Current Effect of gr in a Closed Iron Eing First Cross- Connection of Magnetic Magnetic Energy of Moving Charged Spheres Section 4 Completion of Section 5 and jx of Self-Conjugate ; its Transfer - - - - - - 450 so - - 4 61 H - k not necessarily Impressed Magnetic Force 449 4 51 Intrinsic Magnetisation... 4,000, 666| we have P= p= P- ,, -7 5 -3 8 -0 25 -2 19 -0 27 -1 4 -2 03 P= -0 15 P= -0 27 p= 1 nearly equality between a and b, and the sensitiveness increases rapidly, reaches a maximum, and afterwards diminishes less rapidly In the next four, a:b::l :10 , and the sensitiveness is There is a falling off in sensitiveness in greater and behaves similarly the following four, where a b -. -. lO I, and a great falling . 9. Impressed Electric Force in Dielectrics. - - - 4 71 Section 10 . Dielectric Displacement and Absorption. - - - 476 Section 11 . The Principle of Thermal Resistance. - - - 4 81 Section 12 . Electrisation and Electrification. Natural Electrets. 488 Section. PROPAGA- TION OF CURRENT IN WIRES. 14 1 Art. 21. DIMENSIONS OF A MAGNETIC POLE. - - - - 17 9 Art. 22. THEORY OF MICROPHONE AND RESISTANCE OP CARBON CONTACTS. 18 1 Art. 23. THE EARTH AS A RETURN CONDUCTOR. - - - 19 0 Art. 24 - - - 549 Internal and External Energies. - 550 Mechanical Force between two Regions. - - - 5 51 The External P in terms of the Surface P^^. - - 553 Annihilation of the Surface Current. - - - 554 Annihilation