Electromagnetic Theory, Vol. 1 (1893) — part 25 of 31

During the fourth stage, the place of maximum negative displacement shifts itself to nearly the middle of the region between the tip and the node, beyond which the positive dis- placement has a nearly similar distribution, with a maximum. But this positive distribution is only three-parts formed, as the bead is still of some importance. The fifth stage completes the formation of the tail, with the displacement negative in one half and positive in the other, and nearly symmetrical with respect to the middle. Finally, we come to a state of perfect symmetry, with one maximum of induction and two

BB2

372 ELECTROMAGNETIC THEORY. CH. IV.

(or a maximum and a minimum) of displacement. This may be readily understood by considering that in the expression for the true electric current, &E + cE, the second term finally be- comes a small fraction of the first, or the current is practically the conduction-current only. Then B, and therefore the dis- placement, is proportional to the space-variation of the induc- tion. Owing to the tail being made up of a complicated mix- ture of infinitesimal electromagnetic waves going both ways, we lose sight of the fundamental property of elastic wave-pro- pagation in a dielectric, the resultant effect being propagation by diffusion now, or very nearly so. The approximation to this result is closest in the middle of the tail. At the tips, on the other hand, we still have elastic waves, but, of course, of insensible strength.

Now, suppose that the initial state is one of induction only, though still in a plane sheet. It may be regarded as the coin- cidence of two plane electromagnetic sheets of half the strength, with similar inductions and opposite displacements. The sub sequent history may, therefore, be deduced from the preceding. The initial induction splits into two halves, of which one moves to the right at speed v, and the other to the left. The history of the first wave as distorted by the conductance has been given. That of the second wave is the same, if we allow for the changed direction of motion and sense of the displacement. So we have only to superpose the two systems to show how an initial distribution of induction in a plane sheet splits and spreads, and the accompanying electric displacement. We have two equal heads, separating from one another at speed 2v, whilst the two tails unite to make a stouter kind of tail (referring to the induction) joining the two heads. This tail is always thickest in the middle. In fact, the distributions of induction and displacement are symmetrical with respect to the initial position from the first moment. In the final state that is tended to, when the tails have vanished, the induction is distributed in the same way as in the former case, in spite of the remarkable difference in the initial phenomena.

If, again, the initial state is one of displacement alone in a plane sheet, this generates two oppositely-travelling electro- magnetic waves in which the displacements are similar and the inductions are opposite. The result is therefore to be got by

THEORY OP PLANE ELECT ROMAGNET1C WAVES. 53

combining the previous solution for a wave of positive induc- tion and displacement moving to the right, with a similar wave of negative induction and positive displacement moving to the left.

Whatever be the magnitude of the conductivity, if finite, the same phenomena occur during the conversion of elastic waves to diffusion waves, and they may be represented by the same dia- grams, if suitably altered in the relative scale of ordinates and abscissae. But by increasing the conductivity from the small value which makes the above-described process take place over a considerable interval of time, we make it occur in a small interval only; and when we come to what are usually con- sidered good conductors, viz., metals, then the interval of time is so small that we may say practically that the heads vanish almost at 6nce, and before the propagation has proceeded any notable distance. The rest of the story is the spreading by diffusion or mixture. Thus we have an important practical distinction between the very good and the very bad conductor, as regards the manner of propagation of induction. In the former the electric displacement is of no account hardly, except perhaps for very short waves, and the practical theory is the theory of diffusion. At the other extreme are perfect non-conductors, in which the propagation is entirely by elastic waves. Between the two we have, in bad conductors, a mix- ture of the two kinds, though with a continuous transition from one kind to the other. The mathematical treatment of elastic waves is the easiest. The next in order of difficulty are the waves of pure diffusion, by the ordinary Fourier mathe- matics. The most troublesome are the intermediate forms of changing type, and the full analytical results are best got by a generalised calculus. But the general nature of the results may be obtained approximately by easy numerical calculations, with diagrammatical assistance. This is explainable without difficulty, and may be entered upon later on, when we come to detailed problems.

la the above the conductivity has been electric only, and we have seen that there is a great difference between its effects on the electric and on the magnetic flux, which arises from the positive reflection and persistence of the latter, and the nega- tive reflection and subsidence of the former flux. The corres-

374 ELECTROMAGNETIC THEORY. CH. IV.

ponding effects in a magnetic conductor may be readily deduced by the proper transformations, and exhibit analogous differences. Thus, starting with a pure plane wave sheet moving to the right, it is now the displacement that is con- served and divided between the head and tail, being initially all in the head, and finally all in the tail. On the other hand, the induction is negative in the tail at first and until the attenua- tion of the head is considerable. After this, the region of posi- tive induction extends from the head into the tail. As time goes on, and the head disappears, we have negative induction in one half and positive in the other (forward) half, whilst the displacement is positive all along it, with its maximum nearly in the middle. In the final state of diffusion we have symmetry with respect to the initial position of the wave, and the induc- tion is proportional to the space-variation of the displacement. The second term of the magnetic current ^H + juH is then a small fraction of the first term.

Now, in the medium with duplex conductivity we must take the distortionless condition for the standard state. When this obtains there is no tailing, although the head (displacement and induction together) attenuates according to the time-factor c~^/c. But if tho electric conductivity be in excess (that is, greater than is required to make k/c=g/fjL) there is tailing of the kind de- scribed as due to electric conductivity, with the induction posi- tive and the displacement negative in the tail at first. And if the magnetic conductivity is in excess, the tailing is of the other kind, with positive displacement and negative induction at first. But of course the details are not the same, because both the fluxes now attenuate to zero in time, in the tail as well as in the head.

Application to Waves along Straight Wires.

§ 199. We must now endeavour to give a general idea of the tailing of waves when they run along conducting wires, in co-ordination with the previous. In the first place, observe that although the lines of electric and magnetic force in a pure electromagnetic sheet in ether must be always perpendicular to one another, yet they need not be straight lines, except in their elementary parts. On the contrary, we may have an infinite

THEORY OP PLANE ELECTROMAGNETIC WAVES. 375

variety of distributions of the electric and magnetic fluxes in curved lines in sheets which shall behave as electromagnetic waves. Keeping to plane sheets, we may take any distribution of displacement in a sheet that we please, and make it an electromagnetic wave by introducing the appropriate distribu- tion of induction, also in the sheet. If left to itself it will move through the ether one way or the other at speed v, according to the directions of the fluxes. It should be noted, however, that the fluxes should have circuital distributions. For if not, and there is electrification or its magnetic analogue, they, too, should be moved through the ether so as to exactly keep up with the wave. If the electrification does not move thus, we have a changed state of things. But there is a way out of this difficulty. Let the displacement, when it is discon- tinuous, terminate upon infinitely-conducting lines, or surfaces of cylinders, according to circumstances. The interference produced by holding back the electrification will then be done away with. It is sufficient for explicitness to take a single definite case.

Let the displacement in a plane electromagnetic sheet ter- minate perpendicularly upon a pair of perfectly-conducting cylindrical tubes placed parallel to one another. One is for the positive and the other for the negative electrification. The induction will then go between and round the tubes. One tube may enclose the other, but, preferably, we shall suppose this is not the case, so that our arrangement resembles a pair of parallel wires. Now the wave will run through the ether in the normal manner, without distortion or change of type, and will carry the electrification (on the conductors) along with it.

Similarly if the tubes be magnetic conductors, if the induc- tion terminates perpendicularly upon them, whilst the displace- ment goes between and round them. So far, then, the theory of the propagation of the wave is unaltered.

But this property may be greatly extended. The medium outside the tubes through which the wave is moving may be made electrically conducting. The theory is then identical with that of §198, as regards the distortion produced and the gradual destruction of the wave, ending in the process of diffu- sion. Thid will be true whether the tubes are perfect electric

376 ELECTROMAGNETIC THEORY. CH. IV.

or magnetic conductors, provided we have the displacement terminating upon them in the first case and the induction in the second.

Again, if the medium is magnetically conductive, the theory still holds good, with the same reservations as regards termi- nating the fluxes. But now the distortion is of the other kind, with persistence of the displacement and subsidence of the induction.

Finally, the theory is still true when the medium is a duplex conductor, whilst the tubes are either perfect electric or mag- netic conductors, according to choice. The distortion is now of the electric or of the magnetic kind, according as kfc is greater or less than g/fj..

But not one of these cases is quite what we want to repre- sent propagation along real wires. The nearest approach is the case of finite electric conductivity in the medium combined with perfect electric conductance of the tubes. If we make their conductance be imperfect, we then come close to the real problem. Now when we do this, the theory, when done pre- cisely, becomes excessively difficult, for two reasons. First, the waves in the ether outside the tubes are no longer plane ; and next, the penetration of the disturbances into the sub- stance of the tubes in time by means of cylindrical waves has to be allowed for. The latter is, of course, more necessary when solid wires are in question. A practical working theory is seemingly impossible of attainment by strictly adhering to the actual conditions. But one is possible by taking advantage of the fact that the waves in the ether are very nearly plane under ordinary circumstances. Of this we may assure ourselves by considering that the tangential component of the electric force at the surface of a wire (upon which the penetration depends) is usually a very small fraction of the normal com- ponent of the same outside it. The practical course, then, is to treat the waves in the dielectric as if they were quite plane. This does not prevent our allowing for the distorting effect of the resistance of the wires. It has the effect of making our solutions approximate instead of complete. But the important thing is to have a theory that, whilst sufficiently accurate, is practically workable, and harmonises with more rudimentary theories. This is precisely v;hat we do get, as done by me in

THEORY OP PLANE ELECTROMAGNETIC WAVES. 377

1886. The theory is brought to such a form that we may employ it in several ways, according as circumstances allow us to ignore this or that influence. We may, for example, treat the wires as mere resistances (constant), and this is quite suffi- cient in a variety of applications. Or, using the same equations with more general meanings attached to the symbols, we may find the effects due to the imperfect penetration of the mag- netic induction and electric current into the wires when sub- jected to varying forces at their boundaries, either simple harmonically or otherwise. Furthermore, the theory is in such a form that it admits readily and without change of the intro- duction of terminal or intermediate conditions of the kind that occur in practice, whose effects are brought in according to the usual equations of voltage and current in arrangements of apparatus.

What we are immediately concerned with here, however, is the connection between this theory and the general theory of waves in a medium of duplex conductivity. When the tubes are so thin that penetration is practically instantaneous (for the wave-length concerned), a constant ratio is fixed (assisted by Ohm's law) between the intensities of magnetic and of tan- gential electric force at the boundary of the tube, where it meets the dielectric. And when we incorporate this result in the second circuital law applied to any section of the circuit formed by the parallel tubes, we find that the result is to turn it to the form expressing the existence of magnetic conductivity in the outer medium, without resistance in the wires. That is, the resistance of the wires has the same effect in distorting and dissipating the waves outside as the fictitious magnetic conductivity in the medium generally. This is true in the first and most important approximation to the complete theory. It is a point that is not altogether easy to understand, because the magnetic conductivity is fictitious, whereas the resistance is real. This, however, may be noted, that the theory of propagation of plane waves in a medium of duplex conduc- tivity bounded by perfect conductors for slipping purposes, professes to be a precise theory, whereas the other, although concerning the same problem in most respects, professes to be only approximate so far as the influence of the wires is con- cerned.

378 ELECTROMAGNETIC THEORY. CH. IV.

Transformation of Variables from Electric and Magnetic Force to Voltage and Gaussage.

§ 200. In the consideration of the transmission of waves along wires, especially, of course, in practical calculations, it is more convenient to employ the line-integrals of the electric and magnetic forces as variables rather than these forces them- selves. This transformation of variables involves other neces- sary transformations, and the connection between the old quantities, suitable for waves in general, and the new, should be thoroughly understood, if something more than a super- ficial knowledge of the subject be desired.

Thus, commencing with a circuit consisting of a pair of parallel straight conducting tubes, of no resistance in the first place, take for reference any plane which crosses the tubes per- pendicularly. It is the plane of the wave at the place, and the lines of electric and magnetic force lie in it, the former starting from one (the positive) and ending upon the other (the negative) tube, the latter passing between and round them. These lines always cross one another perpendicularly, but their distribution in the plane may be varied by changing the size and form of section of the tubes by the plane and the distance of separation. Now there is no axial magnetic induction — that is, induction parallel to the tubes. Nor is there any tangential component of electric force on the surface of the tubes. It follows, by the second circuital law, that the line-integral of the electric force from one tube to the other is the same by any path in the reference plane. It is not the same by any path with the same terminations if we depart from the reference plane, but that is not yet in question. Call this constant line-integral V. It is the transverse voltage. In another form, we may say that the circuitation of the electric force in the reference plane is zero.

Next consider the magnetic force. Its circuitation is also zero in the reference plane, provided the one tube or the other is not embraced, because there is no axial electric current in the dielectric. But when a tube is embraced the circuitation is finite. Call this quantity C. It is " the current " in the tubes, or, at any rate, is the measure thereof, and is positive for one tube and negative for the other. These two quantities,

THEORY OP PLANE ELECTROMAGNETIC WAVES. 379

V and C, the transverse voltage and the current, are the new variables. They are as definite as E and H themselves in a given part of the reference plane under the circumstances.

Next, as regards the fluxes, displacement and induction. Intro- duce a second reference plane parallel to the first and at unit distance in advance of it along the tubes. Anywhere between the planes we have D = cE, where E and D vary according to position. Now, as we substitute the complete voltage for E, so we should substitute the complete displacement for D. The complete displacement is the whole amount leaving the posi- tive and ending upon the negative tube, between the reference planes. That is, it is the " charge " per unit length of the tubes, say Q. As D is a constant multiple of E, so is Q a con- stant multiple of V. Say Q = SV. This S is the permittance of the dielectric, per unit length axially. It is proportional to c, the permittivity, but of course involves the geometrical data (brought in by the tubes) as well.

Similarly, we have B = /xH everywhere between the reference planes, H and B varying from place to place. But if we sub- stitute the complete circuitation of H, we should also substi- tute the complete integral of B. This means the total flux of induction passing between and round the tubes, between the reference planes. Call it P. It is the magnetic momentum per unit length axially. As B is a constant multiple of H, so is P a constant multiple of C. Thus P = LC, where L is the inductance per unit length axially. This L varies as the induc- tivity /*, and involves the geometrical data.

In virtue of the relation /*cv2 = 1, the inductance and permit- tance are reciprocally related. Thus, LSv2 = 1. Since v is con- stant, it might appear that the inductance was merely the reciprocal of the permittance, or the elastance, with a constant multiplier, only to be changed when the dielectric is changed. But there is much more in it than this. Different physical ideas and effects are conditioned by inductance and permit- tance. Inductance and inductivity involve inertia, whilst permittance and permittivity involve compliancy or elastic yielding.

Observe that P and Q are analogous, being the total mag- netic and electric fluxes, whilst V and C are also analogous, the voltage and gaussage respectively. So the ratios L and S axe

380 ELECTROMAGNETIC THEORY. CH. IV.

also strictly analogous, in spite of their quantitative recipro- cality. They are both made up similarly to conductance, only in the case of permittance it is reckoned across the dielectric from tube to tube, and in the case of inductance round and be- tween the tubes — that is, the magnetic circuit is closed, and the electric circuit unclosed. There are other ways of setting out the connections, but the above way brings out the analogies between the electric and magnetic sides in a complete manner. The quantity C may be quite differently regarded when the magnetic field penetrates into the substance of a tube (then to be of finite conductance), viz., as the total flux of conduction current, or sometimes of displacement current as well, in a tube. But the above method is independent of the penetration, holding good whether there is penetration or not, saving small corrections due to the waves not being quite plane. The mag- netic view of C, as the gaussage, is also the nearest to expe- rimental electrical knowledge. The other view is darker, because we cannot really go inside metals and observe what is going on there, or the forces in action. They must be inferential in a greater degree than the external actions.

The electric energy per unit volume being JED or JrE2, when this is summed up throughout the whole slice of the medium contained between the reference planes at unit dis- tance apart, we obtain the amount JVQ or JSV2, which is therefore the electric energy per unit of length axially.

Similarly, the magnetic energy density JHB or J/*H2, when summed up throughout the slice, amounts to JCP or JLC2. The reader should, in all these transformations, compare the transformed expressions with the original, and note the proper correspondences. It is all in rational units, of course, to avoid that unmitigated nuisance, the 4?r factor of the present B.A. units.

The density of the flux of energy, which is the vector pro- duct of E and H in general, is of amount simply EH when the forces are perpendicular, as at present. When this is summed up over the reference plane, the result is VC, the pro- duct of the voltage and current. The flux of energy is parallel to the guiding tubes, and EH is the amount for a tube in the dielectric of unit section, whilst VC is the total flux of energy.

THEORY OP PLANE ELECTROMAGNETIC WAVES. 381

The relation E = p>H, or, which means the same, H = cvE, which obtains in a pure electromagnetic wave, becomes converted to V = LvC, or C = SvV. We see that Lv is of the dimensions of electric resistance, and Sv of electric conductance. Similarly, because V and C are the line-integrals of E and H, we see that fjiv is of the dimensions of resistance, and cv of conductance. The activity product VC is what engineers have a good deal to do with, now that " electrical energy," or energy which has been conveyed by electromagnetic means, is a marketable commodity. However mysterious energy (and its flux) may be in some of its theoretical aspects, there must be something in it, because it is convertible into dollars, the ultimate official measure of value.*

Transformation of the Circuital Equations to the Forms involving Voltage and Gaussage.

§ 201. Now, still under the limitation that the guiding tubes have no resistance, and that the dielectric has no conductance, consider the special forms assumed by the circuital laws. These are, in general,

curlH=cE, .,,,.. (1)

(2)

Now, in applying these to our plane waves, we may observe at the beginning that E and H and their time- variations are in the reference plane, and have no axial components. So the only variations concerned in the operator " curl " are such as occur axially, or parallel to the guides, f Let x be distance along them, then the circuital laws become

-^? = cE, (3)

dx v '

~ = ^ (4)

dx x '

• See the Presidential Address to the Institution of Electrical Engineers, January 26, 1893.

t That is, curl reduces from Vv, where y is i?i +JV^ + kVs in general, to %riVi simply. It is now more convenient to use the tensors E and H, as in (3) and (4), ignoring their vectoiial relations.

382 ELECTRO MAGNETIC THEORY. CH. IV.

which apply anywhere in the reference plane. Transforming to V and C, we obtain

(5)

By (3) and (4) we see that the space-variation of H is the electric-current density, and the space- variation of E the mag- netic-current density in the dielectric. By (5) and (6) we express the same truths for the totals concerned.

We may see the meaning of (5) thus. Draw a closed line in the reference plane on and embracing the positive guiding tube. The circuitation of H there is the total surface current C. Let the closed line be shifted to the next reference plane at unit distance forward. Then — dC/dx is the amount by which C decreases during the shift, and (5) tells us that it is equiva- lently represented by the time-rate of increase of the charge on the tube between the reference planes, or the rate of increase of the total displacement outward. This process may be applied to any closed line in the reference plane, provided it embraces the tube. When it is shifted bodily forward through unit distance to the next reference plane, the amount by which the circuitation of H, that is, C, decreases from the first to the second position measures the displacement current through the strip of the cylinder swept out by the closed line. This is the time-rate of increase of the charge per unit length of the tubes, and is the total transverse current from one tube to the other.

The meaning of (6) requires a somewhat different, though analogous, elucidation. Join the tubes by any line in the first reference plane. The line-integral of E along it is V, the transverse voltage. Shift the line bodily along the tubes to the next reference plane. The transverse voltage becomes V + dV/dx, still reckoned from the positive tube to the negative. But if we join the two starting points on the positive tube together, and likewise the ending points on the negative tube, we make a complete circuit, having two trans- verse sides and two axial sides. Now reckon up the voltage in this circuit. The axial portions contribute nothing, because

THEORY OP PLANE ELECTROMAGNETIC WAVES. 383

the electric force is perpendicular to them. The voltage required is, therefore, simply the difference in the values of the transverse voltage in the other two sides, or dNjdx. By the second circuital law, it is also measured by the rate of decrease of induction through the circuit, that is, by -LC. Whence follows equation (6).

The reader who is acquainted with the (at present) more "classical" method of treating the electromagnetic field in terms of the vector and scalar potentials cannot fail to be impressed by the difference of procedure and of ideas involved. In the present method we are, from first to last, in contact with those quantities which are believed to have physical signi- ficance (instead of with mathematical functions of an essentially indeterminate nature), and also with the laws connecting them in their simplest form. Notice that V is not the difference of potential in general. It sometimes degenerates to difference of potential, viz., in a perfectly steady state. But when the state changes, we cannot express matters in terms of an electric potential.

Now, still keeping the guiding tubes perfectly conducting, let the medium in which they are immersed be slightly con- ducting. The electric-current density, which was cE before, now becomes &E + cE, where the additional &E is the conduction- current density. Along with this there is waste of energy at the rate £E2 per unit volume. This waste of energy may also be regarded as a storage of energy, viz., as heat in the medium. But as it is not recoverable by the same means (reversed) which stored it, it is virtually wasted, and we have no further concern with it.

We have next to consider the total waste in the slice of the medium between the two reference planes, in terms of the trans- verse voltage. It sums up to KV2, where K is the transverse conductance of the medium per unit length axially. This is proportional to the conductivity k, and involves geometrical data in the same way as the transverse permittance, as may be readily seen without symbolical proof, on considering that the conduction-currrent lines and the displacement lines are simi- larly distributed, whilst both are controlled by the transverse voltage. In fact, the conduction-current density &E sums up

384 ELECTROMAGNETIC THEORY. CH. IV.

to KV, the complete transverse conduction-current, in the same way as the displacement density cE sums up to the total trans- verse displacement SV, in terms of the permittance and voltage.

The first circuital law (1) becomes changed to

(7)

by the addition of the conduction current. And its modified form (3) for our plane waves becomes

(8)

whilst in terms of the new variables, voltage and gaussage, we have this extended form of the equation (5),

(9) dx

Next, let the medium become magnetically conductive as well, no other change being made. The conduction-current density is ^H, and its distribution resembles that of the induc- tion itself, whilst its amount is controlled by the quantity G, the gaussage. The total magnetic-conduction current may therefore, be represented by KG, where E, is the magnetic conductance, which varies as the magnetic conductivity g, and involves the geometrical data in the same manner as the inductance does.

Similarly, the rate of waste of energy due to the magnetic conductivity is 0rH2 per unit volume, and the total rate of waste in the slice of the medium between the two reference planes amounts to RC2 correspondingly.

Finally, the effect of the magnetic conductivity is to turn the second circuital law from the elementary form (2) to

-curlE = #H + /iH, .... (10)

and, correspondingly, the special form (4) for our plane waves becomes turned to

..... (11)

THEORY OP PLANE ELECTROMAGNETIC WAVES. 385

This, in terms of V and C, is equivalent to

-4^ = RC + LC, ...... (12)

dx

which is the proper companion to the equation (9) expressing the first circuital law. These equations (9) and (12) are the practical working equations.

We have already remarked that the structure of the electric permittance and conductance are similar. From this it follows that the time-constant c/k is identically represented by S/K. Similarly, the time- constant njg is identical with L/R.

The density of the flux of energy is, with the two conduc- tivities, still represented by the vector product of E and H, and therefore the total flux is still the activity product VC. To corroborate this statement, and at the same time show the dynamical consistency of the system, consider that if the quan- tity VC is really the flux of energy across a reference plane, the excess of its value at one plane over that at a second plane further on at the same moment represents the rate of storage of energy between the planes. Therefore, the rate of decrease of VC with x is the rate of storage between two reference planes at unit distance apart. Now,

_i(VC)=-vf-cf ..... (13) dx dx dx

On the right side use the circuital equations (9) and (12), and it becomes . .

V(KV + SV) + C(RC + LC), . . . (14)

or, which is the same,

. . (15)

By the previous, the first term is the total electric waste, the second is the total magnetic waste, the third is the increase of total electric energy, and the fourth is the increase of total magnetic energy, per unit of time, between the two reference planes. This proves our proposition, with a reservation to be understood concerning the circuital indeterminateness of the flux of energy. There is, therefore, no indistinctness anywhere, nor inconsistency. We have now to show in what manner the above is affected by the resistance, &c., of the guides.

oo

386 ELECTROMAGNETIC THEORY. CH. IV.

The Second Circuital Equation for Wires in Terms of V and C when Penetration is Instantaneous.

§ 202. Let the parallel conducting tubes be of finite resist- ance. As a result, the external disturbance penetrates into them, and a waste of energy follows. As a further result, the external disturbance becomes modified. This occurs in a two- fold manner. By cumulative action, the nature of a wave sent along the leads may be profoundly modified as it progresses. Besides this, there is what we may term a local modification, whereby the wave at any place is no longer strictly a plane wave. It is merely with this effect that we are now concerned. The departure from planarity requires a considerably modified and much more difficult theory, no longer expressible in terms of V and C simply, in order to take it into account. But in the construction of a practical theory, we take advantage of the fact that the departure from planarity is slight, to which we have already referred towards the end of § 199. A line of elec- tric force does not now start quite perpendicularly from the positive lead and end similarly on the negative lead. It has a slight inclination to the perpendicular, and therefore curves out of the reference plane to a small extent. In the dielectric itself, this peculiarity is of little importance. But the slight amount of tangentiality of the electric force at the surface of the leads, which is conditioned thereby, is of controlling import- ance as regards the leads themselves, and eventually through them, to the waves in the dielectric, by attenuating and altering the shape of waves (considered axially), as just referred to. We should now consider what form is assumed by the second cir- cuital law in terms of V and C, when we admit that there is tangential electric force on the surface of the leads, but on the assumption that the minor effects in the dielectric itself, which are associated with the presence of the tangential force, are of insensible influence.

We may still regard V as the transverse voltage in the reference plane. Whether we go straight across from the positive to the negative lead, keeping in the reference plane, or leave it slightly in order to precisely follow a line of force in estimating the voltage, is of no moment, because the results differ so slightly. The quantity C is subject to a similar slight

THEORY OF PLANE ELECTROMAGNETIC WAVES. 387