SECTION XI. THE PRINCIPLE OF THERMAL RESISTANCE.
Suppose we, with a sure faith in the truth of the principle of Conservation of Energy, and a knowledge of the equivalence of work and heat,
H.E.P.—VOL. I. 2 II
observe that the motion of a magnet in the neighbourhood of a closed circuit generates heat in it, that this ceases when the motion ceases, and that we satisfy ourselves that this heat is the only final result of the motion, so far as energy is concerned. This heat must be the equivalent of work done. As we have no reason to suppose that the magnet is in a different state at the end from what it was in at the beginning of the motion, we cannot attribute the heat to a loss of potential energy by the magnet. Hence we may conclude that the heat is the equivalent of work done against resistance to the motion of the magnet. That is to say, however the magnet move, its motion is resisted. Here we have Lenz’s law, without any reference to the direction of the current induced in the circuit, but merely as regards conservation. In fact we made no mention of current. If now we bring in other knowledge, that there is a current induced in the circuit, the heat being proportional to the square of its strength, we are still left without means of determining its direction in a given case. Only finally, when we utilise Ampere’s determination of the mutual forces between magnets and currents, can we exactly say in which direction the induced current will be in a given case, for it must always be such as to resist the motion.
Lenz’s law is not, however, the subject of the present section, but is used to point out the distinction between the above kind of resistance to motion and the kind involved in the Principle of Thermal Resistance, to which we now proceed. It is such a large subject, and there are so many ways of treating it, that it is difficult to know how to begin ; after consideration I adopt a method which I have not met with, and which is therefore, if not novel, at least unusual; believing that, whether it be better or worse than other methods, there is advantage in viewing a truth from all possible sides, to allow another law, that of the survival of the fittest, which, like that of thermal resistance, results from averages, to have a chance of operating.
We cannot, in general, alter the configuration of a body -without doing work upon it, or letting the body do work. Considering any solid elastic body, for example, a straight wire : we cannot twist it without doing work. The motion is therefore resisted. On the other hand, if we let the wire untwist, it can do work itself against external resistance. Here, of course, we must have conservation of energy when all actions are taken into account, and nothing novel, so far, is presented. But there is, during the changes of configuration of an elastic body, another kind of resistance brought into play, depending upon the rate of change of configuration, at least, usually. Suppose we twist a wire slowly, and at every stage of the process note exactly the amount of the applied force, or do the same by small instalments. By summation we know the total work done in producing a given twist. Similarly, if we let the wire untwist slowly by small instalments, the force will be the same in the same configuration as during the twisting, and the same amount of work will be done by the body. But if we twist the wire suddenly to the same extent as before, more work will have to be done, or there will be an additional resistance to the motion at everv moment; and if the wire suddenly untwist, it will do less work, or again, there is a resistance to the motion which did not exist before.
We may put it this way: If we measure statically the forces in different configurations we do not by summation get the total work done. There is always an additional resistance to the motion, opposing the change of configuration, never assisting it. I call this thermal resistance, because, whenever it occurs, there is a thermal effect also produced. It is not, however, necessarily a heating, as one might expect, but may be either a heating or a cooling. But, whichever it be, the change of configuration is always resisted. That is the cardinal fact that must be remembered.
If, during the change of configuration, the body be allowed to part with or to receive heat so as to neutralise the thermal effect if it be a heating or a cooling respectively (as by conducting the operation very slowly, and not thermally insulating the body), the thermal resistance itself is evanescent. On the other hand, if the body be thermally insulated, so that heat cannot leave or enter it, it will be of the full amount. This will serve to elucidate the effect above mentioned of suddenly changing the configuration. We then give 110 time for heat to escape or be taken in appreciably. Now given this law, or principle of thermal resistance, and a statement cf one effect upon a body, deduce another. I give a few examples.
(1) . W e observe that heat lengthens and cold shortens a bar. What should be the thermal effect of suddenly stretching it ? It must be such as to oppose the stretching; that is, to shorten the bar, and is therefore a cooling.
(2) . What should be the effect of removing the stretching force ? To oppose the return to the unstretched length, therefore to lengthen the bar; therefore a heating.
(3) . Water above 4° C. expands by heating, what is the effect of compression 1 Such as to oppose the compression; therefore a heating.
(Notice there are two effects—the thermal resistance and the heating or cooling effect.)
(4) . Water below 4° C. contracts by heating. The effect of compression is, therefore, to cool it, as the compression is thereby resisted.
(5) . Water expands in vaporising. Hence pressure raises the boiling point.
(6) . Water expands in freezing. Hence pressure lowers the freezing point.
(7) . An india-rubber band at ordinary temperatures, stretched by a weight, lifts it when heated. What should result from suddenly stretching the band ? The motion must be resisted ; that is the invariable fact. The thermal effect is therefore a heating, for that, by the previous, lifts a weight.
(8) . A twisted wire is suddenly twisted further. Is it a heating or a cooling effect that is produced ? Whichever it be, it must increase the torsional rigidity, so as to oppose the twisting. If, then, heat lessens the torsional rigidity, it is a cooling effect, and conversely.
(9) . Compressing a gas heats it. Hence heat applied to a gas increases its pressure, for this is the only way to make the compression be additionally resisted. This property of a gas is so well known that it is more difficult to recognise the principle. It is hard to imagine the possibility of a gas being cooled by compression. Yet the principle involved is identically the same as in the less obvious illustrations.
Other mechanical illustrations may be multiplied indefinitely, but the above will be sufficient. When we wish to go further, and make applications to electricity or magnetism, it is necessary to be very particular that the necessary conditions are complied with. The thermal resisting force is always opposed to the motion, and so far resembles a frictional force ; but the thermal effect, unlike that of friction, is reversible with the direction of motion, and the motion produced by heat is reversed if cold be applied. Thus, during the stretching of a spring, the pull of the spring is F+ f] if F be what it would be if there were 110 thermal effect, and/ is the small increase produced by, or accompanying it. On the other hand, when the spring shortens, the strength of force is F-f at a corresponding stage, and at the same temperature, / being the same quantity as before. Regarding the forces as vectors, F is constant in direction, f changes, being with F in the stretching, and against it in the unstretching. That is, it is- always against the direction of motion. The heat effect, if it be a cooling during the stretching, is a heating during the unstretching. And as regards the effect of heat, if heating increases, cooling must decrease the elasticity. If in some peculiar state, both heating and cooling produced the same effect 011 the elasticity, or if either stretching or unstretching produced the same thermal effect, we could not immediately apply the principle without reservation. Further investigation would be needed.
(10) . Given a circuit of two metals A and B, at one temperature initially. We observe that slightly heating one junction causes a current from A to B, and cooling it causes a current from B to A. Here is a perfectly7 reversible thermal effect, although accompanied by other strictly irreversible effects in the circuit. In accordance with the principle, what should be the thermal effect at the same junction on passing a current from an external source from A to B ? The current must be made weaker than it would otherwise be, hence the current due to the thermal effect is from B to A, hence it is a cooling, by the previous knowledge. At the other junction it must be a heating, for the current is there from A to B. Thus there is a transfer of heat from the first junction to the second. Notice the peculiarity that the thermoelectric force in the circuit, due to both junctions, weakens the main current; that is the first conclusion. But the E.M.F. at both junctions is from A to B, whence it follows that at the first junction the E.M.F. is weaker than at the second. Further inquiries would lead us to the full theory of thermoelectricity. [See pp. 309 to 327.]
(11) . Given that the capacity of a condenser is increased by heating and decreased by cooling the dielectric, what should be the thermal effect of suddenly charging it 1 The charging must be opposed, hence a decrease of capacity or a cooling effect. Similarly, suddenly discharging the condenser should produce a heating effect, which we may conclude thus:—The discharge must be resisted. Now, the rate of discharge depends upon the difference of potential of the ends of the discharge wire. This must be reduced, therefore. But a reduction of potential with the same charge means an increase of capacity, for which there must be, by the above, a heating effect. Observe here that these thermal effects are independent of the absorption phenomenon.
(12) . A curious result of thermal resistance is that a perfectly elastic solid, if such really existed, vibrating, would come to rest without any external or internal friction to cause it. Thus in a vibrating spring, thermal resistance opposes its motion whether it be moving to or from the equilibrium position. Now, if the spring could give out and take in heat instantaneously, so as to keep its temperature constant, the effect would vanish, and, without friction, the spring go on for ever. Similarly, if the spring could be thermally insulated, the cooling and heating effects produced when moving to or from the equilibrium position would balance, and the spring could go on for ever. But practically, neither one nor the other condition can be complied with, and the spring must be brought to rest without friction entirely by the thermal effect being practically radiated or conducted away in one motion, without an exactly counterbalancing receipt of heat in the opposite motion.
There is another way of looking at the principle which is useful, viz., to direct attention to the flow of heat into or out of the body when it is being strained, supposing that the body is, in the first place, in thermal equilibrium with its environment, and that it can receive or lose heat instantaneously. Thus, in example (9), compressing a gas drives heat out of it. Now, go to the other end of the operation. Take heat out of a gas. It compresses itself. Or take example (1). Stretching a bar draws heat into it, and sending heat into a bar makes it stretch itself. From this point of view we regard two events as being invariably connected—motion of matter of a certain type, and a flow of heat in a certain way. But the method is less general than the preceding, assuming, as it does, that the flow of heat is always permitted, likewise the motion.
In all cases whatever in which the principle of thermal resistance has been experimentally tested, it has been found to be correctly followed by Nature. Can we then assert its invariable truth, and apply the principle unhesitatingly to hitherto unverified cases, possibly unverifi- able 1 Is it possible to give a rigid demonstration of its truth ? The first question may [within certain limits] be answered in the affirmative, the latter not. We cannot prove it even as we prove the truth of conservation of energy, viz., by seeing that it is a necessary truth in pure dynamics, and extending our notions to all the operations of nature; observing that it is experimentally true in a great many cases, and convincing ourselves of its universal application with the assistance of a little faith. For we cannot deduce the principle of thermal resistance from the laws of dynamics. It would be no breach of conservation of energy were it to be exactly reversed, were strains to be assisted by thermal effects. All we can really do is to convince ourselves that, being true in all observed cases, and its negation leading to extraordinary consequences which are not observed, though not dynamically impossible d, priori, it must, by faith, be generally true.
First, put the principle in a mathematical form. Let an elastic body be strained from one configuration to another, keeping it always at one temperature t, to allow which a quantity of heat H leaves the body, whilst IF is the work done upon the body. Then the principle asserts simply that dJVjdt ± H is positive, or that dJFJdt and // are either both positive or both negative, which is easily verified by the above examples. More work is caused to be done by the thermal effect if it be not than if it be allowed to escape.
Next, to see the consequences of its negation, follow the example of the founder of thermodynamics, Sadi Carnot, when, in the most masterly and scientific manner, he set to work to find the cause of the motive power of heat. After the above strain, bring the body back exactly to its original state through the same series of intermediate configurations, but at a slightly different temperature t + dt. This requires there to be two other operations (2nd and 4th), viz., to raise the temperature by dt in the second configuration, and to lower it by dt when it has got back to the first. Now the body does work IF + (dJF/dt)dt at the higher temperature. Hence, in the complete cycle, the body does work (d1Fjdt)dt. What else happens is that an amount of heat H is lowered in temperature by the amount dt. Now, without any experience to guide us, H and dJFjdt may be algebraically positive or negative, and not of the same sign necessarily. But if they could be of opposite signs, work would be obtained through a substance raising the temperature of heat. If then wTe take it as axiomatic that it is impossible by conveying heat from a cold to a hot body to obtain mechanical effect, then we prove that the law of thermal resistance is universally true, at least for bodies in mass, and inanimate.
Now this axiom, so called, is really the principle Carnot was led to, viz., work is derived from heat by lowering its temperature. Carnot’s principle is thus a consequence of the principle of thermal resistance. In Carnot’s cycle for a gas engine heat is taken in at a higher and given out at a lower temperature, and the pressure is greater at the higher than at the lower, so that the gas does more work in expanding than is done in compressing it back to its original state, and thus we have finally heat lowered in temperature, and work done by the gas. In his water and vapour cycle it is similar, so he concluded that the necessary condition of obtaining work by thermal agency was the lowering the temperature of heat. He was only wrong as regards the quantity of heat lowered in temperature.
Now this being a consequence of the principle of thermal resistance, and the various examples above given being commonly regarded as consequences of the Second Law of Thermodynamics, with which Carnot was most assuredly not acquainted, how is it that Carnot was acquainted with them ? I do not say he thought of them all. But he knew some of them, and, had he been asked what would be the result in a given case, according to his principle, he would have given the correct answer. Onty his reason for the answer would have been erroneous. For the only legitimate reason that Carnot could have for applying his principle universally would be that he had proved that all reversible engines had the same efficiency between the same temperatures, so that his principle, known to be true for one body, must be true for all.
But this proof was vitiated by his erroneous assumption of the materiality of heat, then universally believed in, and was only put right long after, by Clausius and Thomson, oil the basis of the equivalence of heat and work, fully established experimentally by Joule. Carnot founded his proof on a perfectly unexceptionable axiom (011 his view of the nature of heat). Work could be obtained without any thermal agency if all reversible engines had not the same efficiency between the same temperatures. The substituted axioms of Clausius and Thomson are by no means so satisfying, considered as axioms. They express truths, they involve Carnot’s principle, they’ involve the principle from which Carnot’s is derived, but they are not axioms, unless a law of Nature, only to be learnt by experience, is an axiom.
Clausius said that heat will not pass from cold to hot by itself, or without compensation. True enough, by definition of cold and hot, if the cold and hot bodies be in contact. Otherwise not self-evidently true, though a law of Nature. Thomson said we cannot get work out of a body by cooling it below the lowest temperature of surrounding objects. This I also admit to be true, knowing that it involves Carnot’s principle, and believing that; but it is not self-evidently true. A third axiom that can be used is that we cannot convert heat into work without lowering the temperature of heat. This I believe to be the best of all, being the simplest expression of the truth of Carnot’s principle that work is got by lowering the temperature of heat.
Of course Carnot was wrong in his quantitative estimate of the efficiency of reversible engines, but his principle remains unaltered. When, with the principle of thermal resistance, we combine that of the equivalence of work and heat, and use Carnot’s criterion of the perfectness of an engine, that it must be reversible, we arrive at an absolute measure of temperature independent of any particular substance, and using it, give quantitative expression to the principle of thermal resistance, viz. :—
d IVj dt = H/t,
where t is temperature according to the scale of equal dilatations of an imaginary perfect gas under constant pressure, whose energy, at constant temperature, is independent of its volume. It now becomes a form of the Second Law of Thermodynamics.
There is a reason for everything, and therefore (if that be an axiom) for the axioms of thermodynamics, better called laws. The reason of conservation of energy the student of dynamics can understand, if he can grant that the laws of matter in motion observed in masses are true for the smallest parts of bodies. The reason of thermal resistance is also, by the aid of the kinetic theory of gases, becoming evident, and will no doubt some day be established for all bodies. It arises from the irregular nature of the motions we call heat. We cannot control single molecule?!. Could we do so, down would go the law of thermal resistance and heat could he converted straight into work, molecular irregular motions to an equivalent amount of motion of a mass, the irregularity disappearing, without at the same time having to lower the temperature of another quantity of heat, as at present required. But what we J cannot do with inanimate matter may be going on always to a certain J extent in living matter; not because living matter is exempt from any law of Nature, but that we do not yet know all its laws. The perpetual running down of the available energy of the universe is a matter that must be cleared up. It is incredible that it can always have been going on, and dismal in its final result if uninterrupted. It is therefore the duty of every thermodynamician to look out for a way of escape.
SECTION XII. ELECTRIZATION AND ELECTRIFICATION. NATURAL ELECTRETS.
A dielectric, not including ether, the universal medium, when under the influence of electric force, may be said to become electrized. We must not say electrified, as that refers to something entirely different from electrization, namely, electrification, or discontinuity in the displacement. The proper measure of the intensity of electrization will appear presently ; in the meantime we need only observe that electrization is the analogue of magnetization, and, like it, requires the presence of matter.
Electrization may lie approximately perfectly elastic with reference to the standard zero state, as in a dielectric in which absorption does not occur, so as to disappear on the removal of the exciting cause. This must be an impressed electric force somewhere, except in a transient inductive state, when there may be none, unless we choose to consider the electric force of induction as impressed; or else there must be electrification somewhere. There may also be residual electrization, namely, when absorption occurs. This may tend naturally to wholly subside, or a part of it may remain. The body is then permanently electrized. But, apart from artificial production of permanent electrization, it exists naturally in pyro electric crystals, if nowhere else.
A word is evidently wanted to describe a body which is naturally permanently electrized by internal causes. Noticing that “magnet” is got from “ magnetism ” by curtailment at the third joint from the end, it is suggested that we may get what we want by performing the same operation upon electricity. An “ electric,” which is what results, would be a very good name for an intrinsically electrized body, but for two reasons. First, it was once used to signify what we should now call a dielectric or an insulator; and secondly, electric is now used as an adjective, or, equivalently, electrical. The former of these objections is of hardly any weight, that use of the word as a substantive being wholly obsolete. The latter is heavier, but still of no great importance. Another word that suggests itself is electret, against which there is nothing to be said except that it sounds strange. That is, however, a mere question of habit. Choosing, at least provisionally, the second word suggested, to avoid collision with the adjective, we may then say that certain crystals, if no other bodies, are natural electrets; that solid insulating substances may be made electrets artificially, with a greater or less amount of permanency; that liquid insulators can only be electrets for a minute interval of time, if at all; whilst gases, whose particles are always vigorously wandering about, are never electrets. But all insulators can be electrized; they are only electrets when the electrization, or part of it, is intrinsic.
Let J be the intensity of intrinsic electrization anywhere, and let
J = ce 147r,
c being the specific capacity at the place. Then e is the intrinsic electric force, and the displacement D and actual electric force E are determinable fully by the conditions
D = cE/47r, div D = p, curl (e - E) = 0;
supposing that there is no conductivity. Here p is the volume-density of electrification, if there be any. Should there be conductors present, charged or uncharged, their effect is determinable by the additional conditions of there being no force in any conductor (unless there be impressed force therein), and every separate conductor to have a given total charge, zero or finite, according as the conductor is not or is electrified when placed by itself in a space previously free from electric force.
The physical explanation of electrization is of course matter of speculation, as it depends not only on the nature of molecules, but also on their relation to the ether. Weber’s theory of induced magnetization obviously suggests its analogue. Accordingly, we might say that the molecules of a body are always permanent electrets; but should there be, in a very small portion of it, but still large enough to contain a very great number of molecules, an average uniformity of distribution in all directions of their axes of electrization, there will be no external signs of its being electrized, or the intensity of electrization is zero. Next, the molecules admit of rotation, so that under the influence of externally caused electric force the axes of electrization are turned to a greater or less extent towards coincidence with the direction of the electrizing force. This preponderance of electrization in one direction of the molecules in our small volume naturally causes there to be a finite intensity of evident electrization, or of electric moment. Should this angular displacement of molecules be only elastically resisted, the evident electrization will wholly disappear on the removal of the electrizing force, or the body was only inductively electrized. Should there be a slow slipping, the phenomenon of absorption will occur, there being no longer a complete return of the molecules to their original state on the removal of the electrizing force. The body is then made an electret. Should the slip be permanent, the body is an artificially made permanent electret.
Such a theory is of course very empirical, and admits of considerable variation according to the hypothesis we adopt regarding the angular displacement and its tendency to subside. There is of course a maximum intensity of electrization which cannot be exceeded, viz., when all molecules are turned the same way. It is only one way of accounting for evident electrization, and may be utterly unlike the reality. Weber’s theory of rotation of molecules has a great many recommendations, but I do not think they go further than to make it anything better than a working hypothesis, even with the support of Professor Hughes’s experiments.
But, admitting that molecules are electrets, we may go a step further, and, in a manner, account for it. In the first place, the state of electric force and displacement determined by the above equations would be identically given in another manner. Do away with e, that is, do away with J, the intrinsic electrization, and substitute for it an arrangement of circuital magnetic current of density
- curl e 477- = G, say.
The magnetic current G and the intrinsic electric force e are equivalent, so far as the distribution of displacement is concerned. In the case of a bar uniformly electrized longitudinally, the current G will be entirely superficial, going round the bar. But by Ampere’s device, we may substitute a network of currents for a current bounding the network. Thus we may get down to the molecules, and ascribe their electrization to molecular magnetic currents whose moments equal their electric moments. But I do not put this forward as being at all a physical explanation of why molecules are electrets, if they be. There is merely a mathematical equivalence.
Nor do I, in the parallel case of explaining the magnetism of a molecule by means of a molecular electric current, admit that any step towards a physical explanation of magnetism has been made. Not that I attach much importance to the common objection originating from the fact that an electric current in a conductor requires a continual supply of energy to keep it up, owing to the Joule-heating that goes on. For we observe the effects of heat in mass, and ascribe heat to the kinetic energy of agitation of the molecules. Not of the parts of molecules, but of their wholes. Now, considering a molecule as an atom, for simplicity, if an atomic current generated heat as a current in a wire does, the heat would be the energy of agitation of the parts of the atom, which, though indivisible, may jet not be of unchangeable form. This communication of energy would surely alter the nature of the atom. In fact, it is assuming an atom to be like a bod}' in the ordinary sense. In brief, conduction as we know it is an affair in which molecules or atoms as wholes are involved, so far as the heat is concerned, and it would be far more wonderful if atoms had internal resistance than, as we are obliged to suppose, that atoms are perfect conductors, if we choose to have atomic currents.
My reason for not considering a molecular electric current to be a physical explanation of magnetism is, that although the more closely we look at the matter, apart from old-fashioned notions of an electric current as something going round and round, the more we are led to the conclusion that the magnetized molecule with its field of force and the Amperean current with its field of force are really one and the same thing, yet we are brought no nearer to an understanding of either, so that explaining the former by the latter is futile, regarded physically. But let us return, for the present, to electrization.
Dismissing altogether all ideas concerning the possible electrization of single molecules, keep to evident electrization. To illustrate it with sufficient comprehensiveness, without any great complication, or difficult calculations, let the electrized body be of spherical shape, and let us perform a series of simple operations upon it.
(1) . Let there be a uniform field of electric force of strength E, and displacement D, in air of specific capacity c, practically equal to unity. Then bring into the field a perfectly neutral solid spherical dielectric of radius a and specific capacity cr Call those points on its surface where it is cut by a straight line through its centre parallel to the force of the undisturbed field the poles, the negative pole being that where the line of force enters, and the positive where it leaves the sphere. This line is the axis. Supposing cx greater than c, the lines of force will be drawn in by the sphere symmetrically with respect to the axis. The displacement is continuous, and is therefore made greater within the sphere than in tlie field previously. On the other hand the electric force within the sphere is weakened. Both are parallel to the axis and uniform within the sphere. Let Dl and Ex be the displacement and force, then
Dx = c^EJA-rr, and lJl = '3Dcl/(2c + c1).
Thus it is not possible to make the displacement more than three times as great as before, and that is when cx = cc. It is twice as great when cx = 4/\ There is no real electrification, but the sphere appears electrified, to surface-density D1 at the positive pole, and elsewhere proportional to the sine of the latitude. The potential, not counting that of the undisturbed field, which increases uniformly parallel to the axis, is err cos 6 within, and a-a3cos 6jr2 without the sphere, where o- = E(cv - c)/(2c + c^), at a point distant r from its centre, 6 being the co-latitude. If the sphere be taken out of the field, its apparent electrification will disappear, unless absorption or conduction have occurred.
(2) . Whilst in the field, as above, let absorption occur, and go on until the displacement has become increased by 1)2, such an amount as would be caused by an impressed force e2 uniformly distributed parallel to the axis. We must not write JJ2 = ; D2 is less than this amount. For continuity in the displacement requires that when the displacement in the sphere increases from some cause in itself, there must be a corresponding change in the external displacement. The effect of the intrinsic electrization of intensity
J2 — Cfo/iTT
is to make D2 = 2J2cj(2c + cY)
within the sphere. The apparent charge is increased in density to (7^ + D2) cos 0.
(3) . Next, remove the sphere from the original field of force. There is left the field due to the intrinsic electrization. The displacement within the sphere is D2, and the apparent charge is of density Z>2 cos 6. The potential is pr cos 6 within and (pa3jf2) cos 6 without the sphere, where p = cvc2/(2r, + rT).
In fact, the field is exactly similar to the magnetic field of a uniformly magnetised sphere. If the absorption gradually disappear, so will the external field.
(4) . But whilst the intrinsic electrization remains appreciably steady, let us cover the sphere with a metal coating, or in any other way produce surface-conduction. There will be a current through the sphere from the negative to the positive side, and oppositely in the conducting coating. Thus the displacement within the sphere must be increased by the short-circuiting; it becomes
D.A = CjC2/4w = J.2,
the greatest displacement possible without external aid. The external field is done away with altogether. There is now a real distribution of electrification on the surface of the sphere, of density - J2 cos 6; i. e., negative on the positive side and positive on the negative side. The potential is zero inside and outside. The electret is now in the state of the dielectric of a condenser in which absorption has occurred, and a first discharge taken, the remaining charge not being accompanied by difference of potential between the metal plates.
(5) . On removing the metal coating, the surface-charge gives no signs of its presence.
(6) . Keeping the sphere insulated, let the intrinsic electrization subside. As it does so, an external field of the opposite character to before appears, the real surface-electrification remaining constant, whilst the displacement it exactly neutralized becomes less. When the intrinsic electrization has gone, supposing it to wholly disappear, the displacement, inside and outside, is merely that due to the surface- charge. Inside, the displacement is
and the potential is - 47rJ2r cos dj(2c + Cj).
Outside, the potential is - AirJ2as cos 6/r2(2c + cl).
The apparent electrification has density
- 2J2c cos 6f(2c + cj.
(7) . Finally we may get rid of the surface-charge by again putting on the conducting coating, after which the sphere will be in its original neutral state.
The study of the theory of electrization is in some respects more important than of magnetization; on account of its greater generality it is more instructive. We have conductors and dielectrics, real and apparent electrification. The magnetic problems are less general on account of the absence of magnetic conductors, with corresponding absence of an}' magnetic representative of real electrification, as in (4) above, the magnetic matter or free magnetism being, except as regards a constant factor, the representative of the apparent or imaginary electricity, as in (3), where there was no real electrification.
As regards natural electrets, Sir W. Thomson’s theory of pyroelectricity is (so far as is known to me) contained in a short article in “Nicol’s Cyclopaedia,” reprinted in Vol. I. of “Mathematical and Physical Papers,” Art. 48, p. 315. Being only a few lines in length,
I can scarcely be quite certain that, when fully developed, it would be exactly as I state it—that is to saj1, when details are gone into, similarly to the above—though, generally speaking, there is no room for ambiguity. A pyro-electric crystal is a natural electret. In its neutral state, however, its external field of force has been done away with by surface-conduction and convection. Disregarding eolotropy, it is then in the condition of the sphere in (5) above, the surface being charged so that its density equals tlie divergence of the internal displacement. Warming or cooling the electret, by altering the internal stresses, alters the intensity of electrization, whereby the surface-charge no longer exactly balances it. If, for example, warming decreases the intensity of electrization, the positive end appears to acquire a negative charge, the negative end a positive, like the sphere in (6) above. If it be kept at the higher temperature, and surface-conduction occur, its surface- charge will readjust itself to again balance the internal displacement. Evidently we cannot get rid of the surface-charge as in (7), unless we can make the intensity of electrization zero.
The electret may, however, without change of temperature, produce external force by breaking it across its axis of electrization, when, evidently, the two pieces will not be neutral, unless the act of fracture should cause exactly the right amount of electrification on the fractured surfaces, which is highly improbable. But by surface-conduction again the two pieces will each become neutral.
Thermodynamic principles have also been applied to the case by Sir W. Thomson, under date 1879. This is the easiest part of the matter, considered qualitatively, if we know exactly the influence of heat. Applying the principle of thermal resistance discussed in the last section, we see that moving a natural electret about in an electric field must cause thermal effects. If, for example, as above, heat decrease the intensity of electrization, by decreasing the capacity, if we suddenly put the electric in an electric field so as to increase the displacement, the increase must be resisted, and this requires the electret to be heated. If now, we suddenly invert it, so as to decrease the displacement, this decrease must be resisted, hence a cooling effect. Should e vary as well as r, the application will be more complex. The phenomenon is no doubt very insignificant, but is very curious when we consider that it must occur in neutral crystals showing no external signs of their electrization.
SECTION XIII. SIMULTANEOUS CONDUCTION CURRENT AND ELASTIC
DISPLACEMENT.
In the ordinary intercourse of man with his environment he is more or less accustomed to overlook, ignore, or treat as non-existent all phenomena whose recognition is not of immediate practical utility to him. For instance, very few people are even aware, until their attention is forcibly called to it, of the multitude of most singular optical phenomena that occur in the everyday use of the ej'es, some of them very difficult of explanation. Their perception would not be of immediate utility to the average man. They are therefore left unnoticed, as if they were not, until even their recognition becomes difficult. The phenomena occur, and are seen by the eye mechanically, but the mind’s eye is blind to them.
Most electricians, in a somewhat similar manner, are accustomed to confine their attention to only one part of the wonderful phenomena occurring during the existence of a conduction current. The}' may think of the wire, of its resistance, the current in it, and the E.M.F. causing it. Or possibly, in a more advanced stage, they may think also of the heating of the wire, and of external work done, as in driving a motor, with the necessary consideration of conservation of energy as regards the amount of work done. Still, however, it is the current in the wire that is the first object of attention, and what goes on outside it is ignored. This is perfectly natural, for what we call the strength of the current is the one thing that is, in the steady state, practically altogether independent of the external conditions ; whilst, again, the steady state is of great importance, and may be brought under calculation with comparative ease. We are aware of the existence of an external magnetic field, and also of an electric field, or of an electromagnetic field having two sides, the electric and the magnetic. The}’ vary under different external circumstances. But as the steady conduction current is independent of them they are ignored. This independence, which is really a fact of an extraordinary character, is, by habit, taken for granted, and ceases to have anything remarkable about it.
If we have a closed circuit with a steady impressed force in it, there results a steady current in the circuit, whose strength is calculated from the data relating to the wire itself, and is independent of the distribution of matter around, provided we do not disturb the insulation at the immediate surface of the conductor. If we bring a mass of iron near the circuit, or in general alter the distribution of external matter as regards its permeability to magnetic induction, there results merely a momentary disturbance of the conduction current. When this has ceased, the current is just as it was before, although the external state has been considerably altered, there being in particular a different magnetic field, and a change in the distribution of the rays of energy converging to the wire. This, with the similar transient disturbance of current by motion of a magnet or another closed current, comprises what is ordinarily understood bjT the induction of currents. Only the magnetic energy is concerned, in the main.
There are also the inductive effects due to alterations, not of the magnetic permeability, but of the electric capacity outside the wire, or by altering the external electric field. The current induced, or the alteration made in the previously steady current, is, as in the case of magnetic induction, transient only, and 110 disturbance is produced finally in the distribution of current. (Except indirectly, as by altering the structure of the wire itself, and its conductivity.)
The effect of surrounding a wire supporting a current with soft iron is to decrease the current temporarily. On the other hand, the effect of surrounding it with gutta percha is to increase the current on the whole temporarily, it being increased in that part of the circuit next the source more than it is decreased in the other part. If in the former case T be the increase made in the magnetic energy, the battery does 2T less work than if the iron had not been brought to the wire. In the latter case, if U be the increase made in the electric energy, the battery does 2 U more work than if the gutta percha had not been brought to the wire.
Both the inductions, electric and magnetic, are in simultaneous action always. In a large class of cases, however, especially with condensers, the magnetic induction is of comparative insignificance. In another large class, as of coils containing iron, and circuits closed, the electric induction is negligible. In intermediate cases, when neither is negligible, we have very complex effects. But in all cases there is no permanent effect on the distribution of conduction current, however greatly the electric and magnetic fields outside the circuit are permanently altered, and likewise the manner of transit of energy from the source to the parts of the circuit away from the source. In brief, when the impressed forces are given, the conductivity conditions alone determine the steady state of the current. And, although it is not at once seen, the same is true whatever electric capacity or permeability the conductor itself may possess, so that there is interior electric or magnetic energy. The latter is recognised. Also, the electric capacity of very bad conductors is known, but there is little or no information regarding the capacity of good conductors, though for various reasons, I believe in its existence. The coefficient of permeability jx is known to never vanish. That of conductivity k has an enormous range, and may perhaps also vanish altogether—for instance in planetary space, if nowhere else. Similarly, the coefficient of electric capacity c may have a larger range than it is at present supposed to have in bad conductors. We tacitly assume c = 0 in good conductors in general.
It will be understood that in thus speaking of the electric capacity (specific) of a conductor, we do not in any way refer to the Capacity of a Conductor of electrostatics, which is really the capacity of the surrounding dielectric, by all analogy with usage in the corresponding conduction problems. We refer to the specific capacity of the material of the conductor for elastic displacement in itself. This has, of course, nothing to do with the nonconducting dielectric outside. It is sometimes said that the specific capacity of a conductor is infinite. This is a mischievous delusion. It is tolerable, for mere mathematical purposes, sometimes to assume c = oo in a conductor, and ignore its conductivity altogether ; ie., do away with the conductor, and substitute for it a dielectric of infinitely great specific capacity. We may, similarly, sometimes conveniently replace a voltaic cell (mathematically) by a condenser of infinite capacity and constant difference of potential. Or, we may consider a modern accumulator as a condenser, for limited purposes, in spite of the obvious radical difference in their physical nature. But the absurdity of considering a conductor as of infinite specific capacity is readily seen. For although we may, by increasing greatly the specific capacity of a dielectric, imitate, in some respects only, a conductor, yet as specific capacity refers to elastic displacement, such displacement must afterwards subside when allowed to, which is wholly different from the behaviour of conductive displacement, which has no tendency to return. The proper condition is not c = go , but c = 0 (unless we know there is elastic displacement in the conductor, as we do when it is badly conducting), with the auxiliary condition, k finite. Conductive and elastic displacement are wholly distinct things, and may be combined in any proportion. The former dissipates energy, the latter stores it.
In the following is an investigation of the general problem of steady impressed electric force and electrification in space with any distributions of electric capacity and conductivity. So far as I know, it has not hitherto been fully treated, but only piecemeal, and under limitations. For instance, Maxwell’s treatise does not consider impressed force in the dielectric at all, and the internal capacity of conductors but imperfectly. Nor, again, is the determination of the state of the dielectric due to currents in conductors treated, which presents some curious peculiarities. Also, it is of advantage to deal comprehensively with the matter, since in practice we really have this simultaneous conduction and elastic displacement, either at the same or at different places, but connected.
Given k and c the electric conductivity and capacity, for conduction current C and elastic displacement D, at every point of space, and also the distribution of impressed force e; find the steady state, or, rather, show how it is determinate. E being the electric force, we have
C = /i,E, divC=0, curl (e - E) = 0, (86)
D = cE/47r, div D = /□, curl (e - E) = 0, (87)
for the full statement of conditions at a point. First, the linear relations, viz., Ohm’s law and Faraday’s (or Thomson’s or Maxwell’s). The second of (86) expresses that the conduction current is continuous everywhere, and the second of (87) defines the electrification density p in terms of the displacement, viz., its divergence, the proper measure of the discontinuity of the elastic displacement. The third conditions are identical. They express that the actual force E is the sum of impressed e and of a polar force F, such that curlF = 0. If c= 0 at a certain place we have only (86) to consider ; if k = 0, then only (87); if neither vanishes, then both. Also, there will usually be surface conditions, which will come later.
First imagine e = 0 everywhere, and k finite everywhere, that is, an infinite conductor whose conductivity nowhere quite vanishes, and of no capacity. Then we may write either
div £(e + F) = 0, curl F = 0, (88)
and determine F; or else
div C = 0, curl (C/k - e) = 0, (89)
and determine C. It is obvious that the first of (88) has any number of solutions for F; we must then show that one (and only one) of them satisfies the second of (88), thus fixing F. Or, it is obvious that the second of (89) has any number of solutions for C; we must then show that one (and only one) of them satisfies the first of (89), thus fixing C. Thus, since the first of (88) expresses that C is circuital, choosing any circuital C settles F. Or,
IcF = curl A - ke
is the general solution, wherein A is any vector whatever. Similarly the general solution of the second of (89) is
C/k = e - VP,
wherein P is a perfectly arbitrary scalar.
Now, whatever F and f may be, if Fx — F + f, then
2 FjZ-F, = 2 (F + f)/t(F + f) - 2 F/.'F + 2 f/.f + 2 2F/jf,
the 2 indicating summation through all space. But if F satisfies both of (88) whilst F1 satisfies only the first, the third summation on the right vanishes. For F is polar, and ki is circuital, so that, bv the elementary property of the polar force, 2F/.f=0 for an}7 one infinitely slender tube of ki, and therefore for all. Thus we are reduced to the first and second summations on the right side. The first is fixed, the second may vary, but is necessarily positive, every element of it being positive. Hence f=0 makes the summation ^F^Fj a minimum. But Fx is any solution of the first of (88), whilst F is a solution of both, so we prove that any solution FT of the first, which also satisfies the second of (88), makes 2 F1/l,F1 a minimum. But this quantity has a minimum, for it is necessarily positive, unless F} = 0 everywhere. Flence there is a solution of (88), viz., F, when f=0. There cannot be two solutions. For, if F and F + f be both solutions of (88), we must have
curl f = 0 and div ki = 0,
and therefore ^ f/t:f = 0, which can only be by f=0, making F the sole solution.
We may treat (89) similarly. Let C satisfy both conditions, whilst C-t- y satisfies only the second, so that y satisfies curl y/k= 0. Then the total Joule-heat per second is
^ (C + y)k~'(C + y) = 2 C/.-,C +1’ yk~ly + 2 2C/-1 y.
Here the third summation 011 the right vanishes, because C is circuital and yjk polar. The second summation is necessarily positive, therefore y = 0 makes the heat a minimum. This clones the current, making C + y become C, satisfying both conditions. As the minimum is necessary, the necessity of a solution of (89) follows, and that it is unique is
H. E.P. VOL. I. 2 I
shown by S yk~ly vanishing when y is circuital and k~ly polar, requiring that 7 = 0, if we assume that C and C + y are both solutions of (89). As regards (88), we may state the result thus. There is one distribution of polar force F, a linear function of which, namely, IF, has a given distribution of convergence, viz.,
conv kF = div ke.
Or, there is one distribution of force E differing from e by a polar force alone, a linear function of which, namely, /t;E or C, is continuous everywhere. As regards (89), we may similarly state that there is only one circuital current C, a linear function of which, namely, Cfk or E, has a given distribution of curl, viz.,
curl C Jk = curl e.
Or, assuming that Ohm’s law and the continuity of the current are always obeyed, let C vary. Then ^ FZ'F is made a minimum by that one distribution of C which makes F polar. And, assuming that Ohm’s law is obeyed, and also that F is always polar, let F vary. Then -EC, the heat, is made a minimum by that one distribution of the current that is completely circuital.
The gist of the above, and also of the more abstruse and complex demonstrations that may be given, is expressed by the theorem 2FC = 0, or, a polar force, F, does no work on a circuital flux, C, arising from the property of a polar force that its total round any closed curve is zero. Given first Ohm's law, the linear connection between the actual electric force and the current-density, and also that the current must be circuital. If the impressed force be so distributed that it alone is sufficient to satisfy Ohm’s law and continuity (that is, when div£e = 0), the force e is the actual force, and the current is ke. But should this condition not be complied with by the impressed force, it is clear that an auxiliary or complementary force, F, is required which, together with e, shall make up the actual force E to satisfy continuity and Ohm’s law. The question, then, is, how is this complementary force to be found ? Any number of F’s may be made up, but only one of them is completely polar. We then have ^FC = 0, and the total heat expressed by ^ EC = 2 eC - eAe - ^ FkF.
It is naturally suggested by this making of the total heat equal to the activity of the impressed forces, that we examine the effect of subjecting the complementary force, not to being polar, but to satisfying SFC = 0, or 2 eC = - EC. which is possible in other ways than by a polar force. Thus, knowing that F uniquely determined polar satisfies this condition, and supposing that E, C, and F represents the real solution, let us alter the current in any manner, keeping it circuital, also obeying Ohm’s law, and finally keeping the heat per second always equal to the activity of the impressed forces. Let f be the alteration in the complementary force, then kf is the alteration in the current, and is circuital. Also the alteration of 2 eC is - ekf. But, by our final restriction above,
2 (F + f )(C + kf) = 0, or Sf(C + Xf) = 0, which reduces to ~ e/f = - - fkf
on putting C = k(e + F), and then 2 FH=0, F being polar and H circuital. But is positive, hence is negative. Hence ^eC is reduced. That is to say, any change made in C from the real current, subject to Ohm’s law, continuity and conservation, decreases the heat and the (equal) activity of the impressed forces. Hence, in the real case, the impressed forces work as fast as ever they can, and this is by having the auxiliary force polar.
Taking a simple linear circuit, to exemplify, we find that there are only two solutions, the real, and C = 0. Both satisfy conservation (meaning 2 eC = 3 EC), but no other current will. But we need not be misled by C = 0 being, in this case, the only other solution than the real. For if we introduce a shunt, making three wires joining two points, and have an impressed force in only one of the wires, we may create additional solutions without number. For two simple circuits can be made containing the impressed force, furnishing two extreme solutions, in which no change from the real current is made in the current in one or the other branch not containing the impressed force. Between these extremes, by changing the current in all three branches suitably, we get any number of other solutions, in all of which continuity, Ohm’s law, and conservation are satisfied. The heat varies from zero up to a certain maximum, which occurs in the real case; then the auxiliary force is completely polar, or has a potential. Of course, did we ever find a seeming departure from the proper distribution of force related to the known impressed force, we should naturally ascribe it to other impressed forces, and so come back to potential again.
SECTION XIY. CONDUCTION AND DISPLACEMENT (Continual).
Supposing now that we have, by the previous, satisfied ourselves that the distribution of current in the steady state is uniquely determinate by the subjection of the current to Ohm’s law and continuity, and of the complementary force to being wholly polar, which last condition makes the activity of the impressed forces the greatest possible subject to conservation. The question then arises, how is this affected by the conductor being a dielectric as well, so that there must be elastic displacement in it 1 If we compare the dielectric conditions (67) with those for the conduction current (86), we see that they are of exactly similar form, except in the presence of p, the volume-density of electrification. If, then, we start with p = 0, and proceed to find the displacement produced by the impressed force on the supposition that there is no conductivity anywhere (just as before, in the conduction problem, we assumed the specific capacity to be zero everywhere), we know that the displacement is uniquely determinate, as was the current before. Like the current, it is circuital, and is so distributed as to make (not the heat, but) the potential energy of displacement a maximum. The total work done by the impressed force is twice this. The other half is not accounted for. In the transient state the energy is partly electric, partly magnetic; in the absolutely steady state it is wholly electric; practically there is always conductivity somewhere, so that half of the work is spent in the Joule-heat of induced conduction currents, and is not radiated away, and, so to speak, lost. But with that we have no concern at present.
Now this distribution of displacement will not in general be consistent with that of electric current determined by the conductivity conditions. To be consistent, c the electric capacity, and k the electric conductivity, must be eveiywhere in the same ratio, or capacity x resistance (both specific) must be a constant. Hence, in general, either only one of the solutions is correct, or else neither. A little consideration will show that it is the conductivity solution that is correct, and that we must, after finding the distribution of electric force E from the conductivity conditions without any reference to those of capacity, determine the displacement and electrification to correspond to the thus-found electric force by the first two conditions of (87), viz.,
D = cE/47t, p = div D.
For we thus satisfy all the conditions, which we could not do by starting witli the dielectric problem, finding E to suit, and from it, finally, the distribution of current. Why we are able to solve the problem by the conductivity conditions alone is mathematically accounted for by the presence of p in the continuity condition relating to the displacement. The distribution of displacement D, and the distribution of electrification p, are such as to have no polar force when existing together; or the polar force of the displacement alone and that of the electrification alone are equal and opposite. Thus the electric field is not in any way disturbed, and the current is therefore also undisturbed and independent of the existence of elastic displacement.
On this point it should be remembered that whereas in considering a magnet as a collection of very small magnets, or any very small portion of the magnet as being polarised, the positive end of a polarised particle is that end to which the vector polarisation points, and is the end upon which positive magnetic matter may be assumed to be collected ; yet on the other hand, if we do the same with electric displacement, it is the negative electrification that is on the positive side of a particle, that to which the vector displacement points. We cannot, in the magnetic problem, imitate the above state of electric displacement ami electrification having no polar force. For the magnetic analogue of D, which is B/4Tr, B being the magnetic induction, is always continuous.
Now, to go further. At the commencement we assumed that the conductivity nowhere quite vanished, so that we have been, so far. considering the current and displacement produced by given impressed force in an infinite conductor, in which k is finite ev erywhere, whilst c is quite arbitrary, and may vanish in certain portions if we please; it being the conductivity conditions alone that determine the field of force. But, practically, we must regard certain parts of space as being wholly nonconducting. This, though apparently included in the preceding, viz., by taking the special value k = 0 in certain spaces, really, to a great extent, necessitates a changed mode of treatment. It is not sufficient to find E from the conductivity conditions, putting / =0 in noneon- ducting space. For this will not give the proper field in the nonconductors, but only in the conductors, unless at the same time c = 0 in the former, or, in special cases, c = constant.
Also, it may happen that when the conductivity is finite, though very small, in certain parts of space, it will be practically necessary to suppose it to be quite zero, on account of there being a first approximately steady state, and then, a long time after, a wholly different, quite steady state; which last, though it is not what is practically wanted, is what the conductivity conditions give. Consider, for example, a submarine cable whose ends are earthed through condensers. Put a battery in circuit at one end. The practical steady state is reached quickly, in a few seconds (of course not counting disturbances or absorption). But as the dielectric is slightly conducting, if we keep on the battery, the “ charge of the cable ” will in time—minutes or hours, according to the insulation— nearly disappear; theoretically, in an infinite time. This is the real final state, but the first approximately steady state is what is practically wanted, given, very closely, by assuming k = 0 exactly in the dielectric.
Now divide all space into two sets of regions, the conducting and the wholly nonconducting, including in the latter the dielectric in such a case as just mentioned. All conducting matter which is continuously connected must be regarded as a single conductor. Thus all the wires 011 a line of pules, if they are earthed, together with the earth, and all that is in conducting connection with it, form strictly a single conductor. But if we loop two of the wires, removing earth, they form a separate conductor. (Leakage ignored.) There may thus be any number of distinct conductors, each self-continuous, but wholly separated from all the rest by nonconducting matter, or else unbounded parti}'; though the last is, as regards conductivity, practically unrealisable. Similarly the rest of space, the nonconducting space, forms a number of self- continuous regions, each either wholly bounded by conducting matter, or else partly unbounded; which last is practically realised.
Now, selecting any one of the conducting regions, let the impressed force e be given in it. It is readily shown, by the simplest modification of the conduction problem for all space, that C (and of course E) are definitely fixed by the distribution of e and of conductivity. The modification consists in applying the space-integrals to the finite space occupied by the conductor, at the same time introducing the surface condition that the current is confined to the conductor, or is tangential at its boundary, which condition is expressed by CN = 0, if C be the current-density and N the unit vector normal to the surface. But this modification may be wholly avoided by integrating through all space as before, thus including surface terms, 011 the assumption k = 0 outside the conductor. The solution obtained applies to the conductor only; the part for the external space must be wholly rejected. There is an exception, namely, if c = 0 as well as k= 0 in the whole external space, when the external solution for the electric force will be the correct one, as will also be the case sometimes if c = constant.
Thus the internal electric state of any conductor depends solely on the conductivity and the impressed force in it, and is independent of all external conditions. (This is not true for the magnetic state of the conductor, which will be influenced by the external conditions not only as regards permeability, but as regards current and magnetisation; but at present it is onty the electric state that is in question.) We therefore settle the electric state of the whole of the conducting parts of space, one conductor at a time. Not only that, but in the same manner as before done in the case of the infinite conductor, we settle the displacement and electrification in every conductor, one at a time, from the known electric force in them, making no alteration whatever in the electric field.
There remain, finally, the nonconducting regions, and now the matter gets rather more complex. So far, our knowledge ceases at the boundaries between the conducting and the nonconducting regions. But at every point on these boundaries, on the nonconducting side, the tangential component of the electric force is fixed. For the second equation of induction is
curl (e - E) = 4ttG = pK, (23) bis
where G is the magnetic current, H the magnetic force, and /j, the permeability. Here E - e = F1} the electric force of the field by itself, not counting impressed force. Applying the Version Theorem to this, interpreting it for a mere surface, we see that the tangential component of Fj must be the same on both sides of the surface. Otherwise the surface would be the location of a magnetic current-sheet, of strength G per unit area. This continuity of the tangential component of the electric (similarly of the magnetic) force of the field is true whether the state be steady or not. When not steady, Fj is not polar. At present Fj = F, and is polar. We know, then, the tangential component of F (including its direction) at every part of all the boundaries. In terms of potential, we know its surface-variation over every boundary, but not its absolute magnitude. The last is unknown, not merely as regards a constant, but as regards n constants, one for each conductor, which, if we choose to employ potentials, must be mutually reconciled, so as to leave only one of them arbitrary. Now we cannot do this without knowing the state of the dielectric nonconductors. The potential must be found last of all, even in the conductors, whose electric state is independent of one another. Obviously, even w£re there no other reason to give, this would be a powerful argument against potential representing any physical state, in the same manner as electric force, current, etc., do.
As we took the conductors one at a time before to completely find their electric state independently, so now we may take the nonconducting regions one at a time, and find the electric state of each independently of the others. Selecting, then, a single nonconducting region, the conditions to be satisfied within it are,
D = rE/4/r, div D -=p, curl (e - E) = 0, (87) bis
where now, of course, e is the impressed force at the point considered. Or, eliminating D, and putting E = e + F, F being the polar force,
div ^(e + F) = p, curl F = 0 (90)
The electrification density p is now not to be found, as in the con- f I ductors previously, but must be given. Thus e, p, and r are given at each point, and D, E, F have to be found, or simply F, since that fixes the others.
There is also the before-stated boundary condition; FN being the normal component of F, the vector normal component is (FN)N, so that it is
F - (FN)N = YNYFN (91)
that is given over the bounding surface or surfaces. Or, simply YFN, the tangential F turned through a right angle 011 the surface, may be regarded as given. A preliminary examination, as regards energy, of the three conditions (90) and (91), shows that they are insufficient to determine F. We need to know, also, the boundary representatives of p, the electrification. But it is not the electrification at every point of the boundaries, but only their totals, that is needed, i.e. the charges. Let <jv q2, etc., be the charges of the separate surfaces, then
yj-SDNj, </2=SDN2, etc (92)
must be given, D being the displacement at a surface, and N1? IN2, etc., unit normals from the conductors to the nonconductor. For it is clear, by the continuity of the electric current, that if a conductor, wholly surrounded by a nonconductor, had a charge before tlie impressed force in the conductor began to act, such charge will not be altered in amount, though its distribution may be changed, when the impressed force acts. Similarly, impressed force in the dielectric cannot alter its amount, but only its distribution, nor can the introduction of external electrification. The same applies partly when the conducting surface wholly surrounds a dielectric \ neither e in the conductor nor in the dielectric can charge it. But, on the other hand, the introduction of any interior electrification, by continuity of the current, requires the surface to have a total charge equal to the whole interior electrification taken negatively. In any case, then, qv q2, etc., must be known, independently of external electrification, in the one case, or depending on interior electrification in the other.
It will be convenient to break up the problem into two, thus :—
(a) . Given e, p, and qv q2, etc., and that YFN = 0. Find F.
(b) . Given YFN, and that e = 0, p = 0, qY = 0, etc. Find F.
We take (b) first, as it is the simpler, and is also directly connected with the conduction current. That is to say, we suppose that before the impressed forces in the conductors began, the whole system was free from electrification or impressed force in the nonconductors. We have
div cF = 0, curl F = 0, SDN-0, YFN = 4-g, ...(93)
F being the force in the region. The first two apply to every point in the region, the third to any conducting bounding surfaces as a whole, whilst in the last g is given over the whole boundary. We have
2F1cF1 = ^F<-F + Sfcf+2 2Fcf, (94)
if Fj = F + f, whatever they be. Suppose now F to be a solution of (93), and F + f to satisfy the second and fourth conditions, so that F, is a solution of the second and fourth, but is otherwise unrestricted. By the first, D is continuous within the region, and, bj7 the third, at its boundaries as much leaves as enters the region. We may therefore close D completely outside the region without disturbing D within it.
D, thus extended, is a circuital fiux, and therefore may be represented by
4TTD = curl Z,
where Z is determinate in various ways. This makes
2 Fcf - 2 f curl Z/4jt = 2 conv VfZ/47r, (95)
This summation extends through the region. By the Convergence Theorem, the last form is at once expressible as a surface-integral over the boundary, viz.,
2 N VfZ/47r = I- ZVNf/4ir = 0,
the vanishing taking place because YfN = 0. Thus 2Fcf=0. Therefore, by inspection of (94), SFjdFj is made a minimum by f^O, making Fa = F, and proving F to exist. That it is unique follows by the same reasoning as in the conductivity problem.
The physical interpretation of the above, as regards Z, which is best taken as the magnetic impulse, or time-integral of the transient inductive magnetic force, is interesting, but must be left over. In the above, however, Z may be anjr vector whose curl is D in the nonconductor, its curl in the conductors being arbitrary. Thus, if we close D in any manner through the conductors, and call the complete system D1? then
Z = curl 2 Dj 4-r + any polar vector
satisfies the requirement. Or, instead of the curl of the vector-potential of D1? we may form the vector-potential of its curl [p. 206, ante]. As regards the transformation used in (95), if f and Z are any vectors, we have
conv YfZ = f curl Z - Z curl f;
and (95) follows because f has no curl.
Lastly, the assumption made in the above demonstration that Ft is possible, to satisfy the second and fourth of (93), though nearly obvious, is made evident by constructing any scalar potential whose tangential variation at the boundaries is of the required amount, its value elsewhere being arbitrary, and letting Fx be its slope.
SECTION XV. CONDUCTION AISD DISPLACEMENT (Condition).
As regards other causes influencing the electric field in the space external to the conductors, there rtmains the problem {a) of the last section. If, in it, we put e = 0, we reduce it to the common electrostatic problem :—Given the volume electrification within, and also the charges upon equipotential surfaces bounding the nonconducting region (at least when they require to be independently stated, apart from the volume electrification), show that the field of force is determinate. We might take this for granted, and confine our attention to impressed force only; but as it makes scarcely any difference to include electrification, we shall do so. We have then the conditions div c(e + F) = -iirp, curl F = 0, to be satisfied by F within the nonconducting region; and at any bounding surface,
- DN = q, YFN = 0,
the first of these expressing that the given charge (/ on the surface is the total normal displacement from it into the nonconductor, and the second that the force has no tangential component at the surface, or is normal thereto, or that the surface is equipotential.
Assuming F to satisfy the above, alter it to Fx = F + f in any way that does not alter the electrification or the charges. That is, subject f to
divt-f=0, 2Nd*=0
within the region, and at a surface, respectively. We have merely to show that this change increases the electric energy - FcF/87r, a quantity that must have a minimum, to show that F is determinate, by reasoning used before, which need not be repeated; and, since
2 F^Fj = - FcF + - i<f + 2 2Frf,
wherein the second sum on the right is positive, we have only to show that SFrf, the third sum, vanishes. Now, by the first condition for f, we have cf perfectly continuous within the region ; and, by the second, as much enters as leaves any bounding surface. That is, cf is an}' circuital flux whatever, either closed entirely within the region, or else partly or wholly through the rest of space. Therefore
cf = curl Z,
Z being any function whose curl is cf, and
2Fef=SZVNF = 0;
the first summation extending through the region, the second over the bounding surfaces, and vanishing because VNF - 0 at every point thereof.
This finishes the question of the determinateness of F. In any conductor it is settled by the distribution of impressed force and of conductivity therein, and the interior displacement and electrification follow. Then, knowing the tangential force set up at the boundaries of the nonconductors due to impressed force in tlie conductors, this is sufficient to settle the force all through the nonconductors when the distribution of capacity is given, provided there be 110 electrification in them, or impressed force, or boundary charges. No electrification or charges will be produced, as the displacement in the nonconductors can 1‘2 closed through the conductors. Should there be electrification, c ’ etc., in the nonconductors, these have no effect in the conductors, produce 110 boundary tangential force, and their effect in the nonconductors may with convenience be separately considered. Knowing F to be determinate through all space, we may construct a potential to suit it, fixing its value arbitrarily at any one point. Should we, however, make the potential the direct object of attention in our investigations, instead of the force, we should greatly complicate certain parts of them. For example, to show that e in the conductors fixes F in the nonconductors when there is no electrification, etc. If there are n conductors, n fields of force (namely in them) are found by the conductivity conditions. The potential is therefore known over the surface of all the conductors, except as regards a constant for each. If we assume these constants to be known, we may prove that the thus- fixed boundary potential fixes the potential throughout all the nonconducting regions. But the resulting fields of force in them will be (except by extraordinary accident) entirely wrong. For the potential thus determined gives charges to the bounding surfaces, whereas there should be none. We must, therefore, to get matters right, communicate to the bounding surfaces exactly equal charges of the opposite kind, and distribute them in such a manner as not to disturb the already correctly settled tangential force, i.e., distribute them equipotentially. The field of force will then get right, and the potential will be the sum of the wrong potential and that due to the charges added to cancel those given by the wrong potential.
As regards e in the nonconductors, it may be due to thermal or chemical causes, or be any other impressed electric force. Intrinsic electrisation is also included under e; thus, if J be its intensity, J = ce/Air gives us the value of e to correspond. p. 489.] So far as producing polar force is concerned, a distribution of e acts in precisely the same manner as would a distribution of electrification of volume-density cr = convce/47r. Here, however, the similarity ceases. It is quite different as regards the displacement and the energ)1', as e contributes to the displacement equally with polar force, thus, D = c(e + F)/4TT. Sometimes F acts to assist e, but more frequently it is the other way. We should also remark that this false electrification cr may, like p, the real electrification, have surface distributions over conductors ; whenever in fact, ce is not tangential. The surface-density is then — Nce/47r. But, unlike a real charge, which can be varied in distribution by influence, the false charge is fixed under the same circumstances. It (cr) should never be referred to as electricity or electrification without the prefix false, or some other qualification to distinguish it from Maxwell’s electrification or free electricity, which is always discontinuity produced in the elastic displacement, only to be got rid of by conduction, with dissipation of energy in producing heat. In fact, cr is as false electrification as the distribution of electric current round a bar magnet, which would correspond to the same magnetic field, is a false electric current.
Various Expressions for the Electric Energy.
There being any steady state, with impressed forces both in the conductors and in the nonconductors, also electrification and surface charges, let us obtain expressions for the electric energy. We may consider the whole field of displacement as made up of four fields, viz., that in the nonconductors due to impressed force in the conductors; ditto, due to impressed force in the nonconductors; ditto, due to electrification and surface-charges; and lastly, that in the conductors. The total electric energy will be the sum of the energies of the separate fields, together with their mutual energies, if they be not conjugate. We shall denote the four fields by the suffixes 1, 2, 3, 4; in all other respects employing the same notation.
1. First, let Ux be the electric energy in the nonconductors due to e4 in the conductors. Then,
Ux = 2 -i-EjDj = 2 iF/F^tt = 2 IPjDjN = 2 (96\
Here the first form of U1 is the form common in all cases, the energy per unit volume being half the scalar product of the force and the displacement. The second form is virtually the same, only putting the displacement in terms of the force, which is polar. These summations of course extend throughout the whole nonconducting regions. The third form is a surface-summation over the whole boundary, of half the product of the scalar potential and the normal displacement. The fourth form is also a summation over all the boundary, of half the scalar product of the vector-potential Zx and the false magnetic current gr We have
- curl Zj = 47^, VFjN =
Zx is the magnetic impulse, or time-integral of the transient magnetic force during the variable states which would occur if the impressed force were suddenly cancelled, ending in the removal of the displacement Dx in the nonconductors. Or, without altering the value of the summation, we may take Zx to be the time-integral, of the actual magnetic force, taken negatively, arising from putting on the impressed force, though this is less convenient in general. Should there be no electric current finally, e4 + F4 = 0, if F4 is the polar force in the conductor, and therefore 471^ = VNe4. In this case also,
Ui = -ie4Jr dt,
r being the true current at time I after starting e4, so that J^dt is the total displacement, elastic and conductive. The other half of the work done by e4 is spent in heat. Observe that it is necessary for the impressed force in the conductor to reach to its boundary, and to have a tangential component there, for it to be able to set up displacement outside without at the same time producing current in the conductor.
2. Next, let U.2 be the electric energy in the non-conductors due to impressed force in them only. Then,
U2 = 2 J E,D0 = 2 J e0D, = 2 JZ,g\, 1
=2Je./e2/41r-2p2rP2/4^ = 2ie/e;!/47r-2|/>, J l*’°«
The first form requires no remark. The second results because E2 = e2 + F2, and 2FoD2 = 0. In the third form Z., is the magnetic impulse (of e2 now, of course), and
- curl Z,2 = 47tIX, - curl e.j = ;
the object of introducing 47r’s being to harmonise as well as possible with the usual electromagnetic formulae. In the fourth form we have the difference of two sums, the first a constant, the second the energy of the field of the polar force by itself, as if due to electrification. In the fifth form the energy of the polar force is expressed in terms of its potential and the false electrification o\ All the summations are taken throughout the nonconductors. There may possibly be surface summations as well, if e exists at the boundaries. All the formulae in (96)2 are the same as if there were no conductors present.
3. Let U3 be the common electric energy due to electrification p and surface charges qv q2, etc.
Ua = 2 P3D3 = 2 £F3D3 = 2 ^ (96)a
The first and second forms are identical, the force being polar. In the third form the energy is expressed in terms of the potential and electrification. We may also here employ the magnetic impulse, by a device, but with no utility.
4. Let Ui be the electric energy in the conductors due to e4 in them. Then
U, = 2 p4D4 = 2 h(e4 + F4)D4 = 2 ie4D4 + 2 i/>4, (96)4
Here F4 is the polar force in the conductors due to e4, and 1\ its potential; is the interior electrification, if any. Should there be none, D4 is closed within the conductor, and we may write
U4 = ?hZ,g4,
if curl Z4 = - 47rD4, and curl e4 = - 47rg4.
It is clear that a similar form exists for the energj* in any volume in which there is no electrification, g being represented by the tangential boundary force turned through a right angle, besides the part depending on e4, if it exist in the volume. (Here e4 is the same as in the previous concerning Ur)
5. Lastly, there are the mutual energies. We get rid of Uu, U2i, and t/s4 at once by observing that the fourth field is in a different place from the first, second, and third, so that these quantities are zero. We can also get rid of UVA and Uos; the first and third fields, though coexistent, are conjugate, and so are the second and third. Thus,
Z7w = 2DlF3 = 2Z1VNFs = 0,
Zx being the same quantity as before. The second summation extends over the boundaries, and vanishes because F3 is normal to them.
Similarly show that Un vanishes. There remains only U12, which does not necessaril}’ vanish.
^ = 2^ = 2 BA ( ,
= 2 Zjg., = 2 Zog'j = 2 e.^Dj. | ^ h
Here the quantities are the same as before used in treating Ul and U2. Finally, U being the complete electric energy in all space,
u U. + U.+ U^+U^+U,.
Here the first three components of U may be taken together conveniently. Let the impressed forces in the conductors and the nonconductors start at the same moment, and Z be the magnetic impulse, = former Zx + Z2, then
U1+ U2 + £/j2 = 2 -|-Z g, where g* = former gt + g.„ that is,
■±7Tg = - curl (Ej + E2), with the equivalent boundary representative.
Finally, we may, by means of the various conjugate properties, put together the potential results. Let P be the resultant potential from all causes; p the electrification density whether in conducting or nonconducting matter, defined by p = div D, including in it surface electrification; cr the density of false electrification, defined by cr = conv re/4:ir = conv J, if J be the (equivalent) intensity of electrisation, i.e., J = re/4-, whether in the conductors or not; then
U=?beJ + ?},P(p-<r) (97)
expresses the total electric energy, the summations being through all space. Notice the minus sign before the false electrification. (97) may also be easily proved directly.
Here, and in any of the summations containing a scalar potential, any constant may be added to the potential, owing to the sum of the electrification being zero, ditto of false. There is an apparent exception when, as may be imagined, tubes of displacement go out to infinity, though this cannot really occur. In such a case, we must choose P to vanish at infinity. But we may get over this by regarding the end <>f space as an electrified surface. The surface-density will be infinitely small, but the charge on it finite, being the exact negative of all the other electrification. Counting it, the constant in P again becomes arbitrary.
AVe may extend the results to apply to transient states by including in e the electric force of induction, = - A, if A be such that curl A = B, the magnetic induction. Considering fTx during the transient state, we find
l\ = 2 ^0, + 2 JZjgj
at any moment, Z: being the magnetic impulse that would arise from removing e at that moment, Gj the real magnetic current B/47T at the moment, and g^, as before, the false magnetic current at the boundary. The first summation extends through the nonconductors. AVhen the steady state is reached, €1^ = 0. and Zx and arrive at their before-used steady values. Similarly for U2, Uv and t\2 during their transient state.
Section XVI. Magnetic and Electric Comparisons.
(a) . Let two large masses of iron be united by a thin iron wire. For simplicity of exposition let the two masses and the wire be homogeneous throughout, and only “ elastically ” magnetisable. Call the masses 1’ and Q. and the wire t>\ Let air surround them to a great distance, to get rid of foreign influences. The masses may be conveniently imagined brought near one another, and the wire not to join them straight, but to be led round through the air, although these assumptions are immaterial.
Now let a portion of the wire, which we shall call x, be intrinsically magnetised to intensity I parallel to its length. Here, again, we may conveniently imagine x to be very short, to be a mere disc, in fact, perpendicular to the axis of the wire. The magnetisation I is the same as an impressed magnetic force h similarly distributed, if I = /j.h/47r, and /x is the permeability of the iron. Instead of this impressed force in x, we may equivalently have an electric current round x, say in a coil of very small depth. Or, instead of this real current, we may refer to a false electric current round x.
The magnetic induction passing through x due to h (or its equivalents) is both altered in amount and in distribution from what it would be were x alone in the air. Its amount is increased by the permeance of the magnetic circuit being increased. And its distribution is altered, on account of the permeability being changed unequally in different places. It is, to a certain extent, led along the wire to the mass P (of course with a large amount of leakage on the way), through the air to Q, and through the wire back to x, completing the circuit. But if we increase the permeability of the iron we increase the induction, and at the same time cause relatively more of it to pass through P and Q, with relatively less loss from the wire (out on one side of x and in on the other). Indefinitely increase the permeability. The total induction in the circuit reaches its greatest value. The leakage becomes relatively insignificant, and by making the wire thinner may be reduced indefinitely. Then nearly all the induction is led through the wire to P, through the air to Q, and back to x through the wire. The total induction depends on the form and relative position of P and Q (with which, to be exact, we must count the two portions of w on opposite sides of x). In this final state the magnetic force in the air is everywhere perpendicular to the surface of P, Q, and w, except at x, where it is partly tangential.
(h). Replace the iron by a solid dielectric nonconductor of high electric capacity or low electric elasticity. Keplace the impressed magnetic force h by impressed electric force e; or by intrinsic electrisation of intensity J = ce/47r, if c is the electric capacity; or, finally, by a magnetic current round x, similar to the before-used electric current, except that it must circulate the reverse way. The flux is now elastic electric displacement. Its distribution is similar to that of magnetic induction in (a). On indefinitely increasing c, or doing away with the electric elasticity in the solid dielectric, the greatest displacement is obtained. It then passes through the air mainly from P to Q. The electric force in the air is normal to the surface of the solid dielectric, except at x again.
(c) . If in the case (b), before making the capacity infinite, and when, consequently, the final state (b) is not assumed, we communicate to the solid dielectric throughout its whole substance any finite degree of electric conductivity, we shall cause the displacement in the air to immediately (practically) assume the final state (b). The conductivity may vary anyhow, and likewise the electric capacity, which last may in fact be zero now. This is the common case of char^ins: a condenser,
C D '
except that it is usual to consider the conductor as having no electric capacity.
(d). If, in the case (a), of magnetic induction, we could communicate to the iron any finite degree of what, by analogy, we may call magnetic conductivity, rendering it impossible for the iron to support magnetic force without a magnetic conduction current, with dissipation of energy, it would be unnecessary to make the permeability infinite to obtain the final distribution in the air there mentioned. The permeability might be finite or zero, and the magnetic conductivity have any distribution in the iron. The final state would be as in (c), with magnetic induction instead of electric displacement. The examples (a), (b), and (c) are real, except as regards the assumption of infinite values of and c in (a) and
(b) respectively, which may be conveniently imagined to become very great, though not really infinite, thus letting us approximate to the required results without using impossible conceptions. The example (jl) is unreal. It is introduced to show the analogy.
In examples (b) and (c) the displacements out from P to the air, and in from the air to Q are identical. (The final state (b) is referred to.) But there is no electrification in case (b), whilst there is in case (c). If in case (b) we remove the wire from P and Q, to a distance, we shall at the same time do away with the former apparent charges of P and Q. There will be left merely the small displacement from the wire on one side to that on the other side of x where the impressed e is, some small part of which will of course go through P and Q rid the air. But if, in case (r), we remove the wire, P and Q will be left charged as before ; besides that we have an insignificant wire-charge. Herein lies the difference between the elastic and conductive displacement.
Similarly, if in the example (a) we remove the wire from P and Q, we at the same time do away with their apparent magnetic charges. AVliilst, if we could realise the case (d), on removal of the wire the bodies P and Q would be left really charged magnetically. They would apparently be unipolar magnets, though without any interior polarisation, it being a matter of the terminal induction of the medium between them. In the final states of (a) and (b) there is finite interior induction or displacement, being the exact complements of those outside in the air, although there is no interior magnetic or electric force, and consequently no interior magnetic or electric energ}'. But in examples (d) and (c), having the same external induction or displacement as in (a) and (b), we have no interior induction or displacement, as well as no magnetic or electric force and no interior energy.
(icc). Going back to (c) with an impressed electric force in the wire, let us join P to Q by a second conducting wire, attaching it to them anywhere, but for distinctness, away from the first wire. We shall now have a steady conduction current in the closed circuit made, as well as elastic displacement in the air, having a different distribution to the previous; the amount of change depending materially upon the resistance of the second wire compared with that of the rest of the circuit, being small when it is great, and conversely. The electric force is no longer normal to the conductors at any part of their boundary. There may now be any amount of internal electric energy, existing independently of the external, depending upon the internal capacity of the conductor, in which there is electric force, and consequently elastic displacement if there be capacity for it.
(bb). If, in the final state (b), we unite P to Q by a second dielectric wire also of infinite capacity, we make a closed dielectric circuit which has no elastance (or elastic resistance to displacement), and in which infinite displacement corresponds to finite impressed force, and likewise infinite electric energy. Clearly this is an impossible example, and we should never reach the final state. Let us therefore modify the conditions somewhat. Do away with the impressed force e at a-, and substitute a bodily distribution of impressed force similar to that of the polar electric force (reversed) in the conductor in example (cc). Or, substitute equivalent intrinsic electrisation. Or, finally, a distribution of magnetic current 011 the boundary, given by g = VFN/47T, if F is the external electric force in example (cc), and N a unit vector normal from the solid to the air. This magnetic current goes round the wire at x in the same direction as in (b), and oppositely round all the rest of the circuit. It is strong at x and weak elsewhere, of such strength that total at x equals that elsewhere.
With this changed distribution of impressed force or equivalents, the external electric field will be identically that in (re). The impossible infinite internal displacement is abolished, the total impressed force round the circuit (of the solid dielectric) is zero; there is a finite internal displacement which exactly closes the external; and there is no internal energ}'', because there is 110 electric force. (That is to say, as c is increased the internal energy becomes less and less without limit, the electric force going down to zero, the displacement assuming a finite value.)
(cia). If in the final state (a) we unite P to Q by a second iron wire, also of infinite permeability, we make a circuit of infinite permeance in which is a finite impressed force, so that we have infinite internal induction and magnetic energy to correspond. Modify as in (bb). Do away with the impressed magnetic force h at x, and substitute a bodily distribution of impressed magnetic force (or intrinsic magnetisation) similar to that of impressed electric force in (bb) or of the polar auxiliary force (reversed) in (cc). Or, substitute a distribution of electric current on the boundary given by y = VNH/4~, if H is the external magnetic force, when similarly distributed to the external electric force in (cc) and (bb). This boundary false electric current is similarly arranged to that ot magnetic current in (bb), except in reversal of direction of circulation everywhere.
Under the changed circumstances, there will be no internal magnetic force, and no magnetic energy, but finite magnetic induction, which will exactly close the external induction, which, again, has the same distribution as the displacement in the air in cases (cc) and (bb).
(dd). We should cause the same external induction as just arrived at, without modifying the distribution of impressed force, if we could impart magnetic conductivity to the iron-circuit. That is to say, in example
(d) unite P to Q by a second magnetically conductive iron wire. If the distribution of magnetic conductivity be the same as that of electric conductivity in (cc), we shall have the external magnetic induction distributed like the electric induction in (cc) and (bb), and like the magnetic induction in (aa). There would be a steady magnetic conduction current in the iron-circuit, similar to that of electric current in (cc), and the internal magnetic energy (like the internal electric energy in (cc)) would be arbitrary, depending upon the permeability of the iron. (In this replacement of electric conduction current by magnetic conduction current, the electric conductivity of the iron-circuit is ignored. Similarly in (ddd), later).
Any magnetic field is accompanied either by true electric current T or by impressed magnetic force h. For the latter we may substitute a distribution of false electric current y, such that if it were real the induction would be the same. Similarly, in a nonconducting dielectric in which there is no electrification, the elastic displacement is accompanied either by a true magnetic current G = B/47r, if B is the induction, or by impressed electric force e (or equivalent electrisation). For the latter we may substitute a false magnetic current g.
Also, U being the whole electric and T the whole magnetic energy, we have
47ry = curlh, 47rg=-curie, T=^\ A(F + y), iZ(G + g>).
Here A and Z are the electric and the magnetic impulses that would arise on the sudden removal of h and of e respectively (also partly due to r or to G), when the induction 01 the displacement would wholly subside and spread.
Now, if we confine ourselves to a limited region, we shall still have the electric energy within it expressed in the same manner, provided we include in the false magnetic current g a boundary magnetic current given by g = VFN/47J-, if F be the electric force (not counting impressed e) and N a unit normal from the boundary to the region ; and further, provided that it be possible to close the displacement outside the region, thus not interfering with that within it. This is not alwaj's possible. And the magnetic energy is expressible in the same manner, if we include in y a boundary false electric current given by y = VNH/47T, if H is the magnetic force, not counting impressed h ; with no reservation as to the possibility of closing the induction outside the region, because it is always possible.
But in neither case is it generally true that these distributions of current (including those on the boundary), would, if they existed alone, set up the identical displacement in the one case and induction in the other, independent of what may be beyond the region. (To illustrate this, we need merely take a piece of a round tube of induction, with plane ends cutting the tube perpendicularly. If the induction is uniform and steady, the only electric current is the false current round the round part of the tube, and this solenoid will clearly not produce h.e.p.—vol. i. 2 K uniform induction, but induction that spreads out in passing from the middle to either end of the tube.) But we may very easily produce independence of the external state, by short-circuiting the unclosed displacement or induction, as the case may be, by making either the electric capacity or the permeability infinitely great beyond the region. Or, merely by making cor JJ, infinite in a mere boundary-skin, or over enough of it to completely close the displacement or the induction. When this is done, the currents, magnetic or electric as the case may be, produce the exact given displacement or induction within the region, without external aid.
In examples (b) and (c) we have identical displacement outside the solid dielectric or the conductor. Since the states are steady, G = 0. And, as there is no impressed force in the air, which is our region of electric energy, g is reduced to the surface-current g = VFN/47T. This is zero except round x (because F is normal elsewhere), and there it is g = VNe/47r, which is the value of g in the expression Z7 = 2 ±Zg for the electric energy in the air.
In examples (a) and (d) the total magnetic energy in the air in the identical distributions of induction is T= 2 A Ay, since there is no true electric current. And the false electric current (or it might be a true current in a thin coil) is confined to round :r, being given by y = VhN/4tt.
But in example (cc), with electric current in the wire, and in (bb) as modified, the magnetic current is no longer confined to be round :c, but extends over the whole conductor in (rc), and solid dielectric in (bb); being g = VFN/47r, wherein F is no longer normal to most of the surface. This g, used in Z7 = 21Zg, gives us the electric energy in the air in both cases. But there is a perfectly arbitrary amount in the conductor in example (cc), and none in example (bb). In order, therefore, that in (bb) the magnetic current should exactly correspond to the external displacement, we must, as before explained, short- circuit the latter beyond the air; i.e., make c = co outside the air- region. We then get the case (bb) as modified, and see the reason of the modification.
Similarly, in examples (aa) and (del) the energy of the identical external magnetic inductions is T = 2|Ay, wherein y is the boundary electric current given by y = VNH/47T. But in (del) there is also an arbitrary amount of internal magnetic energy, so that the boundary- curreut y would not generally, existing alone, produce the actual external magnetic induction. It can be made to do so by abolishing the impressed force h at x in (dd), and making fx = oo throughout the iron. Then we obtain (aa) as modified.
(ccc). In example (cc), with dissipation of energy by conduction, there is also magnetic energy, not before mentioned, as it would have introduced some confusion. For all space its amount is T= 2 A AC, if A is the electric impulse and C the conduction current in the wire. That part of it in the air is 2-i Ay where y = YNH/4tt. This is a boundary-current, not round the wire, but along it, of the same total amount as the real current in it, as if it were all pressed to the surface. It would not alone set up the external field, but would do so if we short-circuit as much of the induction of the field as is unclosed (at the boundary).
(ddd). Similarly, in the unreal example (dd) with dissipation by the magnetic conduction current in the iron-circuit, there is, besides the magnetic energy, also electric energy. In all space, the amount is U= 2 ^ZG, if G be the magnetic current in the wire. That in the air alone amounts to 2 |Zg, where g is a surface magnetic current given by g = VFN/47r. It is along the wire, like G inside, and of the same total amount, as if G were pressed to the boundary. It would exactly correspond to the external displacement, if the latter were short- circuited at the boundary. The unreal (d), (dd), and (ddd) are merely brought in because they, to a certain extent, assist the other comparisons.
SECTION XVII. THE MAGNETIC FIELD DUE TO IMPRESSED E.M.F.
Sections XIII., XIV., and XV. were principally devoted to the consideration of the electric field set up by impressed electric force, also as modified by previously existing electrification. There is also simultaneously a magnetic field, if there be electric current. This will depend on how the impressed forces are distributed, which question we need not return to further than to say that should they be wholly in nonconducting regions there can be no steady current, but merely a transient one producing elastic displacement; and that if there be impressed force in a conducting region, which is the first condition fur there to be a steady current, it must not be polar in its distribution therein, and with its lines perpendicular to the boundary, or there can be no current again. (The term polar force is borrowed from magnetism to signify any force whose distribution is such that its integral round any closed curve is zero. This is the most useful property by which to identify a polar force. The lines of force start from certain places and terminate at others; these places are the poles, in an extended sense ; any pole, positive or negative, may be conceived to send out a definite amount of “force,” uniformly in all directions, i.e., according to the inverse square law. The mathematical expression is curlF = 0, if F be polar; the boundary representation of curl F being the tangential component of F turned through a right angle on the surface. That the lines of F must be perpendicular to a series of surfaces does not sufficiently identify a given force with a polar force, as this is not inconsistent with the integral force in a circuit being a finite quantity, and so giving rise to the corresponding flux.) Thus to have current in a conductor the impressed force must have a finite value in at least one closed path entirely within the conductor. The current thus depends upon the “ curl ” of the impressed force. This is of great importance in the theory of the Volta-force or other boundary forces. The curl of any force is always arranged in closed lines, e.g., the closed line at the common meeting-place of zinc and copper (in contact) and a medium surrounding them. [As stated, the current produced by impressed force depends upon its curl, but this does not necessitate that the impressed force should be in the conductor. If the curl is the same, the force may equally well be outside it, and yet produce the same fluxes. Of this, more later J
Supposing now the arrangement is such that there is current, the determination of the state of the magnetic field from it is, in comparison with the determination of the state of the electric field anywhere, a comparatively simple matter, the former being a reduced and greatly simplified form of the latter, with changed meanings of the quantities concerned. The flux is the magnetic induction. That has no divergence, to begin with—one simplification as compared with electric; displacement. JNext, there is no magnetic current, as it ceases when the state becomes steady. And, finally, the ratio of the flux to the force (magnetic), or the permeability, is everywhere finite, so that there is no division of space into permeable and impermeable regions. In brief, the full statement of the conditions is contained in
B = /u,H, divB = 0, curl (H - h) = 4TTC ; (98)
the first being the linear connection between the flux and the force, the second expressing the continuity of the flux, and the third the relation between the magnetic force H and the current C, in which h is the impressed force of intrinsic magnetisation I = fih/Av. As, by the linearity of the equations, the field due to C and to h is simply the sum of their separate fields, we may put h - 0 at once, and therefore deal entirely with induced magnetisation (quasi-elastic).
If we integrate the third equation (98) on this understanding, we see that II may be any vector whose curl is 4-n-C, and is therefore indeterminate as regards a polar force, F. We have then H = h, + F, wherein, on account of the presence of F, we may choose hT to have no divergence. It is then definitely given by
hj = curl of vector potential of C = curl 2 Cfr,
if r be the distance from an element of C to the place where hj is reckoned.
As thus defined, hj is what is usually called the magnetic force of the current. It is the magnetic force of the current if there be no variation of permeability anywhere; otherwise it is what the magnetic force would be if there were no such variation. The absolute value of the permeability, provided it be the same everywhere, is a matter of indifference, so far as hT is concerned. The polar force F therefore represents the change made in the magnetic force by variations of permeability, due, of course, principally to the presence of iron.
For limited purposes, hj may be regarded as the impressed magnetic force due to a distribution of intrinsic magnetisation I1 = /xh1/47r. That is, if we abolish the electric current and substitute I1? the magnetic force and the induction would be unchanged. But whilst h of real intrinsic magnetisation may be arbitrary, and is usually in very limited portions of matter, the lines of as we have chosen it, are closed, and extend over all space in general, though by particular arrangements of current they may be shut out from certain spaces.
Let T be the total magnetic energy due to C. The two most noteworthy forms for T are the first summation being of the scalar product of the magnetic force and induction (-f Sir), and probably representing the real distribution of the magnetic energy; the second of half the scalar product of the current and the electric impulse A, which is excessively unlikely to be anything near the real distribution. In (99)j A may be any vector whose curl is B; but to give it the most physical significance, it is best to take it to be the electric impulse arising from inertia, or the time- integral of the electric force of induction on sudden removal of the impressed force e keeping up C. It is a scientific concept which does not express any physical state or condition. The electric impulse A at a given place does not depend upon the magnetic state there, but upon its condition everywhere; as, 011 removal of e, disturbances are propagated to the place, these determine the electric force of induction whose time-integral is A.
Other forms are useful in showing the influence of variable permeability. Thus
T— 2 ^hjB/4?r (because 2 FB = 0), 'j = 2 - 2 iF/xF/4tt = 2 Ihjlj - 21 Slp. f
Here hj and I2 are definitely known, by the preceding. The new quantities 12 and p are the magnetic potential and the volume-density of magnetic matter; the polar force being the slope of 12, or F = - Y12, whilst p is given by
p = conv Ij = - (4:r) /a, (when //. is scalar).
Thus p exists only at places where the permeability varies, therefore mainly at the bounding surfaces of different kinds of matter, or, disregarding perfectly abrupt changes, in thin layers at the bounding surfaces.
If V/x be perpendicular to h15 we have p = 0. Hence, starting with p constant, when h: is the real magnetic force, if we select a complete tube of force, or any region bounded wholly by lines of force, and alter its permeability to any other value (constant throughout the region), the magnetic force will be unaltered, whilst the induction will be altered within the region in the same ratio as the permeability. If now we choose to ignore the changed permeability, we may ascribe the altered induction to an additional electric current, on the surface of the region, perpendicular to the magnetic force and of the proper strength to produce the increased induction. In (99)1} A will be altered by a quantity depending upon this false current. By adding more and more tubes of force to the region, wTe finally include all space, or alter the permeability everywhere in the same proportion. Then, as might be expected, the false current to account for the increased induction 011 ignoring the altered permeability, occupies the same situation as the real, in fact increasing its strength in the same ratio as the permeability was increased.
There is another form for T which is very curious, related to the electric energy due to the same impressed electric force e. Let U be the total electric energy, then U= 2 ^ED, if E be the electric force and D the elastic displacement. Now this last is the time-integral of D, the transient displacement current during the charge. Let Dj be the simultaneous transient conductive current, so that their sum is the transient current of induction during the charge, a circuital system of current which finally ceases, and which, when added to the final current C, makes up the actual current Y at any moment during the charge. Thus,
r = C + D + Dr
The time-integral of Dj is D1} the complement of D ; the two together being circuital. Then, to match U= 2 A ED, we shall have
T= -2PD, (99),
This we may verify by a former equation. Assuming it to be true, wre have
U-T = 2 m(D + DJ = 2 Ae(D + DJ,
or, 2( U - T) = 2 Ej(r - C)dt = 2 ej(r - G)dt, j
by definition of D and Dx, which expresses that the work done by e during the transient state is 2(U- T) more than if the current started instantly everywhere in its final distribution. This proof of (99)s therefore rests upon equation (64). But it is easity proved directly, by using the electric impulse A. Thus, by (99)1}
T= 21 AC - 2 AA/LE, if k = conductivity,
.(99),
-ipj
Now, let Ea be the electric force in the transient state, i.e.
Ej = e + f - A, where f is polar.
Then *A = fc(e + f)-(r-D),
therefore, T= 2 |E J{£(e + f) - (C + Dj)}^,
by definition of Dx; and, since 2 keE = 2 eC = 2 EC, and 2 fC = 0, we are reduced to (99)3, as required. The equation (99)4 may be transformed to
r-r=2|-zg, (99)5
if we have
- curl Z = 4tt(D + Dx), and - curl e = 4irg.
Here g is the false magnetic current corresponding to the impressed force, e, going round the lines of e, roughly speaking. And Z may be taken to be the magnetic impulse at a point, the analogue of A, as we may thus verify. By the just-given equation of Z, differentiating to the time, we get
- curl Z = 4tt(D + Dj) - 47r(r - C) = curl (H2 - H),
if Hj be the magnetic force at any moment during the variable period, H being its final value. Therefore
-Z = H1-H + F1I
where F2 is polar, showing that - Z may be taken as the magnetic force of induction. Or thus : remove e suddenly, the time-integral of the magnetic force in the variable period following at any point will be Z. The definition of Z may be extended so that its time-variation shall be the actual magnetic force; but it is simplest, and harmonises best with the electric impulse, to make it refer to the variable period only.
Take e to exist only across a single thin slice of a wire, the most elementary case. Then g will be round the boundary. D + is then the integral current through g during the charge (not counting the final current, if any); or, reversed, it is the integral current through g when e is removed. If this current be oscillatory, it may amount to nothing in the total. If so, the potential and kinetic energies were equal. The value of Z at the place of g is also zero, of course, by (99)5, the magnetic force there reversing with the current. (The place of g is where energy leaves the seat of e when it is-working.)
Another pair of allied expressions for the parts of U and T outside the conductors is Expressed by
T= ^j*f//2conv YE1H/4rr, U = h^di 2 conV VEHj^zr. ..
Here Ex and Hj are the values of the electric and magnetic forces at time t after e was started, and E, H, their final values; the time-integral to include the variable period, and the 2 being summation through the whole space outside the conductors.
By means of the convergence theorem we may at once transform these expressions to surface-integrals over the boundaries of the conductors. Thus, let N be the unit normal out from conductor to nonconductor, and
VElN = 47Tg1, VNH1 = 4jry1, YEN = 4jrg, VNH = 4ry; y and yl being therefore boundary (false) electric, and g, gx boundary (false) magnetic currents. Then
T = - 2 JHfg^ = - 2 iy= 2 JAy,
V = - 2 = - 2= 2 JZg,
wherein of course the S’s are mere surface-integrals, since on the surface only are the quantities g and y. In the last forms, which we had occasion to employ in the last section, A and Z are, as there and in the former part of the present section, the electric and magnetic impulses. Notice, however, that the g in (99)7 is not the same as the g in (99)6, and there defined; the present g’ extending over all the boundary in general; although, should T- 0 (no final current), we see by (99)5 and (99)7 that they are then identical, situated round e, which must reach the boundary.
Here we may notice a peculiarity of interest in connection with the difference of treatment of vectors according as they are circuital or polar. Suppose we have two conducting bodies in air, and charge them oppositely by an impressed force in a connecting wire. The electric energy set up is U = 2 ±Zg, g being round e only. But now suppose we disconnect the wire from the bodies and take it, with e, away to a distance. We have altered the field very slightly, so that Z has nearly the same value anywhere as before, and U also. But g has gone altogether. How then does 2^Zg apply, the force being normal to the conductors ?
Notice that, as stated in the last section, this formula only applies when it is possible to close the displacement outside the region in which U exists, so as not to pass through it and alter the field. Thus tlie formula applies to all the space between the two oppositely charged conductors provided we leave out a little piece joining them, along which to let the displacement return. This little piece may be reduced to a mere line, thus infinitesimally altering the field. The formula then reduces to a line-integral, which will be found to become ^ charge x difference of potentials, the common electrostatic formula.
On the other hand, if we join the conductors by a wire through which they will discharge (unless balanced by impressed force in it), the formula acquires reality at once; g at first moment being at place of contact (round the spark), and thereafter over the conducting surface generally. During the discharge there will be a real magnetic current G in space, and the value of U at any moment will be 21Z(G + g). [See last section, p. 513 ]
SECTION XVIII. NORMAL ELECTROMAGNETIC SYSTEMS. ENERGY CONJUGATE
PROPERTIES.
The specification of the complete state of the electromagnetic field at a given moment requires a knowledge of seven quantities. We must, in the first place, know the electric capacity and conductivity, and the magnetic permeability, c, k, and /x. Next, we require to know the electric and the magnetic force, E and H. From these five data we know, by the linear relations, the conduction current C = £E, the elastic displacement D = cE 47r, and the magnetic induction B=/xH. We also kn ow the electric energy AED, the magnetic energy |HB/47r, and the dissipativity EC; all per unit volume. But, in addition, we require to have given the impressed electric and magnetic forces, e and h. Then, by the two equations of induction,
curl (H - h) = 47tF = 47r(C + D), ^ curl (e - E) = 4-G = B, J
we know the true electric current T, and therefore the displacement current D ; and also the magnetic current G. As for the electrification, it is known because D is known, of which it is the divergence. The seven data may be otherwise stated ; for instance, instead of E and H, we may have I) and B, or C and B. As regards the number of distinct numbers on which these seven quantities depend, if we take any three rectangular axes of reference, the four vectors E, H, e, and h require three numbers each, making twelve, and the three operators e, k, and /x require six numbers each (if there be no rotatory k), making eighteen. Thus altogether thirty numbers are concerned ; or thirty-three, if there be rotatory conductivity. In case of isotrop}^ the number is fifteen, owing to c, k, and /u, being then simply scalars.
We not only know the complete state of the system, but the rate at which it is changing; for E and H are known, and therefore, if the impressed forces be given at every moment, we can find the changes it goes through under their influence; or, if they be absent, in settling down to a state of equilibrium.
The equation of activity at any moment is ^er + ?hG = Q+U+T-,
the left member being the total activity of the impressed forces in all space, and the right its equivalent, the sum of the dissipativity Q (Joule-lieat per second) and the rates of increase of the electric and magnetic energies.
Now, suppose no relative motion of masses is permitted, thus making c, k, /a functions of position only, and excluding the impressed forces brought into play by such motions. If now e and h be constant with respect to the time, the system will settle down to a stead}' state, in which 2eC = Q simply. If, further, there be no e in conductors, or, more generally, only such distributions as may cause elastic displacement, but no steady conduction current, the final field is simply that due to h and to the e left, and to electrification and its surface equivalent, the charges of conductors.
But, by the linearity of equations, the inductive phenomena during the subsidence to the final state under the influence of steady e and h may be got by superimposition. We may therefore, in investigating subsidence, take e^O, h = 0, and no electrification or charges on conductors ; so that the subsidence is to a final state of no E or H anywhere.
We then have, at every moment after removal of impressed forces, Q+U+i'= 0; (100)
the rate of decrease of the sum of the electric and magnetic energies being equal to the dissipativity. Q, U, and T are all necessarily positive, being sums of squares, or else of positive scalar products. (For instance EC = EZE ; if C is parallel to E, it is a square, kW; if not parallel, their mutual angle must be always acute.) This necessary positivity is of the greatest importance, as it excludes the possibility of indefinite increase of normal systems of force left to themselves, making them always subside, either without or with oscillations. Let, next, there be two systems of electric and magnetic force dis tinguished by the suffixes j and 2, so that their equations of induction are
curl Hj = 471-1^, - curl = 47rG.,, curl H2 = 47rr2, - curl E2 = 47rG2.
Using the third and second of these, we find by space-integration,
2 Ejr2 = 2 Ej curl H2/4tt = 2 H2 curl E^tt = - 2 H2Gr
Similarly, by the first and fourth, we shall get 2E2F1 = -SHjGgj so that we have
2 (E^g + HA) = 0, and 2 (E^ + HjGa) = 0; or, 2 EjC2 + 2 EjDg + 2 H2B1/4?r = 0,1 (101)
2 EjjCj + 2 EgDj + 2 H1B3/4tt = 0. J
If we add these, we shall obtain
Qls+Uls+f12 = 0, (102)
the equation of mutual activity, UV1 and Tvl are the mutual electric and magnetic energies, and Qv2 the mutual dissipativity, or the excess of the total dissipativity when the two fields co-exist over the sum of the separate dissipativities.
Let, next, the arrangement of E and H be such that in subsiding they change in magnitude only, not in distribution. Let E0 and H0 be the distributions at time t = 0, and, at time /,
E = E0e"e, H = H0e’“ (103)
The constant n is the reciprocal of a time, and is of course negative, if the subsidence be real. The larger n, the more rapid the subsidence. E and H thus defined constitute a normal system of electric and magnetic force.
Now,
Q = 2EA-E, tf=2E('E/87r, r= 2 H/xH/Btt, = =U0t™,
by (103), if Q0, U0i T0, be the initial values. From this, t=2nT, and U = 2nUy which makes the equation of activity (100) become
Q+2>i(U+T) = 0;
or the ratio of the energy left at any moment to its rate of leaving is constant, = - (2m)-1.
If then the two systems to which equations (101) refer be both normal, with rates of subsidence ?ij and n2, we shall have
Ql + 2n1(L\ + T1) = 01 \ and Q2+2n2(U2+T.2) = 0, J
when they exist separately; and, in addition, when they co-exist, the equations (101), in which d/dt = n1 or ?i2, according to whether it operates on the first or second system. Now
EA-EA. ~ ^2®i» and E1C2 = E0Cj,
if there be no rotatory conductivity coefficient; so that (101) becomes
l2 + ?l2^12 + ?i1^12 = ^> \ /i/)K\
i(?i2+«i^12+7i2r12=o, j ^ '
where
Q12 = 22 E^, U12 = 2 EjD2, T12 = 2 H1B2/4tt.
Between the equations (105) we may eliminate in succession either the Q, or U, or T. Thus we get, if we leave out the common factor (wi - n2),
0-^12 ^12»
0 = ^>Q\o + (ni + w2^12>
which are the universal conjugate properties of normal systems.
The first tells us that the mutual potential and kinetic energies of two normal systems are equal. This being true at every moment during the subsidence, it follows, by (102), that the mutual dissipativity is derived from them equally. This is the interpretation of the second and third of (106), the second saying that the mutual dissipativity equals twice the rate of decrease of the mutual kinetic energy, and the third that it equals twice the rate of decrease of the mutual potential energy.
Any one of (106) enables us to decompose a given initial state of electric and magnetic force into the sum of normal distributions, when the nature of the latter has been found, and hence determine the manner of subsidence. Thus, let E0, H0, be the initial state, and Ej, Hj, E„ H2, E3, H3, etc., the various normal distributions, with con .(106)
stants 1^, ?i2, etc. Let
E0 =^41E1 + ^40E0 + ^3E3 + ...,
H0 = ^1H1 + ^X“ + ^sH8 +
where the A’s are coefficients fixing the absolute magnitudes of the normal solutions. If the A’s can be found, then, at time t later, we shall have
E = Ax Eje’*1* + A2 E./,2< + ...,
E being what E0 then becomes; and a similar equation for H.
The A’s are found thus:—E0 and H0 being given, and the rth coefficient Ary being required, calculate the mutual potential and the mutual kinetic energy of the given state with respect to the ?th normal distribution. Let them be Uor and Tnr. Their values are
Uor = A1crlr + A2Uir + AsUSr+... +ArUrr+
Tor = A1Tlr +A^Tir + A3l\r + ... + ArTrr + ....
Subtract the second line from the first, and there results Uor - Tor = Ar(Urr - Trr) ■ or, Ar = (Uor - Tcr)/(Urr - Trr\ (107), since, by the first of (106), all the remaining terms cancel. Thus, the rth coefficient equals the excess of the mutual potential over the mutual kinetic energy of the given state and the r1h normal state, divided by twice the excess of the potential over the kinetic energy of the normal state itself.
The second and third of (106) give two equivalent relations, by widely different processes, viz.,
Ar=(Qor + inTor) '(Qrr + inTrr), (107), and Ar = (Qor + ±nU„)f(Q„ + 4 nU„) (l°7)s
As the first of (106) is the easiest to remember, so it is in general the most readily applied, giving (107)r But some special cases should be noticed. If there be no potential energy, but only kinetic energy and dissipativity ; that is, in all problems in which dielectric displacement is not taken into account, as, for instance, any combination of conductors between which there is electromagnetic induction, but with no condensers, we have
c = 0, T12 = 0, #12 = 0; and Ar = Tor/Trr= QJQ^ ...(108^
If there be no kinetic energy, but only potential energy and dissipativity ; that is, in all cases in which electromagnetic induction is ignorable, as in any combination of conductors and condensers, but not coils, we have
ix = 0, U12 = 0, Qu = 0; and Ar = UJ Urr = QJ Qrr. (108).2
If there be potential and kinetic energy, but no dissipation,
k = 0, UV = T1S; and Ar = (Uor-Tor), (Urr - Trr). ...(108)s
In this last case conductors are excluded. We have a strictly conservative system, from which all radical friction is excluded. It goes on oscillating for ever, but never does any useful work. We must therefore abolish it. A peculiarity connected with (108)3 will be noticed in the next section. Also that the general properties (106) are true whether the rates of subsidence of the two systems be unequal or equal, although in the latter case special procedure is required.
The nature of the normal distributions themselves depends upon the distribution of c, k, and /x throughout space. We have
curl H = (47rk + cn)E, - curl E = (109)
in a normal arrangement. Hence, either E or H being found, the other follows. Thus, eliminate H to get the equation of E,
curl /a-1 curl E + n(iirk + ?»)E = 0 (110)
Any solution of this is a normal E, and the corresponding H is definitely fixed by the second of (109). Not counting the simple cases of linear circuits and similar problems, (110) has been solved in three dimensions in a very few cases.
Presuming we have obtained the normal solutions, the question arises, what values of n shall we take ? We must take all that satisfy all the conditions of the problem. One form of the determinantal equation, whose roots give all the admissible values of 71, is the equation of activity itself,
Q+fr+T = 0; or, Q+2n(U+T) = 0,
I
Si' n
applied to the normal solutions. It is an equation in n only, with various constants, but independent of z, y, z, and t. That is, if in some special case of (110), we know the normal solution, we can find the equation of n by writing out the equation of activity extended over the whole system. But the equation of n is usually to be obtained from the boundary conditions, when the normal functions are known through bounded spaces. [This is the proper way. The other way may be accidentally, but is not generally true.]
If we extend our calculation of the excess of the mutual potential over the mutual kinetic energy of two normal systems through a bounded space, instead of all space, we shall obtain, not Ul2 - T12 = 0, but
47T(ni _ ».2)(tf12 - T12) = 2N(YEoH1 - VEjiy, (Ill)
the summation being over the boundary, N being the unit normal drawn inward. Hence U-T for a single normal system n is given by
8TT(U- T) = 2 (ErE - H/xH) )
= 2 N< VEH' - VE'H), / K '
where the accent means differentiation with respect to H. The first is a volume-, and the second a surface-snmmation.
There are, of course, corresponding bouudaiy forms for the second and third of (106).
The general properties of normal systems (100), (102), (104), (106), (107), and (108), are not peculiar to the special dynamical connections which are involved in the electromagnetic equations, but belong to any dynamical system in which forces of reaction are proportional to displacements, and resistances to velocities, with reciprocal relations amongst the coefficients which are equivalent (in the electromagnetic case) to the three linear relations between forces and fluxes being of a symmetrical nature; or' c, k, and p self-conjugate, with no rotatory power. Conservation of energy requires this to be true for c and fx; and (106) ar^ not true unless k be also self-conjugate.
SECTION XIX. REMARKS ON NORMAL ELECTROMAGNETIC SYSTEMS. CONDITIONS OF
POSSIBILITY OF OSCILLATORY SUBSIDENCE. EQUAL ROOTS, AND THEIR EFFECTS.
In the last section I omitted to define the three symbols, U^, T^, and except by implication. They express the doubles of the potential energy, kinetic energy, and dissipativity of the /ltL normal system Er, Hr; being quantities formed in the same manner as 19, Ul2, and Tu, defined just after equation (105).
As a preparation for what follows it will be useful to bear in mind the general character of the subsidence to equilibrium of a displaced elastic body, which, for our purpose, may be simply a stretched elastic thin wire fixed at its ends. Let it be bent into the form of the arc of a bow, or, more accurately, into the form of the sinusoidal curve, and then be left to itself. If there be no resistance the wire will go on vibrating for ever with uniform frequency, always preserving the sinusoidal form. But if there be resistance to its motion, proportional at every moment to its speed, its amplitude of vibration will continuously decrease, although the frequency (lowered) will be still uniform. By a sufficient increase in the coefficient of resistance (say, by motion in a viscous fluid), we shall ultimately stop the vibrations, the displaced wire returning to, but never crossing its equilibrium position. The displacement at time t in the original frictionless vibration was represented by
(ax sin + bY cos)cxt.
Friction makes it €~Clt(a2 sin + Z>2 cos)c3^,
showing the oscillatory subsidence to equilibrium. When, by sufficient increase of resistance, the oscillations are iust stopped, it is
£"C4'(«3+^0; and finally, further increase makes it
«4c_C6f + Z>4e“C6‘,
the sum of two independent non-oscillatory subsidences.
In general, if inertia be altogether negligible, but not elastic yielding, or the friction, there can be no oscillations. Similarly, if the elastic yielding be negligible, but not inertia and friction, there can be no oscillations. To have oscillations we require both inertia and elastic yielding; besides that the resistance must not be too great.
Coming now to the electromagnetic applications, we shall expect the subsidence of normal systems to come under these four types. If there be no elastic displacement., and therefore no potential energy, the subsidence of a normal system must be non-oscillatory; and it must l>e real subsidence, not indefinite increase according to the same law. Similarly, if there be no inertia (fx = 0, no magnetic induction, no magnetic or kinetic energy), the subsidence must also be real and non- oscillatory. But if neither elastic displacement nor inertia be negligible, there will be either non-oscillatory or oscillatory real subsidence, according to the relative importance of the resistance. In these three cases there is supposed to be always resistance. But if there be none but only elasticity and inertia to consider, the normal systems will be simple harmonic with respect to the time, and go on vibrating for ever. Cases in which two of the three quantities c, k, and /a are non-existent, scarcely belong to the present subject. And the fourth case above (vibrations in dielectric media, with no dissipation) does not occur in ordinary problems, as it requires unrealisable conditions.
Now the equation of activity of a normal system is
2n(U+T) + Q = 0, where *7= SEcE/Stt, T= 2 H/xH/Stt, Q = S E£E.
Here E and H constitute a normal system, E being a solution of (110), and H derived from it by the second of (109); or else H being a solution of the H equation corresponding to (110), and E derived from it by the first of (109). (If there be no inertia, the electric force is polar. Then the single scalar, the electric potential, will serve for variable.) If n be real, the normal functions E and H are real, or may be so chosen as to be real. Then also Q, U, and T are real. Further, it is necessary in the electromagnetic applications that they cannot be negative. This is secured by the angle between a force and a flux being less than 90° at the most.
(c--= 0.) First, ignore elastic displacement. Then c = 0, U is non- existent, and ■2nT+Q = 0.
We see at once that if n be real, it must be negative. If, then, we show that it cannot be imaginary, we prove that all the n’s are real and negative, when c vanishes, but not [i and L Thus, if i stand for
( - 1)£, let , • 7 •
' ’5 nv = a + bit n.2 = a- bi,
be a pair of imaginaries. They turn H into L + Mi and L — Mi respectively, and E into Lj + Mji and Lj-Mrespectively, L, M, etc., being real. Using ??,, the expressions for T and Q become
T= 2 (LfiL - M/xM + 2iLfjM) 8tt, )
Q = 2 (L^-Lj - M1/;M1 + 2/LjZ-Mj). f
Using these in the equation of activity, with n = and separating the real from the imaginary parts, we get
0 = 2 {2rt(L/xL - MfjM) - 4fcL/xM + 8^(1^ - M^Mj)},
0 = 2 {2b(LfjJj - M/xM) + 4«L/xM + 1 GttL/Mj}.
But also, by the conjugate properties in (108)^ we have the mutual T and Q of the two s}rstems hx and n.2 both zero; or
2(L/xL - M/xM) = 0, 2 (Lj^Lj - M^.M,) = 0;
the imaginary parts cancelling. These bring the previous equations to
£2 L/xM = 0, r/2 L/xM/47t + 2 L1Z*M1 = 0.
From the first of these we conclude that b = 0, unless L and M are the magnetic forces of two normal systems, which is not the case here. The imaginary parts are therefore non-existent, which brings us to h-j =«, ??2 = a, T = 0, Q = 0.
WThat we wanted to show was that imaginaries could not exist. In addition, we show that if there be a pair of equal n’s, they will make the kinetic energy and the dissipativity of the (equal) normal systems both zero. The only way this can happen, T and Q being the sum of quantities that cannot be negative, is for each of their elements to vanish, and, therefore, E = 0, H = 0. That is, if n be double (or repeated any number of times), that value of n will make the normal functions vanish over all space.
(n = 0.) Next, ignore magnetic induction. Then ju. = 0, T is nonexistent, and 2nU+Q = Q
We can show that the n’s are all real and negative, excluding oscillatory subsidence, and that the first conditions of a repeated n are tLo, Q=0, which necessitate the vanishing of the normal functions for that value of n. But, owing to the peculiarities arising from the division of space into conducting and non-conducting regions, the matter cannot be shortly treated, and will be returned to. (c, k and fx.) Take next the general case of T, U, and Q all existent. Write the activity equation thus, 2n\U+T) + nQ = 0, and solve as a quadratic. Then,
{-Q±{Q2 - ^UT)h}jiC, n={-Q + {Q‘2-in‘2UT)\}l±T.
Remembering that if n is real, Q, U, and T are all positive (if not zero), we see that Q2>4n'2UT, or Q2>UT, or the dissipativity greater than the geometrical mean of the rates of decrease of the potential and kinetic energies, must be true. And n is negative. The limit of reality is reached when Q2 = -in- UT; or
V = T, inU+ Q = 0, 4nT+Q = 0.
Thus U12 = Tv2, the general conjugate property of two normal systems (ul and n2) when they are unequal, is also true when they are equal, i.e., when n is a double root of the determinantal equation of n. This includes the previous special cases of either c = 0, or else /x = 0. Further information regarding imaginary n's may be obtained by separating the real from the imaginary parts in the above.
(k = 0.) When we take k = 0, in the equation of the normal E functions, we have
curl/A-1curlE + f?i‘-’E = 0, /xnK = -curlE (H^)
If, on the other hand, we take c= 0, we have an equation for E of the same form, but containing n instead of n2. Hence the same normal functions serve in both cases, if Airkn and cnr be exchanged. Former conclusions regarding n in the case c = 0 are therefore now true of it2. That is, every n2 is real and negative, making the us, pairs of oppositely signed equal imaginaries, as
n.2=-ui, n.^ = bi, //4= -hi, etc., ...(114)
where a, b, etc., are real, indicating simple harmonic oscillations without subsidence.
The property lTy> = Tv2 is true for any two roots, whether naturally associated or not; i.e., for nx with respect to all the rest, including its companion n.2, whose square is the same. But also, the second and third conjugate properties (106), keeping in the (nx-n2) factor there omitted, are
(n 2 -ni)UV2 = 0, (V - n*)T12 = 0,
hence Ul2 = 0 = Tl2, except if ?^2 = /i22; that is, in case of the naturally paired n’s. Also, the equation of activity becomes
U+T= 0,
for every single root n.
As the first of (113) contains n2, if we take the E normal functions from it, they will be identical in pairs, E,=E9, Eo = E,, etc., for the roots (114).
But then the second equation (113) shows that the corresponding H functions are the negatives of one another in pairs, thus Hx = - H3=- H4, etc. Thus the expansions of E0 and H0, the initial states of electric and magnetic force, become
E0 = Mi + ^2)E1 + (^3 + ^4)E3+...,
H0 = (^i -^2)^1 + (^3 “^4)^3 +
The mutual potential energy of any two double normal systems is zero, and the same is true of the mutual kinetic energy. We therefore have
2 E0cE1 = (Ax + A2)2 E^E^
^ H0/xHj = (AX - A2)2 Hj/xHj ;
giving Ax and A2 in terms of the initial state E0, H0. Using these, and putting the solutions in the appropriate real form, taking H1i = M}, H st = M3, etc., we find that
E° = E°fEl C°S ~ 2 W sin)f^ +
H0 = (2 E0c-E1 sin + 2 H0/xM, cos)at + ...
Z, JCjjCJCjj J
express the values E0 and H0 reach at time t later.
The proof that there cannot be any imaginary w2’s requires some modification from the proof of absence of imaginary n’s in the case c = 0, owing to the changed conjugate properties. It also shows that a repeated n2 makes the normal functions vanish. (See Thomson and Tait 011 Cycloidal Motion with no Dissipative Forces, “Natural Philosophy,” vol. I., part 2).
(Equal roots.) This remarkable property of the vanishing (with equal roots) of the normal functions in case any one of the three c, k, and y. is zero, is closely connected with another property, viz., that of shutting out the hnt term from the solutions. Looking to the formulae at the commencement of this section, we see that on the boundary between oscillatory and non-oscillatory subsidence we have, instead of the form «,€"'* + «2e”2f, that of (a + bt)cnt. Also, when by a change in the value of some constant, two roots are made to approach one another, and then again diverge imaginary, between the two states we have a pair of equal roots. If, then, the oscillatory form of solution is possible we have the te11* term on the very verge of oscillation. Now, in certain cases we know that oscillations are impossible; they require both kinetic and potential energy to be concerned; so, if either be absent, something must happen to prevent solutions taking the oscillatory form. That something is the vanishing of the normal functions, thus excluding the hnt terms, and making the solution in case of a double root take the form cunt, the same as if the root were not repeated.
h.e.p.—vol. 1. 2l
Let nx and n2 be a pair of ns, and write down the corresponding terms of the E0 expansion. They are
— „ t ^ (E/E, — H0/i.H2)-r, „2t n 15) S (E,- E, -Hj/jH,)1 +S(E^17-"B‘23l7) 2 (116)
Here there are no restrictions put upon c, k and Ex, Ha, and E.„ Ho, are the normal functions corresponding to %and n2, and the Ul2 = Tu formula has been employed to find Ax and A2, the coefficients of E1 and E„ (the fractions).
At first glance it might appear that if, by some change in the value of some electrical constant concerned, or generally, by a changed distribution of c, k, fx, the roots nx and n2 are made to approach and finally reach equality, making Et and Hj also approach to and finally be the same as E2 and H2, their coefficients Ax and A2 will also approach and ultimately be equal. But, in general, nothing could be further from the truth, and instead of equality, we shall have infinite inequality, on account of the denominators approaching zero from opposite sides, sending one A up to positive and the other down to negative infinity. For if nl = n + h, and n2 = n- h, where h is very small, we shall have
Ej = E + /i.E', Hj = H + IiHf,
E2 = E-AE', H2 = H - hW,
if the accent denote differentiation to n, provided the functions E, H, and their differential coefficients do not vanish. These make
2 (E^-Ej - H^Hj) = 27^2 (EcE' - H/xH').
This is the value of the denominator of Ax in (115), and that of A2 is the same taken negatively, thus showing the infinite divergence of the A’s. [There is an example worked out on p. 90.] The two terms, when united, and h made to vanish, give rise to a solution of the form
Cj Fcni + C^ten\
where (\ and C» are new constants, E is the old E, and F is a new function derived from E. From this we see that when the repeated ii makes the normal functions vanish, as when any one of c, k, /x is zero, the second term goes out altogether.
The double-root solution in these cases of vanishing E and H is
2 (Ej/E7 — Hp/aH^-ti, nt /I I C\
2 (EVE' - H'/xH') ’ K }
differing from the original form only in this, that instead of the normal functions E and H, we take their differential coefficients with respect to n. This single term takes the place of the former two terms.
If the root n be triple, E' and H' will also vanish. Then take E" and H" instead; and similarly go on to further differentiations in case of further repetitions of n.
If iV=0 be the determinantal equation of the ??’s, the function U- T of a normal system contains N', the differential coefficient of N with respect to n, as a factor. X' = 0, in addition to N- 0, is the condition that n is a double root. Similarly, the function U - T, not of E and H, but of E' and H', contains N", the second differential coefficient of TV, as a factor, and so on.
Before leaving this curious subject of the effect of equal rates of subsidence, we should notice that when the duplicity of an n, making the conjugate properties of unequal n’s hold good for two of the same value, necessitates the simultaneous vanishing of the normal functions, it does so in virtue of the positivity of Q, U, T, as before mentioned. But should they be allowed to be negative, although, for example, in the case c = 0, we still have T= 0, Q = 0, when an n is double, there is no longer any necessity for E and H to simultaneously vanish. Then we have the tent term, and the t2ent term if a triple root, and so on. The vanishing of T will then depend on its expression containing N, for the special value of n, as a factor. As our expressions for Q, U, and T are in the form of the sum of scalar products, we can only make any one of them negative by allowing that the force and the flux, in some parts of space at least, can make an obtuse angle with one another; that is, be opposed, which is a contradiction to common sense. In special applications, involving only a limited number of degrees of freedom, the positivity of U, T, and Q will require that certain functions of the electrical constants, usually determinants, cannot be negative for any values of the electric variables.
SECTION XX. SOME CASES OF SUBSIDENCE OF DISPLACEMENT.
In the electromagnetic scheme we have the equations of a dynamical system, involving the potential energy of elastic displacement (or of electric polarisation, if that very vague term be preferred; any vector function may be made up of polarised elements, whether it be circuital or polar, so it is as well not to attach too much importance to the idea of polarisation), the kinetic energy (or magnetic energy), probably of a rotational motion, and dissipation of energy by forces analogous to frictions proportional to velocities, when the electric current in a conductor is taken as a generalised velocity. There is nothing peculiarly electrical until we specify the connections of the different magnitudes. It is one out of the infinite number of dynamical systems subject to
Q+U+T=0t
the general equation of activity when energy is neither communicated to the system nor allowed to be withdrawn except through the irreversible frictional forces. The three qualities to which c, and k refer, relate to the potential energy, the kinetic energy, and the dissipativity. In order to render practically simple the theory of special cases, it is necessary to place restrictions upon their values, restrictions that we may know to be untrue. This is perfectly legitimate, as it is the common-sense procedure in all matters of reasoning to simplify as far as possible. But it becomes necessary to be careful in the interpretation of the extreme results of a limited theory.
Consider, for example, the discharge of a condenser through a wire. The first approximation to its theory is got by ignoring inertia. If q0 be its initial charge, that left at time t later is q0€~tlu, Avhere tA is the time-constant, the product of the capacity of the condenser and the resistance of the wire. An appropriate mechanical illustration is the restoration to equilibrium of a bent spring of negligible mass in a viscous fluid.
But if we push this to extremes, by shortening the discliarge-wire indefinitely, this theory says that the discharge will always be of the same character, though finally instantaneous. This is entirely wrong. The influence of inertia may be negligible when the resistance is great, but is not when it is small. We allow for inertia by introducing the inductance of the circuit, bringing in an electromotive force proportional to the rate of decrease of the current. Then we find that when the resistance of the wire is below a certain value the discharge becomes oscillatory. This is quite correct, and the theory as amended is then true within a far wider range than before. But it, again, must not be pushed to extremes. It shows that if the resistance be reduced to nothing, whilst the inductance of the circuit is finite, as it must be, the oscillations continue for ever undiminished in strength, with frequency (27r)-1(s^)_i, if s be the inductance and p the capacity of the condenser. I.e., short-circuiting a condenser would never get rid of its charge, except momentarily, when the energy is all kinetic. Here, of course, the objection is that we cannot indefinitely reduce the resistance in circuit, on account of the resistance of the metallic coatings, previously neglected, when the external resistance was great in comparison. Allowing for that, we still have oscillatory subsidence. But when we consider further that a short-circuited condenser can scarcely be treated as a linear circuit, and that we have ignored the dissipation of energy by the oscillatory phenomenon in the magnetic field producing vibratory electric currents in neighbouring conductors, we see that the complete theory of a short-circuited condenser may be only roughly represented by taking into account three constants, the capacity of the condenser, the inductance, and the resistance. What is true enough within certain limits (or uncertain, because no definite line can be drawn between the true and the false) may be wholly untrue beyond them, owing to circumstances of the minutest previous significance becoming then of (relatively) paramount importance.
Retardation in a medium in which yu. = 0, c/k = constant.
This is a very singular case, and of considerable interest. In general, we have
div (47rk 4- )E = 0,
at
to express the continuity of the true electric current, E being the electric force, k the conductivity, and r the specific capacity.
It will be convenient to put oj^iT=p. The new quantity p is the capacity per unit volume considered as a condenser. We may avoid ambiguity by using the word “specific” in connection with in the absence of a better nomenclature.
In any normal system djdt = n, a constant, so that the above becomes
div(£+;m)E = 0, (117)
or, since the displacement is D=^E,
div (Jc/p + w)D = 0 (118)
Now, if there be no inertia, or /A = 0, and no impressed forces, we shall also have
curlE = 0, or curl D/p = 0 (H9)
In (117), (k+pn) is a function of position, k and p being variable from place to place, whilst n is constant. Apply Sir W. Thomson’s theorem of determinancy. If (k +pn) be everywhere positive, the only solution of (117), subject to the first of (119), is E = 0. Similarly, if (k+pn) be everywhere negative, the only solution is E = 0. In both cases the point of the demonstration is that 2 (k +JM)'E2 is necessarily positive if (k+pn) be everywhere positive, and negative if it be everywhere negative. This quantity 'I (k+pn)E2 is Q + U, and is therefore zero. It follows that, in any normal system, (k +pn) must be positive in some parts of space and negative in others (unless it be zero everywhere). Therefore, if kl/pl be the least, and k.2 p.-, the greatest value of k/p, it follows that (kljp1 + n) is negative and (kjp., + n) is positive. Hence the values of n for all the normal systems lie between -k1jpl and - h2jp2- Or, their time-constants all lie between the greatest and least values of pjk. If then kfp is the same everywhere, there is only one rate of subsidence for any initial state, given by (kjp + n) = 0. (To show that there cannot be imaginary n s, make use of (J12 = 0, Qu = 0, applying them to the solutions corresponding to a supposed pair of imaginaries. It follows that the unreal part of the roots is zero, and that the normal functions vanish in the case of equal roots.)
Given that the initial displacement is D0 in a medium in which plk is constant, and /x = 0, and that it is left without impressed force, we therefore obtain the subsequent state in the following manner. Let
div D 0 = p0,
so that p0 is the initial electrification. Find Dp such that
div Dj = p0, curl D Jp = 0.
Dj is uniquely determinable by these conditions. Then D0 = Dx + (D0 - Dj), where (D0 - DT) is a system of circuital displacement. It will subside instantaneously, leaving D1? which will then subside so that the displacement D at time t later is given by
D = Dl€-*^. (120)
The conduction current is /.'D/p, and the displacement current the negative of the same, so that the true current is zero. It is not a case of propagation at all, every elementary condenser discharging through its own resistance. It is the instantaneous vanishing of the circuital displacement that is connected with propagation, it being what would happen if k — 0 with the same distribution of p. First the displacement readjusts itself to make the electric force polar with the same electrification; and then, what is left subsides everywhere at the same rate, according to (120).
Now, any distribution of impressed force sets up a corresponding distribution of circuital conduction current, and, therefore, since k and p are everywhere in the same ratio, of circuital displacement, without electrification. But it is only displacement with electrification that has a finite rate of subsidence. Hence there is no retardation whatever in connection with impressed force. However it vary with the time, the corresponding displacement will vary with it instantaneously. Evidently this is a case in which inertia is not negligible. Maxwell (Vol. I., chap. X.) treats of the case p = constant, k = constant. The extension to pjk = constant allows us to distribute capacity as we please, and so obtain immediately the solutions of various problems connected with shunted condensers.
Now let there be inertia. Although (119) is no longer true, yet (118) is; and, since (k/p + n) is constant, it may be written
(k/p + n) div D = 0.
There is, therefore, no electrification in any normal system, unless (k/p+n) = 0. It follows that if there be electrification initially, the above process of dividing D0 into DL and D0 - Dx is applicable to give us the part of the subsequent state depending on electrification. Thus (120) is true whether there be magnetic induction or not, the left member, however, being not the complete displacement, but only that depending upon the initial electrification. The other part of the initial displacement, D0 - Dp will subside, not as before, instantaneously, but according to the nature of the normal distributions other than the (kjp + n) = 0 solution, depending upon the distribution of k and p. in space, and also upon the initial state of magnetic induction.
Why (120) is true in spite of inertia, is because there is no true current, the force being polar, and therefore no magnetic induction in connection with the electrification solution. As before, no electrification can be produced by any impressed forces, so that the (k/p + n) = 0 solution may be left out of account. The retardation in connection with the effects of impressed force will depend wholly upon the other ri s.
As a simple application of the preceding, let k = 0 everywhere except in a single wire, forming a closed circuit. It is a perfectly insulated dielectric wire whose conductivity may vary as we please along it, provided its capacity vary in the same ratio. Let now k and p signify the conductance and capacity per unit length of wire, and D the total displacement across its section. Then Djp is the electric force per unit length, and is the equation of continuity, if x be distance measured along the wire. Or,
(kjp + n) clD/clx = 0,
in a normal system. If then D0 be the initially given displacement, divide it into Dx and D0 - Dx, such that
dDJdx = dDJdx = p0)
the initial electrification of a cross-section, and such that the E.M.F. round the circuit is zero. At time t later, the displacement is
D = Dxe-^f (121)
the part I)Q - Dx, which is circuital, and is the mean initial displacement all round the circuit, instantly vanishing (if there be no inertia). As before, there is no true current during the subsequent subsidence. By the “mean” displacement is meant the quotient of the total initial E.M.F. round the circuit by the total elastance, that is, (2 Djp) ~ (2 1 jp), the summation extending round the circuit. To eorroborate, insert a conductor in the circuit having no capacity. There will now be two normal rates of subsidence, one of which is the previous. If be the new u, it is given by
0 = A' ■+ Kx + Snv
where K and Kx are the conductances of the old and of the new wire, and S is the total capacity of the old wire, i.e., S' = (Zp*1)-1, the reciprocal of the sum of the elastances round the circuit. The solution will now be
D= (D0 -Dx)^+ Dxe-1a/p (122)
As the auxiliary wire is shortened, nx goes out to negative infinity. Then we return to the former instantaneous subsidence of the mean displacement when the whole wire has capacity, (122) becoming (121). When Kx is finite, there is necessarily electrification somewhere; if not in the old wire itself, then at its ends, where it joins on to the new one. If the circuit be open instead of closed, there will be no instantaneous subsidence in any case; the solution is then (121) with 7>0 instead of Dx on the right side.
Any impressed force in the circuit will only suffer retardation in its effects as regards the nx term. It would be very convenient, as well as wonderful, if some ingenious inventor could construct a telegraph cable whose electrostatic capacity should be in the conductor instead of outside it. Having it there, however, a first approximation towards lessening the retardation is to give greater conductivity to the insulating covering. Even a leakage-fault raises the speed of working considerably. Nothing is worse for rapid signalling (when pushed to limiting speeds) than the most perfect insulation. The lower it can be made (natural high conductivity, not due to faults which by getting too bad would stop communication) consistent with getting enough current at the receiving end, the better, and much better, it is for the signalling. Of course there are other considerations, but we must return to the immediate subject.
We can easily obtain the effect of inertia in modifying the solution (121) . Let s be the inductance per unit length of wire, constant for purposes of calculation, and really so if the wire be circular. Let also k and p be constant. We have only to examine how the circuital displacement, (D0 - Z^), subsides, in which alone magnetic induction is concerned. Let V be the true current over the cross-section, like D. The electric force to correspond is that of inertia, viz., - sr. Hence
r = (k +P^j ( - or T = (k +pn)( - snT),
in a normal system; and the determinantal equation is
spn2 + skn +1=0,
giving the two n*s,
or n2 = -k/'2p± (k2jip2 - 1 /ps)K
Besides the circuital displacement, the initial current may be arbitrary. Let it be ro. Then at time t later,
r _ r0+HjP, — Dj^t r0+n2(i)0 — )
2 + ksnx 2 + ksn2 ’
D = - spnx x ditto - spn2 x ditto + Dxe~kt,p.)
The current is oscillatory if
k/p<'2/(ps)1-,
and non-oscillatory if it be greater. This differs completely from the condenser and coil theory; for now we get oscillations by reducing the inductance, whereas in the other case, it is by reducing the inductance that we get rid of oscillations.
Although in this solution we take into account the magnetic field, yet we only regard that part of the electric field that is within the conductor, so that the specific capacity c in the wire must be much greater than in the surrounding air to render the latter negligible.