Electromagnetic Theory, Vol. 1 (1893) — part 8 of 31

Then, supposing the external field to be due to charges upon insulated conductors or to internal electrification in the air (kept at rest for the purposes of the argument), no change will occur in their amounts ; but there will be merely an alteration in the distribution of the electrification on the conductors, caused by the displacement becoming denser within the foreign body than before (or less dense, if its permittivity can be less than that of the air). The intrinsic electrification of the body itself, if any, must also be allowed for; but should it have none previously, it will remain unelectrified when introduced into the field, and the displacement will pass freely through it, and out again in a solenoidal manner. This is expressed by the surface condition

where D: and D2 are the displacements in the air and foreign body respectively at their interface, and N1? N2 are the unit normals from the interface to the two media.

The only quite simple case (excepting that of infinite plane sheets) is that of a sphere of uniform permittivity brought into a previously uniform field. The ultimate displacement in the sphere is then parallel to the original displacement in the air. It may vary between zero and three times the original displace- ment, as the permittivity of the sphere varies from zero up to infinity. It is certainly a little surprising that the ultimate displacement with infinite permittivity should be only three times the original (and it is not much less when the permittivity is only 10 times that of the air) ; whilst, on the other hand, the zero displacement when the permittivity is zero (a quite ideal case) is obvious enough, because the displacement never enters the sphere at all, but goes round it. The original uniform field may be conveniently that between the parallel plates of a very large air condenser. There is, however, a double action taking place. When the transverse voltage of the condenser is maintained constant by connection with a suitable source, the insertion of the foreign body, which increases or reduces the permittance of the condenser, will increase or reduce its charge under the action of the constant source, besides concentrating the displacement within the body, or the reverse. With constant charges, however, when the plates are insulated, the insertion

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 95

of the foreign body will reduce the transverse voltage when its permittivity exceeds that of the air ; and conversely. Increase of permittivity also increases the stored energy when the trans- verse voltage is constant, but reduces it when the charges are constant.

Now, since the electric force does not terminate on the boun- dary of the foreign body, but extends all through it, so does the electric stress. So far, however, as the resultant force and torque on the body, when solid, is concerned, we may ignore the internal stress altogether, and consider only the external, or stress in the air. This is a particular case of a somewhat impor- tant and wide prope^ in abstract dynamics, which we may state separately thus.

Dynamical Principle. Any Stress Self-equilibrating.

§ 78. The resultant force and torque due to any stress in any region is zero. Or, any stress in any region forms a self-equili- brating system.

Imagine any distribution of stress to exist in a region A, and to terminate abruptly on its boundary. Or, equivalently, imagine a piece of a stressed solid to be removed from its place without altering the stress. The stress-variation, when esti- mated in a certain way, constitutes mechanical force tending to move the body. This will be, in the case of an ordinary irrota- tional stress, entirely translational force. But there are two kinds. First, there is internal force, reckoned per unit volume, due to :he continuous variation of the stress in the body. Next, ;here is superficial force, reckoned per unit area, due to the abrupt cessation of the stress. This surface traction is repre- sented simply by the stress vector itself, acting on the inner side of the surface of the body. Now, the resultant effect of these two forcives, over the surface and throughout the volume of the solid respectively, in tending to translate and rotate it, is zero. Or, in other words, the force and torque equivalent to the surface forcive are the negatives of those due to the internal forcive. If it were otherwise, the differential action would cause indefinite increase in the translational and rotational energy to arise out of the internal mutual forces only of a body.

96 ELECTROMAGNETIC THEORY. CH. II.

If the stress be of the rotational type, there will be an in- ternal torque (per unit volume) as well as a translational force. Still, however, the resultant force and torque due to the surface tractions will cancel those due to the internal forces and torques.

In case there be any difficulty in conceiving the traction exerted on the surface of a body by the stress within itself, we may replace the sudden cessation of the stress by a gradual cessation through a thin skin. The solid is then under the influence of continuous bodily force only (and torque also, if the stress be rotational) conveniently divisible into the internal force all over, and the force in the skin. Otherwise it is the same.

Now put the solid piece back into its place again. Since its own stress balances itself, we see that whatever the forcive on the piece may be it must be statically equivalent to the action upon its boundary of the external stress only, constituting an external surface traction. There need be no connection between the external and the internal stress. The latter may be any- thing we like, so far as the resultant force and torque on the piece are concerned. The difference will arise in the strains produced, or in the relative internal motions, when for one stress another is substituted*-

Electric Application of the Principle. Resultant Action on Solid Body independent of the Internal Stress, which is statically indeterminate. Real Surface Traction is the Stress Difference.

§ 79. Returning to the electric field, we see that whatever be the nature of the reaction of the foreign body on the original state of the field, the resultant mechanical action on the body as a whole is fully represented by the stress in the air just outside it, in its actual state, as modified by the presence of the body, and that we need not concern ourselves with the internal state of stress. Nor are we limited, in this respect, by any assumed proportionality of electric force to displacement in the body, or assumed absence of absorption, or other irregu- larities and complications. That is, we need not have any theory to explicitly account for the change made in the electric field by the body. Nor do we gain any information regarding

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 97

the internal stress from merely a knowledge of the external stress, although that involves the reaction of the body on the electric field. This is the meaning of the statical indeter- minateness of the stress before referred to, and the principle applies generally.

The air stress vector P will usually have both a normal and a tangential component at the surface of a body, viz.: —

U cos 26 normal, U sin 28 tangential,

if 0 be the angle between the normal N to the surface (drawn from the body to the air) and the electric force E, and U the density of the electric energy, or the tensor of the stress vector, In only one case, however, will this external surface traction represent the real forcive in detail (as well as in the lump), viz., when there is no stress at all on the other side of the boundary, that is, in the case of a conductor in static equilibrium. In general, the real surface force is represented by the vector P1-P2, the stress difference at the boundary, P: being the external and P2 the internal stress vector, which two stresses may, if we please, be imagined to be united continuously by a gradually changing intermediate stress existing in a thin skin, an idea appropriate to molecular theories. (It may be remem- bered that P when positive means a pull.) Each of these may be split into a normal and a tangential component. Now, the tangential components are

Uj sin 20j and U2 sin 202,

where the suffix x relates to the air, and 2 to the other medium. Or ExDj sin 0l cos 6l and E2D2 sin 02 cos 6%,

But here we have normal continuity of the flux D, and tangential continuity of the force E; (otherwise the surface would be electrified and covered with a magnetic current sheet); that is

D1cos^1 = D2 cos #2, ) and E1sin^=E2sm02. /

These relations make the tangential tractions equal and opposite, so that there is no resultant tangential traction, and

H

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the actual traction is entirely normal, being the difference of the normal components of Pl and P2, or,

.... (6) which is the same as (being subject to (5),)

^Dx cos Ol (Ex cos 01 - E2 cos 02)

• P! sin ^ (Dx sin 6^-0,3 sin 02) . . . (7) The coefficient J comes in for a similar reason to before, § 75.

This formula (6) or (7) being the real surface traction when E varies as D in the body, and the stress is of the same type as in air, furnishes a second way of calculating the resultant force and torque on the body, when its permittivity is uniform ; and it is noteworthy that the surface traction is, as in the case of an electrified conductor in equilibrium, entirely normal, although it may now be either a pull or a push.

Translational Force due to Variation of Permittivity. Harmonisation with Surface Traction.

§80. Noting that the mechanical force on the elastically electrizable body is situated where the change of permittivity occurs, and is in the direction of this change, it may be inferred that when the permittivity varies continuously there is a bodily translational force due to the stress variation which is in the direction of the most rapid change of permittivity. This is, in fact, what the stress vector indicates when c varies continuously, viz., the force represented by

-JEV=-VJJ, ..... (8)

where yc means the vector rate of fastest increase of c round- about the point considered. Since U = JCE2, the second form in (8) will be understood, meaning the vector- slope of U as dependent upon the variation of c only.

This bodily force, and the previous surface force may be har- monised by letting c vary not abruptly, but continuously from the value Cj on one side to c2 on the other side of the surface of discontinuity, through a thin skin, and summing up the trans- lational forces in the skin by the formula (8). Thus, if x be

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 99

measured normal to the skin, the translational force per unit volume is - JE2 (dc/dx) in the direction of the normal ; or

,™ dc

~ 2-kf -j-

dx dx

where Eft and Dn are the normal components of E and D, and E4 the tangential component of E. Now Dn and Ef are con- stant ultimately, for the reason before given, so we can inte- grate (9) with respect to x immediately, giving

..r id*.-

between the limits ; or, since En and Et are proportional to the cosine and the sine of 6 respectively.

[JcE2cos2<9] ..... (11)

between the limits, which is the same as (6), which was to be verified.

Movement of Insulators in Electric Field. Effect on the Stored Energy.

§ 81. Since a sphere of uniform permittivity placed in a uniform field causes the external lines of electric force to be symmetrically distorted fore and aft, it has no tendency to move, but is merely strained. But if the body be not in an initially uniform field, or be not spherical, complex calculations are usually needed to determine the effect. If, however, it be only a small piece, the tendency is for it to move in the direction in which the energy, or the stress, in the field increases most rapidly, inde- pendent of the direction of the electric force, when its permit- tivity exceeds that of the gaseous medium. The total electric energy will be diminished by permitting the motion when the charges are constant ; but increased should the field be kept up by constant sources of voltage.

These properties are rendered particularly evident by taking the extreme cases of infinite and zero permittivity of a small body placed in a widely varying field, that surrounding a charged sphere, for example, the electric force varying in intensity as

H 2

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the inverse square of the distance from its centre. Here the non-permittive body is repelled, for the lines of force go round it, and the lateral pressure comes fully into play, and is greater on the side next the charged sphere. On the other hand, with the infinitely permittive body, concentrating the displacement, it is the tension that comes fully into play, and this being greater on the side next the charged sphere, the result is an attraction. The permittance of the sphere, also, is increased in the latter case, and decreased in the former — that is, when the natural motion of the body to or from the sphere is allowed ; so, since the total electric energy is JSV2 where S is the permit- tance and V the voltage, or, equivalently, JVQ, if Q is the charge, we have always a diminution of energy when the natural motion is allowed, whether resulting from attraction or repulsion, if the charge is constant ; but an increase of energy if the voltage is constant.

In intermediate cases the tension is dominant when the c of the body exceeds, and the pressure is dominant when it is less than that of the air, there being perfect equilibrium of a piece of any shape in any field if there be equality of permittivity, and therefore no disturbance of the field. Here we see the part played by the lateral pressure in the case of conductors in equilibrium. It has no influence on them immediately, and might be thought wholly unnecessary, but it is equally important with the tension in the non-conducting dielectric itself.

Magnetic Stress. Force due to Abrupt or Gradual Change of Inductivity. Movement of Elastically Magnetised Bodies.

§ 82. Passing now to the corresponding magnetic side of the stress question, we may observe that the analogy is an imperfect one. Thus, proceeding as at the beginning of § 74, to have a stationary magnetic field without impressed forces, we shall find that there must first be no magnetic conduction current ; and next, that the gaussage in any circuit must be zero. The magnetic force we then conclude to be confined entirely to the magnetically non-conducting regions, and to terminate perpen- dicularly upon their boundaries. Thus we come to the conception of a number of detached magnetic conductors im-

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 101

mersed in a magnetically non-conductive medium, these con- ductors having magnetic charges on them measured by the amount of induction leaving or terminating upon them, with possible associated volume magnetification in the non-conduct- ing medium.

The magnetic stress will exert a normal traction

. . , . . (12) per unit area on the conductors, and a force

Hcr = Hdiv.B ..... (13)

per unit volume on tridimensional magnetification, of density cr, measured by the divergence of the induction. These are analo- gous to the forces on surface and volume electrification.

Also, when the inductivity /* varies, we shall have a normal surface traction of amount,

T1cos2^1-T2cos2^, .... (14)

per unit area, analogous to (6), when p changes value abruptly at the interface of two media ; and a force

-iH2W ...... (15)

per unit volume, when /z varies continuously.

The forces (12), (13), however, are absent, because of the ab- sence of magnetic conductivity, and, in connection therewith, the absence of " magnetification." But (12) may be sometimes used, nevertheless, when it is the stress across any surface that is in question, and we create surface magnetification by regard- ing one side only.

We are, therefore, left with the forces (14), (15), depending upon variation of inductivity, abrupt or gradual. These ex- plain the mechanical action upon elastically magnetised media, e.g., the motions of bodies to or from a magnet pole, according as they are paramagnetic or diamagnetic, which, it should be remembered, depends fundamentally upon the varia- tion in the intensity of magnetic force near the pole ; and the axial equilibrium of a paramagnetic bar, and equatorial equi- librium of a diamagnetic bar. Faraday's remarkable sagacity led him to the essence of the explanation of these and other allied phenomena, as was later mathematically demonstrated by

102 ELECTROMAGNETIC THEORY. CH. II.

Sir W. Thomson. Questions relating to diamagnetic polarity are, in comparison, mere trifling.

Force on Electric Current Conductors. The Lateral Pressure becomes prominent, but no Stress Discontinuity in general.

§ 82a. But the magnetic stress has other work to do than to- move elastically magnetised matter under the circumstances stated. Wholly independent of magnetisation, it produces the very important moving force on conductors supporting electric- current, first mathematically investigated by Ampere.

The lateral pressure of the stress here comes prominently into view, when we ignore the stress in the interior of the con- ductors, so that the stress vector in the air at the boundary of a conductor represents the moving force on it per unit area. Thus, two parallel conducting wires supporting similar currents attract one another, because their magnetic forces are additive on the sides remote from one another, rendering the lateral pressure on them greater there than on the sides in proximity. But when the currents are dissimilar the magnetic force is- greater on the sides in proximity, and, therefore, the lateral pressure of the stress is greater there, producing repulsion.

Proceeding further, and considering the stress within ih& conductor also, according to the same law, we find this pecu- liarity. In the case of unmagnetisable conductors (typified practically by copper), there is no superficial discontinuity in the stress, and therefore no surface forcive of the kind stated. This may be easily seen from § 79, translating the results from the electric to the magnetic stress. There is no tangential discontinuity in the stress because the normal induction and tangential magnetic force are continuous ; and there is no dis- continuity in the normal component of the stress because (since there is no difference of inductivity) the normal magnetic force and the tangential induction are also continuous.

In (5), (6), (7) turn E to H, and D to B, and U to T, and note that (5), as transformed, are true when the force and flux are exchanged, so that the transformed expressions (6) or (7) vanish.

In the case of a real conductor, therefore, with finite volume density of electric current, the moving force is distributed

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 103

throughout its substance ; and the variation of the stress indi- cates that the translational force per unit volume is expressed

by F = VOB, . ..... . (16)

where 0 is the current density. This is what Maxwell termed "the electromagnetic force," and it is what is so extensively made use of by engineers in their dynamos, motors, and things of that sort. It is probable, I think, that in grinding away at the ether they also stir it about a good deal, though not fast enough to produce sensible disturbances due to etherial dis- placement.

Force on Intrinsically Magnetised Matter. Difficulty. Max- well's Solution probably wrong. Special Estimation of Energy of a Magnet and the Moving Force it leads to.

§ 83. There is next the force on intrinsically magnetised matter to be considered in connection with the magnetic stress. This is, perhaps, the most difficult part of magnetic science. Although, so far as the resultant effect on a magnet is con- cerned, we need not trouble about its internal state, but, as before, merely regard it as being pushed or pulled by the stress in the surrounding air, such stress being calculable from the distribution of magnetic force immediately outside it, as modified by the magnet itself, yet it is impossible that the real forcive can be represented merely by the surface traction PN.

Now, it is possible to find a distribution of magnetification over the surface, which shall be externally equivalent to the interior magnetisation, or to whatever other source of induction there may be. Then we may substitute for the surface trac- tion PN, another traction, namely, upon the surface magnetifi- cation.

Or, we may find a distribution of fictitious electric current upon the boundary of the magnet, which shall be externally equivalent to the interior sources, and then represent the forcive by means of fictitious electromagnetic force on this current, § 82a.

Or we may combine these methods in various ways. Evi- dently, however, such methods are purely artificial, and that to obtain the real forcive we must go inside the magnet. This can only be done hypothetically, and with precarious validity.

104 ELECTROMAGNETIC THEORY. CH. II.

One way of exhibiting the mechanical action on a magnet, or on magnetised matter generally, is that given by Maxwell in his chapter on the stresses (Vol. II.). This I believe to be quite erroneous for many reasons, the principal being that it does not harmonise with his scheme generally, and that it lumps to- gether intrinsic and induced magnetisation, which have essen- tial differences and are physically distinct. There are many other ways of exhibiting the resultant force and torque as made up of elementary forces, acting upon magnetisation, or on free magnetism, or on the variation of magnetism estimated in different ways. It is unnecessary to enter into detail regarding them. Nobody would read it. It will be sufficient to point out the particular way which harmonises with Maxwell's scheme in general, in the form in which I display it, with a special esti- mation of magnetic energy. Proportionality of force and flux is assumed. The want of this proportionality is quite a separate question. Given a definite relation between force and flux, the accompanying change in the stress vector, in accordance with the continuity of energy, can be estimated. This I have re- cently shown how to do in another place, which shall not be more explicitly referred to.

Intrinsic magnetisation possesses the peculiarity that it is, in a manner, outside the dynamical system formulated in the electromagnetic equations, inasmuch as it needs to be ex- hibited in them through the medium of an impressed force, although this is disguised in the ordinary mode of representa- tion. Calling this intrinsic magnetic force h0 (any distribution), as before, the induction due to it in a medium of any inductivity (varying continuously or abruptly, if required) is found in the same way as the displacement due to intrinsic electric force in a non-conducting medium of similarly distributed permittivity ; or as the conduction current due to the same in a medium of similar conductivity ; or, to make a fourfold analogy, as the magnetic conduction current due to h0 in a (fictitious) medium of similar magnetic conductivity.

I may here point out that a clear recognition of the correct analogies between the electric and magnetic sides of electro- magnetism is essential to permanently useful work. Many have been misled in this respect, especially in comparing Maxwell's displacement with magnetic polarisation. The true

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 105

analogue of D is B. Investigations based upon the false foundation mentioned can lead to nothing but confusion. There are enough sources of error without bringing in gratuit- ous ones.

Now, presuming we have induction set up by h0, how is the energy to be reckoned ? I reckon its amount per unit volume to be JHB generally, whether outside or within the magnet ; or, in the usual isotropic case, J/*.H2 or J/ir^B2. This makes the total work done by h0 to be h0B per unit volume, if h0 is suddenly established ; of which one half is wasted, and the rest remains as stored magnetic energy. That is,

, .... (17)

if the 2 indicates space-summation. If h0 be suddenly des- troyed, the energy 2T is set free and is dissipated, mainly by the heat of currents induced within the magnet itself and sur- rounding conductors. This is the meaning of JHB being the stored energy per unit volume. But it may not be immediately available. To take an extreme case, if we have a complete magnetic circuit, so magnetised intrinsically that there is no external field, the energy, as above reckoned, is the greatest possible, since H = h0. But it is now not at all available, unless h0 be destroyed. On the other hand, Maxwell would appear to have considered the energy of a magnet to be 2 J/*(H - h0)2, which is zero in the just mentioned case. This reckoning does not harmonise with the continuity of energy, although it has significance, considered as energy more or less immediately available without destruction of h0. The connection of the two reckonings is shown by

lio); . (18) j or, when H and B are parallel,

2|h0B = 2iJuh02-2^(H-li0)2. . . (19).

Now, according to the reckoning (17) of the stored energy, and the consequent flux of energy, the stress vector derived therefrom indicates that the moving force per unit volume is

F = Vj0B ....... (20),

where j0 = curl hn. . . , , . . (21).

106 ELECTROMAGNETIC THEORY. CH. II.

The interpretation is that the vector J0 represents the distri- bution of (fictitious) electric current, which would, under the same circumstances as regards inductivity, set up, or be asso- ciated with, the same distribution of induction B as the intrinsic magnet is. It may be remarked that the induction due to an intrinsic magnetic force does not depend upon its distribution primarily, but solely upon that of its curl.

Substituting this current system for the intrinsic magnetise tion, equation (20) indicates that the moving force is " the electromagnetic force " corresponding thereto, according to (16), § 82. The accompanying surface distribution of current, repre- senting the abrupt cessation of h0, must not be forgotten; or we may let it cease gradually, and have a current layer in the skin of the magnet. It may be far more important than the internal current, which may, indeed, be non-existent. For instance, in the case of a uniformly longitudinally mag- netised bar, the equivalent current forms a cylindrical sheet round the magnet. In general, the surface representative of

Jois

VNh0, (22)

where N is a unit normal drawn from the boundary into the magnet. It should also not be forgotten that if the inductivity changes, there is also the moving force (15) or (14) to be reckoned, besides that dependent upon the intrinsic magnetisa- tion.

Force on Intrinsically Electrized Matter.

§84. The electric analogue of intrinsic magnetisation is in- trinsic electrisation, represented in a solid dielectric in which "absorption " has occurred, and perhaps in pyroelectric crystals. It seems very probable that there is a true electrisation, quite apart from complications due to conduction, surface actions, and electrolysis. When formulated in a similar manner to in- trinsic magnetisation, by means of intrinsic electric force e0> producing displacement according to the permittivity of the medium (which displacement, however, is now also affected by the presence of conducting matter), and with a similar reckon- ing of the stored energy, viz., JcE2 per unit volume, where R

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 107

includes e0, we may expect to find, and do find, a moving force analogous to (20), viz.,

(23)

where g0 = - curl e0 . ... . . . (24).

That is, magnetoelectric force on the fictitious magnetic cur- rent g0 which is equivalent to the intrinsic force e0. And, similarly to before, we may remark that the flux due to e0 depends solely upon its curl.

Summary of the Forces. Extension to include varying States in a Moving Medium.

§ 85. Now bring together the different moving forces we have gone over. On the electric side we have

, . (25), and on the magnetic side

Fm = [H.r]-iH2vM + VOB + Vj0B, . . (26),

where the third term on the right of (25), and the first on the right of (26), in square brackets, are zero ; Ho- being force on magnetification and VDK the magnetic analogue of VOB. The other component forces can all separately exist, and in stationary states.

Passing to unrestricted variable states, with motion of the flux-supporting media also, the electric and magnetic stress vectors indicate that the moving forces arising therefrom are obtainable from those exhibited in (25), (26), by simply changing the meaning of the electric current and the magnetic current symbols in the third terms on the right. Above, they stand for conduction current only, and one of them is fictitious. They must be altered to G0 and J0, the " true " currents, as explained in § 66. Thus

. . (27) . . (28)

express the complete translational forces due to the electric and magnetic stresses (31), (32), § 72.

108 ELECTROMAGNETIC THEORY. CH. II.

The division of the resultant translational force into a number of distinct forces is sometimes useless and artificial. But as many of them can be isolated, and studied separately, it is not desirable to overlook the division.

The equilibrium of a dielectric medium free from electrifica- tion and intrinsic forces, which obtains when the electric and magnetic forces are steady, is upset when they vary. There is then the electromagnetic force in virtue of the displacement (electric) current and the magnetoelectric force in virtue of the magnetic current ; that is,

F = VDB + VDB (29)

(30), dt v* dt

where W is the flux of energy. Here we neglect possible small terms depending on the motion of the medium.

That there should be, in a material dielectric, moving force brought into play under the action of varying displacement and induction does not present any improbability. But it is less easy to grasp the idea when it is the ether itself that is the dielectric concerned. Perhaps this is, for some people, because of old associations — the elastic solid theory of light, for in- stance, wherein displacement of the ether represents the disturbance.

But if we take an all round view of the electromagnetic con- nections and their consequences, the idea of moving force on the ether when its electromagnetic state is changing will be found to be quite natural, if not imperatively necessary. We do know something about how disturbances are propagated through the ether, and we can, on the same principles, allow for bodily motion of the ether itself. Further, reactions on the ether, tending to move it, are indicated. But -here we are stopped. We have no knowledge of the density of the ether, nor of its mechanical properties in bulk, so, from default of real data, are unable to say, except upon speculative data, what motions actually result, and whether they make any sensible difference in phenomena calculated on the supposition that the ether is fixed.

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

Union of Electric and Magnetic to produce Electromagnetic Stress. Principal Axes.

§ 86. From the last formula we see that to have moving force in a non-conductor (free from electrification, &c.) requires not merely the coexistence of electric and magnetic force, but also that one or other of them, or both, should be varying with the time. That is, when the energy flux is steady, there is no moving force, but when it varies, its vector time-variation, divided by -y2, expresses the moving force.

The direction of W is a natural one to choose as one of the axes of reference of the stress, being perpendicular to both E and H, which indicate the axes of symmetry of the electric and magnetic stresses. The two lateral pressures combine together to produce a stress on the W plane

Pw^-W^U + T), .... (31)

where Wx is a unit vector parallel to W. (Take N = Wl in the general formula (30) § 72 for PN).

If, further, E and II are perpendicular to one another, the stresses on the planes perpendicular to them are

PE1=E1(U-T)J (32)

P^H^T-U), (33)

which are also entirely normal. Here EL and Hx are unit vectors. Thus, Wx is always a principal stress axis, while E: and Hj_ become the other pair of principal axes when they are perpendicular, as in various cases of electromagnetic waves. Thus when a long straight wire supports a steady current, or else is transmitting waves, the principal axes of the stress at a point near the wire are respectively parallel to it and perpen- dicular to it, radially and circularly. The first one has a pressure (U + T) acting along it, the second (parallel to E) a pressure (T — U), and the third (parallel to H) a pressure (U-T). (This legitimate use of pressure must not be con- founded with the utterly vicious misuse of pressure to indi- cate E.M.F. or voltage, by men who are old enough to know oetter, and do.) In general, U and T are unequal. But in the

110 ELECTROMAGNETIC THEORY. CH. II.

case of a solitary wave or train of waves with negligible distor- tion U and T are equal, and there is but one principal stress, viz., that with axis parallel to W, a pressure 2U or 2T. It is this pressure (or its mean value) that is referred to as the pressure exerted by solar radiation, and its space-variation con- .stitutes the moving force before mentioned. This matter is .still in a somewhat speculative stage.