Electromagnetic Theory, Vol. 1 (1893) — part 7 of 31

arranged to show the flux of energy, and can always be obtained when the equations of motion are known and also the nature of the stored and wasted energies.

Electromagnetic Application. Medium at Eest. The Poynting Flux.

§70. Now, in the electromagnetic case, the "equations of motion " are the two circuital laws, and to form the equation of activity, we may multiply (1) by (H - h0 - h) and (2) by (E - e0 - e) and add the results ; or else multiply (3) by (H - h0) and (4) by (E - e0) and add the results. The equation of activity thus obtained has then to be dynamically interpreted in accord- ance with the principle of continuity of energy.

When the medium is stationary, there is no difficulty with the interpretation. We obtain

e0J + h0G = Q + U + T + divW, . . (12) where W is a new vector given by

W = V(E-e0)(H-h0), . . . (13)

and U is the electric energy, T the magnetic energy, and Q the waste per unit volume.

On the left side of (12) is exhibited the rate of supply of energy from intrinsic sources. On the right side it is accounted for partly by the waste Q, and by increase of the stored energy U and T. The rest is exhibited as the divergence of the flux W given by (13). We conclude, therefore, that W expresses the flux of energy in the electromagnetic field when it is stationary.

This remarkable formula was first discovered and interpreted by Prof. Poynting [Phil. Trans., 1884, Pt. 2], and independently by myself a little later. It was this discovery that brought the principle of continuity of energy into prominence. But it should be remembered that there is nothing peculiarly electro- magnetic about a flux of energy. It is here made distinct, because the energy is distinctly localised in Maxwell's theory.

The flux of energy takes place in the direction perpendicular to the plane containing the electric and magnetic forces (of the field), say

W = VE1H1, (14)

if E = e0 + E1; H = h0 + Hr

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The formula is not a working formula, in general, for to know W we must first know the distribution of the electric and magnetic forces. But it is a valuable and instructive formula for all that. Its discovery furnished the first proof that Max- well's theory implied that the insulating medium outside a conducting wire supporting an electric current was the medium through which energy is transferred from its source at a dis- tance, provided that we admit the postulated storage of energy. For inside a wire the electric force is axial, and the magnetic force circular about the axis ; the flux of energy is therefore radial. When the current is steady, it comes from the boundary of the wire, and ceases at its axis. It delivers up energy on the way, which is wasted in the Joule-heating. But when the current is not steady, the magnetic and electric energy will be also varying.

If we work out the distributions of electric and magnetic force outside the wire, according to the conditions to which it is subjected and its environment, we can similarly fully trace how the energy is supplied to the wire from its source. It is passed out from the source into the dielectric medium, and then converges upon the wire where it is wasted. The flux of energy usually takes place nearly parallel to the wire (because the electric force is nearly perpendicular to its boundary) ; its slight slope towards the wire indicates its convergence thereupon. If the wire had no resistance, there would be* no convergence of energy upon it, the flux of energy would be quite parallel to it. Details are best studied in the concfete application, and it is only by the con- sideration of variable states, and the propagation of electro- magnetic waves, that we can obtain a full understanding of the meaning of W considered as a flux of energy.

In the case of a simple progressive plane wave disturbance, in which a distribution of E and H (mutually perpendicular) in the plane of the wave is propagated unchanged through a medium at constant speed, it is a self-evident result that the energy of the disturbance travels with it. The flux of energy is, therefore (since e0 = 0, h0 = 0),

W = v(U + T) = VEH, ..... (15) where v is the velocity of the wave, and U, T are the densities

80 ELECTROMAGNETIC THEORY. CIL II,

(which are here equal) of the electric and magnetic energies. In this example, which is approximately that of radiant energy from the sun, the idea of a flux of energy, and the conclusion that its proper measure is the density of the energy multiplied by the wave velocity, are perfectly plain and reasonable. Now, many cases of the propagation of waves along wires can be reduced to this simple case, with a correction for the resistance of the wire, and other cases can be represented by two or more oppositely travelling plane disturbances. In a very complex electromagnetic field, the flux of energy is necessarily also very complex, and hard to follow ; but the fundamental principles concerned are the same throughout.

The flux of energy W arises from the internal structure of the ether. It is somewhat analogous to the activity of a stress. But the only dynamical analogy that is satisfactory in this respect is that furnished by Sir W. Thomson's rotational ether, when interpreted in a certain manner, so that 2E shall repre- sent a torque, and H the velocity of the medium, with the result that (on this understanding) VEH is the flux of energy, whilst U is the potential energy of the rotation, and T the kinetic translational energy. But it is very difficult to extend this analogy to include electromagnetic phenomena more com- prehensively. [See Appendix at the end of this chapter.]

Extension to a Moving Medium. Full interpretation of the Equation of Activity and derivation of the Flux of Energy.

§ 71. Passing now to the case of a moving medium, we shall obtain, from equations (1), (2), § 66, the equation of activity,

(e0 + e) J + (h0 + h)G = E J + HG

• div V(B - 0o - e)(H - ho - h). (16)

Or from equations (3), (4), §66, we may get the form

e0 J0 + h0G0 = E J0 + HG0 + div V(E - e0)(H - h0), (1 7)

and from these a variety of other forms may be derived. The dynamical interpretation in accordance with the prin- ciple of continuity of energy is not so easy as in the former case. I have recently given a full discussion of these

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equations in another place. The following gives an outline of the results : —

Since there is a flux of energy when the medium is stationary, there will still be a flux of this kind (i.e., independent of the motion) when it is moving, whatever other flux of energy there may then be. I conclude that the Poynting flux W still preserves the form

W = V(E-e0)(H-h0)5 .... (18)

from the fact that disturbances are propagated through a medium endowed with a uniform translational motion in the same manner as when it is at rest. Otherwise we could only know that it must reduce to this form when at rest. Next, there is the convective flux of energy

...... (19)

where q. is the velocity of the medium.

Thirdly, there is a flux of energy representing the activity, A, of the electromagnetic stress. It is given by

. . . (20)

Fourthly, there is, in association with this stress, a convective flux of other energy, say,

1(U0 + T0) ....... (21)

The complete flux of energy is the sum X, of these four vectors, i.e. : —

X = W + q(U + T) + A + q(U0 + T0), . . (22)

W and A being given by (18) and (20).

Finally, the activity of the intrinsic forces is

OoJo + VJo, ..... (23)

so that J0 is the true electric current.

We, therefore, have the equation of activity brought to the standard form

e0J0 + h00-0 = (Q + U + T) + (Q0 + U0 + T0) + div. X, (24)

which is a special form of equation (11), with the convergence of the energy flux in it replaced by its divergence (the negative of convergence) on the other side of the equation.

82 ELECTROMAGNETIC THEORY. CH. II.

But the unknown terms, on the right side of (24), with the zero suffix, may be entirely eliminated, including those in X, by making use of the secondary equation of translational activity

T0), . . (25)

where F is the translational force due to the stress. * When this is done, equation (24) takes the form

e0J0 + h0G0 = Q + U + T + Fq + div.[W + A + q (U + T)] (26)

where the energy flux represented includes the Poynting flux, the stress flux, and the convective flux of electric and magnetic energy.

We may observe that in the expression (20) for A occurs the term - q (U + T), so that this term may be eliminated from (26), making the energy flux in it become

V(B - e0)(H - ho) - VeH -, VEh. . . (27)

But these changes, with a view to the simplification of expres- sion, cause us to altogether lose sight of the dynamical signi- ficance of the equation of activity, and of the stress function. Equation (26) is, therefore, the best form.

It should be understood that it is an identity, subject to the two laws of circuitation and the distribution of energy accord- ing to that of £ and H in the field. But it should be also mentioned that in the establishment of (26) it has been assumed that the medium, as it moves, carries its intrinsic properties of permittivity and inductivity with it unchanged. That is, these properties do not alter for the same portion of the medium, irrespective of its position, although within the same unit volume these properties may be changing by the exit of one and entry of another part of the medium of different permit- tivity and inductivity ; understanding by " medium" whatever is supporting the fluxes, whether matter and ether together, or ether alone ; and it is also to be understood that the three velocities q, u, and w are identical, or that electrification, which is always found associated with matter, moves with the medium, which is then the matter and ether, moving together. In other respects equation (26) is unrestricted as regards either homogeneity and isotropy in respect to permittivity, inductivity, and conductivity (electric, and fictitious magnetic).

[* I regret to have misrepresented Dr. Burton's notion of a moving strain figure. He does it entirely by conservative elastic forces, and my objection does not apply.]

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 83

Whether, when matter moves, it carries the immediately surrounding ether with it, or moves through the ether, or only partially carries it forward, and what is the nature of the motion produced in the ether by moving matter, are questions which cannot be answered at present. Optical evidence is difficult of interpretation, and conclusions therefrom are con- flicting.

But in ordinary large scale electromagnetic phenomena, it can "make very little difference whether the ether moves or stands stock-still in space. For the speed with which it pro- pagates disturbances through itself is so enormous that if the ether round a magnet were stirred up, artificially, like water in a basin, with any not excessive velocity, the distortion in the magnetic field produced by the stirring would be next to nothing.

Strictly speaking, when matter is strained its elastic and other constants must be somewhat altered by the distortion of the matter. The assumption, therefore, that the permittivity and inductivity of the same part of the medium remain the same as it moves is not strictly correct. The dependence of the permittivity and inductivity on the strain can be allowed for in the reckoning of the stress function. This matter has been lately considered by Prof. Hertz. But the constants may also vary in other ways. It is unnecessary to consider here these small corrections. As usual in such cases, the magnitude of the expressions for the corrections is out of all proportion to their importance, in relation to the primary formula to which they are added.

Derivation of the Electromagnetic Stress from the Flux of Energy. Division into an Electric and a Magnetic Stress.

§ 72. From the form of the expression for A in equation (20), viz., the flux- of energy due to the stress, we may derive the expression for the electromagnetic stress itself. If the stress were of the irrotational type considered in works on Elasticity, we could do this by means of the formula (10), § 68. ' But for a stress of the most general type the corresponding formula is

(28)

UNIVERSITY

84 ELECTROMAGNETIC THEORY. CH. II.

where Q, is the stress vector conjugate to Pq ; these are identi- cally the same when the stress is irrotational. This gives

QN = D.EN + B.HN-N(U + T), . . (29)

from which we» obtain PN by merely exchanging E and D, and H and B ; thus

PN = E.DN + H.BN-N(U + T). . . . (30)

This is the stress vector for any plane denned by N, a unit vector normal to the plane.

But it is only in eolotropic bodies that we have to distinguish between the directions of a force and of the corresponding flux. Putting^ these on one side, and considering only ordi- nary isotropy, the interpretation is simple enough. It will be observed that the electromagnetic stress (30) divides into an electric stress

E.DN-N.JED, (31)

and a magnetic stress

H.BN-N.JHB (32)

To find their meaning, take N in turns parallel to and perpen- dicular to the force E (or to D, since its direction is the same). In the first case, the stress (31) becomes

N(ED - JED) = N.JED = NU,

indicating a tension parallel to the electric force of amount U per unit area.

In the second case, when N is perpendicular to E, we have DN = 0, so that the stress is

-NU,

that is, a pressure of amount U. This applies to any direction perpendicular to E, so that the electric stress consists of a tension U parallel to the electric force, combined with an equal lateral pressure.

Similarly the magnetic stress consists of a tension T parallel to the magnetic force, combined with an equal lateral pressure.

It will also be found that the tensor of the electric stress vector is always U, and that of the magnetic stress vector is always T. The following construction (Fig. 5) is also useful : — Let ABC be the plane on which the electric stress is required, BN the unit normal, BE the electric force, BP the stress.

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

85

Then N, E, and P are in the same plane, and the angle be- tween N and E equals that between E and P. Or, the same operation which turns N to E also turns E to P, except as regards the tensor of P.

To show the transition from a tension to aniequal pressure, imagine the plane ABC to be turned round, and with it the normal N. Of course E remains fixed, being the electric force at the point B. Start with coincidence of N and E. Then P also coincides with them, and represents a normal pull on the surface ABC. As N and E separate, so do E and P equally, so that when E makes an angle of 45° with N the normal pull is turned into a tangential pull, or a shearing stress, P being

N

FIG. 5.

now at right angles to N. Further increase in the angle E makes with N brings BP to the other side of BC j and when E is at right angles to N, we have P and N in the same line, but oppositely directed. That is, the tension has become converted into an equal pressure.

Uncertainty regarding the General Application of the Electromagnetic Stress.

§ 73. We may now consider the practical meaning of the stress whose relation to the electric and magnetic forces has, unaer certain suppositions, been formularised. Go back to the foundation of electromagnetic theory, viz., the mechanical forces experienced by electrically charged bodies, by conductors supporting currents, and by magnets, intrinsic or induced. It

86 ELECTROMAGNETIC THEORY. CH. II.

is by observation of these forces in the first place, followed by the induction of the laws they obey, and then by deductive work, that the carving out Of space into tubes of force follows; and now, further, we see that the localisation of the stored energies, according to the square of the electric and magnetic force respectively, combined with the two circuital laws, leads definitely to a stress existing in the electromagnetic field, which is the natural concomitant of the stored energy, and which is the immediate cause of the mechanical forces observed in certain cases. But the theory of the stress goes so far be- yond experimental knowledge in some respects, although agree- ing with it in others, that we could only expect it to be true if the theoretical foundations were also rigidly true in all respects. Such is not the case, however. To begin with, the way of ex- pressing the action of ordinary matter merely by altering the values of the two ether constants, and by a fresh property, that of conductivity, is extremely bald. It is, indeed, surpris- ing what a variety of phenomena is explained by so crude a method.

The objection is sometimes made against some modern theo- retical developments that they are complicated. Considered as an argument, the objection is valueless, and only worthy of superficial minds. Whatever do they expect ? Do they not know that experimental knowledge, even as at present existent, shows that the theory of electromagnetism, when matter is present, must, to be comprehensive, be something far more (in- stead of less) complicated than theory as now developed ? The latter is, as it were, merely a rough sketch of a most elabo- rate subject, only small parts of which can be seen at one time.

In the next place, even if we take the stated influence of matter on the ether as sufficient for the purposes of a rough sketch, the theory of the stress should, except in certain rela- tively simple cases, be received with much caution. Why this should be so will be apparent on examining the manner in which the stress has been obtained from the circuital laws. If we inves- tigate the subject statically, and, starting from certain mechanical forces regarded as known, endeavour to arrive at a stress which shall explain those forces, we shall find that the problem is essen- tially an indeterminate one. All sorts of stress functions may

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be made up which are precisely equivalent in their effects in the gross, that is, as regards translating or rotating solid bodies placed in an electric or magnetic field. To remove this indeter- minateness a dynamical method must be adopted, wherein what goes on in the unit volume whilst its electric and magnetic states are changing, and the matter concerned is itself in motion, are considered. If our system of connections is dynamically complete and consistent, and is such that the flux of energy can be traced, then a determinate stress comes out, as we have found. The method is, at any rate, a correct one, however the results may require to be modified by alterations in the data. Besides that, the distribution of energy (electric and magnetic) in bodies is in some cases open to question ; and a really speculative datum is that concerning the motion of the ether as controlled by the motion of matter. Now, this datum appears to be one which is essential to the dynamical method ; the only alternative is the statical and quite indeterminate method. Our attitude towards the general application of the special form of the stress theory obtained should, therefore, be one of scientific scepticism. This should, however, be carefully distinguished from an obstinate prejudice founded upon ignorance, such as is displayed by some anti-Maxwellians, even towards parts of Maxwell's theory which have received experimental demon- stration.

The stress theory can, nevertheless, sometimes be received with considerable confidence, if not absolute certainty. The simplest case is that of ordinary electrostatics.

The Electrostatic Stress in Air.

§ 74. Let there be no magnetic force at all, and the electric force be quite steady, and the medium be at rest, and there be no impressed forces. These limitations bring the circuital laws down to

0 = *E, (1)

-curlE = 0; ....'. (2)

that is, there must be no conduction current anywhere, and the voltage in any circuit must be zero. The first condition (1) implies that there is no electric force in conductors. We may, therefore, divide space into conducting and non-conducting

88 ELECTROMAGNETIC THEORY. C1I. II.

regions, and our electric field is entirely confined to the latter. The second condition (2) implies tangential continuity of E at the boundary of the non-conductor, so that as there is no E in the other or conducting side, there must be no tangential E on the non-conducting side. The lines of force, therefore, terminate perpendicularly on the conducting matter. Whether the conductors are also dielectrics or not is quite immaterial. The displacement also terminates normally on the conducting surface in the usual case of isotropy. Thus D, the tensor of D, measures the surface density of electrification, when the positive direction of D is from the conductor to the insulator. But in general it is the normal component of D, that is DN, where N is the unit normal vector, that measures the density of the electrification. Besides this, there may be interior electrification of the non-conductor, its volume density being measured by the divergence of the displacement. The arrangement of the electric force, so that the circuital voltage shall be zero throughout the non-conductor, and give the proper internal electrification, and the charges on the conductors, is uniquely determinate.

We have thus the ordinary case of a number of charged con- ductors in air, with the difference that the air, or parts thereof, may be replaced by matter of different permittivity. It is also to be noted that one non-conducting region which is entirely separated from another by conducting matter may be taken by itself, and all the rest ignored.

Now, first without replacing the air by matter of different permittivity, we see that there are two entirely different ways of considering the mutual actions of the conductors. The old way is analogous to Newton's way of expressing the fact of gravitation. We may say that any element of electrification p repels any other p' with a force

pp'l 47rcr2,

if r be the distance between the two charges, and that the resultant of all such forces makes up the real forcive.

In the other way, appropriate to the philosophy of Faraday, as developed by Maxwell, these forces acting at a distance are mathematical abstractions only, and have no real existence. What is real is a stress in the electric field, of a peculiar nature,

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being a tension of amount U parallel to E, combined with an equal lateral pressure, and it is the action of this stress that causes the electrified conductors to move, or strains them, according to the way they are supported, when by constraints they cannot appreciably move.

Since the electric force if normal to the conducting surfaces, the stress vector is entirely a normal pull of amount U per unit area, and the motions or tendencies to move of the con- ductors are perfectly accounted for by this pull. They do not move because of electrical forces acting at a distance across the air, but because they are subjected to moving force on the spot by the stress terminating upon them.

Thus a charged soap-bubble is subjected to an external radial tension, and therefore expands; and so, no doubt, does a charged metal sphere to some small extent. The parallel plates of a •condenser are pulled together. When they are very large compared with their distance apart, the force on either is

J ED x area, = J E x charge.

Here E is the transverse voltage divided by the distance be- tween the plates, so that, if the plates be connected to a constant .-source of voltage, the attraction varies inversely as the square -of the distance between them; whereas, if the plates be in- .sulated and their charges constant, the attraction is the same .at any distance sufficiently small compared with the size of the plates.

But by sufficiently separating the plates, or by using smaller ^plates, the displacement, which was formerly almost entirely 'between them, will spread out, and will terminate in appreci- able amount upon the sides remote from one another. By the pull on the remote sides thus produced the attraction will be lessened, and the further the plates are separated the more •displacement goes to their backs, and the less is the attraction. When the distance is great enough it tends to be simply the -attraction between two point charges. Thus the attraction between two distant oppositely charged conducting spheres, which varies closely as the inverse square of the distance, depends entirely upon the slight departure from uniformity •of distribution of the electrification over their surfaces, whereby

90 ELECTROMAGNETIC THEORY. CH. II.

the normal pull on either is made a little greater on the side next the other than on the remote side. Also, the inverse square law itself, which is exactly true for point charges, is merely the ultimate limit of this operation.

Some attacks have been made on the law of inverse squares, especially in its magnetic aspect. But these attacks appear to have been founded upon misapprehension. The law is true, and always will be.

The moving Force on Electrification, bodily and superficial. Harmonisation.

§ 75. In the above electrostatic application of the stress, it will be observed that the tension alone comes into play, at least explicitly, owing to the tubes of displacement terminating perpendicularly on the conductors. Thus each tube may be compared with a rope in a state of tension, pulling whatever its ends may be attached to. But the lateral pressure is needed to keep the medium itself in equilibrium, so that the only places where translational force arises from the stress is where there is electrification. The mechanical force is

F = EdivD = E/> ..... (3)

per unit volume. This is the force on volume electrification,. and is the result of the differential action of the stress round about the electrification, as in the case of the inverse square law between point charges, lately mentioned. The correspond- ing surface force is

F = iE.DN = NU ..... (4)

per unit area. Now, here DN is the surface equivalent of div D, so there is at first sight a discrepancy between the ex- pressions for the force per unit volume (3), and per unit area (4), on bodily and surface electrification. How the coefficient J comes in may be seen by taking the limiting form of the previous expression. Let there be a thin skin of electrification, of amount o- per unit area ; E falling off from E outside to 0 inside the skin. Evidently the mean E is JE, so that the total force on unit area of the skin obtained by summation of the forces on the volume electrification in the skin, is not. Eo-, but JEo-. This is merely a mathematical harmonisation^

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From the point of view of the stress the difficulty does not pre- sent itself.

The harmonisation is simply evident when the layer is of uniform density, for the electric force will then fall off in in- tensity uniformly. It might, however, be suspected that, per- haps, the result would not come out quite the same if we assumed any other law of distribution, and kept to it in pro- ceeding to the limit by making the skin infinitely thin. But a cursory examination will show that it is all right ; for if E is the electric force within the layer, the electrification density will be c (dE/dx) and the translational force will be cE (dE/dx) per unit volume, if x is measured perpendicularly to the skin's surfaces. Integrate through the skin, and the result is

where the suffixes refer to the value just outside the skin, on its two sides. In the present case the second term U2 is zero (within the conductor), so that the result is the single normal pull of the tension on the non-conducting side.

Depth of Electrified Layer on a Conductor.

§ 76. In this connection the old question of the depth of the layer of electrification on a conducting surface crops up. Has it any depth at all, and, if so, how much ? The question is not so superficial as it looks, arid the answer thereto lies in the application thereof. If a powerful mental microscope be applied to magnify the molecules and produce evident heterogeneity, the surface of a conductor would become indefinite ; and unless the molecules were found to be very closely packed, it is evident that the displacement in the ether outside the conductor would not terminate entirely upon those molecules which happen to be most superficially situated, but that a portion of the dis- placement would go deeper and in sensible amount reach molecules beneath the first set, and an insensible amount might penetrate through many layers. Thus in a molecular theory the depth of the layer of electrification has meaning, and could be roughly estimated.

But the case is entirely different in a theory which delibe- rately ignores molecules, and assumes continuity of structure.

92 ELECTROMAGNETIC THEORY. CH. II.

A conductor is then a conductor all through, and not a heterogeneous mixture ; and the surface of a conductor is an unbroken surface. The electrification on it is therefore surface electrification, and has no depth. For it to be otherwise is simply to make nonsense. It is desirable to be consistent in working out a theory, for the sake of distinctness of ideas ; if, then, we wish to give depth to surface electrification, and still keep in harmony with Maxwell's theory, we must change our way of regarding a conductor, and bring in heterogeneity. Each view is true, in its own way ; but as in the mathematical theory continuity of structure is tacitly assumed, we have a simultaneous evanescence of one dimension in the distribution of electrification.

The same question occurs in another form in the estimation of -bodily electrification, when the meaning of volume density of electrification is considered. When air is electrified, it is probable that the electrification is carried upon the foreign particles suspended in the air, and it may be partly upon the air molecules themselves. In either case it is ultimately surface electrification, and quite discontinuous. But, merely for the sake of facility of working, it is desirable to ignore all the dis- continuity, and assume a continuous and practically equivalent distribution of bodily electrification. Thus, as previously we saw surface density to be a kind of volume density, so now we see that volume density is a kind of surface density. When, therefore, we say that the translational force per unit volume is E/>, where E is the electric force and p the volume density of electrification, we really mean that E/> is the average, obtained by summation, of the translational forces on the multitudinous electrified particles, every one of these forces being itself a differential effect, as before seen, viz., the resultant of the unequal pulls on different parts of a particle exerted by the electric stress.

As ether has some of the properties of matter, and as electri- fication is found in association with matter, it is possible, however improbable, that ether itself may become electrified. But of this nothing is known. Nor, more importantly, is it understood why the electric stress appears to act differently on positively and on negatively electrified matter. But, if we begin to talk about what is not understood, we enter illimit-

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able regions. Men who are engaged in expounding practical problems sometimes make the boast that they take and discuss things as they are, not as they might be. There is a sound of specious plausibility here, which is grateful and comforting ; but, as a matter of plain fact, questions of physics never are theorised upon as they are, but always as they might be. It is a necessity to limit the field of inquiry, for to take things as they are, or as they seem to be, would lead at once to a com- plete tangle. For the problem, as it presents itself in reality, there is always substituted a far simpler one, containing certain features of the real one emphasised, as it were, and othera altogether omitted. The juveniles, who take things as they are, do not do it ; they only think so. They may strain out a few gnats successfully, but swallow, quite unawares, all the camels in Arabia. But the principle and practice of limitation and substitution is the same all over; in politics, for instance, where a fictitious British Constitution does brave duty, as a scarecrow, and in other useful ways.

Electric Field disturbed by Foreign Body. Effect of a. Spherical Non-conductor.

§ 77. To further exemplify the significance of the electric stress, let us introduce a foreign body into a stationary electric field. The field will be disturbed by its introduction, and will settle down to a new state ; the change depending upon the nature of the foreign body, whether conducting or non-conduct- ing, in substance or superficially, and upon whether it has a charge itself, or contains any other source of displacement. If it be a good conductor, either charged or uncharged, the final, state, reached very quickly, will be such that the displacement will terminate normally upon its surface, thus reproducing the previous case (§§ 74 to 76). But if it be a non-conductor, the result is somewhat different. If superficially conducting, we may indeed have an ultimate electrification of the surface, so as to come wholly or partly under the same case : but if there be no superficial or internal conduction, or only so little that a long time must elapse for it to become fully effective, what we do is simply to replace the dielectric air in a certain region by another dielectric of different permittivity, usually greater..

94 ELECTROMAGNETIC THEORY. CH. II.