Heaviside 1893
Chapter II. Outline of the electromagnetic connections.
Appendix A: The rotational ether in its application to electromagnetism.
Chapter III. The elements of vectorial algebra and analysis.
Chapter IV. Theory of plane electromagnetic waves.
Appendix B: A gravitational and electromagnetic analogy.
(OCR text) PREFACE.
THIS work was originally meant to be a continuation of the series "Electromagnetic Induction and its Propagation," published in The Electrician in 1885-6-7, but left unfinished. Owing, however, to the necessity of much introductory repetition, this plan was at once found to be impracticable, and was, by request, greatly modified. The result is something approaching a connected treatise on electrical theory, though without the strict formality usually associated with a treatise. As critics cannot always find time to read more than the preface, the following remarks may serve to direct their attention to some of the leading points in this volume.
The first chapter will, I believe, be found easy to read, and may perhaps be useful to many men who are accustomed to show that they are practical by exhibiting their ignorance of the real meaning of scientific and mathematical methods of enquiry.
The second chapter, pp. 20 to 131, consists of an outline scheme of the fundamentals of electromagnetic theory from the Faraday-Maxwell point of view, with some small modifications and extensions upon Maxwell's equations. It is done in terms of my rational units, which furnish the only way ot carrying out the idea of lines and tubes of force in a consistent and intelligible manner. It is also done mainly in terms of vectors, for the sufficient reason that vectors are the main subject of investigation. It is also done in the duplex form I introduced in 1885, whereby the electric and magnetic sides of electromagnetism are symmetrically exhibited and connected, whilst the "forces" and "fluxes" are the objects of immediate attention, instead of the potential functions which are such powerful aids to obscuring and complicating the subject, and hiding from view useful and sometimes important relations.
The third chapter, pp. 132 to 305, is devoted to vector algebra and analysis, in the form used by me in my former papers. As I have at the beginning and end of this chapter stated my views concerning the unsuitability of quaternions for physical requirements, and my preference for a vector algebra which is based upon the vector and is dominated by vectorial ideas instead of quaternionic, it is needless to say more on the point here. But I must add that it has been gratifying to discover among mathematical physicists a considerable and rapidly growing appreciation of vector algebra on these lines; and moreover, that students who had found quaternions quite hopeless could understand my vectors very well. Regarded as a treatise on vectorial algebra, this chapter has manifest shortcomings. It is only the first rudiments of the subject. Nevertheless, as the reader may see from the applications made, it is fully sufficient for ordinary use in the mathematical sciences where the Cartesian mathematics is usually employed, and we need not trouble about more advanced developments before the elements are taken up. Now, there are no treatises on vector algebra in existence yet, suitable for mathematical physics, and in harmony with the Cartesian mathematics (a matter to which I attach the greatest importance). I believe, therefore, that this chapter may be useful as a stopgap.
The fourth chapter, pp. 306 to 466, is devoted to the theory of plane electromagnetic waves, and, being mainly descriptive, may perhaps be read with profit by many who are unable to tackle the mathematical theory comprehensively. It may be also useful to have results of mathematical reasoning expanded into ordinary language for the benefit of mathematicians themselves, who are sometimes too apt to work out results without a sufficient statement of their meaning and effect. But it is only introductory to plane waves. Some examples in illustration thereof have been crowded out, and will probably be given in the next volume. I have, however, included in the present volume the application of the theory (in duplex form) to straight wires, and also an account of the effects of self-induction and leakage, which are of some significance in present practice as well as in possible future developments. There have been some very queer views promulgated officially in this country concerning the speed of the current, the impotence of selfinduction, and other material points concerned. No matter how eminent they may be in their departments, officials need not be scientific men. It is not expected of them. But should they profess to be, and lay down the law outside their knowledge, and obstruct the spreading of views they cannot understand, their official weight imparts a fictitious importance to their views, and acts most deleteriously in propagating error, especially when their official position is held up as a screen to protect them from criticism. But in other countries there is, I find, considerable agreement with my views.
Having thus gone briefly through the book, it is desirable to say a few words regarding the outline sketch of electromagnetics in the second chapter. Two diverse opinions have been expressed about it. On the one hand, it has been said to be too complicated. This probably came from a simpleminded man. On the other hand, it has been said to be too simple. This objection, coming from a wise man, is of weight, and demands some notice.
Whether a theory can be rightly described as too simple depends materially upon what it professes to be. The phenomena involving electromagnetism may be roughly divided into two classes, primary and secondary. Besides the main primary phenomena, there is a large number of secondary ones, partly or even mainly electromagnetic, but also trenching upon other physical sciences. Now the question arises whether it is either practicable or useful to attempt to construct a theory of such comprehensiveness as to include the secondary phenomena, and to call it the theory of electromagnetism. I think not, at least at present. It might perhaps be done ii the secondary phenomena were thoroughly known ; but their theory is so much more debatable than that of the primary phenomena that it would be an injustice to the latter to too closely amalgamate them. Then again, the expression of the theory would be so unwieldy as to be practically useless ; the major phenomena would be apparently swamped by the minor. It would, therefore, seem best not to attempt too much, but to have a sort of abstract electromagnetic scheme for the primary phenomena only, and have subsidiary extensions thereof for the secondary. The theory of electromagnetism is then a primary theory, a skeleton framework corresponding to a possible state of things simpler than the real in innumerable details, but suitable for the primary effects, and furnishing a guide to special extensions. From this point of view, the theory cannot be expressed too simply, provided it be a consistent scheme, and be sufficiently comprehensive to serve for a framework. I believe the form of theory in the second chapter will answer the purpose. It is especially useful in the duplex way of exhibiting the relations, which is clarifying in complicated cases as well as in simple ones. It is essentially Maxwell's theory, but there are some differences. Some are changes of form only ; for instance, the rationalisation effected by changing the units, and the substitution ol the second circuital law for Maxwell's equation of electromotive force involving the potentials, etc. But there is one change in particular which raises a fresh question. What is Maxwell's theory? or, What should we agree to understand by Maxwell's theory ?
The first approximation to the answer is to say, There is Maxwell's book as he wrote it ; there is his text, and there are his equations : together they make his theory. But when we come to examine it closely, we find that this answer is unsatisfactory. To begin with, it is sufficient to refer to papers by physicists, written say during the twelve years following the first publication of Maxwell's treatise, to see that there may be much difference of opinion as to what his theory is. It may be, and has been, differently interpreted by different men, which is a sign that it is not set forth in a perfectly clear and unmistakeable form. There are many obscurities and some inconsistencies. Speaking for myself, it was only by changing its form of presentation that I was able to see it clearly, and so as to avoid the inconsistencies. Now there is no finality in a growing science. It is, therefore, impossible to adhere strictly to Maxwell's theory as he gave it to the world, if only on account of its inconvenient form. But it is clearly not admissible to make arbitrary changes in it and still call it his. He might have repudiated them utterly. But if we have good reason to believe that the theory as stated in his treatise does require modification to make it self-consistent, and to believe that he would have admitted the necessity of the change when pointed out to him, then I think the resulting modified theory may well be called Maxwell's.
Now this state of things is exemplified by his celebrated circuital law defining the electric current in terms of magnetic force. For although he did not employ the other, or second circuital law, yet it may be readily derived from his equation of electromotive force ; and when this is done, and the law made a fundamental one, we readily see that the change it suffers in passing from the case of a stationary to that of a moving medium should be necessarily accompanied by a similar change in the first, or Maxwell's circuital law. An independent formal proof is unnecessary ; the similarity of form and of the conditions of motion show that Maxwell's auxiliary term in the electromotive force, viz., VqB (the motional electric force), where q is the velocity of the medium and B the induction, requires the use of a similar auxiliary term in the first circuital law, viz., VDq, the motional magnetic force, D being the displacement. And there is yet another change sometimes needed. For whilst B is circuital, so that a convective magnetic current does not appear in the second circuital equation, D is not always circuital, and convective electric current must therefore appear in the first circuital equation. For the reason just mentioned, it is the theory as thus modified that I consider to represent the true Maxwellian theory, with the other small changes required to make a fit. But further than this I should not like to go, because, having made a fit, it is not necessary, and because it would be taking too great a liberty to make additions without the strongest reason to consider them essential.
The following example, which has been suggested to me by remarks in Prof. Lodge's recent paper on " Aberration Problems," referring to a previous investigation of Prof. J. J. Thomson, will illustrate the matter in question. It is known that if V be the speed of light through ether, the speed through a stationary transparent body, say water, is V//A, if p is the refractive index. Now what is the speed when the water is itself moving in the same direction as the light waves ? This is a very old problem. Fresnel considered that the external ether was stationary, and that the ether was /a 2 times as dense in the water as outside, and that, when moving, the water only carried forward with it the extra ether it contained (or equivalently). This makes the speed of light referred to the external ether be V//* + v(l -ft~ 2 ), if v is the speed of the water. The experiments of Fizeau and Michelson have shown that this result is at least approximately true, and there is other evidence to support FresnePs hypothesis, at least in a generalised form. But, in the case of water, the additional speed of light due to the motion of the water might be ^v instead of (1 - fir 2 ) v, without much disagreement. Now suppose we examine the matter electromagnetically, and enquire what the increased speed through a moving dielectric should be. If we follow Maxwell's equations literally, we shall find that the extra speed is 1/2ν, provided i?/V is small. This actually seems to corroborate the experimental results. But the argument is entirely a deceptive one. Maxwell's theory is a theory of propagation through a simple medium. Fundamentally it is the ether, but when we pass to a solid or liquid dielectric it is still to be regarded as a simple medium in the same sense, because the only change occurring in the equations is in the value of one or both ethereal constants, the permittivity and inductivity practically only the first. Consequently, if we find, as above, that when the medium is itself moved, its velocity is not superimposed upon that of the velocity of waves through the medium at rest, the true inference is that there is something wrong with the theory. For all motion is relative, and it is an axiomatic truth that there should be superimposition of velocities, so that V//* + v should be the velocity in the above case according to any rational theory of propagation through a simple medium, the extra velocity being the full v t instead of Jv. And, as a matter of fact, if we employ the modified or corrected circuital law above referred to, we do obtain full superimposition of velocities.
This example shows the importance of having a simply expressed and sound primary theory. For if the auxiliary hypotheses required to explain outstanding or secondary phe- nomena be conjoined to an imperfect primary theory we shall surely be led to wrong results. Whereas if the primary theory be good, there is at least a chance of its extension by auxiliary hypotheses being also good. The true conclusion from Fizeau and Michelson's results is that a transparent medium like water cannot be regarded as (in the electromagnetic theory) a simple medium like the ether, at least for waves of light, and that a secondary theory is necessary. Fresnel's sagacious speculation is justified, except indeed as regards its form of expression. The ether, for example, may be identical inside and outside the body, and the matter slip through it without sensibly affecting it. At any rate the evidence that this is the case preponderates, the latest being Prof. Lodge's experiments with whirling discs, though on the other hand must not be forgotten the contrary conclusion arrived at by Michelson as to the absence of relative motion between the earth and surrounding ether. But if the ether be stationary, Fresnel's speculation is roughly equivalent to supposing that the molecules of transparent matter act like little condensers in increas- ing the permittivity, and that the matter, when in motion, only carries forward the increased permittivity. But however this matter may be finally interpreted, we must have a clear primary theory that can be trusted within its limits. Whether Maxwell's theory will last, as a sufficient and satisfactory primary theory upon which the numerous secondary developments required may be grafted, is a matter for the future to determine. Let it not be forgotten that Maxwell's theory is only the first step towards a full theory of the .ether ; and, moreover, that no theory of the ether can be complete that does not fully account for the omnipresent force of gravi- tation.
There is one other matter that demands notice in conclusion. It is not long since it was taken for granted that the common electrical units were correct. That curious and obtrusive constant 4?r was considered by some to be a sort of blessed dispensation, without which all electrical theory would fall to pieces. I believe that this view is now nearly extinct, and that it is well recognised that the 4?r was an unfortunate and mischievous mistake, the source of many evils. In plain English, the common system of electrical units involves an irrationality of the same kind as would be brought into the metric system of weights and measures, were we to define the unit area to be the area, not of a square with unit side, but of a circle of unit diameter. The constant TT would then obtrude itself into the area of a rectangle, and everywhere it should not be, and be a source of great confusion and inconvenience. So it is in the common electrical units, which are truly irrational. Now, to make a mistake is easy and natural to man. But that is not enough. The next thing is to correct it. When a mistake has once been started, it is not necessary to go on repeating it for ever and ever with cumulative inconvenience.
The B. A. Committee on Electrical Standards had to do two kinds of work. There was the practical work of making standards from the experimentally found properties of matter (and ether). This has been done at great length, and with much labour and success. But there was also the theoretical work of fixing the relations of the units in a convenient, rational, and harmonious manner. This work has not yet been done. To say that they ought to do it is almost a platitude. Who else should do it ? To say that there is not at present sufficient popular demand for the change does not seem very satisfactory. Is it not for leaders to lead ? And who should lead but the men of light and leading who have practical influence in the matter ?
Whilst, on the one hand, the immense benefit to be gained by rationalising the units requires some consideration to fully appreciate, it is, on the other hand, very easy to overestimate the difficulty of making the change. Some temporary inconvenience is necessary, of course. For a time there would be two sorts of ohms, &c., the old style and the new (or rational). But it is not a novelty to have two sorts of ohms. There have been several already. Eemember that the number of standards in present existence is as nothing to the number going to be made, and with ever increasing rapidity, by reason of the enormously rapid extension of electrical industries.
Old style instruments would very soon be in a minority, and then disappear, like the pins. I do not know that there is a more important practical question than this one of rationalising the units, on account of its far-reaching effect, and think that whilst the change could be made now with ease (with a will, of course), it will be far more troublesome if put off until the general British units are reformed; even though that period be not so distant as it is customary to believe. Electricians should set a good example.
The reform which I advocate is somewhat similar to the important improvement made by chemists in their units about a quarter of a century ago. One day our respected master informed us that it had been found out that water was not HO, as he had taught us before, but something else. It was henceforward to be H 2 0. This was strange at first, and inconvenient, for so many other formulae had to be altered, and new books written. But no one questions the wisdom of the change. Now observe, here, that the chemists, when they found that their atomic weights were wrong, and their formulae irrational, did not cry " Too late," ignore the matter, and ask Parliament to legalise the old erroneous weights ! They went and set the matter right. Verb. sap.
DECEMBER 16, 1893.
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