Electromagnetic Theory, Vol. 2 (1899) — part 16 of 31

SLBMBKTB OF VEOTOBIAL ALGEBRA AKD ANALYSIS. 233 *

Comparing this with the seoond of (324), and asaaming the equality of f and we see that H is velocity, /i density, er^ rigidity ; and so on. H/p means the time-integnd of H. Or, more explicitly, let Z be the time-integral of Hi so that H « j^; then

f-OfP«-c-iV«)Z, .... (326) which compares with the first of (324). Thus

Z » E/p ooiresponds to O, (spadal dispL)

H=pZ

curlH

D»cE

ti

if

it

q. (velocity), curl q, (2 X spin), curl G, (2 X rotation), /», (density),

(compliancy), f^, (impd. foroe)| Jpq« (kin. energy).

(mag. force) (el. current) (eL displ.) (indactivity) (permittivity) c (mag. source) f —curie (mag. energy) J/H

This is, as far as it goes, an excellent analogy, particularly on account of its directness and ease of application, and on account of the similarity of sources. The disturbances of H generated by fin the dielectric and propagated away at speed (f»:)~^ are precisely represented by the velocity generated by similarly distributed impressed Newtonian force, which is propagated in the same manner through the solid at the equivalent speed (fi/p)^. Of course the correspondence at similar moments also applies to the other quantities compared.

But it is imperative (in general) that the motions in the solid should be small. The disturbances should therefore be of the fluctuating or alternating character, so that Z, the time-integral of H, does not mount up. For, correspondingly, if this were allowed, then G- would mount up, and the elastio solid get too much out of shape to allow us to treat d/dt at a fixed point and S/^t for a moving particle as identical.

Observe also, that the electric energy ^cE^ is not matched by the potential energy of distortion; although their total amounts are equal, their distributions are entirely different. This is a fatal failure in detail. Nor have we any right to expect that a scheme like MaxwelFs, depending upon rotation^ can be perfectly matched by one which depends on shears.

234

BLSOTROMAGKETIO THEORY.

CH. IIL.

Seoond Example: Same as last* bnt Electric Force compared with Velocity.

§ 147. Another analogy of the siime class is got by making: the displacement in the solid represent the time-integral of E ill the dielectric. Thus let e = 0 in (323), and -curl h = + g. It is now g that is the source. Eliminate H between the two equations (323). We get

e;)E + ffBCurI(-curIE)/fip. . . . (327) Or, since div £«0 (when there is no electrification)

^^{f^'^^^i"^'"^)^ ' ' ^^^^^

if A»E/p, making 'E = pA.

Comparing (327) with (324) for the solid, we see that we- have E representing velocity, fx compliancy, c density, and so- on, being a general turning over of all the quantities. Thus

Aȣ/^ corresponds to G (displacement),

c ,f Pi

(eL force) £ Q. (velocity),

H „ n curl G,

— curl h ,, (circuital impd. force).

The conclusions are similar as regards the generation and* propagation of disturbances of the two kinds compared. There is a similar failure to before as regards the energy of dis- tortion. It ought t«' be the magnetic energy ^fj-B^ now, but it is not. This want of correspondence as regards one of the energies will be removed (and can only be removed) by doing away with the rigidity and substituting something else, which must, however, be equivalent in its results as regards the propagation of disturbances from place to place.

Third Example : A Oonductiiig Dielectric compared witb

Viscous Solid. Failure.

§148. As we see from the preceding that the propagation of electromagnetic disturbances through a non-conductor can*

ELEMENTS OF VECTORIAL ALGEIiUA AND ANALY-JIS. 23^

be imitated in an incompressible solid, posaessiug the usual dis- tortional rigidity, let us next introduce electric conductivity on the one hand, and examine what changes are needed in the analogies of §§ 146, 147 to keep them working, if it be possible. Since there is waste of energy associated with the conduction current, it is clear that some sort of friotiooal force requires to be introduced into the elastic solid.

First try distortional friction. The equation of motion of an incompressible viscous solid is

fi = [f5P'-(«o + »iP)v']<^» • • (329)

where f| is the impressed force per unit Yolume, G the displaoe- ment, p the density, Hq the rigidity, and »^ the associated fHo- tionality. Also, p stands for d/dtf for practical convenience in the operations, whether direct or inverse. Thus, p~^q^ or q/p means the time-integral of q, which would in the ordinary

notation of the integral calculus be expressed by jq.dtf a nota-

J 0

tion which is not convenient for manipulative purposes in in- vestigations of this class. We may derive (329) from (324) by changing n to Ji^ + n^p ; or by (306), (313). [See also p. 229.]

Now the circuital equations of E and H in a conducting dielectric (homogeuoous, isotropic, stationary), are

ourl(H-h)-(^ + cp)B, ourl(e-E)=/i;?H, . (330)

comparing which with (323) we see that the effect of introducing conductivity is to change cp to k + cpt bo that the equation of H is got by making this change in (325), giving

where Z = H//3. But on comparing this equation with (329), we see that the viscosity in the one case and the conductivity in the other enter into the equations in ditTereat ways. We can only produce a proper correspondence when the coefficients of are equal ; or, in another form, when the operators p{k + cp)"^ and (uq + n,;;) are equivalent in effect. This is not generally possible. But there are two extreme cases of agree- ment. One we know already} viz., when there is no conduc-

236

BLEOTBOKAGNSnO TEOBOBT.

CH. m.

tivity and no viscosity, as in § 146. The other is when there is no permittivity and no rigidity, which is sufficieiitlj important to separately considered.

Fourth Example: A Pure Oonductor compared with a Viscous Liquid. Useftil Analogy.

§ 149. This is the extreme case of a pure conductor, or a conductor which cannot support elastic displacement. It dissi- pates energy, but does not store it electrically ; though, on the other hand, its magnetio storage capacity is retained. It

is compared with an incompressible yiscous solid with the

rigidity abolished; that is to say, with an incompressible viscous liquid. Putc^O iu (331), and we have the equation

f-(^p-irV*)H; . . (332)

and putting nQ = 0 in (329), with q, the velocity substituted for jdG, gives us the equation

^1 = (PP " ^iV")^ (333)

in the liquid. These admit of immediate comparison. Observe^ however, the curious fact that whereas in § 146 the permit- tivity was the reciprocal of the rigidity, so that the vanishing of one means the infinitude of the other, yet now both the permittivity and the rigidity vanish together, as if they were equivalent This emphasises the incompatibility of (329), (331). Our present analogy must stand by itself all idea of permit- tivity and rigidity being thrown away.

We compare magnetic force H in a pure conductor with the velocity q in a viscous liquid; the inductivity fi with the density of the liquid, so that the magnetic energy and the kinetic energy are compared, and the current is twice the spin. Also the source f= curie in the case of the conductor is compared with circuital impressed force in the liquid. So far is the same as in § 146. But now, in addition, we have the electric resistivity represented by the liquid viscosity. We conclude that the disturbances q generated by in the liquid are similar to the disturbances H in the conductor generated by f, and that the propagation of electrical disturbances in a conductor is like that of motion iu a viscous incompressible liquid. It

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 237

takes place by the process called diffusion. It is the limiting case of elastic wave-propagation, with distortion and dissipation of energy by friction.

We are usually limited to small motions in this analogy, so that the impressed forces should not act in one sense continu- ously, but should fluctuate about the mean value zero, for the reason mentioned before. But we are not always limited to small motiona. In certain symmetrioal distributions of mag- netic force and current we may remove this restriction altogether. In the- liquid case (333) p strictly means hjU, whilst in (332) it means djdt. In the former a moving particle is followed ; in the latter we keep to one place. But in the cases of laminar flow referred to, BjU reduces to djdt readily enough, so that the analogy is carried on to cases of steadily actmg impressed forces.

The analogy is an important one, owing to the readiness with which the setting in motion of water by sliding friction can be followed, as when a wind blows over its surface. Two interest- ing cases are the penetration of magnetic induction and (with it) electric current into a core enveloped by a solenoidal coil, in whose curcuit an impressed force, yariable or steady, acts ; and the penetration of magnetic induction and electric current into a straight wire when an impressed force aotsin its circuit. Consider- ing the latter we may state the analogy thus. Understanding, in the first place, that the circumstances should be such that the influence of electric displacement outside the wire is negli- gible, we may replace the wire by a similar tube of water, and then replace the impressed voltage in the electric circuit by a uniform tangential drag upon the surface of the water in the direction of the length of the pipe. The current of water in the pipe and the electric current in the wire, will vary similarly under the action of similarly varying impressed forces. Thus, a steadily applied force on the water will first pull the outermost layer into motion; this, by the viscosity, will pull the next layer, and so on up to the axis. The initial current is purely superficial ; a little later we have a central core in which the water is practically motionless, surrounded by a tubular portion whose outermost layer is moying rapidly, and innermost very slowly ; later still, the whole mass is in motion, though leas rapidly at the axis than at the boundary;

238

BLBOTROMAONSnO TH80RT.

Cn. III.

finally, the whole mass of water acquires a state of uniform motion. Substituting electric current for current of water, we obtain a representation of the way the electric current is set up in a wire, pstssing through the various stages from the initial surface current to the final uniformly distributed current. If the impressed force be rapidly oscillatory, we stop the penetra- tion as above described in its early stage, so that if the frequency be great enough, there is a practical confinement of the current (in sensible amount) to the skin of the wire (or of tlie water respectively). But if the mean value of the oscillatory force be not zero, it is the same as having two impressed forces, one steady, the other periodic, without any bias one way or the other. Then we have finally a steady enrrent throughout the wire, pltu the oscillatory current with superficial concentration.

When a straight core is magnetised in a solenoid, it is the magnetic induction which is longitudinal, or parallel to the axis. Comparing this with the current of water in the pipe, we have the same state of things as before described, as regards the penetration of magnetic induction into the core.

One effect is to increase the resistance of a wire when the frequency is great enough. It will be remembered that Prof. Hughes (in 1886) brought forward evidence of this increase, and, therefore, in the opinion of others, of the truth of the theory of surface conduction along wires under certain oircum- stances which was advanced by me a year previous. There has since been plenty of confirmatory evidence of a more complete nature ; that is, not merely of an approximation towards, but of an almost complete attainment of surface-conduction.

This action of conductors is sometimes referred to as magnetic screening. It should, however, be noted that the screening is done by the conductor itself, not by the currents *' induced" in the outer part of the wire or core, which are, as it were, merely a sign that the screening is taking place. The screening action often seems to be merely superficial ; but this is accidental, from the disturbauces being communicated to the conductor from without. In reality, any and every part of a conductor is screened from the rest by the portion immediately surrounding it, that is, by its skin, and by that only. It is primarily the con- ductivity that causes the screening, so that it is rather conduc- tive screening than magnetic screening. The result is that dis-

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ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 239

turbances can only pass through a conductor by the (relatively) very slow process of diffusion, so unlike that by which they are transferred through a non-conductor. But the property of con- ductivity acts conjointly with the inductivity, so that although copper would bo the best screener on account of its high conduc- tivity, the great inductivity of iron usually far more than com- pensates for its inferior oonduotivitj, and makes iron take the first place as a screener.*

Fiftli Example : Modification of the Second and Ponrtli.

§ 150. As we were not able to make an analogy with e and k both finite, likewise and it^, when H was taken to be ▼elod^, let us try the effect of taking E to be velocity, as in § 147. Change the source from f to ga -ourlh; then we have merely to alter cp in (328) to k + cp to obtain the required modification. Thus,

g-((*+cp)-g)E=((ti, + cp«)-^')A. . (334)

if A»E//>. Here, again, on comparison with (329), we see that distortional friction will not furnish a proper analogy in the general case. There seem to be but two special cases possible; first, with the conductivity and friotionality both sero (not the resistivity and frictionality), which reproduces the case in § 147 ; and, secondly, the permittivity and the rigidity both zero (not permittivity and compliancy), which brings us to the viscous liquid again. Thus, putting e*0 in (334) brings us to

g=(/(p-^')A, (335)

which compares with (333). We now compare conductivity with density, and inductivity with the reciprocal of the viscosity. This is by no means so useful as the previous form of the viscous fluid analogy, although it is fundamentally of the same nature, involving propagation by diffusion.

As in §§ 146, 147, we had a failure of representation of one of the energies by the energy of distortion, so now, in the viscous fluid analogies, we fail to represent the dissipation ol energy properly. The latter depends, as shown before, on the velocity of distortion, so we might expect a similar failure.

840

moTBOMAorano thiort.

GH. in.

•bcttt Example : A Ckmdnctiiig Dielectric oon^aied with, an Elastic Solid with Tranalatioxial Friotton.

§ 151. But we can get a working analogy to suit the con- ducting dielectric in another way. Thus, still keeping the distortional elasticity or rigidity, give up the viscosity or dis- tortional friction, and Bubatitute translational friction. For the equation of motion of an incompressible solid with the usual constants, rigidity n and densitj pp and with translational Iriotiouaiity pi is

fjo.(^+p^-nv2)a, . . . . (336)

where f, is circuital impressed force. Now operate on (331) by 1 + k/cp, produdng the equation

Comparing this with (336) we see that there is a proper cor- respondence of form on the right sides, though not on the left The circuital impressed force f| is replaced, not by f, but by t+kf/cp. This is awkward for the analogy as regards the gene- ration of disturbances. But away from the sources, there is a useful correspondence. As in §146, Z corresponds to spacial displacement in the solid, H to velocity, /x to density, c to com- pliancy (distortional) ; whilst now, in addition, the translational frictionality in the solid has the complex representative fjJ:/c in the dielectric ; or Jc{fiv)^f if v is the speed of propagation of disturbances. So disturbances of H are propagated in the same way through a conducting dielectric, as are disturbances in an incompressible solid with the usual rigidity, with frictional resistance to displacement superadded. But we do not cor- rectly localise the energy dissipated. The rate of waste per unit volume is kE^ in the dielectric, whilst it is p^q^ in the solid, which corresponds to k{fivYTi^ in the dielectric. This being entirely against Joule's law renders this form of analogy unsatisfaotory. But we may easily make a substantial improve- ment by taking E to be velocity in the solid.

Seventh Example : Improvement of tlie Siztb.

§ 152. Thus, let g » - curl h be the source of disturbance in the dielectric, so that we have the equations (334) to consider.

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SLEBIBNTS OF TIBCTOBIAL ALQSBBA. AND ANALTSIB. 241

Ttike tlie secxmd form and compare it with (33$), and we see at onoe that there is a comdderably more simple oorrespondenoe than in the case last treated. Thus, the source of circuital displacement Q corresponds to the source g of circuital A (the time-integral of £) ; the electric force now means Telocity 3 ft is compliancy ; c is density ; k itself is the frictionality ; the electric energy |cE- is kinetic energy ; and the waste (Joulean) is properly localised in the solid by the f rictional waste of energy /i^Q- per second. The induction mH is twice the rotation. In fact, this case is the same as in §147, "with translational fric- tion added without destroying the analogy. It is clear that the correct localisation of the waste, and the correspondence of sources, makes the analogy be more readily followed. Of course, there is the same failure as regards the distortional energy as in § 147.

Eighth Example: A Dielectric with Duplex Conductivity compared with an Elastic Solid with Translational Elasticity and Friction. The singular Distortionless Case.

§ 153. As a last example in which use is made of distortional elasticity, let us generalise the non-conducting dielectric by the introduction of magnetio conductivity. We have

curl (H - h) = (/; + c/>)E, . . . (338)

curl(e-E) = (y + /x^)H, . . . (339)

where is the magnetic conductiTity. So^ if curl ]|«0, and we eliminate we have

curie = ^(ijf+/xp)-_^^H. . • (340)

Or, if curl e « 0 and we eliminate H, we have

-curlh-^(^ + rp)-^^)E. . . (341)

We see that we cannot, when h and g are both finite, get a satisfiujtory analogy as regards the generation of disturbances by fp circuital impressed force hi a solid. (Consider, therefore, only the free propagation, subject to the equation

0-((^+cp)(5'+/*i>)-v2^Z, . . (342)

242 BLBOTBOMAGNETIO TBBOBT. OH. III.

where for Z we may subBtitute H or E or other eleotrioal quan- tities. Since, in (342), we have a term proportional to Z itself (or its Bubetitate), we must introduce elastic resistance to dis- placement in the elastic solid to make an analogy. Thus,

is the equation of motion in the incompressible solid if there be elastic and frictional as well as inertial resistance to traualation, by (307), (313), §145.

Now expand (342) and divide by c. We get

0=(|^+^</ + ^^)/> + /xp2^Z~^'z, . (344)

which is suitable for comparison with (343). When Z is the spacial displacement, H is velocity, c compliancy, fi density, as in § 146 ; but in other respects the correspondence is complex, for the elastic resistance to translation is k^/c, whilst the frictionality is ^ + kplc.

If we take £ to be velocity there will be another set of corre- spondences, c being density and p compliancy.

However unsatisffiMitory the analogy maybe in details, we have still the result that disturbances in a dielectric with duplex oonductiyity (electric and magnetic) are propagated similarly to motions in an elastic solid constrained in the manner above de- cribed. The eflect of the two kinds of distortion and waste of energy involved in the existence of the two conductivities are therefore imitated by one kind of frictionalily in the elastic solid, assisted by elastic resistance to translation.

The sin^ar distortionless case oimes about when

4/)q/)2 = P]^ in the solid, kjc — gjp in the dielectric.

, (345)

In either case the effect of the frictional resistance on the dis- turbances is made merely attenuative. IMsregaiding the atten- uation with the time, the propagation takes place in the di- electric in the same way as if it were non-conducting, and in the solid in the same way as if it were devmd of frictional and elastic resistance to translation. In the electromagnetic case we may follow this into detail with ease in a symmetrical manner. It is, however, rather troublesome in the elastic

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■MDUNTB OP YBOIOBIAL ALQBBRA ASD ANALYSIS. 243

•dlidy on aocount of the want of correspondence in detail, doe to the emplovment of distortional elastioit^, and for other feaaoDB.

The Botational Ether, OomiirefisiUe or Incompressible.

§ 15^A11 the above analogies, however good or bad they may be in other respects, are deficient in one vital property, masmuch as they involve the elasticity of distortion, and with it, the energy of distortion. Bat the electromagnetic equations^ especially when pat in the daplez form i^mmetrical with respect to the electric and magnetic sides, have nothing in them suggestive of distortional forces, nor can we represent either the electric or the magnetic energy as the energy of distortion. On the other hand, the equations are fully suggestive of rotation. If, then, the elastic solid has still to do duty for purposes of analogy, and yet not fail upon the point mentioned, it is clear that the rigidity must be done away with, at least as an electromagnetically active influence. Its place must then be taken by elastic resistance to rotation, by using a rotational instead of a shearing stress. This brings us to the medium invented by Lord Kelvin, called by him simply Ether, and contrasted with Jelly, which means an incompressible elastic solid with ordinary rigidity. This rotational ether has mass of course, for one thing, which brings in kinetic energy (of translation) ; and, on the other hand, it possesses, by means of internal arrangements with which we are not immediately concerned, the property of •elastically resisting rotation, and consequently stores up -energy of the potential kind, to balance the kinetic. If this were all, there woold be little more to say. Bat Lord Kelvin foond that his Ether enabled him to complete a long-delayed work, viz., to prodace a satisfactory elastio«olid analogy to suit the problem of magnetic induction (" Mathematical and Physical Papers," Vol. I., art. 27, and Vol. III., art. 49) ; and I have pointed out that this rotational ether enables us to construct good all-round analogies for the propagation of dis- turbances in a stationary dielectric (Appendix to chap. II., an^), with correct localisation of both energies and of the flux of energy. As these are the only all-round elastic-solid analogies yet known, we may here consider them ugaiii,

ii2

BLIOTROMAONBIIO THBORT.

OH. m.

espeoially in relation to the prerionsly given analogies, and to show where the changes oome in.

Go back to equation (313), giving the translational foroe F due to the stress in an elastic solid when there is rotational elasticity as well as rigidity. We have

P-n(V20+JVdivO)+*Vdiva-vcnrl«0, (346)

where Q is the displacement, n the rigidity, k the compressive

resistivity, and v the new elastic constant connected with the rotation. Now, we have ourP =» V div - V 2, so we may resirrange (346) thus,

T = (n + v)V'a + {k + in-v)Vdiv(3t, . (347) and this may be split up into circuital and divergent parts, vii.,.

= (» + v) V2ai, (circuital), . (348) P,-(ife+|n)V20^ (divergent). (349)

Now observe that in the circuital equation, n and v occur additively, so that equal parts of each have the same effect, and either may be replaced by an equal amount of the other. This is irrespective of the compressibility, which is not con- cerned in circuital disturbances. "We see, then, that the rigidity may be done away with altogether and equal v substi- tuted, without affecting the propagation of circuital disturb- ances. This is a very remarkable property.

Observe, also, that in the divergent equation v does not enter at all, although n does, and in the same way as in the ordinary compressible rigid solid. In fact, the coefficient k + ^n is the same as the former m+n, of § 142 and later. So the propagation of divergent disturbances is the same as if the rotational elasticity were done away with.

Now, we have already noticed the case in which the speed of the divergent wave is brought down to aero— the contractile ether of g 142, 143. In an ordinary rigid solid, which was then referred to, this requires negative compressibility. But now, if n be abolished and v substituted, so that the circuital propagation is unaffected, the divergent equation will contain only k, so that to make the speed of the divergent wave lero we need only abolish the resistance to compression, or make ib»0. The medium does not now resist either change of

BLBMBKTS OF TBOIORIAL ALQBBRA AND ANALYSIS. 245

shape or of size, but resists rotation only, and the potential energy is the energy of rotation. This is one extrenie form of the rotational ether, made by combining it with the contractile, by the evanescence of the rigidity and of the resisUmce to com- pression. The medium is quite neutral as regards expansion and compression.

Now, it is clear, from equation (348) and matter in previous paragraphs, that we may construct analogies to suit the non- conducting dielectric by abolishing n and using v instead, without concerning ourselves with h at all, which may have any Yalue from zero to infinity. But Lord Kelvin's ether is got by going to the other extreme from the neutral case just men- tioned.

Jirst fiotational Analogy : Magnetic Force compared with

Velocity.

§ 154. Take k=cc (with n = 0) making the medium incom- pressible^ and therefore making disturbances in an unbounded medium be necessarily circuital, whether the impressed forcive he droultal or not. We now have

ri=vy2a, (300)

by (348), whilst the other equation (349) will merely assert that is balanced by difference of pressure. This we are not concerned with, since our impressed foroiye should be circuital jn the electromagnetic comparisons. The stress (310) is now

Pm« -I'VNcurlG, .... (351)

where we ignore the uniform pressure for the reason mentioned. The torque accompanying this stress is, by (311),

S = 2i'curia, (3r)2)

and the translations! force is

P« -curlJS = vv2Q^ . . . (353)

agreeing with (350). So the equation of motion is, by (306),

fi-(ftP*'»'V2)a-(/»p-]^V% . . (354)

where is the circuital impressed force, and q is the velocity. A& before, we keep to small motions in generuL

246

ELECTROMAGNETIC THEORY.

CH. III.

Equation (354) may be at once compared with the electro- magnetic

f«(^p-V!)H=(Av2-^')z. • . (355)

as in (325), (326), our first example, and we therefore deduce the following correspondenoes : —

stands for

(spacial displ.)

(mag. force) H

t>

99 Qi

(velocity).

(el. current) curl H

99

„ curlq,

(2 X spin).

(el. displ.) D

99

„ curlG,

(2 X rotation).

(ind activity) /*

11

99 P9

(density).

(permittivity) c

l

99 l/^',

(rotl. compliancy)!.

(induction) B

99

99

(momentum).

(el. force) £

99

19 JS,

(J X torque).

(mag. energy) JmH^

99

(kin. energy).

(el. energy) JcB*

99

„ ii<curlO)^(rotl. energy).

(source of H) curl e

n

(impd. force).

(energy flux) VEH

(energy flux)

Keferring to the similar list

in § 146, it will be seen that the

correspondences there given are repeated here, except that the elastic constant changes its meaning, and that there are now additional correspondences. The electric energy is correctly localised by the potential eneigy of the rotation. Since the kinetic energy is also coireotly localised, we need not be surprised to find that the energy-fiuz has the same distribution.

The activity of the stress Ph Is Ph^ per unit area, which is the same as $NP,, if P, be the stress on the plane whose normal is q, provided the stress is irrotationaL But if ro- tational, as at present, it is qTS[%y where Q is conjugate to P. This activity means energy transferred from the side of the plane when the stress vector is reckoned, to the other side, or against the motion. The vector expressing the energy flux is therefore -jQq, not counting the oonvective flux. In our present case, by (351), Q is the negative of P, because it is purely a rotational stress, so W = qP^^ is the vector flux of energy; or, by (351),

W « - v^Vai curl G » - vYn curl G, . (356)

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XLBMENTB OF VECTORIAL ALOKBRA AlTD ANALTSI8. 247

which is, bj the above table, the same as Vkh, the electro- magnetic flux of energy. (In the above, q and are the tensor and unit vector of q.)

The above reasoning is applied directly to the unit element of volume. But it may be easier to follow by taking any volume into consideration. If N is the normal outwards from the surface enclosing it, then "P^ is the pull, per unit area, of the matter outside, on the matter inside the surface, across the unit area of the interface ; and P^q is its activity. The total activity of all over the surface is therefore 2 Pj^q, which is the same as - SN^-P,,, and expresses the work done per second by the matter outside on the matter inside the surface, or the rate of transfer of energy from the outside to the inside ; and its equivalent is the rate of increase of the stored energy, potential and kinetic, within the surface. The convergence of the vector qP,^ expresses the same property for the unit volume, so that {P^ is the flux of energy (per unit area) itself. (The small oonveotive flux of energ3f is ignored here.)

Circuital Indetenninateness of the Flux of Energy in general.

§ 156. That any circuital flux of energy may be superadded, without making any difference in the transformations of energy, is a fact which is of importance as evidence against the objectivity of energy, but is of no moment whatever in the practical use of the idea of a flux of energy for purposes of reasoning. We should only introduce an auxiliary curonital flux when some useful purpose is served thereby.

I may here remark that (speaking from memory) when Pro- fessor J. J. Thomson first objected to the VEH formula as representative of enexgy-flux, by reason of the circuital inde- tenninateness, he added that a knowledge of the real flux could only be determined from an actual knowledge of the real dynamical connections involved, that is to say, by a know- ledge of the mechanism. But this is surely an inconclusive argument, because the circuital indeterminateness applies even then. In fact, it is universal in its application, as energy is at present understood. For there to be an absolutely definite flux of energy through space iecuis to require energy to have objec- tivity in the same sense as matter, which is a very difficult notion to grasp, and still more diflicult to accept. But even if it be ac-

248

SLIGTBOlUOKXnC THIOBT.

OH. ni.

oepted, the aigument of circuital indeterminatenen remains in action, when it is demred to find what it the flux in a given case.

In a recent number of the FhU. Mag,, Mr. Macanlay, by the addition of a certain circuital flux to VSH, brings out the result that in stationary states the modified flux reduces to ^C, where p is potential and 0 current-density. I am, how- ever (unless I forget much of what I have leamt in the last 15 years), unable to see that the auxiliary flux proposed servei any useful purpose. In the first place, it greatly complicates the flux of energy in general, and is entirely against the more simple ideas to which we are naturally led in the study of electromagnetic waves, whether in dielectrics or conductors. Besides that, it is implied that the function or electric potential, is a determinate quantity specifying some definite state of the medium. Tiiis seems to me to be an idea which has no place at all in Maxwell's theory. It may have place in other theories, but that is not to the point. To exemplify, put a closed circuit (with battery) supporting a steady current inside a metal box, and electrify the latter from outside. According to the formula, the flux of energy in the wire suffers a very remarkable change, by reason of the raising of the potential of the box and its interior. But, according to the interpretation of Maxwell's scheme which I expound, there is no change whatever in the electrical state of the interior of the box in the steady state (though there may and must be a transient disturbance in the act of charging it, which is a separate question), and the energy distribution and its flux are the same as before, because E and H, which settle the state of • affairs, electric and magnetic, are unchanged. There are, however, I believe, some electricians who will be much gratified with the pO flux in steady states.

The total flux of energy through a wire will be pC, where by C we mean the total current, obtained from pO by integratioii over the cross-section. Now this is somewhat suggestive of my expression YC for the total flux of energy along a circuit of two conductors. But there are radical differences. For VG is the integral form of VEH in the dielectric. Although C in YC means the same as in being the oirouitation of H, or the gaussage, V has no connection with p the potential, for it (Y) is the line-integral of £ across the circuit from one wire

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ELEMENTS OF VECx^ORIAL ALGEBRA AND ANALYSIS. 249

to the other — t.e., the transverse voltage. Again, the VC formula holds good in variable as well as in steady states, whilst the proposed pC holds for steady states only. Even in steady states, when Y specializes itself and becomes difference of potential, it remains different from p.