Electromagnetic Theory, Vol. 1 (1893) — part 4 of 31

where R and K are constants, the resistance and the conduct- ance, taking the place of resistivity and conductivity, when voltage takes the place of electric force, and the current that of current-density. The activity of the impressed voltage is

VC = RC2 = KV2 ..... . . (2)

and represents the Joulean waste per second in the whole con- ductor, or the volume-integral of EC or of &E2 before con- sidered.

Permittance and Elastance.

§ 30. Permittivity gives rise to permittance, and elastivity to elastance. To illustrate, for the conductor, substitute a nonconducting dielectric, leaving the terminals and external arrangements as before. We have now a charged condenser. Displacement, i.e., the time-integral of the current, takes the place of current in the last case, and we now have

D = SV, V = S-!D, .... (3)

if D is the displacement, S the permittance, and its reciprocal the elastance of the condenser. This elastance has been called the stiffness of the condenser by Lord Rayleigh. It is the elastic resistance to displacement. The displacement is the measure of the charge of the condenser. The total energy in the condenser is

(4)

i.e., half the product of the force (total) and the flux (total), between and at the terminals ; it is also the volume-integral of the energy-density, or J2cE2.

As the dielectric is supposed to be a non-conductor, the cur- rent is I) or SV, and only exists when the charge is varying. But it may also be conducting. If so, let the conductance be K, making the conduction current be C = KV. The true cur- rent (that is, the current) is now the sum of the conduction and displacement currents. Say,

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 29"

This is the characteristic equation, of a condenser. It comes to the same thing if the condenser be non-conducting, but be shunted by a conductance, K. In a conducting dielectric the permittivity and the conductivity are therefore in parallel arcr as it were. It was probably by a consideration of conduction in a leaky condenser that Maxwell was led to his inimitable theory of the dielectric, by which he boldly cut the Gordian knot of electromagnetic theory.

The activity of the terminal voltage we find by multiplying, (5) by V, giving

vr=vc-fvi>,

2 + SV2

\ .... (6)

representing the waste in Joulean heating and the rate of in- crease of the electric energy. Each of these quantities is the sum of the same quantities per unit volume throughout the substance concerned.

Permeance, Inductance and Reluctance.

§ 31. Permeability gives rise to permeance, inductivity to inductance, and reluctivity to reluctance.

The formal relation of reluctance to reluctivity with mag- netic force and induction, is the same as that of resistance to resistivity with electric force and conduction current, or of elastance to elastivity with electric force and displacement.

Permeance is the reciprocal of reluctance. In this sense I have used it, though only once or twice. Prof. S. P. Thomp- son has also used tfae word in this sense in his Cantor Lectures with good effect.

If we replace our illustrative conductor by an inductor, supporting magnetic induction, and suppose it surrounded by imaginary matter of zero inductivity, and have an impressed gaussage instead of voltage at the terminals, we shall have a flux of induction which will, if the force be weak enough, vary as the force. If H be the gaussage and B the induction enter- ing at the one and leaving at the other terminal, the ratio H/B is the reluctance, and the reciprocal B/H is the permeance. The energy stored is

30 ELECTROMAGNETIC THEORY. CH. II.

When the relation of flux to force is not linear, we can still usefully employ the analogy with conduction current or with displacement by treating the ratio B/H as a function of H or of B ; as witness the improved and simplified way of consider- ing the dynamo in recent years. I must, however, wonder at the persistence with which the practicians have stuck to " the lines," as they usually term the flux in question.

I am aware that the use of the name induction for this flux, which I have taken from Maxwell, is in partial conflict with an older use. But it is seldom, if ever, that these uses occur to- .gether, for one thing ; another thing is that the older (and often vague) use of the word induction has very largely ceased of late years. It was not without consideration that induction was adopted and, to harmonise with it, inductance and induc- tivity were coined.

Inductance of a Circuit.

§ 32. The meaning of inductance has sometimes been mis- conceived. It is not a synonym for induction, nor for self-induc- tion, but means " the coefficient of self-induction," sometimes abbreviated to " the self-induction." It is essentially the same as permeance, the reciprocal of reluctance, but there is a prac- tical distinction. Consider a closed conducting circuit of one turn of wire, supporting a current Cr As will later appear, this G! is also the gaussage. That is, the line -integral of the magnetic force in any closed circuit (or the circuitation of the force) embracing the current once is Cr Let also BA be the induction through the circuit of Cr Then

BX-LA, ...... (8)

where, by what has already been explained, LH is the perme- ance of the magnetic circuit, a function of the distribution of inductivity and of the form and position of the conducting core. The magnetic energy is

JBA-iW ..... (9)

by using the first expression in (7), remembering that H there is now represented by C^ and then using (8). This

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 31

energy " of the current " resides in all parts of the field, only (usually) a small portion occupying the conductor itself.

Now substitute for the one turn of wire a bundle of wires, N in number, of the same size and form. Disregarding small differences due to the want of exact correspondence between the bundle and the one wire, everything will be the same as before if the above Cj means the total current in the bundle. But if the same current be supported by each wire, practical convenience in respect to the external connections of the coil requires us to make the current in each wire the current. Let this be C, so that Cj = NO. Then we shall have, by (8),

Bi = (L1N)C . . . . (8a) to express the flux of induction ; and by (9) and (Sa),

...... (9o)

if L = N2L1} to express the energy. This L is the inductance of the coil. It is N2 times the permeance of the magnetic circuit.

Again, regarding the coil as a single circuit, B:N is the induction through it — that is, Bx through each winding. Calling this total B, we have, by (8a),

B = B1N=(L1N2)C = LC, .... (86)

which harmonises properly with (9a).

The difference between inductance and permeance, therefore, merely depends upon the different way of reckoning the current in the coil. With one winding only, they are identical. I should here observe that I am employing at present rational units. Their connection with the Gaussian units will appear later. It would only serve to obscure the subject to bring in 47T, that arbitrary and unnecessary constant which has puzzled so many people.

It will be seen that the distinction between permeance and inductance is a practical necessity, in spite of their fundamental identity. But which should be which ? On the whole, I prefer it as above stated, especially to connect with self-induc- tion. Regarding permeability itself, it would seem that this name is more particularly suitable to express the ratio /*//z0 of

32 ELECTROMAGNETIC THEORY. CH. II.

the inductivity of a medium to that of ether, which is, in factr consistent with the original meaning, I believe, as used by Sir W. Thomson in connection with his " electromagnetic defi- nition " of magnetic force. But to inductivity, as before- mentioned, a wider significance should be attached. As has been more particularly accentuated by Prof. Riicker, we really do not know anything about the real dimensions of ft and c ; or, more strictly, we do not know the real nature of the electromagnetic mechanism, so that ft and c are very much what we choose to make them, by assumptions. The two prin- cipal systems are the so-called electrostatic, in which c = 1 in ether, and the electromagnetic, in which ft = 1 in ether. But with these specialities we have no further concern at present.

Cross-connections of Electric and Magnetic Force. Circuital Flux. Circuitation.

§ 33. The two sets of quantities, the electric and magnetic forces, with their corresponding fluxes and currents, and the connected products and ratios, may be considered quite inde- pendently of one another, without any explicit connection being stated between the electric set and the magnetic set, whether they coexist or not. But to have a dynamical electromagnetic theory, we require to know something more, viz., the cross- connections or interactions between E and H. Or, in another form, we require to know how an electric field and a magnetic- field mutually influence one another.

One of these interactions has been already partially men- tioned, though only incidentally, in stating the meanings of permeance and inductance. It was observed that the electric current in a simple conductive circuit was measured by the gaussage in the corresponding magnetic circuit.

A word has been much wanted to express in a convenient and concise manner the property possessed by some fluxes and other vectors of being distributed in closed circuits. This want has been recently supplied by Sir W. Thomson's introduction of the word " circuital " for the purpose.* Thus electric current is a circuital flux, and so is magnetic induction. The fundamental basis of the property is that as much of a circuital flux enters

• "Mathematical and Physical Papers," Vol. III., p. 451.

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 33

any volume at some parts of its surface as leaves it at others, so that the flux has no divergence anywhere. This qualification, "anywhere," should be remembered, for a flux which may diverge locally, as, for instance, electric displacement, is not circuital in general, though even electric displacement may be circuital sometimes. Further, as a flux need not be distributed throughout a volume, but may be confined to a surface, or to a line, we have then specialised meanings of circuital and of divergence. Or a volume-distribution and a surface or line- distribution of a flux may be necessarily conjoined, without, however, any departure from the essential principle concerned. The word " circuital," which will be often used, suggested to me the word " circulation," to indicate the often-occurring operation of a line-integral in a closed circuit ; as, for instance, in the estimation of circuital voltage or gaussage. Now, in the case of a moving fluid, Sir W. Thomson called the line-integral of the velocity in a closed circuit the " circulation." This is curiously like " circulation." But " circulation " seems to have too specialised a meaning to be suitable for application to any vector, and I shall employ " circulation." The operation of circuitation is applicable to any vector, whether it be circuital or not.

First Law of Circuitation.

§ 34. Now in the case of a simple conductive circuit, we have two circuital fluxes. There is a circuital conducting core sup- porting an electric current, and there is a circuital flux of in- duction through the conductive circuit. In the electric circuit we have Ohm's law,

E = RC, (1)

where E is the circuital voltage, C the current, and R the re- sistance. And in the magnetic circuit we have a formally similar relation,

H=L-IB, (2)

where H is the circuital gaussage, B the induction, and Lr1 the reluctance. Or,

B = LH, (3)

where L is the inductance (or the permeance, when there is only one turn of wire).

D

34 ELECTROMAGNETIC THEORY. CH. II.

Now, the cross-connection in this special case is implied in the assertion that H and G are the same quantity, when mea- sured in rational units The expression of the law of which this is an illustration is contained in any of the following alter- native statements.

The line-integral of the magnetic force in any closed circuit measures the electric current through any surface bounded by the circuit. Or,

The circuitation of the magnetic force measures the electric current through the circuit. Or,

The electric current is measured by the magnetic circuita- tion, or by the circuital gaussage.

The terminology of electromagnetism is in a transitional state at present, owing to the change that is taking place in popular ideas concerning electricity, and the unsuitability of the old terminology, founded upon the fluidity of electricity, for a comprehensive view of electromagnetism. This is the excuse for so many new words and forms of expression. Some of them may find permanent acceptation.

The above law applies to any circuit of any size or shape, and irrespective of the kind of matter it passes through, mean- ing by " circuit " merely a closed line, along which the gauss- age is reckoned. By " the current " is to be understood the current ; not merely the conduction current alone, or the dis- placement current alone, but their sum (the convection current term will be considered separately).

It is also necessary to understand that a certain convention is implied in the statement of the law, regarding positive senses of translation and rotation when taking line and surface integrals. Look at the face of a watch, and imagine its circum- ference to be the electric circuit. The ends of the pointers travel in this circuit in the positive sense, if you are looking through the circuit along its axis in the positive sense. Also, you are looking at the negative side of the circuit. Thus, when the current is positive in its circuit, the magnetic induction goes through it in the positive direction, from the negative side to the positive side. Otherwise, the positive sense of the current in a circuit and the induction through it are connected in the same way as the motions of rotation and translation of a nut on an ordinary right-handed screw. This is the " vine " system used

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 35

by all British writers; but some continental writers use the "hop" system, in which the rotation is the other way, for the same translation. It is useless trying to work both systems, and when one comes across the. left-handed system in papers, it is, perhaps, best to marginally put the matter straight, and then ignore the text.

Second Law of Circuitation.

§35. The other cross-connection required is a precisely •similar relation between voltage and magnetic current, with, however, a change of sign. Thus : —

The negative line-integral of the electric force in any circuit (or the electric circuitation) measures the magnetic current through the circuit. Or,

The voltage in any circuit measures the magnetic current through the circuit taken negatively. Or,

Magnetic current is measured by the circuital voltage re- versed ; and other alternative equivalent statements.

Definition of Curl.

§ 36. In the above laws of circuitation the currents are the concrete currents (surface-integrals), and the forces also the •concrete voltage or gaussage. When we pass to the unit volume it is the current-density that is the flux. The circuita- tion of the force is then called its " curl." Thus, if J be the electric current and G the magnetic current, the two laws are

curlH^J, (4)

-cur!E1 = G, (5)

where Ex and Hx are the electric and magnetic force of the •field. We may now say concisely that

The electric current is the curl of the magnetic force.

The magnetic current is the negative curl of the electric force.

There is nothing transcendental about " curl." Any man who understands the laws of circuitation also understands what " curl " means, though he may not himself be aware of his knowledge, being like the Frenchman who talked prose for many years without knowing it. The concrete circuitation is sufficient for many problems, especially those concerning linear

D2

36 ELECTROMAGNETIC THEORY. CH. II.

conductors in magnetic theory. But it does not suffice for mathematical analysis, and to go into detail we require to pass from the concrete to the specific and use curl. How to mani- pulate " curl " is a different matter altogether from clearly understanding what it means and the part it plays. The latter is open to everybody ; for the former, vector-analysis is most suitable.

Let a unit area be chosen perpendicular to the electric cur- rent J. Its edge is then the circuit to which Ha belongs in (4). The gaussage in this circuit measures the current-density. Similarly, regarding (5), the voltage in a unit circuit perpen- dicular to the magnetic current measures its density (nega- tively). In short, what circuitation is in general, curl is the same per unit area.

Impressed Force and Activity.

§ 37. In the statement of the laws of circuitation, I have intentionally omitted all reference to impressed forces. That there must be impressed forces is obvious enough, because a dynamical system comprehending only the electric and mag- netic stored energies and the Joulean waste, is only a part of the dynamical system of Nature. We require means of show- ing the communication of energy to or from our electromagnetic system without having to enlarge it by making it a portion of a more complex system. Thus, taking it as it stands at pre- sent, the activity per unit volume we have seen to be

..... (6)

where the left side expresses the activity of the electric and the magnetic force on the corresponding currents, and the right side what results, viz., waste of energy, Q per second, and increase per second of the electric energy U and the magnetic T ; and, as there are supposed to be no impressed forces, if we integrate through all space, we shall obtain

(7)

where 2 means summation of what follows it. Or, if Q0 be the total waste, and similarly U0 and T0 the total energies, .

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 37

meaning, that whatever energy there be wasting itself is derived solely from the electric or magnetic energy, which decrease accordingly. This is the persistence of energy when there are no impressed forces.

Now, if there be impressed forces communicating energy at the rate A, the last equation must become

T0, .... (8)

and A must be the sum of the activities of the impressed forces f in the elements of volume, in whatever way space may be divided into elements, large or small, and however we may choose to reckon the impressed forces. There may be many ways of doing it ; f may sometimes, for example, be an ordi- nary force, and v, the velocity to match, is then a translational velocity. But for our immediate purpose, it is naturally con- venient to reckon the impressed forces electrically and mag- netically ; so that the corresponding velocities are the electric and magnetic currents. We shall then have, if e be the im- pressed electric, and h the impressed magnetic force,

to represent their activity per unit volume, and in all space,

... (9)

Instead of (7). This is the integral equation of activity. We cannot remove the sign of summation and make the same form do for the unit volume, for this would make every unit volume independent of the rest, and do away with all mutual action between contiguous elements and transfer of energy between them. This matter will be returned to in connection with the transference of energy.

Distinction between Force of the Field and Force of the

Flux.

§ 38. The distinction between Hx and H and between Ex and E is often a matter of considerable importance. We have

(10) (11)

38 ELECTROMAGNETIC THEORY. CH. II.

Now it is E and H that are effective in producing fluxes. Thus E is the force of the flux D and also of 0 ; and H is the force of the flux B. On the other hand, in the laws of circuitation, as above expressed, the impressed forces do not count at all j so that we have, in terms of the forces E and H,

curl(H-h) = J, (12)

-curl (E-e) = G, (13)

equivalent to (4) and (5). To distinguish from the forces of the fluxes, I sometimes call El and H3 the forces " of the field." Of course they only differ where there is impressed force. As the distribution of the energy, as well as of the fluxes, depends upon E and H, it is usually best to use them in the formulae.

Classification of Impressed Forces.

§ 39. The vectors representing impressed electric and mag- netic force demand consideration as to the different forms they may assume. Their line-integrals are impressed voltage and gaussage. Their activities or powers are eJ and hG- respectively per unit volume, and in this statement we have a sort of defi- nition of what is to be understood by impressed force. For, J being the electric current anywhere, if there be an impressed force e acting, the amount eJ of energy per unit volume is communicated to, or taken in by, the electromagnetic system per second ; and this should be understood to take place at the spot in question. It must then be either stored on the spot, or wasted on the spot, or be somehow transmitted away to other places, to be there stored or wasted, according to a law which will appear later on. Similarly as regards h and G-.

But this concerns only the reckoning of impressed force, and is independent of its physical origin, which may be of several kinds. Thus under e we include —

(1.) Voltaic force. (2.) Thermo-electric force. (3.) The force of intrinsic electrisation. (4.) Motional electric force.

(5.) Perhaps due to various secondary causes, especially ia connection with strains.

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 39

And under h we include —

(1.) The force of intrinsic magnetisation.

(2.) Motional magnetic force.

(3.) Perhaps due to secondary causes.

Voltaic Force.

§ 40. Voltaic force has its origin in chemical affinity. This is still a very obscure matter. For a rational theory of Chemistry, one of the oldest of the sciences, we may have to wait long, in spite of the activity of chemical research and of the develop- ment of the suggestive periodic law. Yet Chemistry and Electricity are so intimately connected that we cannot under- stand either without some explanation of the other. Elec- tricity is, in its essentials, a far simpler matter than Chemistry, and it is possible that great light may be cast upon chemical problems (and molecular physics generally) by previous dis- coveries and speculations in Electricity. The very abstract nature of Electricity is, in some respects, in its favour. For there is considerable truth in the remark (which, if it has not been made before, is now originated) that the more abstract a theory is, the more likely it is to be true. For example, it may be that Maxwell's theory of displacement and induction in the ether is far more than a working theory, and is something very near the truth, though we know not what displacement and induc- tion are. But if we try to materialise the theory by inventing a special mechanism we are almost certain to go wrong, however useful the materialisation may be for certain purposes. No one knows what matter is, any more than ether. But we do know that the properties of matter are remarkably complex. It is, therefore, a real advantage to get away from matter when possible, and think of something far more simple and uniform in its properties. We should rather explain matter in terms of ether, than go the other way to work.

However this be, we have the fact that definite chemical changes involve definite voltages, and herein lies one of the most important sources of electric current. Furthermore, there is the remarkable connection between the quantity of matter and the time-integral of the current (or quantity of electricity) produced, involved in the law of electro-chemical

40 ELECTROMAGNETIC THEORY. CI1. II.

equivalents, which is one of the most suggestive facts in physics, and must be a necessary part of the theory of the atom which is to come. That the energy of chemical affinity may itself be partly electromagnetic is likely enough. That even conduction may be an electrolytic process is possible, in spite of the sweet simplicity of Ohm's law and that of Joule. For these laws are most probably merely laws of averages. The well-known failure of Ohm's law (apparent at any rate) when the periodicity of electromagnetic waves in a conductor amounts to billions per second may perhaps arise from the period being too short to allow of the averages concerned in Ohm's law to be established. If so, this may give a clue to the required modification.

Thermo-electric Force.

§ 41. Thermo-electric force has its origin in the heat of bodies, manifesting itself at the contact of different substances or between parts of the same substance differing in temperature. Now heat is generally supposed to consist in the energy of agitation of the molecules of bodies, and this is constantly being transferred to the ether in the form of radiant energy, i.e., electromagnetic vibrations of very great frequency, but in a thoroughly irregular manner. It is this irregularity that is a general characteristic of radiation. Now the result of sub- jecting conductors to electric force is to dissipate energy and to heat them. This is, however, an irreversible process. But when contiguous parts of a body are at different temperatures, a differential action on the ether results, whereby a continued effect of a regular type is produced, reversible with the current, and therefore formularisable as due to an intrinsic electric force, the thermo-electric force. At the junction of different materials at the same temperature it is still the heat that is the source of energy.

The theory of thermo-electric force due to Sir W. Thomson, based upon the application of the Second Law of Thermo- dynamics (the First is a matter of course) to the reversible heat effects has been verified for conductive metallic circuits by the experiments of its author, and those of Prof. Tait and others. With some success the same principle has also been applied by von Helmholtz to voltaic cells, which are thermo-electric as well

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 41

as voltaic cells. There are wheels within wheels, and Ohm's law is merely the crust of the pie.

Intrinsic Electrisation.

§ 42. Intrinsic electrisation is a phenomenon shown by most solid dielectrics under the continued action of electric force. It is the manifestation of a departure from perfect electric elasti- city, and is probably due to a molecular rearrangement, result- ing in a partial fixation of the electric displacement, whereby it is rendered independent of the " external " electrising force. Thus the displacement initially produced by a given voltage slowly increases, and upon the removal of the impressed voltage only the initial displacement will subside, if permitted, imme- diately. The remainder has become intrinsic, for the time, and may be considered due to an intrinsic electric force e. If Ix be the intensity of intrinsic electrisation, and c the permittivity, then

ii-« ......... a)

Ij is the full displacement the force e can produce elastically, all external reaction being removed by short circuiting. It is not necessarily the actual displacement. The phenomenon of 41 residual charge," "soakage," "absorption," &c., are accounted for by this e and its slow variations.

Maxwell attempted to give a physical explanation of this phenomenon by supposing the dielectric to be heterogeneously conductive. This is perhaps not the most lucidly successful of Maxwell's speculations. How far electrolysis is concerned in the matter is not thoroughly clear.

Intrinsic Magnetisation.

§ 43. Intrinsic magnetisation is, in some respects, a similar phenomenon, due to a passage from the elastic to the intrinsic form of induction externally induced in solid materials. Calling the intensity of intrinsic magnetisation I2, we have

where h is the equivalent intrinsic magnetic force, and /* the inductivity (elastically reckoned).

In one important respect intrinsic induction is a less general phenomenon than intrinsic displacement. There is no magnetic

42 ELECTROMAGNETIC THEORY. CH. IL.

conductivity to produce similar results as regards the magnetic current as there is electric conductivity as regards the electric current. But if there were, then we could have a magnetic " condenser," with a magnetically conductive external circuit, and get our residual results to show themselves in it, quite similarly in kind to, though varying in magnitude and perma- nence from, what we find with an electric condenser.

The analogue of Maxwell's explanation of " absorption " would be heterogeneous magnetic conductivity. This is infi- nitely more speculative than the other, which is sufficiently doubtful.

Swing's recent improvement of Weber's theory of magnetism seems important. But as in static explanations of dynamical phenomena the very vigorous molecular agitations are ignored, it is clear that we have not got to the root of the matter. We want another Newton, the Newton of molecular physics. Facts there are in plenty to work upon, and perhaps another heaven- born genius may come to make their meaning plain. Pro- perties of matter are all very well, but what is matter, and why their properties ? This is not a metaphysical inquiry, but con- cerns the construction of a physical theory.

•

The Motional Electric and Magnetic Forces. Definition of a Vector-Product.

§ 44. The motional electric and magnetic forces are the forces induced by the motion of the medium supporting the fluxes. To express them symbolically, it will save much and repeated circumlocution if we first define the vector-product of a pair of vectors.

Let a and b be any vectors, and c their vector-product. This is denoted by

c = Vab, (3)

the prefix V meaning "vector," or, more particularly here, "vector-product." The vector c is perpendicular to the plane of the vectors a and b, and its tensor (or magnitude) equals the product of the tensor of a into the tensor of b into the sine of the angle between a and b. Thus

c = absmO, (4)

if the italic letters denote the tensors, and 6 be the included

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 43

angle of the vectors. As regards the positive sense of the vector c, this is reckoned in the same way as before explained with regard to circuitation. Thus, when the tensor c is posi- tive, a positive rotation about c in the plane of a and b will carry a to b. If the time by a watch is three o'clock, and the big hand be a and the little hand b, then the vector c is directed through the watch from its face to its back. These vector-products are of such frequent occurrence, and their Cartesian representation is so complex, that the above concise way of representing them should be clearly understood.

On this understanding, then, we can conveniently say that the motional electric force is the vector-product of the velocity and the induction, and that the motional magnetic force is the vector -product of the displacement and the velocity. Or, in symbols, according to (3),

....... (5)

h = VDq, ..... . . (6)

where q. is the vector velocity.

Example. A Stationary Electromagnetic Sheet.

§ 45. It should be remembered that we regard the dis- placement and the induction as actual states of the medium, and therefore if the medium be moving, it carries its states with it. Besides this, it usually happens that these states are themselves being transferred through the medium (independently of its translational motion), so that the resultant effect on pro- pagation, considered with respect to fixed space, is a combination of the natural propagation through a medium at rest, and what we may call the convective propagation. Of course we could not expect the two laws of circuitation for a medium at rest to remain true when there is convective propagation.

The matter is placed in a very clear light by considering the very simple case of an infinite plane lamina of E and H travel- ling at the speed of light v perpendicularly to itself through a homogeneous dielectric. This is possible, as will appear later, when E and H are perpendicular, and their tensors are thus related : —

. (7)

44 ELECTROMAGNETIC THEORY. CH. IL

Or, vectorising v to v,

...... (8)

according to the definition of a vector-product, gives the directional relations as well as the numerical.

Now, suppose we set the whole medium moving the other way at the speed of light. The travelling plane electro- magnetic sheet will be brought to rest in space, whilst the medium pours past it. Being at rest and steady, the electric displacement and magnetic induction can cnly be kept up by coincident impressed forces, viz. : —

e = E, h = H

Now compare (8) with (5) and (6) ; consider the directions carefully, and remember that the velocity q. is the negative of the velocity v, and we shall obtain the formulae (5), (6), which are thus proved for the case of plane wave motion, by starting with a simple solution belonging to a medium at rest.

The method is, however, principally useful in showing the necessity of, and the inner meaning of the motional electric and magnetic forces. To show the general application of (5) and (6) requires a more general consideration of the motional question, to which we now proceed.

Connection between Motional Electric Force and " Electromagnetic Force."