Electromagnetic Theory, Vol. 1 (1893) — part 21 of 31

§ 178. Using the same kind of point-source, let it act inter- mittently and alternatingly ; say, on positively for an interval a, then off for an interval /2, then on negatively for an interval a, followed by off for an interval ft, and so on recurrently, like positive and negative applications of a battery with dead in- tervals. The result is, by the last case, to divide the spherical space occupied by the disturbances at a given moment into concentric shells, of depths va and v/3 respectively. In the latter is no disturbance, since they correspond to the dead in- tervals. In the former are the steady displacements appro- priate to their distance from the source, consecutive shells being positive or negative according to the state of the source when they started from it. On their boundaries are electro- magnetic sheets generating these steady states continuously on one side, and destroying them on the other.

THEORY OP PLANE ELECTROMAGNETIC WAVES. 315

Next, do away with the dead intervals, so that the applied force is like that of a common reversing key, first positive of full strength, and then negative of full strength, but without any interval. The result is to completely fill up the sphere of disturbance, the successive shells of depth va containing steady fields — alternately positive and negative — being brought into contact, and only separated from one another by the electro- magnetic sheets. This may be considered rather an unexpected result. Our impressed force is periodic, but expresses the extremest form of variation, namely discontinuities, and is only expressible simple-harmonically by an infinite series of simple- harmonic forces of all frequencies from zero to infinity. But only when the impressed force varies is electromagnetic dis- turbance generated, so that when in the above manner we confine its variations to be momentary, the electromagnetic disturbances are also momentary, with the consequent result that we have our dielectric occupied by quite steady states of displacement separated by infinitely-thin electromagnetic shells. The latter travel. The steady states also appear to travel, but do not really do so. They are being continuously generated and destroyed at their boundaries, but are quite steady elsewhere.

If the impressed force vary simple-harmonically instead of discontinuously, the result is less simple. We find that there is no clean separation into steady states and electromagnetic waves, the two being mixed inextricably. We can see this by substituting for the simply periodic variation of the force a very great number of constant forces of small duration and of different strengths, so as to.roughly imitate the simple- harmonic variation. Every discontinuity in the force will behave in the way described, and the final result of the superimposition of effects, when we proceed to the limit and have continuous simple-harmonic variation of the force, is a train of simple- harmonic waves proceeding from the source. They are not pure electromagnetic waves, and there is no steady state any- where. The wave-length X is given by A = VT, where T is the period of the impressed force, or the reciprocal of the frequency. The electric and magnetic forces are simple-har- monic functions of the time everywhere, right up to the extreme wave-front of the disturbance, where, of course, is an electromagnetic sheet following a different law. But

316 ELECTROMAGNETIC THEORY. CH. IV.

should the source occupy a finite spherical space, then the bounding sheet becomes a shell, of depth equal to the diameter of the sphere, as before described. Now the steady fields become insensible at a great distance in the discontinuous case ; to correspond with this we have the fact that the simple harmonic waves tend to become pure as they expand. The greatest departure from the pure state is round about the source. Here the frequency becomes a matter of importance. When very low, the wave-length is great, and we should, there- fore, have to go a great distance to find any sign of wave- propagation. Moreover, the waves themselves would be exceedingly weak. Only round the source is there sensible disturbance, and it is practically the steady displacement appropriate to the momentary state of the source, accompanied by a weak state of magnetic force connected with the displace- ment nearly in the instantaneous manner. That is, the time- variation of the displacement is the electric current, and the displacement itself is sensibly that of the static theory. Under these circumstances there is next to no waste of energy.

But by increasing the frequency we can completely alter this state of things. For we increase the rate of change of the impressed force, and therefore the strength of the electromagnetic waves generated, for one thing. Besides this, we can bring the region of electromagnetic waves nearer to the source, and so contract the region in which the displacement was approxi- mately static as much as we please. In the limit with very great frequency we have nearly pure electromagnetic waves close up to the source. The waste of energy increases very rapidly with the frequency, varying as its fourth power with simple-harmonic waves. The extremest case of waste is that of a discontinuity, already mentioned, as when setting up the steady state we waste half the work done, and then, when the force is removed, waste the other half.

Self-contained Forced Electromagnetic Vibrations. Contrast with Static Problem.

§ 179. When the sphere of impressed force is of finite size, another effect comes into view of a somewhat striking character, of which the theory of a point-source gives no information. By increasing the frequency sufficiently, we shall reduce the ampli-

THEORY OF PLANE ELECTROMAGNETIC WAVES. 317

tude of the external vibrations, and on reaching a certain fre- quency depending upon the size of the sphere, they will entirely vanish. That is, at a certain frequency, a simple harmonic impressed force, acting uniformly within a spherical space in a dielectric, produces no external effect whatever. The vibrations are stationary or standing, and are purely internal, or confined to the sphere itself. By general experience of statical problems this would seem to be impossible. But we must expect to find strange manners when we go into strange lands.

This strange behaviour arises from the general electromag- netic property that the true source of disturbances due to im- pressed force (electric or magnetic) is the curl thereof. The impressed force itself, on the other hand, is associated with the supply of energy. (See § 87.) In the present case the sur- face of the sphere of impressed force is the seat of its curl, its intensity varying as the cosine of the latitude, if the polar axis be parallel to the impressed force ; and consequently, as already described for the case of setting up the steady state, the elec- tromagnetic waves proceed both ways from the surface at which they originate. The waves going inward contract, then cross at the centre and expand again. We have, therefore, two trains of outward waves, and the external effect may be imagined to be due to their superimposition. That they may sometimes assist and sometimes partly cancel one another, according to relative phase, may be readily conceived, and the theory indicates that at any one of an infinite series of definite frequencies there is complete cancellation externally. This refers to the simple-harmonic state. There is always an external effect initially, at any frequency — namely, the initial electromagnetic shell of depth equal to the diameter of the sphere of impressed force, because it consists entirely of the beginning part of the first outward wave, the second one only reaching to its rear ; but on the inner side of this shell there may be no disturbance, except within the sphere of impressed force. The waste of energy is not continuous at the critical fre- quencies, but consists merely of the energy in the initial shell. The impressed force works continuously, but as much positively as negatively within a period, so that it is inactive on the whole.

Effects of the above described kind are principally remark- able from the contrast they present towards more familiar

318 ELECTROMAGNETIC THEORY. CH. IV.

effects, especially those of a statical kind. The behaviour of electrification on a sphere may be taken as an illustrative con- trast. We know that if we have a spherical surface in a uni- form dielectric covered with electrification, the displacement is partly internal and partly external. Only when the electri- fication is uniformly spread does the displacement vanish internally, and become wholly external. This was for- merly explained by the inverse-square law. Every element of electrification was supposed to exert force equably in all directions round itself, internally as well as externally, and a mathematical consequence of this is the complete cancellation of the internal force when the electrification is uniformly spread on the surface of a sphere. The same reasoning was applied when the sphere was not of the same nature as the external medium — when it was a conductor; to wit. Now the internal force is also zero in this case. More- over, it is zero whatever be the nature of the internal material (without electrical sources). But it is certainly untrue that the electric force due to a point-source of displacement is the same in all directions around it when the medium is not homo- geneous. So the reason for the absence of internal force is entirely wrong, although it seems right in one case, viz., that of homogeneity. The true, and sufficient, and comprehensive reason for the equilibrium, and the absence of internal force is the satisfaction of the second circuital law when the external electric force is perpendicular to the surface. This is independent of the nature of the internal matter, and is obviously to be pre- ferred to a hypothesis that is only valid sometimes. Varying the internal material will make the law of spreading of force be different in any number of ways, and, if we like, different for every element of electrification, by arranging various kinds of heterogeneity. But independently of this, we might vary the law of force in many ways in special cases. Thus, in the case of a uniform spread of electrification on a sphere, we may imagine the displacement from any particle of electrification to radiate in any symmetrical manner that will give zero displacement inside and a uniformly radial displacement outside. The simplest case after the usual uniform spreading in all directions, is a planar spreading. Let every element of electrification on the surface send out its

THEORY OF i'LANE ELECTROMAGNETIC WAVES. 319

displacement in the tangential plane only, though equally in all directions in that plane. This will give the correct static result. It is not altogether a fantastic example, for we can make an electromagnetic problem of it by letting the electrified surface expand at the speed of light. Then the assumed law of spread is the actual law, for every element of electrification is the core of a plane electromagnetic wave. The resultant mag- netic force due to all the waves, however, is zero, and the resultant electric force is as in the static problem. (See §§61 and 164.)

Now when we have vibrating electromagnetic sources on a spherical surface producing electromagnetic waves simple- harmonically, we also in general have both internal and external effects, for there is an internal as well as an exter- nal train of waves. But whilst they cannot cancel internally, they may do so externally. Here is one contrast with the static problem, and along with it is another, viz., that it is the nature of the external medium that is now indifferent (with a reservation), whereas in the static problem we have independence of the nature of the internal medium. The reservation hinted at is connected with the initial uncancelled electromagnetic shell. The external medium should allow it to escape, or absorb it somehow, so that it will not interfere with the effects under consideration. To regard the zero external effect as being actually due to the coexistence of two trains of waves which cancel one another is merely a mathematical artifice, however, because neither train of waves exists. If they did exist we should have to vary their nature to suit the constitution of the external medium, just as in the problem of static equilibrium we require different laws of force to suit the nature of the internal medium. The comparison of the static with the kinetic problems bhows a complete reversal of relations as regards internal and external.

What we can do with a single surface electromagnetic source we can repeat with others inside it. We see, therefore, that the whole sphere of impressed force may be filled up with vibratory sources of the most vigorous nature without producing any (except initial) external effect, if their periods be properly chosen in relation to their situation, which problem admits of multiple solutions. This is suggestive as regards the stores o> energy bound up with matter.

320 ELECTROMAGNETIC THEORY. CH IV.

Relations between E and H in a Pure Wave. Effect of Self- induction. Fatuity of Mr. Preece's " KR law."

§ 180. In the above very little has been said about the dis- tribution of electric and magnetic force from part to part of an electromagnetic wave. This varies greatly in different kinds of waves, and cannot be explained without the formulae. But there is one very important property of pure electromag- netic waves (which also holds good, more or less approximately, in general) which may be described at present. As already mentioned, the electric and magnetic energies are equal in a pure electromagnetic wave. Now the density of the electric energy is -JcE2, if E is the electric force (intensity), and that of the magnetic energy is J/*H2. If we equate these, we obtain a relation between E and H. Thus

(1)

Also /*cv2 = l ..... (2)

Therefore E = ± jj.vR ....... (3)

The positive or negative sign depends on which way the wave is going. Disregarding this, the electric and magnetic forces have a constant ratio. They are therefore in the same phase, or keep time together in all their variations without lag or lead. Along with this, they have the property of being per- pendicular to one another (that is, E and H are perpendicular, not E and H), and their plane is in the wave-front, or the direction of motion of the wave is perpendicular to E and to H. It is the direction of the flux of energy. These properties hold good in all parts of a pure electromagnetic wave, although the magnitudes and directions of the electric and magnetic forces may vary greatly from one part of the wave to another.

The above is also the state of things that obtains, more or less perfectly or imperfectly, in long-distance telephony over copper circuits of low resistance, and by lowering the resistance per mile we may approximate as nearly as we please to the state of pure electromagnetic waves. It is the self-induction that brings about this state of things, showing such a contrast to the more familiar relations between the electric and magnetic forces on circuits in general. Like a kind of fly-wheel, the self-induction imparts inertia and stability, and keeps the waves

THEORY OF PLANE ELECTROMAGNETIC WAVES. 321

going. It is the long-distance telephoner's best friend who was, not many years since, spurned with contempt from the door. Some people thought there was a very absurd fuss made about self-induction, and that it was made a sort of fetish of ; so, knowing no better, they poured much cold water on the idol. But, whatever opinions we may hold regarding their competence as judges, there can be no question about the stern logic of facts. Self-induction came to stay, and stayed it has, and will stay, having great staying power. Whatever should we think of engineers who declined to take into account the inertia of their machinery? There was also some consider- able fuss made about a supposed law of the squares, or KR law as it was or is called, according to which you could not telephone further than KR = such or such a number, because the speed of the current varied as the square of the length of the line, or else inversely. But in spite of the repeated attempts made to bolster up the KR law, the critical number has kept on steadily rising ever since. I see now that it has gone up to 32,000.* But it need not stop there. Make your circuits longer, and it will go up a lot more.

As regards the ether, it is useless to sneer at it at this time of day. What substitute for it are we to have ? Its principal fault is that it is mysterious. That is because we know so little about it. Then we should find out more. That can- not be done by ignoring it. The properties of air, so far as they are known, had to be found out before they became known.

Wave-Fronts; their Initiation and Progress.

§. 181. Still keeping to a simple dielectric, we may always, by consideration of the fact that the speed of propagation is v, find the form of the wave-front due to any collection of point- sources, and trace the changes of shape and position it under-

• See The Electrician, December 30, 1892, p. 251, for data, in article by Mr. Jos. Wetzler. The 32,000 is for the New York-Chicago circuit of 1,000 miles at 4'12 ohms per mile, or 2'06 ohms per mile of wire. Comparison of Mr. Wetzler's with Mr. Preece's figures is interesting. Good telephony is got by Mr. P. at 10,000, and by Mr. W. at 45,000 ; excellent by Mr. P. at 5,000, and by Mr. W. at 31,000 ; and so on. But there is still less of a KR law in the American than in the English cases.

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322 ELECTROMAGNETIC THEORY. CH IV.

goes as It progresses. For obviously, if a point P be at a dis- tance vt from the nearest source, no disturbance can have reached P from it if it started into action after the moment t = 0 ; and generally, the effect at P at a given moment arising from a particular source depends upon its state at the moment r/v earlier (if r be the distance from the source to P), and upon the previous state of the source, causing residual or cumula- tive action at P. We therefore know the limiting distance of action of the sources for every one of their momentary states. The wave-front belonging to a set of disconnected point-sources consists initially of disconnected spheres. But as they expand, they merge into one another to form a continuous extreme wave- front consisting of portions of spheres. When the point-sources are spread continuously over a surface to form a surf ace-source, we have a continuous wave-front from the first moment ; or rather, two wave-fronts, one on each side of the surface. There is an exception, to be mentioned later, when the sources send waves one way only from the surface. This is not the already described case of the cancelling of two trains of waves at par- ticular frequencies, but is a unilateral action obtained by a special arrangement of surface-sources. Passing over this, observe that when we have once got a wave-front we may ignore the sources which produced it, and make the wave-front itself tell us what its subsequent history will be. For, to trace the course of the wave-front from one moment to the next, we have merely to move any element of the surface in the direction of the normal to that element through a small distance a to obtain the new position at the moment a/v later. This being done for the whole surface, gives us the new position of the wave-front, and by continuing this process we may follow the wave-front in its progress as long as we please. Thus the wave- front coming from the surface of a sphere is a sphere ; from a round cylinder, if infinitely long, also a round cylinder ; but if of finite length, then a cylinder with rounded ends ; from a cube, initially a cube with rounded corners, but becoming more and more spherical as it expands ; and so on. It is easily seen that at a sufficiently great distance from a finite collection of sources of any kind, the wave-front tends to become spherical, or the complex source tends to become equivalent, at a great distance, to a point-source of some complex kind.

THEORY OP PLANE ELECTROMAGNETIC WAVES. 323

Effect of a Non-Conducting Obstacle on Waves. Also of a Heterogeneous Medium.

§ 182. Now consider the effect of an obstacle brought into our medium, say a non-conducting dielectric mass of different inductivity or permittivity. The manner of propagation in it is similar in kind to that in the external medium, bat it varies in detail, and the speed will usually be different. Then, when we have a wave coming from (say) a point- source in the first medium, the presence of the obstacle at first makes no dif- ference. It has no immediate action, and, until the wave reaches it, might as well not be there. But immediately the wave does reach it a change occurs. The interface of the two media becomes the seat of sources of fresh disturbances, or they may be considered fictitious sources, in contrast with the original, for they do not bring in any fresh energy. Thence arises a new effect, viz., reflection. Not the whole, but only a part of the wave disturbance enters the new medium. The rest is thrown back, and forms a reflected wave. In the same way as we may trace the course of primary waves from a source in a uniform medium may we trace the course of the secondary waves set up by the obstacle. The disturbance outside it is then due to the superposition of the primary and secondary waves, and, if there be just one obstacle, this state of things continues. Of course the secondary wave- front may itself be complex, because the disturbances going into the second me- dium and transmitted therein according to its nature, may reach the interface again at other parts, and there suffer re- flection and transmission anew. This complication is done away with by making the obstacle infinitely big, with a plane boundary. Then we have just one wave in the obstacle and two in the medium containing the source, viz., the primary and the first (and only) reflected wave.

But if there be a second obstacle, not only will it, like the first, give rise to a secondary wave when the primary meets it, but each obstacle will act similarly towards the secondary wave from the other, whence arises a pair of tertiary waves, and so on. Thus, with only two obstacles, we shall have a succession of infinitely numerous waves crossing and recrossing one another, arising out of the action of a point-source, with

Y2

324 ELECTROMAGNETIC THEORY. CH. IV.

great complications. But, however complex in detail, we have the important fact that the mere knowledge of the speed of propagation allows us to lay down the whole course of the waves generated, and the position of the wave-fronts belonging to a given epoch at the source. The mathematics of the full treatment may be altogether beyond human power in a reason- able time; nevertheless, we can always predict with confidence that the results must have such and such general properties relating to the course of the waves, besides the properties involved in the persistence of energy.

In the above we were concerned with discontinuous he- terogeneity, or an abrupt change in the value of one or both of the constants c and /* at an interface. When the medium is continuously heterogeneous, or the "constants" change in value continuously from place to place, then the speed v is made a function of position. The process of partial reflection and partial transmission which occurred at the interface now takes place in general wherever the value of c or p changes. These changes being continuous, so are the results, so that we do not have distinctly separable trains of waves, but rather a continuous distortion. We can, however, follow the course of a wave expanding from a point by communicating to the wave- front the speed proper to its position, such speed being, as above, definitively known. A spherical wave may remain spherical, only varying in its speed; or more likely, it may change its form as it expands. It may become ellipsoidal, for example, and of course must do so if the speed in different parts of the medium vary suitably.

It is possible for the inductivity and permittivity to change, either abruptly or continuously, without any reflection, or cast- ing behind of the disturbance as a wave progresses, although its speed varies. In such a case it will be found that the ratio IJL/C is constant. The change in p,, or the ratio of induction to magnetic force, then fully compensates the change in c, the ratio of displacement to electric force, either of which changes by itself would cause reflection.

In connection with a heterogeneous medium (including abrupt changes of nature), we may notice its behaviour as regards steady states, in contrast with that of a thoroughly homo- geneous medium. In the latter case, as we have already

THEORY OP PLANE ELECTROMAGNETIC WAVES. 325

described, the introduction of a steady point-source causes the steady state to begin immediately at and spread around it, after which no change occurs. That is, the medium being homogeneous, there is no reflex action. But when it is hetero- geneous the case is quite different. The steady state of dis- placement due to impressed voltage depends upon the per- mittivity in all parts of the medium, and only comes about as the final result of the infinitely numerous reactions between different parts, due to the reflections that take place, which modify the final distribution of displacement as well as (usually) the total amount. There is a similar contrast in the mechanism of the mathematics involved in the calculation of the steady state. When the medium is homogeneous we may find it through the potential of fictitious matter at the source alone. When heterogeneous, we require to have fictitious matter all over the medium, or rather in all parts where it changes its nature, and, owing to the infinitely numerous reactions, we should, in a general or complete solution, require to perform not one space-integration, but an infinite series of successive space-integrations, before we could arrive at the true potential function, allowing for the variations of permittivity every- where. Thus, by electromagnetic considerations we obtain some insight of the true nature of transcendental static prob- lems, involving potential functions and assumed instantaneous action at a distance.

Effect of Eolotropy. Optical Wave -Surfaces. Electromagnetic versus Elastic Solid Theories.

§ 183. When the medium is not isotropic as regards either the displacement or the induction, or as regards both, we have very remarkable effects, known in the science of optics as double-refraction. There can be two distinct wave-speeds for a given position of the wave-front, and these speeds change when the direction of the normal to the wave-front changes. Hence double refraction, or the separation of a wave entering an eolotropic medium into two waves travelling independently of one another at different speeds. If the medium is only elec- trically eolotropic, although the displacement and induction are still in a wave-front (of either wave), and are perpendicular to one another, the electric force is inclined to the wave-front,

326 ELECTROMAGNETIC THEORY. CH. IV.

so that the ray, which indicates the direction of the flux of energy, is inclined to the normal to the wave-front. Similarly, with magnetic eolotropy alone, it is the magnetic force that is inclined to the wave front. When the medium is both elec- trically and magnetically eolotropic, both the electric and the magnetic force are inclined to the wave-front, whilst the dis- placement and induction, though still in the wave-front, are no longer perpendicular to one another.

By imagining plane electromagnetic sheets to traverse the medium in all possible directions about a point, and comparing their positions with respect to the point at equal times after crossing it, we arrive at the conception of the wave-surface. This is obviously a sphere in an isotropic medium. But when it is eolotropic, the two speeds, and their variation according to the direction of motion of the waves, make the wave-surface become a very singular double surface. With electric eolotropy alone, which is the practical case, it is Fresnel's wave-surface. It is also another Fresnel surface with magnetic eolotropy alone. But when the medium is eolotropic as regards both the induction and the displacement, the wave-surface is of a more general and symmetrical character, including the former two as extreme examples. It is still a double surface, however, except in one case. We have already mentioned that in an isotropic medium there is a peculiar behaviour when the ratio P./C is constant, although fj. and c vary. We might anticipate some peculiarity in the wave-surface when /z/c is constant. This constancy now means that the directional properties of //, are exactly paralleled by those of c. That is, the principal axes of p and those of c are coincident, whilst the value of the ratio of the permittivity to the inductivity is the same for the three axes. The result is to reduce the double wave-surface to a single surface, which is an ellipsoid.

Assuming light to consist of vibrations of an " elastic solid " medium, single or ordinary refraction is explained by an alteration in the density or the rigidity of the ether. Not only is the theory quite hypothetical in considering the kinetic energy to be the energy of vibrational motions of displacement, but the alteration of density or rigidity assumed is a further hypothesis on the top of the main one, for there is no evidence that there is such a change. Moreover, the explanation will

THEORY OP PLANE ELECTROMAGNETIC WAVES. 327

not work properly. On the other hand, the electromagnetic theory says that light consists of electromagnetic vibrations in the ether. This, too, is a hypothesis. But the auxiliary part, that refraction is caused by change of permittivity from one medium to another, is not a hypothesis, but a fact. More- over, the theory works.

Similarly, double refraction in elastic solid theories of light is explained by eolotropy as regards elasticity, or by something similar relating to the density. This is also hypothetical, and not without its troubles. But, on the other hand, Maxwell declared that double refraction occurs because the doubly refracting medium is electrically eolotropic. Now this is a fact too, and the theory is a clear one.

These remarks will serve to illustrate what I mean by the far greater intrinsic probability of the electromagnetic theory, apart from the experimental work of late years. It is much less hypothetical than elastic solid theories. We know nothing about the density or the rigidity of the ether, or how they vary in different bodies, or' if they vary. But we do know a good deal about electric permittivity and magnetic inductivity. In- stead of dealing with possibilities, we are dealing with actual facts.

The old objection that a mechanical theory of light was surely to be preferred to an abstract electromagnetic theory was very misleading. The electromagnetic theory is mechan- ical, without, however, a precise specification of the mechanism. An elastic solid theory is merely a special mechanical theory. It cannot satisfy the electromagnetic requirements, but this failure, though immensely important in itself, is not the point here. Even if it did satisfy them, it would probably be less true than the electromagnetic theory, which, being abstract, does not assert so much. There may be many "mechanical" solutions of an abstract theory. Elastic solid theories are a great deal too precise in saying what light consists of, and mechanical speculations in general should be received with much caution, and regarded rather as illustrations or analogies than expressions of fact. We do not know enough yet about the ether for dogmatising.

One can imagine that a clear-headed man might be able to work his way through all the theories of light yet propounded, assimilate what was useful and true, eliminate what was useless

328 ELECTROMAGNETIC THEORY. CH. IV,

or false, and finally construct a purified theory, not professing to explain what light is, but still connecting together in a per- fectly unobjectionable manner, free from hypotheses, all the principal facts, and of so compliant a nature as to readily adapt itself to future discoveries, and finally settle down to a special form expressing the theory of light. It is hardly likely that the clear-headed man will be found for the purpose, and per- haps his theory would not be very popular. It would be so very abstract. What is far more likely is that the electro- magnetic theory (itself abstract and possessing many of the desired qualifications), which has already begun to find its way into optical treatises at the end, will gradually work its way right through them to the beginning, and in doing so, oust out the most of the old-fashioned hypotheses. The result will be to have optical theory expressed throughout in electromagnetic language. To do it properly, it is hardly necessary to say that the preposterous 47r, which the B.A. Committee seem to want to perpetuate, should be ignored from the beginning.

A Perfect Conductor is a Perfect Obstructor, but does not absorb the Energy of Electromagnetic Waves.