Electromagnetic Theory, Vol. 2 (1899) — part 4 of 31

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

VC = KC2 = KV2, (2)

and represents the Joulean waste per second in the whole con- ductor, or the volume-integral of EO or of kK^ 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, t.e., the time-integral of the current, takes the place of current in the last case, and we now have

D-SV, Vi«S-iD, .... (3)

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

JVD = JSV2 = iS-iD^ .... (4)

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

As the dielectric is supposed to be a non-conductor, the our- rent is D or ST, 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^KY. The true cur- rent (that is, the current) is now the sum of the conduction and displacement currents. Say,

(5)

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 arc, 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 y, giving

vr-vc+VD,

-RC« + ^(jSV«), .... (6)

represamtiiig tlie waste in Joulean heating and the rate of in- eieaae of the electric eaergy, Eaoh of these quantities is the sum of the same quantities per unit volume throughout the substance concerned.

Permeance, Inductance and Beluctance.

§ 31. Permeability gives rise to permeance^ induotivity to inductance, and reluctivity to reluctanoe.

The fonnal 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 elastanoe 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. Proi S. P. Thomp- son has also used the word in this sense in his Cantor Lectures with good effect

If we replace our illustratiTe 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 foroe 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

^0

KLBCTROllAGinBIIO THSORT.

CH. II.

iHB = i(|-)H^ = i(?.)B^.. . . (7)

AVheii 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 hannomse with it, inductaace and induo- tivitj were coined.

Inductance of a Circnit.

§ 32. The meaning ci inductance has sometimea been mis- conceived. It is not a synonym for induction, nor for self-induc- tion, but meanft '* 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 C^. As will later appear, this Cj is also the gaussage. That is, the line-integral of the magnetic force in any closed circuit (or the cirouitation of the force) embracing the current once is C^. Let also be the induction through the circuit of C,. Then

Bi = LiC,. (8)

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

P.Ci = iLiC.^ (9)

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

OUTLINE OF BLBCTBOMAGNETIO CONNECTIONS. 31

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

Now substitute for the one turn of wire a bundle of wires, N in number, of the same sisse 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 thisi be C, so that Cj = NC. Then we shall have, by (S),

Bi«(LiN)C (8a)

to express the flux of induction ; and by (9) and (8a),

iBiC,-}(L,N«)C«

-iLC2, (9a)

if L = N-Li, to express the energy. This L is the inductance of the coiL It is times the permeance of the magnetic •circuit.

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

B = Bil!^=(LiN2)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 tn the coil. With one winding only, they are identical I should here observe that I am employmg at present rational units. Their connection with the Gaussian units will appear later. It would only serve to obscure the subject to bring in 4t, that arbitrary and unnecessary constant which has puzzled eo 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 /i/ft^ of

32

■LBOTBOXAGNSnO THEOBT.

OE. IL

the indnotrnty of a medium to that of ether, which is, in fact, oonsistent with the original meaning, I belieye, as used by Sir W. Thomson in oonneotion with his electromagnetic defi- nition" of magnetio force. But to inductivity, as before- mentioned, a wider significance should be attached. As has been more particularly accentuated by Prof. Rttcker, we really do not know anything about the real dimensions of fi and c ; or, more strictly, we do not know the real nature of the electromagnetic mechanism, so that /x 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 = I in ether. But with these specialities we have no further concern at present.

Gross-connections of Electric and Magnetic Force. Oircnltal

Flux. Oircnitation.

§ 33. The two sets of qnantitieB, the electric and magnetio forces, with their corresponding fluxes and ouirents, and the connected products and ratios, may be oonsiderBd 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, yiz., the cross- connections or interactions between E and H. Or, in another form, we require to know how an electric field and a magnetio field mutually influence one another.

One of these interactions has been already partially men* tioned, though only incidentally, m stating the mfanings of penneance 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

• Mftthamaticia and Fhydoia FHpen, " VoL UL, p. 461.

ODTLXNB OF BIiBCTBOMAONimO OONNECfnONS. 33

amy volume at some parts of its surface as leaves it at others, 80 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 " circuitation," 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 ot the velocity in a dosed circuit the " circulation." This is oarioaaly like " oiromtation." But " oiroulation " seems to have too spedalised a meaning to be soitable for application to any vector, and I shall employ ''circuitation." The operation of circaitation 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 S is the oircuital voltage^ G the current, and B the re- sistance. And in the magnetic circuit we have a formally similar relation,

H-L-iB, (2)

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

B«LH, . (3)

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

D

3i ELECTROMAGNETIC THEOUr. Cil. 11.

Now, the cross-connection in this speckil case is implied ia the assertion that H and G are the same quantity, when mea- sured m 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 throuirli the circuit. Or,

The electric current is measured by the magnetic circuita- tion, or by the circuital f^anssage.

The terminology of electromagnctism 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 electroraagnetism. This is the excuse f<^r so many new words and forms of expression. Some of them may find i)ermanent acceptation.

The above law aj)i)lies to any circuit of any size or shapo, 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 t/te 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 lookiug 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 thid 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

OUTLINB 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 tlic ■same translation. It is useless trying to work both systems, -and when one comes across the left-handed s^'stem in papers, it is, perhaps, best to marginally put the matter straight^ and then ignore the text.

Second Law of Oircnitatlon.

§ 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 tlie electric circuitatiou) 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 statAtnents.

Definitioxi of Otirl.

§ 36. In the above laws of cirouitation 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- tiou of the force is then called its " curl." Thus, if J be the electric current and G the magnetic current, the two laws are

curl Hi = J, (4)

-curlBi-O, (5)

where Ej and Hj 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 cirouitation also understands what " curl " means, though he may not himself be aware of his knowledge, bemg like the Frenchman who talked prose for many years without knowing it. The concrete cirouitation is eufficient for many problenui^ especially those concerning linear

d2

86

■UBOIROMAGNBTIO THSOBT.

CH. n.

oondaotors in magnetio theory. But it does not suffioe fov mathematical analysiSi and to go into detail we require to pass from the concrete to the specific and use corL How to mani- pulate "curl" is a different matter altogether from dearly understanding what it means and the part it plays. The latter is open to OTexybody ; 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 drouit to which belongs in (4). The gaussage in this drouit measures the current-density. Similarly, regarding (5), the voltage in a unit circuit perpen- dicular to the magnetic current measures its density (nega- tiyeij). In shorty what cirouitation is in genera], ourl is the same per unit area.

Im|HreB8ed Foroo and iLctlvHy*

§ 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. AVe require means of show- ing the communication of enerLry to or from our electroma<inetic 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 Yolume we have seen to be

BJ+HO-Q + U + f, (6)

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

0-2(Q + U + f), (7)

where 2 means summation of what follows it. Or, if be the- total waste, and similarly and T« the total energies,

OUTLINB OF BLBOTBOMAOingnO OONinECTIONS.

37

meaning, that whatever enezgy there be wasting itself is derived Bolelj from the electric or magnetic enei^gy, which deorease 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

2f7-A«.Qo + Uo+Toi .... (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 translatioual 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- {uressed electric, and h the impressed magnetic force,

eJ+ha

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

2(eJ-hhG) = ^(Q-l-U-ht) ... (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

Flnz.

§ 38. The distinction between Hj and H and between and £ is often a matter of considerable importance. We have

H-h + Hi, (10)

£-e+Ei. (11)

Digitized by Google

ELECTBOIIAGNBTIO THJSOBY.

OH. IL.

N"ow 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 oircoitation, as above expressed, the impressed forces do not oount at all f so that we have, in terms of the forces E and H,

equivalent to (4) and (5). To distinguish from the forces of the fluxes, I sometimes callE^ and H] the forces "of the field."* Of course they only difier 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 formulas.

Olassiflcation 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 O.

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 in

curl (H-h)=J, curl (E-e)«0,

(12) (13)

connection with strains.

OUTLINS OF ELBCTBOMAONBTIO C0MMBCTI0K8.

39

And under h we iucliide-—

(1.) The force of intrinsio 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 rationed theory of Chemistry, one of the oldest of the sciences, we may have to wait long, in •pite 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 fayour. 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 trath, 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 osefttl 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. AVe 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 rcinarkable 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

KLSCTBOMAQMETIO THSORT.

equival^ts, 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 ohemical 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 due to the required modification.

Tlieniio-electric Foroe.

§ 41. Thermo-electrio force has its origin in the heat of bodies, muiifeeting 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, t.«., 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. Tliis is, however, an irreversible process. But when contiguous parts of a body are at different temperatures, a ditferential action on the ether results, whereby a continued effect of a regular type is produced, reversible with the curreut, 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 Ton Helmholta to voltaic cells, which are thermo-electric as well

OUTLINB OF BLECTBOlCAOinETIO 0ONNB0TION8.

41

«B voltaic cells. There are wheels within wheels, and Ohm's law is merely the crost of the pie.

Intrinsic Electrisation.

§ 42. Intrinsic electrisation is a phenomenon shown by most •eoUd dielectrics under the continued action of electric force. It is the manifestation of a departure from perfect electric elasti- •ci^, 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 Ij be the intensity of intrinsic electrisation, and c the permittivity, then

(1)

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

residual charge," " soakage," " absorption," <kc., 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 lieterogeneously oonductive. 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 Ig, we have

l2 = f^^ (2)

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

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

42

ELECTKOMAGNETIG TIlEUiiV.

CII. II.

conductivity to produce similar results as regards the magnetic current as there is electric conductivity as regards the electric current. lUit 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.

fiwing'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- bom 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 oon> oems 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 e 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) cipials the product of the tensor of a into the tensor of b into the sine of the angle between a and b. Thus

e=a&sin0, (4)

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

Digitized by Google

OUTUNE OF £LECTB0MA0N£T1C CONNECTIONS.

43

angle of the yeotois. As regards the positive sense of the ▼ector 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 i8> 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),

e = V4B (;■)>

haaVDc;, , , .... (G)

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 bo 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 eflfect on pro- pagation, considered with res])ect 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 £ 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 £ and H are perpendicular, and their tensors are thus related: —

E-ftvH (7)

44

SLBOTBOMAGNBTIO THEOBT.

OH. IL

Or, TeotoriBing vtov,

B-fiYHv, (8)

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

Now, suppose wo 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 only be kept up by coincident impressed forces, viz. : —

CbE, h»H.

Now compare (8) with (5) and (G) ; consider the directions •carefully, and remember that the velocity q is the negative, of the velocity v, and we shall obtain the formulas (.3), (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 tlie 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."