Electromagnetic Theory, Vol. 2 (1899) — part 23 of 31

making equal angles with it. tn the OTerlapping region, close to the plane, the dispUcement dae to the union of the two waves is normal to the surface (that is, with an eleotrio con- ductor) which is electrified, and the electrification runs along the surface at a speed depending upon the angle of incidence, being v at grazing incidence (of rays) and v/cos 6 at incidence angle 9, varying, therefore, between v and infinity. It may be, perhaps, rather a novel idea to some readers that electrification can run through space at any speed greater than that of light, but the matter is made simple enough by considering the rate of incidence upon the reflecting surface of different parts of the plane sheet. Tn the case of nearly flush incidence of a sheet, its dift'erent parts strike the surface nearly simultaneously, so that there is an immensely great speed of motion of the elec- trification along the surface. The electrification is the same in amount always, and is continuously existent, so we are some- what justified in speaking of the electrification moving ; but we may equally well regard it as a case of continuous genera- tion of electrification at one end and of annihilation at the other end of the part of the conducting surface which is momentarily charged, the generation and annihilation being performed by the different parts of the incident sheet and the leflected sheet as they reach and leave the surface. Details of these simple oases, leading to a plainer understanding, will oome later, when these general notions are got over. In the limiting case of normal incidence of rays, when the incident sheet strikes the sorfaoe flush, the electrification is non-existent* It goes out of existence just as its speed becomes infinite.

The aboTe describes one extreme kind of reflection of a plane sheet at any angles and is what occurs when H in the incident^ and, thereforsb also in the reflected w9Te, is tangential to the refleotor, whilst E is in the plane of inddenoe. Bat when it is E in the incident waye that is in the plane of inddenoe^ and B is tangential, we have quite another kind off oompoution In the OTerlapping part near the refleotor. There is no B within It at all, and also no eleotrifioation ; whilst the H within it is parallel to the reflector, and simply joins together the Ws m the parts off the waves which do not overlap, the E in one wave being directed towards the surface in the plane off ind- dence, and in the other away ffrom the surface. In both waves, of

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course, H ie in the plane of the wave. In other mspeete we have « similarity in propagation. It is now a surfoce onrrent, instead of electrification, that runs along the snrfaoe at speed «/co8 6,

If the reflector be a magnetic condnotor, we have two veiy

similar main cases, in one of which magnetification (the ana- logue of electrification), and in the other a magnetic current, runs, or appears to run, along the surface.

The Effect of Oondoetiug Matter in Diverting External

Induction.

§ 191. The theory of the effect of a finite degree ot con- ductivity in the medium on electromagnetic waves is far more difficult and complicated than that of the effect of infinite con- ductivity. Nevertheless, we may gain a general idea of the nature of the effect by means of the substitution of simple problems for the real ones that present themselves, and also in this way obtain a knowledge of very important properties con- cerned. There are two ways in which we may regard the ques- tion of conductivity. First, starting from the theory of perfect conductors surrounded by perfect dielectric non-conductors, we may examine the effect of introducing a slight amount of resistivity, to be then increased more and more until at last we come to conductors of high resistiYitj, or infinite, when we have dielectrics merely. The other way is to start with electromagnetic waves in a perfect dielectric, and examine the effect produced upon them of introducing first a snudl amount of conductivity, then more and more, until we come to perfect conductors again. Both ways are instructive — none the less because they give very different views of the same matter.

In the first place, it may be readily conceived that if a con- ductor have only a slight amount of resistivity it may behave approximately the same as if it had none^ and may obstruct waves nearly perfectly internally, and likewise reflect and con- duct them superficially nearly perfectly. This is true, but the element of time has to be taken into account, as it becomes of great importance when there is some resistivity, however little^ instead of quite none. Suppose, for example, we have an initially steady magpietic field, and bring a conductor into it very quickly from a distance, and keep it there. If this be done quickly enough, the first effect is nearly the same as if

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the conductor were perfect. That is, the induction of the field will be driven out of the space it previously occupied, which is now occupied by the conductor, and pass round it tangentially, as if the conductor were of zero induo- tivity. It will, therefore, be mechanically acted upon by the stress in the field in a manner resembling the action upon a perfect diamagnetic body — that is, there will be a force tending to drive it from stronger to weaker parts of the field. In another form, we may say that external force must be applied to the conductor to bring it into the field and to hold it there. But this state of things will not continue. The magnetic force, which is at first only skin deep, will pene- trate into the interior in time, according to a law resembling that of the diffusion of heat. Given time enough, it will assume the same distribution as if there were no oonductivity, although our assumption is that there is nearly no resistivity— that is, in the ultimate state tended to, it is merely the induo- tivity that settles how the magnetic induction will distribote itself. Initially, there is a skin current, its total being mea* eured by the difference in the magnetic force just outside and % little way inside the conductor, in the bulk of which there fs practically no magnetic force or electric current. As in the cue of a perfect conductor, Ohm's law is fully obeyed. There is no inUnrnal current, because there is no internal electric or magnetic force. They simply have not had time to get in, on aooonnt of the obstructiye action of the high oondnctiyity. It is as useless and misleading to say that one electric force is <oanoelled by another, as it is to ascribe the absence of magnetio loroe to the counteracting magnetic force of the skin current. As the magnetio force spreads into the conductor, so does the electric current, which is the curl of the former. In the end, when the magnetic force has got steady, the current cettsoo. There may now still be moving force on the conductor, but not of the same kind as before^ it being simply the ordinary para- magnetic attraction or diamagnetic reptdsion according as the Talue of the inductivity of the body exceeds or is less than t^t of the external medium.

It will be Tcry much the same thing if we start with the conductor at rest in a neutral state in a neutral field, and then establish a steady magnetio field ^ some external

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(sanfle. There will still be the same internal obatmotxm offered by the conductor, and consequent delay in the aaaomp- tioQ of the iteady atate throughout it* In both oaaes, too, there is a necessary waste of energy inToWed in the process^ according to Joule's law of the generation of heat by the eaStl- ence of electric current in conducting matter. Thus two things happen when the degree of conductivity is not infinite. First, the reflection is imperfect at the boundary of the conductor, a portion of an incident disturbance being transmitted into it. Next, in the act of transmission and the attenuation involved, there is a loss of energy from the electromagnetic field. We shall see later more precisely the nature of the attenuating process. At present we may note that if the- external field be not steady or do not tend to a steady state, so that the conductor is exposed to fluctuating forces, then the internal part of the conductor need never acquire any sensible magnetic force. Thus, if the external field be the sura of a balanced alternating, and of a field which would be steady in the absence of the conductor, only the latter part will penetrate fully into it. The former alternating part will penetrate imperfectly, the more so the greater the conductivity, and, as before said, not at all when the conductivity is perfect. The other field is then also excluded. With rapid alternations the region of sensible penetration is only skin-deep, consisting of layers of opposite kinds (as regards direction of the magnetio force, which is nearly tangential), with, however, so very rapid an attenuation of intensity in going inward that practically only one wave need be considered (except for short waves, like light). Of course, the heat generation is now confined to the ^n. What goes in further does so by ordinary heat diffusion.

The time-constant of retardation of a conductor varies as the conductivity, as the inductivity, and as the square of the linear dimensions. This refers to the intervals of time required to establish a definite proportion of the steady state under the action of steady forces — in bodies of different sise, conductivity, and inductivity, but geometrically similar. Here the two pro- perties, conductivity and inductivity, act conjointly, so that, lor example, iron is far more obetmotive than copper, although ita conductivity is much inferior. It is different with the heat- generation. There the inductivity and conductivity act in

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opposite senses, for, with the same electric force, the waste varies as the conductivity, or, with the same current-density, as the resistivity. It results that in cases of skin-conduction of rapidly alternating currents, the resistance per unit area of surface varies directly as the square root of the product of the resistivity (not conductivity), inductivity, and frequency. Thus, whilst we may increase the resistance by increasing the fretiuency, with a given material, and also by increasing the inductivity, we decrease it by increasing the conductivity, in spite of the fact that the internal obstruction varies as the con- ductivity and inductivity conjointly. The point to be attended to here is that mere internal obstruction is no necessary bar to efifective skin conduction, although, of course, in a given case the resistance is greater than if the conduction were more wide- quread. It depends upon how it is brought about, whether by conductivity or inductivity. This is how it comes about that with th« complete internal obstruction of a perfect conductor, with the effective skin reduced to nothing, there is still no resistance, and the slip of electromagnetic waves along them is perfect. But it is different when we obtain the internal obstruc- tion by increasing the inductivity, preserving the conductivity ooQStant. Perfect internal obstruction then means infinite resistance, and no proper slipping of waves at all. If the obstrnotion be not complete, it will be accompanied by very rapid attenuation of waves run*iing along the surface when the obstrnotion arises from high inductivity, and by relatively very slight attenoation when it arises from high conductivity.

The repulsive fbroe which was referred to in the ease of a perfect oonduotor brought into a magnetic field, or when a magnetic field is created outside the perfect conductor, arises from its obstructive action, combined with the fact that it is only the lateral pressure of the magnetic stress that acts on the oonduotor, owing to the tangentiatity of the magnetic force. This repulsive force is naturally also operative, though in a less marked form, when the conductivity is not perfect. In fact, it is operative to some extent whenever there is a sufficiently rapid alternation of the field for the conductance to cause a sensible departure from the undisturbed state of the magnetic force, and is, therefore^ strongly operative with ordinary metallic oonduetors with quite moderate frequency of vibration. The

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remarkable experiments of Prof. Elihu Thomson on this " elec- tromagnetic repulsion " will be remembered. The phenomenon has nothing specially to do with electromagnetic waves. It is magnetic repulsion, rather than electromagnetic, using the word "magnetic" in its general sense, apart from the special fact of magnetisation when the induotivity of the conductor is not the same as that of the ether.*

When a conductor is brought into an electric instead f a magnetic field, the case is somewhat different. There is in any case merely skin conduction, for there is an actual destruc- tion of the flux displacement by conductivity. The final result therefore, is that the ultimate permanent state in the con- ductor is a state of perfect neutrality, just as if the conduc- tivity were perfect. Electric conductivity destroys displace- ment, but it cannot destroy induction. Similarly, magnetic conductivity would destroy induction, but would be unable to d^troy displacement. Thus, a magnetic conductor brought into an electric field would, in time, permit its full penetra- tion, but if brought into a magnetic field the final result would be a state of internal neutrality, however low the conductivity.

however, it be very low, then, whether it be of the electric or the magnetic kind, there will be an initial nearly complete penetration (owing to the removal of the obstruction), followed by subsidence to zero of the flax appropriate to the condno- tivity, electric or magnetic respectively. The persistence of magnetic induction, in spite of the presence of electric oon-

• This reference to Elihu Thomsou's experiments must not be under- stood as a full exfdanation, which is sometimes complex. The idea in the text has been of a lump of metal. When made a disc or a lineer drcoit we have special pecoliaritieB, and the theory may be perhaps best done

in terms ()f in<liictAnces and resistances. The principle concerned of the teuipuiary diversion of magnetic induction by conducting; omtter remains in force, however, whether the matter be in a lump or in a close* i line. In the latter case the tendency of the conductance is to keep the total induction through it constant. Consider first an infinitely con- ducting disc which completely diverts induction ; and next^ a ring made by removing nearly all the disc except the outer part. Induction now goes through the circuit, of course, wlien brought into a magnetic field, Lut its total is zero, by reason of the infinite conductance and the current " induced " in the ring. As is well known, Maxwell considered perfectly conducting linear molecular oirouita in applying his views to Weber's theory ojl diamaguetism.

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dnetiTity, is ft Tery importaat and ugnificant fact, of whtob I ahall give a simple proof in a later Section, in amplification of my proof of 1887-8. The present method of passing from perfect to finite condnctiyity is nnsuitable for Uie purpose, because in good conductors the dielectric permittiyity is altogether swamped, and is therefore ignored ; whilst^ on the other hand, in Tery bad conductors the permittivity is a factor of the greatest importance. Now we can pass continuously from a non-conducting dielectric to a conducting oDe, up to perfect conduction, but we cannot pass the other way without having the permittivity in view all the time, which makes the matter difficult.

Parenthetical Bemarks on Induction, Magnetisation, Indue-

tivity and Susceptibility.

§ 192. As people's memories are very short, and there is some discussion on the subject, I may repeat here that the so> frequently-used word inductivity is not intended for use as a mere synonym for permeability. The latter is the ratio of the InductiTitjr of a medium to that of ether, and is therefore a mere numeric. On the other hand, inductivity has a wider meaning, namely, such that hfiU^ is the density of the magnetic energy, irrespeotiTe of dimensions. \Vc can only make it a numeric by assumption. Even then, it has only a fictitioua identity with permeability — ^a forced numerical identity. Similar remarks apply to some other quantities, but they are particularly necessaiy in the case of inductivity, on account of the obscure and misleading manner in which tho connections between induction, magnetic force, and magnetisa- tion were formerly commonly presented (and still are some- timesX together with the misleading connection between tho susceptibility and the permeability. We should write

if f» is the permeability and k the susceptibility, instead of

where the suffix letter refers to the common irrational reckon- ing. The 4*- is, as usual, simply nonsense, unworthy of sdentifie men near the end of the nineteenth centuzy. Now introduce

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Maxwell's ether theory, and make p^i^-Ii^q, where ft^ is the inductivity of ether, and /x that of some other substance (with the usual reservations), then

f*=/*o(l+«) takes the place of the oommon

which is misleading in two respects, first as regards the obstre- perous 47r, and next in making fi and k be quantities of the same kind. But k is always and essentially a numeric, whilst ft is not. We see that the use of inductivity rather than permea- bility is necessary in electromagnetic theory, as a matter of logical common sense as well as for the purpose of scientific clearness. But this need not interfere with the use of permea- bility in its above-described sense of a ratio. If, on the other hand, one of the two words should be abolished, there can, I think, be little doubt as to which should go.

In the case of purely elastic magnetisation (without intrinaio) «e have

B-=/*F«/*o(1 + k)P,

where B is the induotion and F the magnetic force. Here plJP 18 what the induotion would be in ether, so that the additional part /uiqkF expresses the effect of the matter present, which becomes magnetised. The ratio k, therefore, naturally eiqircssos the susceptibility for magnetisation of the matter. Perhaps, in passing, it might be thought that fiQic should express the sna- oeptibilitj. But this will not do, because magnetisation and induction are similar. The induced magnetisation is f^^KV, In strictness it should not be called the intensily of magnetisation, but rather the density, if we properly cany out Maxwell's prin- ciples about forces and fluxes, or intensities and densities. B is a flux, therefore so is ii^qkY^ to be measured per unit area. Now, the common form is

Bi = /iF, = (l +47r/c,)Ft,

or B, = F< + 43rIi,

if li^K^i. Here we have, apart from the 4w absurdity, an irrationality of a different kind, viz., that induotion and mag- netisation are made identical in kind with magnetic force, since we have the difference of two flux densities expressed by an

THBORT OF PLANS BLBOTBOMAONBHO WAVIB. 351

intensity," wbidi is referred to unit length. Now, this may matter yerj little in practical calculations, but it is more than miaehieYoas in theory. Suppose, for example, we are working with a kind of mathematics that takes explicitly into account

the two ways of measuring vector magnitudes, with reference to length and to area respectively, according as they are re- garded as " flux " densities or " force " intensities. Then, if we do not recognise and take account of the radical difference between and in the last equation, we may expect to be led to singular and unaccountable anomalies. This is, I think, what has happened in Mr. MacAulay's recent paper on the theory of electromagnetism. The remedy is easy. There should be no special limitations imposed upon the quantities <M>ncemed such as occur when permeability and iuductivity are made the same.

It is also highly desirable, for the same purpose of obtaining scientific clearness and freedom from distressing anomalies, that the distinction between " induced " and intrinsic magneti- sation should be clearly recognised and admitted in the formula. In the above there has been no intrinsic magnetisation. Let this now be Iq. An equivalent form is /xhQ, where h^ is the cor- responding intrinsic magnetic force. Tiiis is of the same nature as B. The complete induction becomes

B-fi(P + ho) = /iH,

where H is the complete force of the flux B. This is the best way of exhibiting the induction. If H be split at all, let it be into the part involving the intrinsic force h^ and the rest. Or,

B-lo+fiP.

The other separation, namely, of fxF into the ether induction H^T and the additional part due to matter, is less usefuL If done, then

B = I, + I + /ioF.

If we now amalgamate Iq and I, to make, say, I^, the total magueUsation, intrinsic and induced, we have

B«Ii + /ioP, which, translated into irrational units, makes

Bi«4rf,, + /XjF<;

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and lastly, omitting the fig by aagnming it to be unity, w» obtain the oommon

oontaining three faults, the arbitrary iv, the equalising of an intensity and a flux, and the unnatural union of physically dia- tinot magnetisations.

"Different men have different opinions — some like apples, some like inions." But can anyone possibly really like the roundabout and misleading way of presenting the magnetic flux relations which I have above criticised t There is no- excuse for it» except that it was employed by great men when they were engaged in making magnetic theory, before they had assimilated its consequences thoroughly. When the rough work of construction is over, then it is desirable to go over it again, and put it in a better and more practical form. Wo should copy the vurtues of great men, if we can, but not their fiaults.

Men who are engaged in practical work can hardly be expected to fully appreciate the importance of these things, because their applications are of so highly specialised a nature, in the details of which they may become wholly absorbed. They may even go so far as to say that the paper theory of magnetic induction is not of the least moment, because they are concerned with irou^ and although there may be a certain small range of application of the theory even in iron, yet they are scarcely concerned with it, and, therefore, it is of no consequence. There could not be a greater mistake. A complete tlieory of magnetic- mduction, including hysteresis, must necessarily be so con- structed as to harmonise with the limited theory that has already been elaborated, which is understandable when uxhibiLed in a purified form, freed from 47r's and other anomalies. First of all, we have the ether, in which B = jiqT or B = ixqR, because of the absence of intrinsic magnetisation. Next we come to elastically magnetised bodies in which the relation between flux and force is linear. Then B = /xH, where fi differs from /a^, being either greater or smaller, and is either a constant scalar, or else (with eolotropy) a linear operator. If there is no intrinsic mag- netisation, F and H are still the same, and the curl of either ia the current density. But should there be intrinsic magnetisa- tion, then H^h^+F, whilst B»/aH still, and it is the curl

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d 7 that is the oarrent density. Or, B^Io + fiF. The next step is to make fi7 be not a linear, but some other f unetkm of F, to be experimentally determined, if it be possible to express B - Ig, which is the free induction, as a fonotion of 7. Of course, it can be done approximately. Then comes the diffioult question of hysteresis. This involves the variation of Iq with F, with oonsequent waste of energy. If this little matter be satisfactorily determined, we may expect to have a sound mathematical theory of magnetic induction in an extended form which shall properly harmonise with the rational form of the elementary tlieory. The divergence of B is zero all through. The experimental justification of this generalisation is the fact that no unipolar magnets have yet been discovered.

Effect of a Thin Plane Conducting Sheet on a Wave. Persistence of Induction and Loss of Displacement

§ 193. Coming now to the effects produced on electromag- netic waves by a small amount of conductivity, to be after- wards increased, we shall adopt a particular device for simplify- ing the treatment. Imagine, first, the dielectric medium to possess a uniformly-distributed small conductivity. Evidently, the action of the conductivity on a wave is a continuous and cumulative one. Next, localise the conductance in parallel sheets — that is, substitute for the uniform conductivity a g! tat number of parallel plane conducting sheets, between which the medium is non-conducting. If we increase their number suffi- ciently, their action on a wave whose plane is parallel to that ol the sheets will approximate, in the gross, to the efTect of the uniform conductivity which the conductance of the sheets replaces. We have, therefore, to examine the influence of a single very thin conducting sheet upon a wave. This is not difficult.

Imagine, then, a simple plane electromagnetic sheet to be running through the ether at the speed of light. This is the natural state of things ; and, in the absence of conductivity or other disturbing causes, there will be no change. Now insert a plane conducting sheet in the path of the wave. It should be 80 thin that the retarding effect of diffusion within it Is quite insensible. Let the wave strike it flush. The theory

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ehowB that it is immediately split into two similar waTsa, ooa of which, is transmitted beyond the plate, whilst the other n reflected. The transmitted wave differs from the incident In no respect except strength. It is attenuated in a certain ratio depending upon the conductance of the sheeti being great^ attenuated when the conductance is large, and slightly atten- uated when it is small. The formulse are reserved, TUs txansmitted wave moves on just as the incident wave did, and nothing further happens to it. The reflected wave, on tiie • ot^er hand, having its direction of motion opposite to that of the incident and transmitted waves, travels back the way it came, and nothing further happens to it.

The direction of the magnetic force in the three waves is the same. This is one general property, irrespective of the amount of conductance. But a much more striking one connects the intensity of the magnetic force in the three waves. The sum of the intensities in the reflected and transmitted waves equals the intensity in tlie original incident wave. That is, the con- ductance, with dissipation of energy, has had no effect whatever on the total induction. It has merely redistributed it, by splitting it into two parts, which then separate from one another. The *' number of tubes " in the reflected wave may be made to bear any ratio we please to the number in the transmitted wave by altering the conductance of the plate ; but their sum is always exactly the number of tubes in the incident wave. This property exemplifies, in a manner which may be readily understood, the pei^istence of induction, in spite of conduction and waste of energy.

But as regards the displacement, the case is quite different. From the fact that the reflected wave runs back, whilst its magnetic force preserves its original direction, we see that the electric force must be reversed. On the other hand, it is unchanged in the transmitted wave. If, then, their sum equalled the electric force in the incident wave, it would imply that the transmitted wave was of greater amplitude than the incident. Bub it is smaller, invariably. So there is a loss of displacement. Thus, if H in the incident becomes (1 ~fi)H ia the transmitted wave, where n is some proper fraction, it becomes nH in the reflected wave. At the same time E in the incident becomes (1 - »}£ in the transmitted, and - »£ in the

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reflected ware. The loss of £ is, therefore, 2nE, or the loss of displacement is 2nD. By the loss we mean the excess of the displacement in the incident over the sum of the displacements in the two resulting waves, transmitted and reflected. This loss occurs at the very moment the incident wave coincides with the plate. The plate itself may be regarded as a dielectric homogeneous with the ether outside, or perhaps ef diiFerent permittivily, but with the conducting and dissipating property superposed. When thin enough, the permittance of ti&e plate is of insensible influence, and may be disregarded. But strictly, a conducting dielectric is a dielectric which cannot support displacement without wasting it, so that a continuous supply of fresh displacement is needed to keep it up. The rate of waste of energy is proportional to the electric stress.

But it should be carefully noted that the loss of energy and the loss of displacement are entirely distinct things, which are not proportional ; and that the loss of displacement itself may sometimes require to be understood in a somewhat artificial sense. For the loss may bo greater than what there is to lose. It must then be understood vectorially. This occurs when 71 is greater than ^. When n = J, the reflected and transmitted waves are equally strong, and only differ in the direction of the displacement. The loss of displacement is, there- fore, complete. The loss of energy in the plate is simul- taneously at its maximum, being equal to one-half of the energy of the original wav^e. If we reduce the conductance of the plate, we increase the transmitted wave, and reduce the waste of energy in it and the loss of displacement. The amount of the latter still remains positive, therefore, assuming it to be positive in the incident wave. The extreme is reached when the plate has no conductance. Then the incident wave goes right through without any splitting and reflection, and therefore without attenuation, and there is no waste of energy. On the other hand, if we increase the conductance above the critical value making n = |, we reduce the transmitted wave and increase the reflected, whilst we simultaneously reduce the waste of energy in the plate and increase the loss of displace- ment. The extreme Is reached when the plate is a perfect oonduotor. There is then no transmitted wave and no loss of energy, whilst the reflected wave is of full sise, but with the

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displacement reversed as compared with the incident, so that the loss uf displacement, in the sense described, is the greatest possible, viz., 2D.

It will be sueu from these details that whilst the absorption or dissolving of induction by conductance is a myth, the idea of an absorption of displacement is not without its incon- veniences when the conductance is great, and that this becomes extreme when there is really no loss of energy in the plate, when, in fact, the incident wave does nothing in it, but is wholly rejected with its displacement reversed. It seems, then, more natural to consider the waste of energy from the field caused by the plate as loss. This takes place etiually from the electric and magnetic energies, since they are necessarily equal in every one of the three waves. But in the application made later, the plate is to have very slight conductance (in the limit an infinitely small amount), so that the total displacement cannot change sign, but merely suffers a slight loss. Then the idea of loss of displacement by conductance becomes useful again.

The loss of energy takes place as the incident wave is travers- ing the plate. Its successive layers each cause a minute attenua- tion of the wave passing, and this applies equally to the induc- tion and displacement, so that the transmitted wave emerges from the plate a pure electromagnetic wave, a reduced copy of the incident. The Bucoessiye layers, too, each oast back a minute portion of the wave traversing them, with unchanged sign of the induction, but with displacement reversed; and these rejected fluxes make up the reflected wave. There are evidently residual efifects due to the internal reflections of minute portions of the main reflected wave, but these redduals tend to Yanish when the plate is thin enough.

If the plate be a magnetic instead of an electric condaetor, the theory is quite similar. The transmitted wave is an attenuated continuation of the incident. The reflected wave is also a copy of the incident^ also reduced. But it is now the induction that suffers loss, beqause its direction in the reflected wave is opposite to that in the incident and trans- mitted. On the other band, the displacement now fully per- sists, being merely split into two parts by the plate.

Notice that if the plate be both an electric and a magnetic conductor, its attenuating eflect from these two causes on the

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