SECTION XXV. SOME NOTES ON MAGNETISATION.
Although it is generally believed that magnetism is molecular, yet it is well to bear in mind that all our knowledge of magnetism is derived from experiments on masses, not on single molecules, or molecular structures. We may break up a magnet into the smallest pieces, and find that they, too., are little magnets. Still, they are not molecular magnets, but magnets of the same nature as the original; solid bodies showing magnetic properties, or intrinsically magnetised. We are nearly as far away as ever from a molecular magnet. To conclude that molecules are magnets because dividing a magnet always produces fresh magnets, would clearly be unsound reasoning. For it involves the assumption that a molecule has the same magnetic property as a mass, i.e., a large collection of molecules, having, by reason of their connection, properties not possessed by the molecules separately. (Of course, I do not define a molecule to be the smallest part of a substance that has all the properties of the mass.) If we got down to a mass of iron so small that it contained few molecules, and therefore certainly not possessing all the properties of a larger mass, what security have we that its magnetic property would not have begun to disappear, and that their complete separation would not leave us without any magnetic field at all surrounding them of the kind we attribute to intrinsic magnetisation. That there would be magnetic disturbances round an isolated molecule in motion through a medium, and with its parts in relative motion, it is difficult not to believe in view of the partial co-ordination of radiation and electromagnetism made by Maxwell. But it might be quite different from the magnetic field of a so-called magnetic molecule—that is, the field of any small magnet. This evident magnetisation might be essentially conditioned by structure, not of single molecules, but of a collection, together with relative motions connected with the structure, this structure and relative motions conditioning that peculiar state of the medium in which they are immersed, which, when existent, implies intrinsic magnetisation of the collection of molecules, or the little mass. However this be, two things are deserving of constant remembrance. First, that the molecular theory of magnetism is a speculation which it is desirable to keep well separated from theoretical embodiments of known facts, apart from hypothesis. And next, that as the act of exposing a solid to magnetising influence is, it is scarcely to be doubted, always accompanied by a changed structure, we should take into account and endeavour to utilise in theoretical reasoning on magnetism which is meant to contain the least amount of hypothesis, the elastic properties of the body, speaking generally, and without knowing the exact connection between them and the magnetic property. Hooke’s law, Ut tensio, sic vis, or strain is proportional to stress, implies perfect elasticity, and is the first approximate law on which to found the theory of elasticity. But beyond that, we have imperfect elasticity, elastic fatigue, imperfect restitution, permanent set.
When we expose an unmagnetised body to the action of a magnetic field of unit inductivity, it either draws in the lines of induction, in which case it is a paramagnetic, is positively magnetised inductively, and its inductivity is greater than unity; or it wards oft' induction, in which case it is a diamagnetic, is negatively magnetised inductively, and its inductivity is less than unity; or, lastly, it may not alter the field at all, when it is not magnetised, and its inductivity is unity.
Regarding, as I do, the force and the induction—not the force and the induced magnetisation—as the most significant quantities, it is clear that the language in which we describe these effects is somewhat imperfect, and decidedly misleading in so prominently directing attention to the induced magnetisation, especially in the case of no induced magnetisation, when the body is still subject to the magnetic influence, and is as much the seat of magnetic stress and energy as the surrounding medium. We may, by coining a new word provisionally, put the matter thus. All bodies known, as well as the so-called vacuum, can be inductized. According to whether the inductization (which is the same as “the induction,” in fact) is greater or less than in vacuum (the universal magnetic medium) for the same magnetic force (the other factor of the magnetic energy product), we have positive or negative induced magnetisation.
To the universal medium, which is the primary seat of the magnetic energy, we attribute properties implying the absence of dissipation • of energy, or, on the elastic solid theory, perfect elasticity. (Dissipation in space is scarcely within a measurable distance of measurement.) But that the ether, resembling an elastic solid in some of its properties, is one, is not material here. Inductization in it is of the elastic or quasi-elastic character, and there can be no intrinsic magnetisation. Nor evidently can there be intrinsic magnetisation in gases, by reason of their mobility, nor in liquids, except of the most transient description. But when we come to solids the case is different.
If we admit that the act of inductization produces a structural change in a body (this includes the case of no induced magnetisation), and if, on removal of the inducing force, the structural change disappears, the body behaves like ether, so far, or has no inductive retentiveness. Here we see the advantage of speaking of inductive rather than of magnetic retentiveness. But if, by reason of imperfect elasticity, a portion of the changed structure remains, the body has inductive retentiveness, and has become an intrinsic magnet. As for the precise nature of the magnetic structure, that is an independent question. If we can do without assuming any particular structure, as for instance, the Weber structure, which is nothing more than an alignment of the axes of molecules, a structure which I believe to be, if true at all, only a part of the magnetic structure, so much the better. It is the danger of a too special hypothesis, that as, from its definiteness, we can follow up its consequences, if the latter are partially verified experimentally we seem to prove its truth (as if there could be no other explanation), and so rest on the solid ground of nature. The next thing is to predict unobserved or unobservable phenomena whose only reason may be the hypothesis itself, one out of many which, within limits, could explain the same phenomena, though, beyond those limits, of widely diverging natures.
The retentiveness may be of the most unstable nature, as in soft iron, a knock being sufficient to greatly upset the intrinsic magnetisation existing on first removing the magnetising force, and completely alter its distribution in the iron; or of a more or less permanent character, as in steel. But, whether the body be para- or diamagnetic, or neutral, the residual or intrinsic magnetisation, if there be any, must be always of the same character as the inducing force. That is, any solid, if it have retentiveness, is made into a magnet, magnetised parallel to the inducing force, like iron.
Until lately only the magnetic metals were known to show reten- tiveness. Though we should theoretically expect retentiveness in all solids, the extraordinary feebleness of diamagnetic phenomena might be expected to be sufficient to prevent its observation. But, first, Dr. Tumlirz has shown that quartz is inductively retentive, and next, Dr. Lodge (Nature, March 25th, 1886) has published some results of his experiments on the retentiveness of a great many other substances, following up an observation of his assistant, Mr. Davies.
The mathematical statement of the connections between intrinsic magnetisation and the state of the magnetic field is just the same whether the magnet be iron or copper, para- or dia-magnetic, or is neutral. In fact, it would equally serve for a water or a gas magnet, were they possible. That is,
curl (H - h) = 47iT, div B = 0 ;
H being the magnetic force according to the equation B = ftH, where B is the induction and [x the inductivity, T the electric current, if any, and h the magnetic force of the intrinsic magnetisation, or the impressed magnetic force, as I have usually called it in previous sections where it has occurred, because it enters into all equations as an impressed force, distinct from the force of the field, whose rotation measures the electric current. It is h and /x that are the two data concerned in intrinsic magnetisation and its field; the quantity I, the intensity of intrinsic magnetisation, only gives the product, viz., I = /xh/47r. It would not be without some advantage to make h and /1 the objects of attention instead of I and fi, as it simplifies ideas as well as the formulae. The induced magnetisation, an extremely artificial and rather unnecessary quantity, is (/a - 1) (H - h)/47r.
It will be understood that this system, when united with the corresponding electric equations, so as to completely determine transient states, requires h to be given, whether constant or variable with the time. The act of transition of elastic induction into intrinsic magnetisation, when a body is exposed to a strong field, cannot be traced in any way by our equations. It is not formulated, and it would naturally be a matter of considerably difficulty to do it.
In a similar manner, we may expect all solid dielectrics to be capable of being intrinsically electrized by electric force, as described in a previous section. I do not know, however, whether any dielectric has been found whose dielectric capacity is less than that of vacuum, or whether such a body is, in the nature of things, possible.
As everyone knows nowadays, the old-fashioned rigid magnet is a myth. Only one datum was required, the intensity of magnetisation I, assuming //. to be unity in as well as outside the magnet. It is a great pity, regarded from the point of view of mathematical theory, which is rendered far more difficult, that the inductivity of intrinsic magnets is not unity. But we must take nature as we find her, and although Prof. Bottomley has lately experimented on some very unmagnetisable steel, which may approximate to /x= 1, yet it is perfectly easy to show that the inductivity of steel magnets in general is not 1, but a large number, though much less than the inductivity of soft iron, and we may use a hard steel bar, whether magnetised intrinsically or not, as the core of an electromagnet with nearly the same effects, as regards induced magnetisation, except as regards the amount, as if it were of soft iron.
Regarding the measure of inductivity, especially in soft iron, this is really not an easy matter, when we pass beyond the feeble forces of telegraphy. For all practical purposes /z is a constant when the magnetic force is small, and Poisson’s assumption of a linear relation between the induced magnetisation and the magnetic force is abundantly verified. It is almost mathematically true. But go to larger forces, and suppose for simplicity we have a closed solenoid with a soft iron core, and we magnetise it. Let F be the magnetic force of the current. Then, if the induction were completely elastic, we should have the induction B=/aF. But in reality we have B = /z(F + h) = fiH. If we assume the former of these equations, that is, take the magnetic force of the current as the magnetic force, we shall obtain too large an estimate of the inductivity, in reckoning which H should be taken as the magnetic force. This may be several times as large as F. For, the softer the iron the more imperfect is its inductive elasticity, and the more easily is intrinsic magnetisation made by large forces; although the retentiveness may be of a very infirm nature, yet whilst the force F is on, there is h on also. This over-estimate of the inductivity may be partially corrected by separately measuring h after the original magnetising force has been removed, by then destroying h. But this h may be considerably less than the former. For one reason, when we take off F by stopping the coil-current, the molecular agitation of the heat of the induced currents in the core, although they are in such a direction as to keep up the induced magnetisation whilst they last, is sufficient to partially destroy the intrinsic magnetisation, owing to the infirm retentiveness. We should take off F by small instalments, or slowly and continuously, if we want h to be left.
Another quantity of some importance is the ratio of the increment in the elastic induction to the increment in the magnetic force of the current. This ratio is the same as [x when the magnetic force is small, but is, of course, quite different when it is large.
As regards another connected matter, the possible existence of magnetic friction, I have been examining the matter experimentally. Although the results are not yet quite decisive, yet there does appear to be something of the kind in steel. That is, during the act of inductively magnetising steel by weak magnetic force, there is a reaction on the magnetising current very closely resembling that arising from eddy currents in the steel, but produced under circumstances which would render the real eddy currents of quite insensible significance. In soft iron, on the other hand, I have failed to observe the effect. It has nothing to do with the intrinsic magnetisation, if any, of the steel. But as no hard and fast line can be drawn between one kind of iron and another, it is likely, if there be such an effect in steel, where, by the way, we should naturally most expect to find it, that it would be, in a smaller degree, also existent in soft iron. Its existence, however, will not alter the fact materially that the dissipation of energy in iron when it is being weakly magnetised is to be wholly ascribed to the electric currents induced in it.
P.S. (April 13, 1886.)—As the last paragraph, owing to the hypothesis involved in magnetic friction, may be somewhat obscure, I add this in explanation. The law, long and generally accepted, that the induced magnetisation is simply proportional to the magnetic force, when small, is of such importance in the theory of electromagnetism, that I wished to see whether it was minutely accurate. That is, that the curve of magnetisation is, at the origin, a straight line inclined at a definite angle to the axis of abscissae, along which magnetic force is reckoned. I employed a differential arrangement (differential telephone) admitting of being made, by proper means, of considerable sensitiveness. The law is easily verified roughly. When, however, we increase the sensitiveness, its accuracy becomes, at first sight, doubtful; and besides, differences appear between iron and steel, differences of kind, not of mere magnitude. But as the sensitiveness to disturbing influences is also increased, it is necessary to carefully study and eliminate them. The principal disturbances are due to eddy currents, and to the variation in the resistance of the experimental coil with temperature. For instance, as regards the latter, the approach of the hand to the coil may produce an effect larger than that under examination. The general result is that the law is very closely true in iron and steel, it being doubtful whether there is any effect that can be really traced to a departure from the law, when rapidly intermittent currents are employed, and that the supposed difference between iron and steel is unverified.
Of course it will be understood by scientific electricians that it is necessary, if we are to get results of scientific definiteness, to have true balances, both of resistance and of induction, and not to employ an arrangement giving neither one nor the other. He will also understand that, quite apart from the question of experimental ability, the theorist sometimes labours under great disadvantages from which the pure experimentalist is free. For whereas the latter may not be bound by theoretical requirements, and can employ himself in making discoveries, and can put down numbers, really standing for complex quantities, as representing the specific this or that, the former is hampered by his theoretical restrictions, and is employed, in the best part of his time, in the poor work of making mere verifications.
SECTION XXVI. THE TRANSIENT STATE IN A ROUND WIRE WITH A CLOSE-FITTING
TUBE FOR THE RETURN CURRENT.
The propagation of magnetic force and of electric current (a function of the former) in conductors takes place according to the mathematical laws of diffusion, as of heat by conduction, allowing for the fact of the electric quantities being vectors. This conclusion may perhaps be considered very doubtful, as depending upon some hypothesis. Since, however, it is what we arrive at immediately by the application of the laws for linear conductors to infinitely small circuits (with a tacit assumption to be presently mentioned), it seems to me more necessary for an objector to show that the laws are not those of diffusion, rather than for me to prove that they are.
We may pass continuously, without any break, from transient states in linear circuits to those in masses of metal, by multiplying the number of, whilst diminishing the section of, the “ linear ” conductors indefinitely, and packing them closely. Thus we may pass from linear circuits to a hollow, core; from ordinary linear differential equations to a partial differential equation; from a set of constants, one for each circuit, to a continuous function, viz., a compound of the J0 function and its complementary function containing the logarithm. This I have worked out. Though very interesting mathematically, it would occupy some space, as it is rather lengthy. I therefore start from the partial differential equation itself.
Our fundamental equations are, in the form I give to them,
curl H = 4ttC, - curl E = JJ.H, C = &E, (li)
E and H being the electric and magnetic forces, C the conduction current, k and /j. the conductivity and the inductivity. The assumption I referred to is that the conductor has no dielectric capacity. Bad conductors have. We are concerned with good conductors, whose dielectric capacity is quite unknown.
We are concerned with a special application, and therefore choose the suitable coordinates. All equations referring to this matter will be marked b. The investigations are almost identical with those given in my paper on *f The Induction of Currents in Cores,” in The Electrician for 1884. [Reprint, vol. 1., p. 353, art. xxvm.] The magnetic force was then longitudinal, the current circular; now it is the current that is longitudinal, and the magnetic force circular.
The distribution of current in a wire in the transient state depends materially upon the position of the return conductor, when it is near. The nature of the transient state is also dependent thereon. Now, if the return conductor be a wire, the distributions in the two wires are rendered unsymmetrical, and are thereby made difficult of treatment. We, therefore, distribute the return current equally all round the wire, by employing a tube, with the wire along its axis. This makes the distribution symmetrical, and renders a comparatively easy mathematical analysis possible. At the same time we may take the tube near the wire or far away, and so investigate the effect of proximity. The present example is a comparatively elementary one, the tube being supposed to be close-fitting. As I entered into some detail on the method of obtaining the solutions in “ Induction in Cores,” I shall not enter into much detail now. The application to round wires with the current longitudinal was made by me in The Electrician for Jan. 10,1885, p. 180, so far as a general description of the phenomenon is concerned. See also my letter of April 23, 1886. [Reprint, vol. 1., p. 440; vol. II., p. 30.]
Let there be a wire of radius a, surrounded by a tube of outer radius b, and thickness b - a. In the steady state, if the current-density is T in the wire, it is - Ta2/(b2 - a2) in the tube, if both be of uniform conductivity, and the tube or sheath be the return conductor of the wire. Let Hx be the intensity of magnetic force in the wire, and H2 in the tube. The direction of the magnetic force is circular about the axis in both, and the current is longitudinal. We shall have
Hx = 2ttIY, H2 = - 27rIV(r2 - b2)/r(b2 - a2), (26)
where r is the distance of the point considered from the axis. Test by the first of equations (1J). We have
, 1 d curl = —j-r, r dr
when applied to H.
Now let this steady current be left to itself, without impressed force to keep it up, so that the “ extra-current ” phenomena set in, and the magnetic field subsides, the circuit being left closed. At the time t later, if the current-density be y at distance r from the axis, it will be represented by y = 2 AJ^nr)** (36) where 2 is the sign of summation. The actual current is the sum of an infinite series of little current distributions of the type represented, in which A, n, and p are constants, and JQ(nr) is the Fourier cylinder function. We have
— r-j- = ^Trpiky (4 b)
r dr dr
Let djdt =p, a constant, then n is given in terms of p by
n2 = - 47rfxJcp (5b)
We suppose that h and p, are the same in the wire as in the sheath. Differences will be brought in in the subsequent investigation with the sheath at any distance. In (3b) there are two sets of constants, the A’s fixing the size of the normal systems, and the n’s or p’s, since these are connected by (5b). To find the n’s, we ignore dielectric displacement, since it is electromagnetic induction that is in question. This gives the condition
H2 = 0, at r = b\ (6b)
i.e., no magnetic force outside the tube. This gives us
J1(nb) = 0, (7b)
as the determinantal equation of the ri s, which are therefore known by inspection of a Table of values of the Jx function. Find the A’s by the conjugate property. Thus,
^ ^ F^nrydr - JWoW*/(, - a*) ^ ^
J\\nr)rdr ~{ >
The full solution is, therefore,
2aV ^ J^najJ^nry* .
7 nJ*(nb) ’ { '
giving the current at time t anywhere.
The equation of the magnetic force is obtained by applying the second of equations (lb); it is
TT 8™r Jl(na)J1(nr)^,t t /IOTA
j2_a2Z/ nVtfnb) ’ 1 '
and the expression for the vector-potential of the current (for its scalar magnitude A0, that is to saj1-, as its direction, parallel to the current, does not vary, and need not be considered), is
^ _87rfj.gr Ji(na)jQ(nr) (lift)
0 b2-an3JŁ(nb)
This may be tested by
curlA = /xH; (12&)
curl being now = - djdr. In the steady state (initial), t = 0,
A.-?*
in the wire, and A0 = ^--1 - 62 + r2 + 262 log-Y
u b'2-a2\ rj
in the sheath. Test by (126) applied to (136) to obtain (26).
The magnetic energy being [iH2jSir per unit volume, the amount in length I of wire and sheath is, by (106),
T_iil( 87mr \2 v [J?(na)J?(nr)2 , 2jJ{
87t\62 - a2) n*J*(nb)
To verify, this should equal the space-integral of ^A0y, using (116) and (96). This need not be written. They are identical because
f J(nr)rdr = j* J?(nr) rdr = |6‘V02(nb),
Jo Jo
so that we may write the expression for T thus,
T7—ljMz/^7ra^Y s^Ji(na)e2pt (146} \62 - a2) n4JŁ(nb) ( '
The dissipativity being y2/k per unit volume, the total heat in length I of wire and sheath is, if p = Jc~\ the resistivity, and the complete variable period be included,
Q=P{ 2aT/(b2 - «2) }* 2 JKMSA-S)1 (156)
When t = 0, either by (146) or by easj' direct investigation, the initial magnetic energy in length I is
m 7/1 Y, 2x2fi ,b2 + a2 4 62 464 , 6]
0=log-}. <i«)
giving the inductance of length I as
M-n <■»>
which may be got in other ways. This refers to the steady state. In the transient state there cannot be said to be a definite inductance, as the distribution varies with the time. The expression in (156) for the total heat may be shown to be equivalent to that in (166) for the initial magnetic energy, thus verifying the conservation of energy in our system.
I should remark that it is the same formula (96) that gives us the current both in the wire and tube, and the same formula (106) that gives us the magnetic force. They are distributed continuously in the variable period. It is at the first moment only that they are discontinuous, requiring then separate formulse for the wire and tube, i.e., separate finite formulse, although only a single infinite series.
The first term of (96) is, of course, the most important, representing the normal system of slowest subsidence. In fact, there is an extremely rapid subsidence of the higher normal systems ; only three or four need be considered to obtain almost a complete curve; and, at a comparatively early stage of the subsidence, the first normal system has become far greater than the rest. In fact, on leaving the current without impressed force, there is at first a rapid change in the distribution of the current (and magnetic force), besides a rapid subsidence. It tends to settle down to be represented by the first normal system; a certain nearly fixed distribution, subsiding according to the exponential law of a linear circuit.
To see the nature of the rapid change, and of the first normal system, refer to The Electrician of Aug. 23, 1884 [vol. I., p. 387], where is a representation of the J0 and J1 curves. In Fig. 1, take the distance 0C2 to be the outer radius of the tube, 0 being on the axis. Then the curve marked Jx is the curve of the magnetic force, showing its comparative strength from the centre of the wire to the outside of the tube, in the first normal system. And, to correspond, the curve w from 0 up to C2 is the curve of the current, showing its distribution in the first normal system. We see that the position of the point Bx with respect to the inner radius of the sheath determines whether the current is transferred from the wire to the sheath, or vice versa, in the early part of the subsidence. If the sheath is very thin, so that the radius of the wire extends nearly up to Cg, there is transfer of the sheath-current (initial) from the sheath a long way into the wire. On the other hand, if the wire be of small radius compared with the outer radius of the tube, so that the tube’s depth extends from C2 nearly up to 0, there is a transfer of the original wire-current a long way into the thick sheath. In Fig. 2 [vol. I., p. 388] are shown the first four normal systems, all on the same scale as regards the vertical ordinate, but we are not concerned with them at present.
Since -p~l = 47rfxkb2/ (rib)2,
by (56), and -p-1 is the time-constant of subsidence of a normal system, we have, for the value of the time-constant of the first system,
~Pi~l = ’273 irfJcb2,
because the value of the first nb, say n^b, is 3-83. Compare this with the linear-theory time-constant LjR, where L is given by (17b), and Pi is the resistance of length I of the wire and sheath (sum of resistances, as the current is oppositely directed in them). Let a — \b. Then
L = M28 fxl.
We have also
11= 161 j 37rkb2, therefore LjR =-2117rp&b'*,
so that the time-constant of the first normal system is to that of the current in wire and tube on the linear theory as *27 to *21. But it is only after the first stage of the subsidence is over that this larger time- constant is valid.
We may write the expression for L thus. Let x = bja, then
T _ [xlx? (2x2 log x . \ x2~\~ /’
nearly the same as 2/d logic when x is large. The minimum is when b = a; then L = This is the least value of the inductance of a round wire, viz., when it has a very thin and close-fitting sheath for the return current, so that the magnetic energy is confined to the wire.
When bja is only a little over unity,
T _ f^2 3J2 -a2 - 2ab ~W^a2 (b + af '
We have also R = lb2/irka2(b2 - a2),
and therefore L/R = irkfxa2- 1^.
Irrespective of bja being only a little over unity, we have, with ajb = L/R= ■009(47r/J;&2),
>> 2 > » » J
10 -090
» Hi a »> >
whilst the time-constant of the first normal system in all three cases is •068 (47rfikb2).
The maximum of L/R with bja variable is when
{x2-l)(x2~l)jxl = \ogx,
x being bja. This value of x is not much different from the ratio of the nodes in the first normal system, or the ratio of the value of nr making J^nr) = 0 for the first time, to that making J0(nr) = 0. For the latter value makes logx = *4<j5, and makes the other side of the last equation be *486.
In the subsidence from the steady state, the central part of the wire is the last to get rid of its current. But the steady state has to be first set up. Then it is the central part of the wire that is the last to get its full current. To obtain the equations showing the rise of the current and of the magnetic force in the wire and the tube, we have to reverse or negative the preceding solutions, and superpose the final steady states. As these are discontinuous, there are two solutions, one for the wire, the other for the sheath; but the transient part of them, which ultimately disappears, is the same in both. There is no occasion to write these out.
If the steady state is not fully set up before the impressed force is removed, we see that the central part of the wire is less useful as a conductor than the outer part, as the current is there the least. If there are short contacts, as sufficiently rapid reversals, or intermittences, the central part of the wire is practically inoperative, and might be removed, so far as conducting the current is concerned. Immediately after the impressed force is put on, there is set up a positive current on the outside of the wire, and a negative on the inside of the sheath, which are then propagated inward and outward respectively. If the sheath be thin, the initial (surface) wire-current is of greater and the initial sheath-current of less density than the values finally reached by keeping on the impressed force; whilst if it be the sheath that is thick the reverse behaviour obtains.
This case of a close-fitting tube is rather an extreme example of departure from the linear theory; the return current is as close as possible and wholly envelops the wire-current. Except as regards duration, the distributions of current and magnetic force are independent of the dimensions, i.e., in the smallest possible round wire closely surrounded by the return current the phenomena are the same as in a big wire similarly surrounded, except as regards the duration of the variable period. The retardation is proportional to the conductivity, to the inductivity, and to the square of the outer radius of the tube.
When, as in our next Section, we remove the tube to a distance, we shall find great changes.
SECTION XXVII. THE VARIABLE PERIOD IN A ROUND WIRE WITH A CONCENTRIC,
TUBE AT ANY DISTANCE FOR THE RETURN CURRENT.
The case considered in the last Section was an extreme one of departure from the linear theory. This arose, not from mere size, but from the closeness of the return to the main conductor, and to its completely enclosing it. Practically we must separate the two conductors by a thickness of dielectric. The departure from the linear theory is then less pronounced; and when we widely separate the conductors it tends to be confined to a small portion only of the variable period. The size of the wire is then also of importance.
Let there be a straight round wire of radius av conductivity kv and inductivity /xv surrounded by a non-conducting dielectric of specific capacity c and inductivity [x2 to radius a2, beyond which is a tube of conductivity k3, and inductivity fx3, inner radius a2 and outer «8. The object of taking c into account, temporarily, will appear later. Let the current be longitudinal and the magnetic force circular. Then, by (lb), if y is the current-density at distance r from the axis, we shall have
— — 47Tfxky, or = c/xy, (18b)
r dr dr '
in the conductors, and in the dielectric respectively; the latter form being got by taking y = cEj 4TT, the rate of increase of the elastic displacement.
A normal system of longitudinal current-density may therefore be represented by
yx = AxJ0(nxr), from r = 0 to
72 = A?h(n2r) + B2Ko(n2T\ » r = a\ to a2 - (19 b)
7s = A -AM + „ r = a2 to «3
in tlie wire, in the insulator, and in the sheath, respectively, at a given moment. In subsiding, free from impressed force, each of these expressions, when multiplied by the time-factor ept, gives the state at tlie time t later.
J0(nr) is the Fourier cylinder function, and K0(nr) the complementary function. [For their expansions see vol. I., p. 387, equations (70) and (71)]. The A’s and B’s are constants, fixing the size of the normal functions; the n’s are constants showing the nature of the distributions, and p determines the rapidity of the subsidence.
By applying (186) to (196) we find
- i = - 4*PAP> «I=-/*^P2i = - 4irfx.Jcsp; (206)
expressing all the n’s in terms of the p.
Corresponding to the expressions (196) for the current, we have the following for the magnetic force :—
- i= - (%/P\k\P)A \Ji('« i?')> 1 H2= - (i7r>h!fX/P2){A-/l(ll-f) +
B2Kl(n2r)}> ('2lh)
Ih = - (nJhfisP) J
where, as is usual, the negative of the differential coefficient of J0(z) with respect to z is denoted by J^(z); and, in addition, the negative of the differential coefficient of K0(z) with respect to z is denoted by Kx(z). These equations (216) are got by the second and third equations (16), in the case of IIX and H3; and in the case of H2, by using, instead of Ohm’s law, the dielectric equation, giving
E = Airyjcp,
in the dielectric, E being the electric force. Of course d/dt=p, in a normal system.
We have next to find the relations between the five A’s and i?’s, to make the three solutions fit one another, or harmonize. This we must do by means of the boundary conditions. These are nothing more than the surface interpretations of the ordinary equations referring to space distributions. In the present case the appropriate conditions are continuity of the magnetic and of the electric force at the boundaries, because the two forces are tangential; the conditions of continuity of the normal components of the electric current and of magnetic induction are not applicable, because there are no normal components in question. If the magnetic or the electric force were discontinuous, we should have electric or magnetic current-sheets.
Thus Hx and U2 are equal at r = ax, and II2 and Hs are equal at ?‘ = o2. These give, by (216),
{AxnxiixJ:xy)Jx(r\xax) = (iirn2lp2cp-) {A2Jx(n2ax) + B2Kx(n2ax)}, (226)
and (4tm2]ii.2cf){A^(n^L2) + B2Kx(n2aJ)}
= insl fx:JcsP) {^a'^i(n‘da‘2) + BgK^iigCi^)} (23 6
Similarly, E1 and E2 are equal at r = av and E2 and E3 are equal at r = a2. These give, by (196),
(^AW^) = (i7r/CP) {AZJo(n2ai) + ^oM }> (24Z/)
and
{iirjcjj) {A^fQ{n2o>^} + } = ^3 WoO^) + }. (256)
Thus, starting with A1 given, (226) and (246) give A2 and B2 in terms of Av and then (236) and (256) give A3 and B3 in terms of Av Similarly we might carry the system further, by puttiug more concentric tubes of conductors and dielectrics, or both, outside the first tube, using similar expressions for the magnetic and electric forces; every fresh boundary giving us two boundary conditions of continuity to connect the solution in one tube with that in the next. But at present we may stop at the first tube. Ignore the dielectric displacement beyond it, i.e., put c = 0 beyond r = a3) because our tube is to be the return conductor to the wire inside it. We may merely remark in passing that although when such is the case, there is, in the steady state, absolutely no magnetic force outside the tube, yet this is not exactly true in a transient state. To make it true, take c = 0 beyond r = a3 ; requiring H3 = 0 at r^=a3. This gives, by (216),
A^J-^ngCi^) + B3K^in3a^) = 0 (266)
Now A3 and B3 are, by the previous, known in terms of Av Make the substitution, and we find, first, that AL is arbitrary, so that it, when given, fixes the size of the whole normal system of electric and magnetic force; and next, that the n’s are subject to the following equation:—
n3 T (m (1 \^l(nsa2)^l(n3a3) ~ ) ^1 {n3,Ch) _7HT (n a \
^ 0( 2 to l{ 2 ^
^K0(V.2)x -
^3 lX2
(ni/lxi)J^i(nia'i}Jo{n2ai) ~ (?k>/(276)
^hllh)J\(n\a\)K^ha\) ~ (w2 (n2ai)Jo(wi«i)’ **
where, on the left side, to save trouble, the dots represent the same fraction that appears in the numerator immediately over them.
Now, the ft’s are known in terms of p, hence (276) is the determinantal equation of the p’s, determining the rates of subsidence of all the possible normal systems. We have, therefore, all the information required in order to solve the problem of finding how any initially given state of circular magnetic force and longitudinal electric force in the wire, insulator, and sheath subsides when left to itself. We merely require to decompose the initial states into normal systems of the above types, and then multiply each term by its proper time-factor e** to let it subside at its proper rate. To effect the decomposition, make use of the universal conjugate property of the equality of the mutual potential and the mutual kinetic energy of two complete normal systems, UV2 = Tu [vol. I., p. 523], which results from the equation of activity. We start with a given amount of electric energy in the dielectric, and of magnetic energy in the wire, dielectric and sheath, which are finally used up in heating the wire and sheath, according to Joule’s law.
It would be useless to write out the expressions, for I have no intention of discussing them in the above general form, especially as regards the influence of c. Knowing from experience in other similar cases that I have examined, that the effect of the dielectric displacement on the wire and sheath phenomena is very minute, we may put c = 0 at once between the wire and the sheath. We might have done this at the beginning; but it happens that although the results are more complex, yet the reasoning is simpler, by taking c into account.
The question may be asked, how set up a state of purely longitudinal electric force in the tube, sheath, and intermediate dielectric? As regards the wire and sheath, it is simple enough; a steady impressed force in any part of the circuit will do it (acting equally over a complete section). But it is not so easy as regards the dielectric. It requires the impressed force to be so distributed in the conductors as to support the current on the spot without causing difference of potential. There will then be no dielectric displacement either (unless there be impressed force in the dielectric to cause it). Now, if we remove the impressed force in the conductors, the subsequent electric force will be purely longitudinal in the dielectric as well as in the conductors.
But practical^ we do not set up currents in this way, but by means of localised impressed forces. Then, although the steady state is one of longitudinal electric force in the wire and sheath, in the dielectric there is normal or outward electric force as well as tangential or longitudinal, and the normal component is, in general, far greater than the tangential. In fact, the electrostatic retardation depends upon the normal displacement. But electrostatic retardation, which is of such immense importance on long lines, is quite insignificant in comparison with electromagnetic on short lines, and in ordinary laboratory experiments with closed circuits (no condensers allowed) is usually quite insensible. We see, therefore, that when we put c = 0, and have purely longitudinal electric force, we get the proper solutions suitable for such cases where the influence of electrostatic charge is negligible, irrespective of the distribution of the original impressed force. Our use of the longitudinal displacement in the dielectric, then, was merely to establish a connection in time between the wire and the sheath, and to simplify the conditions.
(In passing, I may give a little bit of another investigation. Take both electric and magnetic induction into consideration in this wire and sheath problem, treating them as solids in which the current distribution varies with the time. The magnetic force is circular, so is fully specified by its intensity, say H, at distance r from the axis. Its equation is, if z be measured along the axis,
d 1 d JJ , d2H , 7 fj , „ jj iju
dr I- 7hrll+W=i,r'J:lr+IU'I!' C^IVF^ ^
OF X,
OftNlA.
in which discard the last term when the wire or sheath is in question ; or retain it and discard the previous when the dielectric is considered. The form of the normal H solution is H=Jx(sr)(A sin + B cos)mz ept, for the wire, where s2 = - (iirjjikp + m2). The current has a longitudinal and a radial component, say T and y, given by r = sJ0(sr) (A sin + L cos)mz ept, y — — mJx{sr)(A cos - B &m)mz ept.
In the dielectric and sheath the K0 and Kx functions have, of course, to be counted with the J0 and Jr)
Now put c = 0 in (276). We shall have
J0{n2r) = 1; - n2Jx(n2r) = 0; K0(n2r) = log (n2r); - n./K^iy) - 1 ; which will bring (276) down to
J (n a ) = —? ~ (ll'say)^i(nna‘2)
Ih Jo(n3a2)Kl{7hCh) ~ Jl(n3a3)K(>(n3,a2)
the determinantal equation in the case of ignored dielectric displacement.
To obtain this directly, establish a rigid connection between the magnetic and electric forces at r = ax and at r = a2, thus. Since there is no current in the insulating space, the magnetic force varies inversely as the distance from the axis of the wire. Therefore, instead of the second of (216), we shall have H2 = — {?ii/fJ'ik1p)A1J1(n1a1)(aJr), by the first of (216). Thus H2 at r = a2 is known, and, equated to II3 at r — a2, gives us one equation between Av A3, and B3. Next we have
j (al/r)H1dr = H1a1 log (r/ax) = - (n^ajlog C'/S) '> Jax
HY meaning, temporarily, the value of at r = av This, when multiplied by /x2, is the amount of induction through a rectangular portion of a plane through the axis, bounded by straight lines of unit length parallel to the axis at distances ax and r from it; or the line- integral of the vector-potential round the rectangle; or the excess of the vector-potential at distance r over that at distance a1; so, when multiplied by p, it is the excess of the electric force at ax over that at r. Thus the electric force is known in the insulating space in terms of that at the boundary of the wire. Its value at r = a2 equated to E3 at r — a2 gives us a second equation between Av A3, and B3. The third is equation (266) over again, and the union of the three gives us (286) again. We now have, if yx and y3 are the actual current-densities at time t in the wire and the sheath respectively,
y^AJM^ 1
y2 = ?B{J0(n3r) - J1(n3a3)K0(n3r)/K^n^)} ept j
where 1i = Ak-&. >7o(^igi) ~ )VtiW/ti)/i(l,i«i) log W‘0
in which only the .4 requires to be found, so that when Ł = 0, the initial state may be expressed. The decomposition of the initial state into normal systems may be effected by the conjugate property of the vanishing of the mutual kinetic energy, or of the mutual dissipativity of a pair of normal systems. Thus, in the latter case, writing (296) thus, yx = 2 Au, yH = ^ Av, we shall have
I
“l f«3
u1uj'dr/k1 + I i\v2rdrjk3 = 0,
0 J<12
uv vv and u2, v2 being a pair of normal solutions.
We can only get rid of those disagreeable customers, the K0 and Kx functions, by taking the sheath so thin that it can be regarded as a linear conductor—i.e., neglect variations of current-density in it, and consider instead the integral current. (Except when the sheath and wire are in contact and of the same material, as in the last section.) Let a4 be the very small thickness of the sheath, and evaluate (286) on the supposition that a4 is infinitely small, so that a2 and a3 are equal ultimately. The result is
JvM = axJl(w1a1) jOvV/^) lo§ (aJai) ~ • • • -(306)
the determinantal equation in the case of a round wire of radius ax with a return conductor in the form of a very thin concentric sheath, radius a.2. Notice that /x3, the inductivity of the sheath itself, has gone out altogether; that is, an iron sheath for the return, if it be thin enough, does not alter the retardation as compared with a copper sheath, provided the difference of conductivity be allowed for.
We may get (306) directly, easily enough, by considering that the total sheath-current must be the negative of the total wire-current, which last is, by integrating the first of (296) throughout the wire,
i= (A/n^ir^ '2J1(n1a1)
This, divided by the volume of the sheath per unit length, that is, by 27ra2«4, gives us the sheath current-density, and this, again, divided by ks gives us the electric force at r = a2. Another expression for the electric force at the sheath is given by the previous method (the rectangle business). Equate them, and (306) results.
We have now got the heavy work over, and some results of special cases will follow, in which we shall be materially assisted by the analogy of the eddy currents in long cores inserted in long solenoidal coils.
SECTION XXVIII. SOME SPECIAL RESULTS RELATING TO THE RISE OF THE
CURRENT IN A WIRE.
Premising that the wire is of radius av conductivity Icv inductivity ; that the dielectric displacement outside is ignored; and that the sheath for the return current is at distance o2, and is so thin that variations of current-density in it may be ignored, so that merely the total return current need be considered; that a4 is the small thickness of the sheath, and k3 its conductivity, we have the determinantal equation (306). Let now
A)— ^fl-2 log(^2/^1)’ -^1 (kj7r(ii") 1, li2 — (2act2a4&3) 1.
L0 is the external inductance per unit length, i.e., the inductance per unit length of surface-current, ignoring the internal magnetic field. and B2 are the resistances per unit length of the wire and sheath respectively, and is the internal inductance per unit length, i.e., the inductance per unit length of uniformly distributed wire-current when the return current is on its surface, thus cancelling the external magnetic field. We can now write (306) thus :—
~~ )'j^wifli(A)//xi) (2n\ai) j > (3i^)
aud, in this, we have
- n?a? = 4-n-fjL^pa^ = 4p/i1IBv
(326)
From (316) we see that the two important quantities are the ratio of the external to the internal inductance, and the ratio of the external to the internal resistance, i.e., the ratios LJ/J^ and B2jBv Suppose, first, the return has no resistance. Draw the curves
V\=Jo(x)lJi(x) and =i W/hK
the ordinates y, abscissas x, which stands for njOj. Their intersections show the required values of x. The JJJ1 curve is something like the curve of cotangent. If LJ^ is large, the first intersection occurs with a small value of x, so small that J^(x) is very little less than unity, so that a uniform distribution of current is nearly represented by the first normal distribution, whose time-constant is a little greater than that of the linear theory. The remaining intersections will be nearly given by «7j(a;) = 0. On the other hand, decreasing LJ^ increases the value of the first x; in the limit it will be the first root of J0(x) = 0. Thus, if the wire be of copper, and the return distant (compared with radius of wire), the linear theory is approximated to. If of iron, on the other hand, it is not practicable to have the return sufficiently distant, on account of the large value of fiv unless the wire be exceedingly fine. Even if of copper, bringing the return closer has the same effect of rendering the first normal s}rstem widely different from representing a uniform distribution of current. It is the external magnetic field that gives stability, and reduces differences of curreut-density.
Next, let the return have resistance. The curve y2 must now be ing L0. It increases the first x, and tends to increase it up to that given by Jx{x) = 0 (not counting the zero root of this equation). Thus there is a double effect produced. Whilst on the one hand the rapidity of subsidence is increased by the resistance of the sheath, on the other the wire-current in subsiding is made to depart more from the uniform distribution of the linear theory. The physical explanation is, that as the external field in the case of sheath of no resistance cannot dissipate its energy in the sheath it must go to the wire. But when the sheath has great resistance the external field is killed by it; then the internal field is self-contained, or the wire-current subsides as if Jx(x) = 0, with a wide departure from uniform distribution. This must be marked when the wire-circuit is suddenly interrupted, making the retum-resistance infinite. Now, let there be no current at the time t = 0} when, put on, and keep on, a steady impressed force, of such strength that the final current-density in the wire is ro. At time t the current-density T at distance r from the axis is given by
r = 1 _ y 2 (A\ + B.2)J0(nlrl)/Jl(n1al)El ^
I o nitli ^ — {^o(^i®i)AA(wiai)}2
where the ?i1a1’s are the roots of equation (316). And the total current in the wire, say Cv and with it the equal and opposite sheath-current, will rise thus to the final value C0,
JZ=i _ y JL &.+R^IRi
It will give remarkably different results according as we take the resistance of the wire very small and that of the sheath great, or conversely, or as we vary the ratio LJfx1. Infinite conductivity shuts out the current from the wire altogether, and so does infinite inductivity; the retardation to the inward transmission of the current being proportional to the product Similarly, if the sheath has no resistance, the return current is shut out from it. In either of these shutting- out cases the current becomes a mere surface-current, what it always is in the initial stage, or when we cannot get beyond the initial stage, by reason of rapidly reversing the impressed force, when the current will be oppositely directed in concentric layers, decreasing in strength with great rapidity as we pass inward from the boundary. But if both the sheath and the wire have no resistance, there will be no current at all, except the dielectric current, which is here ignored, and the two surface-currents.
The way the current rises in the wire, at its boundary, and at its centre, is illustrated in “ Induction in Cores.” For the characteristic equation of the longitudinal magnetic force in a core placed within a long solenoid, and that of the longitudinal current in our present case, are identical. The boundary equations are also identical. That is, (316) is the boundary equation of the magnetic force in the core, excepting that the constants and BJB1 have entirely different meanings, depending upon the number of turns of wire in the coil, and its dimensions, and resistance. If, then, we adjust the constants to be equal in both cases, it follows that when any varying impressed force acts in the circuit of the wire and sheath, the current in the wire will be made to vary in identically the same manner as the magnetic force in the core, at a corresponding distance from the axis, when a similarly varying impressed force acts in the coil-circuit (which, however, must have only resistance in circuit with it, not external self-induction as well). Thus, we can translate our core-soiutions into round-straight- wire solutions, and save the trouble of independent investigation, in case a detailed solution has been already arrived at in either case.
Refer to Fig. 3 [p. 398, vol. I., here reproduced]. It represents the curves of subsidence from the steady state. The “ arrival ” curves are got by perversion and inversion, i.e., turn the figure upside down and look at it from behind. The case we now refer to is when the sheath has negligible resistance, and when we take the constant L0 = 2fiv which requires a near return when the wire is of copper, but a very distant one if it is iron.
Regarding them as arrival-curves, the curve h-Ji 1 is the linear-theory curve, showing how the current-density would rise in all parts of the wire if it followed the ordinarily assumed law (so nearly true in common fine-wire coils).
The curve HaHa shows what it really becomes, at the boundary, and near to it. The current rises much more rapidly there in the first part of the variable period, and much more slowly in the later part. From this we may conclude that, when very rapid reversals are sent, the amplitude of the boundary current-density will be far greater than according to the linear theory; whereas if they be made much slower it may become weaker. This is also verified by the separate calculation in “ Induction in Cores ” of the reaction on the coil-current of the corecurrents when the impressed force is simple-harmonic, the amplitude of the coil-current being lowered at a low frequency, and greatly increased at high frequencies [p. 370, vol. I.].
The curve H0H0 shows how the current rises at the axis of the wire. It is very far more slowly than at the boundary. But the important characteristic is the preliminary retardation. For an appreciable interval of time, whilst the boundary-current has reached a considerable fraction of its final strength, the central current is infinitesimal. In fact the theory is similar to that of the submarine cable; when a battery is put on at one end, there is only infinitesimal current at the far end for a certain time, after which comes a rapid rise.
Between the axis and the boundary the curves are intermediate between HaHa at the boundary and H0H0 at the axis, there being preliminary retardation in all, which is zero at the boundary, a maximum at the axis. It is easy to understand, from the existence of this practically dead period, how infinitesimally small the axial current can be, compared with the boundary current, when very rapid reversals are sent. The formulae will follow.
The fourth curve h0h0 shows the way the current rises at the axis when the return has no resistance, but when at the same time there is no external magnetic field, or L0/Py = 0. The return must fit closely over the wire. We may approximate to this by using an iron wire and a close-fitting copper sheath of much lower resistance. There is preliminary retardation, after which the current rises far more rapidly than when LJis finite.
That is, the effect of changing LJfix from the value 2 to the value 0 is to change the axial arrival-curve from H0H0 to h0h0. Suppose it is a copper wire. Then L0 = 2 means log(ajax) =1, or = 2-718. Thus, removing the sheath from contact to a distance equal to 2-7 times the radius of the wire alters the axial arrival-curve from h0h0 to H0H0. Now this great alteration does not signify an increased departure from the linear theory (equal current-density over all the wire). It is exactly the reverse. We have increased the magnetic energy by adding the external field, and, therefore, make the current rise more slowly. But the shape of the curve H0H0, if the horizontal (time) scale be suitably altered, will approximate more closely to the linear-theory curve lijiv By taking the sheath further and further awaj% continuously increasing the slowness of rise of the current, we (altering the scale) approximate as nearly as we please to the linear-theory curve, and gradually wipe out the preliminary axial retardation, and make the current rise nearly uniformly all over the section of the wire, except at the first moment. In fact, we have to distinguish between the absolute and the relative. When the sheath is most distant the current rises the most slowly, but also the most regularly. On the other hand, when the sheath is nearest, and the current rises most rapidly, it does so with the greatest possible departure from uniformity of distribution.
If the wire is of iron, say = 200, the distance to which the sheath would have to be moved would be impracticably great, so that, except in an iron wire of very low inductivity, or of exceedingly small radius, we cannot get the current to rise according to the linear theory.
The simple-harmonic solutions I must leave to another Section. We may, however, here notice the water-pipe analogy [p. 384, vol. J.]. The current starts in the wire in the same manner as water starts into motion in a pipe, when it is acted upon by a longitudinal dragging force applied to its boundary. Let the water be at rest in the first place. Then, by applying tangential force of uniform amount per unit area of the boundary we drag the outermost layer into motion instantly; it, by the internal friction, sets the next layer moving, and so on, up to the centre. The final state will be one of steady motion resisted by surface friction, and kept up by surface force.
The analogy is useful in two ways. First, because any one can form an idea of this communication of motion into the mass of water from its boundary, as it takes place so slowly, and is an everyday fact in one form or another; also, it enables us to readily perceive the manner of propagation of waves of current into wires when a rapidly varying impressed force acts in the circuit, and the rapid decrease in the amplitude of these waves from the boundary inward.
Next, it is useful in illustrating how radically wrong the analogy really is which compares the electric current in a wire to the current of water in a pipe, and impressed E.M.F. to bodily acting impressed force on the water. For we have to apply the force to the boundary of the water, not to the water itself in mass, to make it start into motion so that its velocity can be compared with the electric current-density.
The inertia, in the electromagnetic case, is that of the magnetic field, not of the electricity, which, the more it is searched for, the more unsubstantial it becomes. It may perhaps be abolished altogether when we have a really good mechanical theory to work with, of a sufficiently simple nature to be generally understood and appreciated.
In our fundamental equations of motion
curl (e - E) = /AH, curl H = 4^-r,
suppose we have, in the first place, no electric or magnetic energy, so that E = 0, H = 0, everywhere, and then suddenly start au impressed force e. The initial state is E = 0, H = 0, curl e = /«H.
Thus the first effect of e is to set up, not electric current (for that requires there to be magnetic force), but magnetic current, or the rate of increase of the magnetic induction, and this is done, not by e, but by its rotation, and at the places of its rotation. [A general demonstration will be given later that disturbances due to impressed e or h always have curl e and curl h for sources.]
Now, imagine e to be uniformly distributed throughout a wire. Its rotation is zero, except on the boundary, where it is numerically e, directed perpendicularly to the axis of the wire. Thus the first effect is magnetic current on the boundary of the wire, and this is propagated inward and outward through the conductor and the dielectric respec tively. Magnetic current, of course, leads to magnetic induction and electric current.
Now, in purely electromagnetic investigations relating to wires, in which we ignore dielectric displacement, we may, for purposes of calculation, transfer our impressed forces from wherever they may be in the circuit to any other part of the circuit, or distribute them uniformly, so as to get rid of difference of potential, which is much the best plan. It is well, however, to remember that this is only a device, similar in reason and in effect to the devices employed in the statics and dynamics of supposed rigid bodies, shifting applied forces from their points of application to other points, completely ignoring how forces are really transmitted. The effect of an impressed force in one part of the circuit is assumed to be the same as if it were spread all round the circuit. It w'ould be identically the same were there no dielectric displacement, but only the magnetic force in question. When, however, we enlarge the field of view, and allow the dielectric displacement, it is not permissible to shift the impressed forces in the above manner, for every special arrangement has its own special distribution of electric energy. The transfer of energy is, of course, always from the source, wherever it may be. The first effect of starting a current in a wire is the dielectric disturbance, directed in space by the wire, because it is a sink of energy where it can be dissipated. But the dielectric disturbance travels with such great speed that we may, unless the line is long, regard it as affecting the wire at a given moment equally in every part of its length; and this is substantially what we do when wre ignore dielectric displacement in our electromagnetic investigations, distribute the impressed force as we please, and regard a long wire in which a current is being set up from outside as similar to a long core in a magnetising helix, when we ignore any difference in action at different distances along the core.
SECTION XXIX. OSCILLATORY IMPRESSED FORCE AT ONE END OF A LINE. ITS
EFFECT. APPLICATION TO LONG-DISTANCE TELEPHONY AND TELEGRAPHY.
Given that there is an oscillatory impressed force in a circuit, if this question be asked—what is the effect produced 1 the answer will vary greatly according to the conditions assumed to prevail. I therefore make the conditions very comprehensive, taking into account frictional resistance, forces of inertia, forces of elasticity, and also the approximation to surface conduction that the great frequency of telephonic currents makes of importance.
Space does not permit a detailed proof from beginning to end. The results may, however, be tested for accuracy by their satisfying all the conditions laid down, most of which I have given in the last three Sections.
The electrical system consists of a round wire of radius av conductivity Jcv and inductivity /xx; surrounded by an insulator of inductivity /12 and specific dielectric capacity c, to radius a2; surrounded by the return of conductivity ks, inductivity and outer radius ay The wire and return to be each of length Z, and to be joined at the ends to make a closed conductive circuit.
Let S be the electrostatic capacity, and L0 the inductance of the dielectric per unit length of the line. That is, the dielectric.
Let V be the surface-potential of the wire, and C the wire-current, or total current in the wire, at distance x from one end, at time t. The differential equation of V is
(346)
where R! and U are certain even functions of p, whose structure will
be explained later, and p stands for djdt. That of C is the same. The connection between C and V is given by - -j- = (R' + L'p)C. (356)
Cl/tC
Both (346) and (356) assume that there is no impressed force at the place considered. If there be impressed force e per unit length, add e to the left side of (356), and make the necessary change in (346), which is connected with (356) through the equation of continuity
(366)
But as we shall only have e at one end of the line, we shall not require to consider e elsewhere. Now, given (346) and (356), and that there is an impressed force V0 sin nt at the x = 0 end, find V and C everywhere. Owing to Rf and V containing only even powers of p, and to the property p2 — - n2 possessed by p in simple-harmonic arrangements, R! and L' become constants. The solution is therefore got readily enough. Let
P = (i S?i)i{(R'2 + L'2n2)i - L'n}h,\ Q = (bSn)i{(R'2 + L'2n2)l + L'n}i. J
These are very important constants concerned. Let also
tan = (L'nP - R'Q)/(R'P + L'nQ),\ tan 6^ — sin 2Ql/(e~*pl - cos 2 Ql). )
These make 6X and 62 angles less than 90°. Then the potential V at distance x at time t is
63 and the current C is
tPx sin (nt + Qx - 6X + 02) + *~rx sin (nt - Qx -0l + 62)~1 ,.~j.
€«(€s« + €-s« _ 2 cos 2Ql)i J* { '
Each of these consists of the sum of three waves, two positive, or from Łc = 0 to x — lt and one negative, or the reverse way. If the line were infinitely long, we should have only the first wave. But this wave is reflected at x = l} and the result is the second term. Reflection at the x~0 end produces the third and least important term.
The wave-speed is n/Q, and the wave-length 2TT/Q. As the waves travel their amplitudes diminish at a rate depending upon the magnitude of P. The angles 61 and 02 merely settle the phase-differences. The limiting case is wave-speed = v, and no dissipation.
The amplitude of the current (half its range) is important. It is
C - ro (Sn)h + 2 cos 2 Q(l - a)~)i
0 \lifi + L'%2)i[_ cn + - 2 cos 2Ql J ’
at any distance x. At the extreme end x = I it is
C°°(1~ 2 C0Smyi' {i]i)
As it is only the current at the distant end that can be utilised there, it is clear that (416) is the equation from which valuable information is to be drawn.
It must now be explained how to get IV and L', and their meanings. Go back to equation (286), Section XXVII. [p. 54], which is the determinantal or differential equation when dielectric displacement is ignored. We may write it
0 = Ł « + ftfL - Ł&
2m' JlM
When p is d/dt it is the differential equation of the boundary magnetic force, or of C, since they are proportional. Separating into even and odd powers of p it will take the form, if we operate on C,
0 = (R' + L'p)C, (426)
where Rf and U are functions of p2. To suit the oscillatory state, put
- w2 for p2, making R' and L' constants. They will be of the form
R' = R{ + R'2, L' = L0 + L{ + L(2; (436)
where R{ depends on the wire, R!± on the return; L[ on the wire, L'2 on the return, and L0 on the intermediate insulator. The forms of R[ and L[ have been given by Lord Rayleigh. They are, if g2 = fJ-ln/Rv where Rx = steady resistance of the wire per unit length,
B! = 11 f 1 + — - ff2 + Hff3 _ \ 'j
1 1 V 12 12.15 12.28.80 [
T, t f, g 13/ 73rfi \ |
1 2/Xl\ 24 + 122.30 122.28.80+”7’ J
to the last of which I have added an additional term. The getting ol the forms of R'2 and L'2, depending upon the return, is less easy, though only a question of long division. I shall give the formulas later. At present I give their ultimate forms at very high frequencies. Let p = resistivity, and </ = frequency = n/'lir, then
= L[ = R[jn (45 6)
ai
These are also Lord Rayleigh’s. For the return we have
R'2 = feMl* U = RUn (466)
«2
I express R[ and R'2 in terms of the resistivity rather than the resistance of the wire and return because their resistances have really nothing to do with it, as we see in especial from the R'2 formula. The R'2 of the tube depends upon its inner radius only, no matter how thick it may be, that is to say upon extent of conducting surface, varying inversely as the area, which is 2im2 per unit length. The proof of (466) will follow.
Now, as regards the meanings. Let us call the ratio of the impressed force to the current in a line when electrostatic induction is ignorable the Impedance of the line, from the verb impede. It seems as good a term as Resistance, from resist. (Put the accent on the middle e in impedance.) When the flow is steady, the impedance is wholly conditioned by the dissipation of energy, and is then simply the resistance Rl of the line. This is also sensibly the case when the frequency is very low; but with greater frequency inertia becomes sensible. Then (R2 + L2n2)H is the impedance.
Here R and L are, in the ordinary sense, the resistance and inductance of unit length of line, including wire and return. When, further, differences of current-density are sensible, the impedance is (Rr2 + L/2n2)H. This is greater or less than the former, according to the frequency, becoming ultimately less, especially if the wire is of iron, owing to the then large reduction in the value of L' as compared with L.
Now, when we further take electrostatic induction into account we shall have the above equations (346) and (356), in which R' and U are the same as if there were no static charge. The proof of this I must also postpone. It is the only thing to be proved to make the above quite complete, excepting (466), which is a mere matter of detail. The proof arises out of the short sketch I gave in Section XXVII. of the general electrostatic investigation, used there for illustration.
The impedance is made variable; it is no longer the same all along the line, simply because the current-amplitude decreases from the place of impressed force, where it is greatest, to the far end of the line, where it is least. The question arises whether we shall confine impedance according to the above definition to the place of impressed force, or extend its meaning. If we confine the use, a new word must be invented. I therefore, at least temporarily, extend the meaning to signify the ratio of V0 to C0 anywhere.
It is very convenient to express impedance in ohms, whatever may be its ultimate structure. Thus the greatest impedance of a line is what its resistance would have to be in order that in steady-flow the current should equal that arriving at the far end under the given circumstances. It will usually be far greater than the resistance. But there is this remarkable thing about the joint action of forces of inertia and elasticity. The impedance may be far less than in the electromagnetic theory. That is, VJC0 according to (41 6) may be far less than (R^ + L/2n?)U. This is clearly of great importance in connection with the future of long-distance telegraphy and telephony.
(In passing I will give an illustration of reduction of impedance produced by inertia. Let an oscillatory current be kept up in a submarine cable and in the receiving coils. Insert an iron core in them. The result is to increase the amplitude of the current-waves. More fully, increasing the inductance of the coil continuously from zero, whilst keeping its resistance constant, increases the amplitude up to a certain point, after which it decreases. The theory will follow.)
To get the submarine cable formulae, ignoring inertia, take Z/ = 0 and R' = R. To get the more correct formulae, not allowing for variations of current-density, but including inertia, take Lf = L the steady inductance, and R! = R. To get the linear magnetic theory formulae, take S = 0, and L' = L, R' = R. Finally, using R7 and Lbut with S= 0, we have the complete magnetic formulae suitable for short lines. Thus 5 = 0 in (41 J) brings it to
6'0 =F0 + (Rn + Z7%2)4.
Equations (346) to (366) are true generally, that is, with R' and U the proper functions of djdt. The solution in the case of steady impressed force will follow, including the interior state of the wire. Also the interior state in the oscillatory case.
A great deal may be dug out of (416). In the remainder of this Section, however, we may merely notice the form it takes at very high frequencies, so high as to bring surface conduction into play, and show how much less the impedance is than according to the magnetic theory. Let n be so great as to make R'/L'n small. Then we may also take L' = L0. Then
P = R'/2L0v, Q = n/v.
Also, if e~n is small, as it will be on increasing the frequency, we need only consider the first term under the radical sign in (416), which becomes
n 0--I?II2LQV
VQ J €
L0V
Take for R' its ultimate form
R'=2(fipq)i/a,
H.E.P.—VOL. II. E
got from (45b) and (466) by supposing wire and sheath of the same materia], and 2/a= 1/aj + 1 ja2.
Then the impedance is
W = iVxexp'iw#
J-JQUCV
where exp is defined by = exp x, convenient when x is complex. Here L0 is a numeric, and #=30 ohms (i.e., when we reckon the impedance in ohms); p=1600 and fx= 1, if the conductors are copper; and 1= 10HV if is the length of the line in kilom.; therefore
FJC0 = lbL0 x exp(4/15i/304flio).
To see how it works out, take L0= 1, a = 1 cm, and q= 104; then
V0/C0 = 15 x exp 4^/300 ohms.
If the line is 100 kilom., PI is made 1^, which is too small for our approximate formula. If 1,000 kilom., it is made 13^, which is rather large. Pl= 10 is large. If it is 500 kilom., then VJCQ = 15 x exp 6§ = 1,178 ohms.
So the impedance is only 1,178 ohms at 500 kilom. distance at the enormous frequency of 10,000 waves per second. It is of course much less at a lower frequency, but the more complete formula will have to be used if it be much lower.
Now compare this real impedance with the resistance of the line in the steady state, its effective resistance according to the magnetic theory, and the impedance according to the same. The resistance of the line we may take to be twice that of the wire, by choosing the return of a proper thickness, or
Rl = 2 x 500 x 105 x 1600/7r = 50 ohms, say.
L will be a little more than 1^, say 1‘6, therefore
Lin = "8 x 27r x 104 = 5060,
so that the linear-theory impedance is nearly 5,100 ohms. •
But, owing to the high frequency, we should use R' and U instead of R and L; here take U = L0 + R/n, then
R'l = 400 ohms.
This large increase of resistance is more than counterbalanced by the reduction of inductance, so that the impedance is brought down from the above 5,100 to about 3,500 ohms, the magnetic theory impedance; and this is about three times the real impedance at its greatest, viz., at the distant end of the line.
It is further to be noted that the wire and return need not be solid, as we see from the value of RJ compared with R. What is needed at very high frequencies is two conducting sheets of small thickness, of the highest conductivity and lowest inductivity; i.e., of copper.
SECTION XXX. IMPEDANCE FORMULA FOR SHORT LINES. RESISTANCE OF TUBES.
In the case of a short line, a very high frequency is needed in general to make it necessary to take electrostatic induction into account in estimating the impedance. Keeping below such a frequency, the impedance per unit length is simply
(R'2 + L'2n2)i.
This is greater than the common (R2 + L2n2)l at first, when the frequency is low, equal to it at some higher frequency, and less than it for still higher frequencies. Thus, for simplicity, let the return contribute nothing to the resistance or the inductance; then, using (446), we shall have
(R'i + L'2n2)-(IP + LW)
R2g RY(L 1\ Wg* (ytiL 79\
- e (47i)
R and L being the steady resistance and inductance of the line per unit length (the latter to include L0 for the external medium), R' and U the real values at frequency n/27r per second, [x the inductivity of the wire, and g = (fxnJR)2.
Thus the first increase in the square of the impedance over that of the linear theory is \ix2n2, independent of resistance; large in iron, small in copper. But as the frequency is raised, the g2 term becomes sensible; being negative, it puts a stop to the increase. We can get a rough idea of the frequency required to bring the impedance down to that of the linear theory by ignoring the g3 term. This gives
n2 = 4E2 ± (486)
The real frequency required must be greater than this, and taking the g3 term into account, we shall obtain, as a higher limit,
»”=4 (496)
approximately. We see that the simpler (486) is near enough.
If the wire is of copper of a resistance of 1 ohm per kilom., making i2 = 104, we shall have, using (486),
n = 204 -i-
If the return is distant, we can easily have L0 = 9. Then the frequency required is about 100 waves per second. This is a low telephonic frequency, so that we see that telephonic signalling is somewhat assisted by the approximation to surface conduction.
If the wire is of iron, then, on account of the large value of fx, a much lower frequency is sufficient to reduce the impedance below that of the linear theory; that is, an iron wire is not by any means so disadvantageous, compared with a copper wire of the same diameter, as its higher resistivity and far higher inductivity would lead one to expect.
But it is not to be inferred that there is any advantage in using iron, electrically speaking, from the fact that the impedance is so easily made much less than that of the linear theory. Copper is, of course, the best to use, in general, being of the highest conductivity, and lowrest inductivity. Nor is any great importance to be attached to the matter in any case, for, on a short line, to which we at present refer, it will usually happen that the telephones themselves are of more importance than the line in retarding changes of current.
We also see that in electric-light mains with alternating currents there may easily be a reduction of impedance if the wires be thick and the returns not too close. On the other hand, the closer they are brought the less is the impedance, according to the ordinary formula. It should be borne in mind that we are merely dealing with a correction, not with the absolute value of the impedance, which is really the important thing.
Now take the frequency midway between 0 and the second frequency which gives the linear-theory impedance. Then It2 + L2n2 becomes wherein use the value of n2 given by (48b). The increase of impedance is not, therefore, in a copper wire, anything of a startling nature. Impedances are not additive, in general. We cannot say that the impedance of a wire is so much, that of a coil so much more, and then that their sum is the impedance w7hen they are put in sequence, at the same frequency.
In passing, I may as well caution the reader against the false idea somewhere prevalent. The increased resistance of a wire is not in any way caused or evidenced by the weakness of the current in the variable period compared with its final strength, a result due to the back E.M.F. of inertia. No matter how great the inertia, and how slowly it makes the current rise, there is no change of resistance, unless there be changed distribution of current. There must always be some change, but it is usually negligible. When, however, as notably in the case of iron, the central part of the wire is inoperative, of course this changed distribution of current means a large increase of resistance, though not of impedance, which is reduced. It is a hollow tube, not a solid wire, that must, to a first approximation, be regarded as the conductor. There cannot be said to be any definite resistance unless the current distribution is definite.
Thus, in the rise of the current from zero to the steady state there is, presuming that there is large departure from the regular final distribution, no definite resistance, and it is clearly not possible to balance a wire in which the above takes place against a thin wire, a conclusion that is easily verified. But the case of simple-harmonic impressed force is peculiar. The distribution of current, though not constant, goes through the same regular changes over and over again in such a manner that the total current at every moment is the same as if a true linear circuit of definite resistance and inductance were substituted. This is very considerably departed from when mere rapid makes and breaks are employed.
Consider now the resistance of a tube at a given frequency. It depends materially upon whether the return-current be within it or outside it. Let there be two tubes, a0 and ax the inner and outer radii of the inner, and a2 and a3 of the outer. By an easy extension of equation (286), the form quoted in the last Section, the differential equation of the total current is
•^o(siai) ~ ~Ąf(siao)Ko(siai)
- : r,
'oM — T'' (S3a3)^o(S3a2)
-.g*?, A*, ■ <5w>
the dots indicating repetition of what is above them. The first term is for the insulator between tubes, the second for the inner tube, the third for the outer. Or,
0 = LQP + (R[ + L[p) + (li'2 + L'2p),
where R[, E», L[, U2 are functions of p2, and therefore constants when the current is simple-harmonic. The division of the numerators by the denominators, a simple matter in the case of a solid wire, becomes a very complex matter in the tube case. I give the results as far as p2. It is not necessary to do the work separately for the two tubes, for, if we compare the expressions carefully, we shall see that they only differ in the exchange of the inner and outer radii, and in changed sign of the whole.
For the inner tube we have
(5U)
where llx is the steady resistance per unit length. This is the coefficient of p, and is therefore nothing more than the inductance per unit length of the tube in steady flow, the first correction to which depends on p3. This may be immediately verified bjr the square of-force method.
The resistance of the inner tube per unit length is
K' = M+B f’gi'l r L _ M ++ 2< l0g «0 , K(log S1
‘ 1 \ Pi /l_12 3la,* 12a,4 o^a? - a*)+ a?(a? - atf-
To obtain, from (516) and (526), the corresponding expressions for the outer tube, change to E2, /Oj to ps, to /x3, to a2, and a0 to a3. The change of sign is not necessary, because it is involved in the substitution of R2 for Rv Or, simply, (516) and (526) holding good when the return is outside the tube, exchange ax and a0, and we have the corresponding formulae when the return is inside it.
Let a0 = \ar This removes a fourth part of the material from the central part of a solid wire of radius ar The return being outside, the resistance is
R[ = Hl + Ii^nfjijTra?/’pi)* x -012.
If solid, the '012 would be ‘083; or the correction is reduced seven times by removing only a fourth part of the material.
But if the return is inside, all else being the same, the resistance is
= jRj + i?1(?i)u17r«02//3i) x *503 = Iij + E^nfj^Tra^/pj) x '031,
so now the correction is reduced less than three times instead of seven times, as when the return was outside.
This difference will be, of course, greatly magnified when the ratio (IJCLQ is large; for instance, consider a solid wire surrounded by a very thick tube for return; the steady resistance of the return will be only a small fraction of that of the wire, but the percentage increase of resistance of the outer conductor will be a large multiple of that of the wire. Thus the earth’s resistance, which, in spite of the low conductivity, is so small to a steady current, will be largely multiplied when the current is a periodic function of the time.
Now, as regards the resistance of the tube at high frequencies. If the return is outside it is
= (53&)
q being the frequency. But if the return is inside, it is
= ibih)
thus depending upon the inner radius when the return is inside, and on the outer when it is outside, for an obvious reason, when the position of the magnetic field where the primary transfer of energy takes place is considered.
Suppose we fix the outer radius, and then thin the tube from a solid wire down to a mere skin. In doing so we increase the steady resistance as much as we please. But the high-frequency formula (536) remains the same. Now, as it would involve an absurdity for the resistance to be less than that in steady flow, it is clear that (536) cannot be valid until the frequency is so high as to make E{ much greater than Ev which is itself very great when the tube is thin. That is to say, removing the central part of a wire, when the return is outside it, makes it become more a linear conductor, so that a much higher frequency is required to change its resistance; and when the tube is very thin the frequency must be enormous. Practically, then, a thin tube is always a linear conductor, although it is only a matter of raising the frequency to make (536) or (546) applicable.
To get them, use in (506) the appropriate J0(x), etc., formulae when x is very large. They are
J0(x) = - K^x) = (sin a: + cos x) ± (7rz)*,j Jx(x) = K0(x) = (sin x - cos x) -f (irx)l. j
These, used in (506), putting the circular functions in the exponential forms, reduce it to
<>=/>+&&•+*** 27ra1 27r«2
where i = ( - 1 )*. Here
- s?a? = 47rfakjafp, therefore %sxaxi = (^l\kxp)hax, and similarly for s3; so we get
0 = L0p + */«! + (/x^/ttkH)*/a2.
Here, since p2 = - n2, jpi = Q?2,)i(l +«); which brings us to
0 = (Ii[ + JR0 + (-Z>o + L[ + Lf>)p, (566)
where ^' = ZŁ = ift/w,\
^2 = (/V’sff)* av Lt = J as before given, except that the inner tube was a solid wire.
If, however, the frequency were really so high as to make these high- frequency formulae applicable when the conductors are thin tubes, it is clear that we should, by reason of the high frequency, need, at least in general, to take electrostatic induction into account even on a short line, and therefore not estimate the impedance by the magnetic formulae, but by the more general of the last Section, in which the same RJ and V occur. As for long lines, it is imperative to consider electrostatic induction. There is no fixed boundary between a “short” and a “ long ” line; we must take into account in a particular case the circumstances which control it, and judge whether we may treat it as a short or a long-line question. To the more general formula I shall return in the following Section, merely remarking at present that there is a curious effect arising from the to-and-fro reflection of the electromagnetic waves in the dielectric, which causes the impedance to have maxima and minima values as the speed continuously increases; and that when the period of a wave is somewhere about equal to the time taken to travel to the distant end and back, the amplitude of the received current may easily be greater than the steady current from the same impressed force. And, in correction of the definition in Section xxix. of V as the surface potential of the wire, substitute this definition, Q = SF, where Q is the charge and S the electrostatic capacity, both per unit length of wire.
SECTION XXXI. THE INFLUENCE OF ELECTRIC CAPACITY. IMPEDANCE FORMULAE.
Let us now return to the more general case of Section xxix., the amplitude of the current due to a simple-harmonic impressed force at one end of a line. Although the formula (416) for the amplitude at the distant end is very compact, yet the exponential form of the functions does not allow us to readily perceive the nature of the change made by lengthening the line, or making any other alteration that will cause the effect of the electric charge to be no longer negligible, by causing the magnetic formula to be sensibly departed from. Let us, therefore, put (416) in the form VJC0 = etc., and then expand the right member in an infinite series of which the first term shall be the magnetic impedance itself, whilst the others depend on the electric capacity as well as on the resistance and inductance. On expanding the exponentials and the cosine in (416), we obtain a series in which the quantities P4 - Q4, P6 - Q6, etc., occur, all divided by
P* + Q2.
To put these in terms of the resistance, etc., we have, by (376),
F* + Q* = SnI, 2 PQ = SnR', Q2-P2 = Sn2U, ...(586)
wThere I = (R'2 + U2n2)^. (5 96)
I being the short-line impedance per unit length. Using these, we convert (416) to the following form,
(Sn)2(lR'2 + Lt2n2) “ • • • J * (60b)
Here we may repeat that V0 and C0 are the amplitudes of the impressed force at one end and of the current in the wire at the other end of the double wire of length Z, whose “constants” are Rf, V, and S, the resistance, inductance, and electric capacity per unit length, Rf and L' being functions of the frequency already given. I do not give more terms than are above expressed, owing to the complexity of the coefficients of the subsequent powers of S. To go further, it will be desirable to modify the notation, and also to entirely separate the terms depending upon resistance in the [ ] from the others. Let
SL' = v~2, f= (R'jUn)2, h = nljv. ...'. (616)
Here v is a velocity, / and h numerics. The least value of the velocity is (SL)~i, at zero frequency, L being the full steady inductance per unit length, as before. As the frequency increases, so does v. Its limiting value is (SL0)~k or (/*2c2)"Ł, the speed of undissipated waves through the dielectric. The ratio / falls from infinity at zero frequency, to zero at infinite frequency. See equations (436) to (466). The ratio h is such that is the ratio of the time a wTave travelling at speed v takes to traverse the line, to the wave-period.
In terms of I, /, and h, our formula (416), or rather (606), when extended, becomes
W = 77[(!i^)2+f{l-7^ + TI-^(l +
- rekK1+iV) ■'To^LrA1+y+mf2) - •••}]■(626)
From this, seeing that in the [], resistance appears in / only, we see that the corresponding no-resistance formula is simply
r0/(70 = Il^A=L0v sin^, (636)
where, of course, v is the speed corresponding to L0, or the speed of un dissipated waves. The sine must be reckoned positive always. To check (636), derive it immediately from (416) by taking E — 0. We shall find the following form of (416) in terms of / and h useful later:—
VJCf0 = \L'v( 1 + /)l{c2« + c-2™ - 2 cos 2Ql}\ (646)
where PI — /i(|-)*{(l + /)* - 1 }*> ’ } (65i)
C^/i(i)J{(l +/)» + l}».
Let us now dig something out of the above formula. This arithmetical digging is dreadful work, only suited for very robust intellects.
I shall therefore be glad to receive any corrections the following may require, if they are of any importance. It will be as well to commence with the unreal, but easily imaginable case of no resistance. Let the wire and return be of infinite conductivity. We have then merely wave propagation through the dielectric, without any dissipation of energy, at the wave-speed which is, in air, that of light-waves. Any disturbances originating at one end travel unchanged in form; but owing to reflection at the other end, and then again at the first end, and the consequent coexistence of oppositely travelling waves, the result is rather complex in general. Now, if we introduce a simple-harmonic impressed force at one end, and adjust its frequency until the wave-period is nearly equal to the time taken by a wave to travel to the other end and back again at the speed v, it is clear that the amplitude of the disturbance will be enormously augmented by the to-and-fro reflections nearly timing with the impressed force. This will explain (636), according to which the distant-end impedance falls to zero when nljv = 7r, or 2tt, - or 3TT, etc.
Here 27r/w is the wave-period, and 2l/v the time of a to-and-fro journey. The current-amplitude goes up to infinity.
If, next, we introduce only a very small amount of resistance, we may easily conclude that, although the impedance can never fall to zero, yet, at particular frequencies, it will fall to a minimum, and, at others, go up to a maximum; and that the range between the consecutive maximum and minimum impedance will be very large, if only the resistance be low enough.
Increasing the resistance will tend to reduce the range between the maximum and minimum, but cannot altogether obliterate the fluctuations in the value of the impedance as the frequency continuously increases. In practical cases, starting from frequency zero, and raising it continuously, the impedance, which is simply El, the resistance of the line, in the first place, rises to a maximum, then falls to a minimum, then rises to a second maximum greater than the first, and falls to a second minimum greater than the first, and so on, there being a regular increase in the impedance on the whole, if we disregard the fluctuations, whilst the fluctuations themselves get smaller and smaller, so that the real maxima and minima ultimately become false, or only tendencies towards maxima and minima at certain frequencies.
By this to-and-fro reflection, or electrical reverberation or resonance, the amplitude of the received current may be made far greater than the strength of the steady current from the same impressed force, even when the electrical data are not remote from, but coincide with, or resemble, what may occur in practice. To show this, let us work out some results numerically.
As this matter has no particular concern with variations of current- density in the conductors, ignore them altogether; or, what comes to the same thing, let the conductors be sheets, so that B! = R, the steady resistance, and L' = L0 very nearly, the dielectric inductance, both per unit length. Then, in (646), let
/= 1, Ql = TTy v = 30 ohms (666)
Then, by the second of (656), we find that
h = 2-85;
and, by (646), that
VJCQ = IL0v. 21[€’82^ + €-82^ - 2]i = 60 -6 Z0 ohms (676)
The ratio of the distant-end impedance to the resistance is therefore
60-6 x 109Z0 60-6 xlO9 20*2 202 /fiolA
m— = —a— = To/t = 285’ <68i>
by making use of the data (666). That is, the amplitude of the received current is 42 per cent, greater than the steady current, when (666) is enforced.
But let Ql = -j-tt, then
VJC0 = + €-*«»]* = 28 L0 ohms;
and the ratio of impedance to resistance is
28 202 , 4 , —— x —— x 4 = - nearly,
60-6 285 3
or the amplitude of current is only 3/4 of the steady current.
And if Ql = |?r, we shall find
VJCQ = 43-5 ohms,
and that the impedance is slightly greater than the resistance. Whilst, if Ql = t, we shall have VJC0 = 47*8 ohms, and find the ratio of impedance to resistance to be 63/85, making the received current 35 per cent, stronger than the steady current. The above data of/= 1, and QI = \TT, ^7r, |7t, and 7r, have been chosen in order to get near the first maximum and minimum of impedance. The range, it will be seen, is very great. Let us next see how these data resemble practical data in respect to resistance, etc. Remember that 1 ohm per kilom. makes R = 104, (resistance per cm. of double conductor). Also, that /= 1 means R = nl= 105nlv if lx is in kilometres. Then, in the case to which (666) to (686) refer, we shall have, first assuming a given value of B, then varying L0, and deducing the values of n and lv the following results :—
LQ = 1, LQ = 10,
E= 103, n= 103, n = 102,
^ = 856, = 8568.
This is an excessively low resistance, ohm per kilom.; the frequencies are rather low, and the lengths great. Next, 1 ohm per kilom. :—
LQ=i, A>=io, z0=ioo,
104, n= 104 M = 10a, 102,
lx = 85. lx — 856. lx = 8568.
The LQ = 100 case is extravagant, requiring such a very distant return current (therefore very low electric capacity). Next, 10 ohms per kilom.:—
LQ—If L0—\0, Z0 = 100,
i2=105, n = 10®, w=104, 7i —10^,
lx = 8’5. lx = 85. Zx = 856.
Lastly, very high resistance of 100 ohms per kilom.:—
R = 106, X0=10, n = 105, lx = 8-5.
In all these cases the amplitude of received current is 42 per cent, greater than the steady current.
In the next case, Ql = \TT, the quantity TII/V has a value one-fourth of that assumed in the above; hence, wTith the same R and Z0, and same frequency, the above values of lx require to be quartered. Then, in all cases, the current-amplitude will be three-fourths of the steady current. Similarly, to meet the Ql = \TT case, use the above figures, with the lxs halved; and in the Ql = fir case, with the lx s multiplied by
A consideration of the above figures will show that there must be, in telephony, a good deal of this reinforcement of current strength sometimes ; not merely that the electrostatic influence tends to increase the amplitude all round, from what it would be were only magnetic induction concerned, but that there must be special reinforcement of certain tones, and weakening of others. It will be remembered that good reproduction of human speech is not a mere question of getting the lower tones transmitted well, but also the upper tones, through a long range; the preservation of the latter is required for good articulation. The ultimate effect of electrostatic retardation, when the line is long enough, is to kill the upper tones, and convert human speech into mere murmuring.
The formula (626) is the most useful if we wish to see readily to what extent the magnetic formula is departed from. In this, two quantities only are concerned, / and h, or (R'jL'n)2 and nljv, and if both / and h are small, it is readily seen that the first form of (636) applies, the factor by which the magnetic impedance is multiplied being (sinh)jh. Even when h is not small the/ terms in (626) may be negligible, and the first form of (636) apply. For example, suppose h = and/small, then (sin^)//i = 3 x *3272 = *9816, showing a reduction of 2 per cent, from the magnetic impedance.
Now, this h = \ means nlx = 105, or the high frequency of 105/2tt on a line of one kilom., 104/2ir on 10 kilom., and so on, down to 10/2^- on 10,000 kilom., always provided the / terms are still negligible. This may easily be the case when the line is short, but will cease to be true as the line is lengthened, owing to the n in / getting smaller and smaller. Thus, in the just-used example, if the resistance is 10 ohms per kilom., and L= 10, we shall have /=T^ on the line of 1 kilom., and /= 1 on 10 kiloms. So far, the / terms are negligible, and the first form of (636) applies. But / becomes 100 on 100 kiloms., which will make an appreciable, though not large, difference; and /= 10,000 on 1,000 kilom. will make a large difference and cause the first (636) formula to fail. It is remarkable, however, that this formula should have so wide a range of validity.
In the above we have always referred to the distant end impedance. But at the seat of impressed force there is a large increase of current on account of the “charge.” Thus, at x = 0, by the formula preceding (416), we have
^ = L'v( 1 + /)*[«“* 1~ 2 cQBjgfj* (69j)
The term impedance is of course strictly applicable at the seat of impressed force. As the frequency is raised, this impedance tends to be represented by VJC0 = L'v(l+f)K
and, ultimately, by VQ/CQ = L0v = 3Q L0 ohms, (706)
if the dielectric be air. L0 is usually a small number.
SECTION XXXII. THE EQUATIONS OF PROPAGATION ALONG WIKES. ELEMENTARY.
In another place (Phil. Mag., Aug., 1886, and later) the method adopted by me in establishing the equations of J^and C, Section xxix., was to work down from a system exactly fulfilling the conditions involved in Maxwell’s scheme, to simpler systems nearly equivalent, but more easily worked. Remembering that Maxwell’s is the only complete scheme in existence that will work, there is some advantage in this ; also, we can see the degree of approximation when a change is made. In the following I adopt the reverse plan of rising from the first rough representation of fact up to the more complete. This plan has, of course, the advantage of greater intelligibility to those who have not studied Maxwell’s scheme in its complete form; besides being, from an educational point of view, the more natural plan.
Whenever the solution of a so-called physical problem has been obtained, according to which, under such or such conditions, such or such effects must happen, what has really been done has been to solve another problem, which resembles the real one more or less in those features we wish to study, which we regard as essential, whilst it is of such a greatly simplified nature that its solution is, in comparison with that of the real problem, quite elementary. This remark, which is of rather an obvious nature, conveys a lesson that is not always remembered ; that the difference between theory and empiricism is only one of degree, even when the word theory is used in its highest sense, and is applied to legitimate deductions from laws which are known to be very true indeed, within wide limits.
It is quite possible to imagine the solution of the general problem of the universe. There does not seem to be anything against it except its possible infinite extent. Stop the extension of the universe somewhere; then, if its laws be fully known, and be either invariable or known to vary in some definite manner, and if its state be known at a given moment, it is difficult to see how it can be indefinite at any later time, even in the minutest particulars in the history of nations or of animal- culse, or in the development of a human soul (which is certainly immortal, for the good and evil worked by a soul in this life live for ever, in the permanent impress they make on the future course of events).
But if this be imagined to be all done, and the universe made a machine, no one would be a bit the wiser as to the reason why of it. (Even if we ask what we mean b}r the reason why, we shall in all probability get into a vicious circle of reasoning, from which there is no escape.) All that would be done would be the formulation of facts in a complete manner. This naturally brings us to the subject of the equations of propagation, for they are merely the instruments used in attempts to formulate facts in a more or less complete manner. The first to solve a problem in the propagation of signals was Ohm, wThose investigation is a very curious chapter in the history of electricity, as he arrived at results which are, under certain conditions, nearly correct, by entirely erroneous reasoning. Ohm followed the theory of the conduction of heat in wires, as developed by Fourier. Up to a certain point there is a resemblance between the flow of heat and the electric conduction current, but after that a wide dissimilarity.
Let a wire be surrounded by a non-conductor of heat, in imagination ; let the heat it contains be indestructible when in the wire, and be in a state of steady flow along it. If C is the heat-current across a given section, and V the temperature there, C w’ill be proportional to the rate of decrease of V along the wire. Or
-dF/dx = RC,
if x be length measured along the wire. The ratio R of the fall of temperature per unit length, to the current, is the “resistance” per unit length, and is, more or less, a constant. Or, the current is proportional to the difference of temperature between any two sections, and is the same all the way between.
The law which Ohm discovered and correctly applied to steady conduction currents in wires is similar to this. Make C the electric current in the wire, and V the potential at a certain place. The current, which is the same all the way between any two sections, is proportional to their difference of potential. The ratio of the fall of potential to the current is the electrical resistance, and is constant (at the same temperature). But V is, in Ohm’s memoir, an indistinctly defined quantity, called electroscopic force, I believe. Even using the modern equivalent potential, there is not a perfect parallel between the temperature V and the potential V For a given temperature appears to involve a definite physical state of the conductor at the place considered, whereas potential has no such meaning. The real parallel is between the temperature gradient, or slope, and the potential slope.
Now, returning to the conduction of heat, suppose that the heat- current is not uniform, or that the temperature-gradient changes as we pass along the wire. If the current entering a given portion of the wire at one end be greater than that leaving it at the other, then, since the heat cannot escape laterally, it must accumulate. Applying this to the unit length of wire, we have the equation of continuity,
- dCjdx = q,
t being the time, and q the quantity of heat in the unit length. But the temperature is a function of q, say
q=SV
where S is the capacity for heat per unit length of wire, here regarded, for simplicity of reasoning, as a constant, independent of the temperature. This makes the equation of continuity become
-dC/dx = SV.
Between this and the former equation between C and the variation of V, we may eliminate C and obtain the characteristic equation of the temperature,
d2Vldx2 = RSV,
which, when the initial state of temperature along the wire is known, enables us to find how it changes as time goes on, under the influence of given conditions of temperature and supply of heat at its ends.
Ohm applied this theory to electricity in a manner which is substantially equivalent to supposing that electricity (when prevented from leaving the wire) flows like heat, and so must accumulate in a given portion of the wire if the current entering at one end exceeds that leaving at the other. The quantity q is the amount of electricity in the unit length, and is proportional to V’ their ratio S being the capacity per unit length. With the same formal relations we arrive, of course, at the same characteristic equation, now of the potential, so that electricity diffuses itself along a wire, by difference of potential, in the same way as heat by difference of temperature.
A generation later, Sir W. Thomson arrived at a system which is formally the same, but having a quite different physical significance. Between the times of Ohm and Thomson great advances had been made in electrical science, both in electrostatics and electromagnetism, and the quantities in the system of the latter are quite distinct. We have
dV
- = RC, q = SV,
dx
_ dC _dq_gdV dx dt dt’
dW pqdF -
m=ESk' (71i)
where on the left appear the elementary relations, and on the right the resultant characteristic equation of V.
Here C is the current in the wire, R its resistance per unit length, and V the electrostatic potential. So far there is little change. But S is the electrostatic capacity per unit length of the condenser formed by the dielectric outside the wire, whose two coatings are the surface of the wire and that of some external conductor, as water, for instance, which serves as the return conductor. Thus S, from being in Ohm’s theory a hypothetical quantity depending upon the nature of the conducting wire, its size and shape, has become a definitely known quantity depending on the nature of the dielectric, and its size and shape. Here is the first step towards getting out of the wire into the dielectric, to be followed up later. The equation q = SV is the electrostatic law expressing the relation between the charge of a condenser and its potential-differ- ence, q being the charge on the wire per unit length, and V its potential. It is assumed that V= 0 at the outer conductor, which requires that its resistance must be very small, theoretically nothing. This makes V definitely the potential at the surface of the wire, and it must be the potential all over its section at a given distance x, if the current is uniformly distributed across the section.
The meaning of the equation of continuity is now, that when the current entering a given length of wire on one side is greater than that leaving it on the other, the excess is employed in increasing the charge of the condenser formed by the given length of wire, the dielectric, and the outer conductor. In the wire, therefore, comparing the electric current to the motion of a fluid, such fluid must be incompressible. It can, however, accumulate on the boundary of the wire, where it makes the surface-charge. This is exceedingly difficult to understand. But in any case, whether electricity accumulates in the wire or only on its boundary, is quite immaterial as regards the form of the equation of continuity, and of the characteristic equation. (Of course it is the equations which give rise to it, and their interpretation, that are of the greatest importance.)
There is very little hypothesis in this system. We unite the con- denser-law with Ohm’s law of the conduction current, on the hypothesis, which is supported by experiments with condensers and conductors, that the equation of continuity is of the kind supposed. But it is assumed that the electric force is entirely due to difference of potential. As, when the current is changing in strength, this is not true, there being then also the electric force of inertia, or of magnetic induction, this should also be taken into account in the Ohm’s law equation, making a corresponding change in the characteristic equation. What difference this will make in the manner of the propagation will depend upon the relative magnitude of the electric force of inertia and of the charge, and materially upon the length of the line. The necessary change will be made in the next Section. At present we may only remark that electrostatic induction is most important on long submarine cables, and that the (716) equations are those to be used for them for general purposes, as the first approximate representation of the facts of the case.
Now, as regards the accumulation difficulty. This is entirely removed in a beautifully simple manner in Maxwell’s theory. The line- integral of the magnetic force round a wire measures the current in it, a fact that cannot be too often repeated, until it is impressed upon people that the electric current is a function of the magnetic field, which is in fact what we generally make observations upon, the electricity in motion through the wire being a pure hypothesis. Maxwell made this the universal definition of electric current anywhere. There is no difference between a current in a conductor or in a dielectric ,as a function of the magnetic field, though there is great difference in the effect produced, according to the nature of the matter. All currents are closed, either in conductors alone or in dielectrics alone, or partly in one and partly in the other. In a conductor heat is the universal result of electric current, and energy is wasted; in a dielectric, on the other hand, the energy which would be wasted were it conducting is stored temporarily, becoming the electric energy, which is recoverable. In a conductor, the time-integral of the current is not a quantity of any physical significance; but in a dielectric it is a very important quantity, the electric displacement, which can only be removed by an equal reverse current. The electric displacement involves a back electric force, which will cause the displacement to subside when it is permitted by the removal of the cause that produced it. Put a condenser in circuit with a conductor and battery. The current goes right through the condenser. But it cannot continue, on account of the back force of the displacement; when this equals the impressed force of the battery, there is equilibrium. Remove the battery, and leave the circuit closed. The back force of the displacement can now act, and discharges the condenser. As for the positive and negative charges, they are numerically equal to the total displacement through the condenser. They are located at the places of, and measure the amount of discontinuity of the elastic displacement, and that is all.
If we must have a fluid to assist (keep it well in the back-ground), then this fluid must be everywhere, and be incompressible, and accumulate nowhere. I am no believer in this fluid. Its only utility is to hang facts together. But when one has obtained an accurate idea of the facts it has to hang together, it has served its purpose. A fluid has mass, and when in motion, momentum and kinetic energy. But the facts of electromagnetism decidedly negative the idea that the electric current per se has momentum or energy, or anything of that kind; these really belong to the magnetic field. It is therefore well to dispense with the fluid behind the scenes.
But when one thinks of the old fluids (of surprising vitality), and of their absurd and wholly incomprehensible behaviour, their miraculous powers of attracting and repelling one another, of combining together and of separating, and all the rest of that nonsense, one is struck with the extremely rational behaviour of the Maxwell fluid. When, further, one thinks of the greatly superior simplicity of the manner in which it hangs the facts together (it is remarkably good in advanced electrostatics, impressed forces in dielectric, etc.), one wonders why it does not take the place of the commonly used two-fluid hypothesis, merely as a working hypothesis, and nothing more.
Returning to the wire. It is important to remember that there are two conductors, not one only, with a dielectric between. When we put an impressed.force in the wire we send current across the dielectric as well as round the conducting circuit. The dielectric current ceases as soon as the back force of the elastic displacement supplies that difference of potential which is appropriate to the distribution of impressed force (which difference of potential depends entirely on the conductivity conditions). The equation of continuity means that when the current entering a unit length of wire on one side is greater than that leaving it on the other, the excess goes across the dielectric to the outer conductor, in which there is a precisely equal variation in the current. The time-integral of this dielectric current q is q, which is the total displacement outward per unit length of wire. The quantity V is the back E.M.F. of the displacement. On removing the impressed force, there is left the electric energy of the displacement, which is \Vq per unit length of wire; the back forces act, discharge the dielectric, and this energy is used up as heat in the conductors.
We can now make some easy extensions of the system (716). E must be the sum of the resistances of the wire and return, per unit length, thus removing the restriction that the return has no resistance.
S, of course, remains the same. But V cannot be the potential of the wire, because V cannot = 0 all along the return. We may, however, call V the difference of potential (although that is not exactly true, on account of inertia, unless we agree to include a part of the E.M.F. of inertia in V). It is, however, definitely the E.M.F. of the condenser, given by q = SV. We need not restrict ourselves, in these first approximations, to round wires, or to symmetrically-arranged returns. The return may be a parallel wire. Of course the proper change must then be made in the value of S.
SECTION XXXIII. THE EQUATIONS OF PROPAGATION. INTRODUCTION OF
SELF-INDUCTION.
The next step to a correct formulation of the laws of propagation along wires is, obviously, to take account of the electric force of inertia in the expression of Ohm’s law. This appears to have been first attempted by Kirchhoff in 1857. According to J. J. Thomson (“Electrical Theories,” The Electrician, June 25, 1886, p. 138) this was his system. Let e = X sin ns,
where e is the charge per unit length, and s is length measured along the wire. The equation of X is
c^tPX=r^dX d?X
2 ds2 16ly dt dt?’
where r is the resistance of the wire in electrostatic units, I its length, y = log (l/a), where a is its radius, and c is a quantity occurring in Weber’s hypothesis, the velocity with which two particles of electricity must move in order that the electrostatic repulsion and the electromagnetic attraction may balance.
As it stands, I can make neither head nor tail of it. But, by extensive alterations, it may be converted to something intelligible. Turn X into e, in the second equation; or, what will come to the same thing, take V as the variable, since e and V are proportional. Then ignore the first equation altogether. Turn s into our variable x. rp, d2F_ r dV 2 d2F
611 da? ~8ly dt + c2 dt2'
Clearly this should reduce to (716) by ignoring the last term. Therefore r/8ly = BS.
Here rjl is the resistance per unit length. Therefore (8y)-1 should be the capacity per unit length, or {8 log (//a)}-1. This is clearly wrong. The I should be a2, the resistance of the return, a far smaller quantity than I; and the 8 should be 2, if the dielectric is air. This last correction may, however, be merely required by a change of units. Making it, we get this result dW_ pqdV^ , qd2F dx2 dt 0 dt*’ in our previous notation, with the addition that L0 is the inductance per unit length of the dielectric only. That is,
L0 = 2fx log (ajaj,
with unit inductivity; a2 distance of return, ax radius of wire. This estimate of the inductance is, of course, too low. The change of units makes it doubtful whether L0 or some multiple of it was meant, but it is clearly a wrong estimate. Notice that L0S is the reciprocal of the square of a velocity, which is numerically equal to the ratio of the electromagnetic and electrostatic units, and is the velocity of light, or close to it.
It is clear that there is room for considerable improvement here in several ways, such as the establishment of the equations independently of such a very special hypothesis as Weber’s; also in the estimation of L; and, in interpretation, to modernise it in accordance wTith Maxwell’s ideas. Having observed that Maxwell, in his treatise, described the system (716) of the last section, with no allowance for self-induction, and knowing this system to be quite inapplicable to short lines, I (in ignorance of Kirchhoff’s investigation) made the necessary change of bringing in the electric force of inertia (Phil. Mag., August, 1876), [vol. I., p. 53], converting the system (716) to the following :—
-tr-no+rtg,
dx dt I d r , t nd r /rTOJ»\
dC_dq_gdV j dx2 dt dt2' dx dt dt’ J
The equations on the left side show the elementary relations, and that on the right the resultant equation of V.
The difference from (716) is only in the first equation of electric force, and in the characteristic equation of V. To the electric force due to V is added the electric force of inertia - LC, where L is the inductance of the circuit per unit length, according to Maxwell’s system of coefficients of electromagnetic induction. That is, L consists of three parts, say L0 for the dielectric, Lx for the wire, and L2 for the return. Their expressions will vary according to the size and shape of the conductors and their distance apart. In case of symmetry about an axis, their determination is very easy by the square-of-force method. The magnetic energy per unit length is }LC2. It is also 2 ^H‘2/87r, if , H is the magnetic force, and the ^summation extends over the region of space belonging to the unit length. As H is a simple function of C and of the distance from the axis, the integration is very easily effected. L is calculated on the hypothesis that the current-density has always the steady distribution, just as R is the steady resistance. As it is, strictly speaking, impossible to have the Faraday-law of induction true in all parts of the conductors without some departure from the steady distributions, it is satisfactory to know that more exhaustive investigation shows that L, not L0, should be used in a first approximation.
In connection with this matter I may mention that, rather singularly, just as I was investigating it, my brother, Mr. A. W. Heaviside, called my attention to certain effects observed on telegraph lines, which could be explained by the combined action of the electrostatic and electromagnetic induction, causing electrical oscillations which made the pointers of the old alphabetical indicators jump several steps instead of one. When freed from practical complications, and worked down to the simplest form, the matter reduced to this, that the discharge of a condenser through a coil is of an oscillatory character, under certain circumstances, and I described the theory in the paper I have mentioned. It had been given by Sir W. Thomson in 1853, but it is a singular circumstance that this very remarkable and instructive phenomenon should not be so much as mentioned in the whole of Maxwell’s treatise (first edition), though it is scarcely possible that he was unacquainted with it; if for no other reason, because it is so simple a deduction from his equations. I lay stress on the word simple, because it is not to be supposed that Maxwell was fully acquainted with the whole of the consequences of his important scheme.
Mr. Webb, the author of a suggestive little book on “ Electrical Accumulation and Conduction,” had very early practical experience of electrical oscillations in submarine cables, when they were coiled up on board ship, ceasing, more or less, as they were submerged.
It is far more difficult to obtain a satisfying mental representation of the electric force of inertia - LC than of that due to the potential, or - dVjdx, as described in the last section. The water-pipe analogy is, however, simple enough. Let L be the mass of the fluid per unit length, C its velocity, then \LC2 is its kinetic energy, LC its momentum, LC the force that must be applied to increase it, - LC the force of reaction. A mental representation of many of the phenomena connected with electrical oscillations is also very simply got by the use of the fluid analogy. It is, however, certainly wrong, as we find by carrying it out more fully into detail. Remark, however, that, as \LC'2 is the magnetic energy per unit length, LC is the generalised momentum corresponding to C as a generalised velocity, LC the generalised externally applied force, an ele trie force, of course, and - LC the force of reaction—that is, the electric force of inertia. This is by the simple principles of dynamics, disconnected from- any hypothesis as to the mechanism concerned.
The magnetic energy must be definitely localised in space, to the . amount Ł/aH2/47t per unit volume, and be regarded as the kinetic energy of some kind of motion in the magnetic field. When steady, there is no force of inertia. But when H changes, and with it C, since these are rigidly connected (in our first approximation) there is necessarily a force of inertia, which, reckoned as an electric force appropriate to C as a generalised velocity, is - LC per unit length.
In the discharge of a condenser through a coil, if we start with a charge, but no current, there is in the first place only the potential energy of the displacement in the condenser. The discharge cannot take place without setting up a magnetic field, proportional in intensity to the current at any moment, so that the original electrical energy is employed in heating the wire, and also in setting up the magnetic energy. When the condenser is wholly discharged, the inertia of the magnetic field keeps the current going, and it will continue until the whole energy of the magnetic field is restored to the condenser (less the part wasted in the wire) in the form of the energy of the negative displacement there produced. Except that the charge is smaller, and of the opposite sign, everything is now as when we started, so that we majr begin again and have a reverse current, continuing until the condenser is again charged in the same sense as at first, with no magnetic field. This is the course of a complete oscillation. But if the resistance be of or above a certain amount, depending on the capacity of the condenser and the inductance of the coil, the oscillations cease, and the discharge is completed in a single current which does not reverse itself.
Similar effects take place, in general, in any circuit, when a change is made which involves a redistribution of electric displacement, or its total discharge, but the full theory is usually very difficult to follow in detail. The so-called “ false discharge ” of a submarine cable is, however, easily comprehensible by the last paragraph.
If, in the characteristic equation of V in (72b), we take L = 0, reducing it to that of (716), we have simple diffusion of the static charge. If, for instance, the ends of the lines be insulated, any initial state of charge will settle down to be a uniform distribution, in a non-oscillatory manner, the smaller inequalities (smaller as regards length of line over which they extend) being wiped out rapidly, the larger more slowly; the law being that similar distributions subside similarly, but in times which are proportional to the squares of the lengths concerned.
If, on the other hand, we take B = 0 in the characteristic equation of V in (726), we have an entirely different order of events. As there is no waste in the wire, it is clear that the total energy of any initial state, electric and magnetic, remains undiminished. We can definitely divide the initial state into two distinct states travelling in the manner of waves in opposite directions, and being continuously reflected at the ends. Or, more simply, set up a charge at a single point of the line. It will divide into two, which will go on travelling backwards and forwards for ever. But into details of this kind we must not be tempted to enter at present, the immediate object being to lay the foundations for a more general theory.
When both terms on the right side of the characteristic equation are counted, propagation takes place by a mixture of diffusion and wave-transfer. A wave sent from one end of the line which would, were there no resistance, travel unchanged in form, and be reflected over and over again at the ends, in reality spreads out or diffuses itself, as well as, to a certain extent, being carried forward as a wave. The length of the line is an important factor. Wave characteristics get rapidly wiped out in the transmission of signals on a very long submarine cable, so that the manner of variation of the current at the distant end approximates to what it would be in the case of mere diffusion.
On the other hand, coming to a very short line, there are, every time a signal is made, immensely rapid dielectric oscillations, before the steady state is reached, due to to-and-fro reflection. As a general rule, this oscillatory phenomenon is unobservable, but it is none the less existent. It is customary to ignore it altogether in formulation, regarding the matter as one in which magnetic induction alone is concerned. Of course the magnetic energy is then far more important than the electric, and the current in the wire rises nearly in accordance with the magnetic theory.
The immense rapidity of the dielectric vibrations is one reason why they are unobservable, except indirectly, and under peculiar circumstances. Sometimes, however, they become prominent, especially when a circuit is suddenly interrupted, when we shall have large differences of potential. Mr. Edison discovered a new force. The enthusiasm displayed by his followers in investigating its properties was most edifying, and thoroughly characteristic of a vigorous and youthful nation. But it was only the dielectric oscillations, it is to be presumed ; unless indeed it be really true, as has been reported, that the renowned inventor has kept the new force concealed on his person ever since.
How is it, it may be asked, that in the rise of the current in a short wire, according to the simple magnetic theory, the potential at any point in the wire is regarded as a constant, viz., its final value when the current has reached the steady state 1 Thus, as we have
e dV 7dC A d2F n
- =ltf]+L and =
I dx dt dx1
if e is the total impressed force in the circuit, and I the length, the potential variation dV/dx must be constant. Supposing then e to exist only at x = 0, the current will rise thus :—
C'=®(1-Ł‘W)’
and the value of -dVfdx must be e/Z, from the very moment e is started, and so long as it is kept on.
When we seek the interpretation, in the more general theory, we find that although the current oscillations become so insignificant on shortening the line that the well-known last formula becomes valid, practically, yet the potential oscillations remain in full force during the variable period. A wave of potential travels to and fro at the velocity (LS)~l, making the potential at any one spot rapidly vibrate between a higher and a lower limit, though not according to the S. H. law, but in such a manner that its mean value is the final value, whilst the limits between which the vibration occurs continuously approach one another; the vibration, on the whole, subsiding according to the exponential law, with 2L/R as time-constant. The quantity e/l, which in the above rudimentary theory is taken to be the actual potential variation, is really the mean value of the real rapidly vibrating potential variation, at every point of the circuit and during the whole variable period, at whose termination, on subsidence of the vibrations, it becomes the real potential variation. [See vol. I., pp. 57 and 132 for details.]
To get rid of this vibration, we have merely to distribute the impressed force so as to do away with the potential variation.
Having now got the elementary relations established, we can proceed to the simplest manner of extending them to include the phenomena attending the propagation of current into the conductors from the dielectric.
SECTION XXXIV. EXTENSION OF THE PRECEDING TO INCLUDE THE PROPAGATION
OF CURRENT INTO A WIRE FROM ITS BOUNDARY.
The first step to getting out of the wire into the dielectric occurs in Sir W. Thomson’s theory, Section xxxii. We certainly get as far as the boundary of the wire. To some extent we make progress in adopting (same Section) Maxwell’s idea of the continuity of the conduction and the dielectric current, when the conduction current is discontinuous itself. Further progress is made (Section xxxili.) in introducing the electric force of inertia and the magnetic energy, so far as dependent on the first differential coefficient of the current with respect to the time, assuming the magnetic field to be fixed by the single quantity C, the wire-current, just as the electric field is fixed by the single quantity V, the potential-difference of the two wires at a given distance.
But the magnetic machinery does not move in rigid connection with the wire-current, as is implied in the specifications of the magnetic energy by like that of the electric energy by L and S being the inductance and the electric capacity, per unit length of line. In going further, I believe the following to be the most elementary method possible, as well as being pretty comprehensive.
To fix ideas, and simplify the nature of the magnetic field, let the line consist of two concentric tubes, separated by a dielectric nonconducting tube. The dielectric is to occupy our attention mainly, in the first place. Let
and a2 be its inner and outer radii, a0 the inner radius of the inner tube, and a3 the outer radius of the outer. Find the connection between the longitudinal electric force at the inner and outer boundaries of the dielectric tube, and the E.M.F. of the condenser, and the E.M.F. of inertia, so far as it depends upon the magnetic field in the dielectric.
Let ABCD in the figure be a rectangle in a plane through the common axis of the tubes, AB being on the inner and CD on the outer boundary of the dielectric, both of unit length. Let the current be from A to B in the inner tube, in which direction x is measured, and therefore from C to D in the outer tube. These currents are not precisely equal under all circumstances, but are so nearly equal that we can ignore the longitudinal current in the dielectric in com- A r = a parison with them; then the current C in the inner necessitates the same current C in the outer tube. The lines of magnetic force are directed upward through the paper, and the intensity of force is 2Cjr at distance r from the common axis of the tubes.
The total induction through the rectangle is therefore
,J““?Sfr= (V2 log “2V=Ł„C, Ja, r \ axJ
if fx2 be the inductivity of the dielectric, and L0 the inductance of the dielectric per unit length of line.
Now, the rate of decrease of the induction with the time, or - L0C, is the E.M.F. of inertia in the circuit ABCD in the order of the letters. But if E and F are the longitudinal electric forces in AB and DC, and
V and W the radial forces in BC and AD, another expression for the E.M.F. in the circuit is E-F+ V — W. But as AB and CD are of unit length, V— W=dVjdx. Hence
E- F+dV/dx= - L0C, or - dVjdx = L0C + E-F. (73b)
Next, let rx and T2 be the longitudinal current-densities at the boundaries of the conductors, px and p2 their resistivities, and ev e2 the impressed forces, if any, in them. Then, by Ohm’s law,
e1 + E = PlTi, >
and therefore E - F=p1T1- p2T2 -e, (746)
if e = eA-e2. Thus e is the impressed force in the circuit per unit length, irrespective of how it is divided between the inner and the outer conductor. Also, e is supposed to be longitudinal.
Now use (746) in (736), making it become
e-dVldx = L0C+p1T1-p2r2. (756)
We now require to connect 1^ and T2, the current-densities at the boundaries of the conductors, with the total currents in them. Representing these connections thus,
PtT^mC, - p2V2 = R%C, (766)
we require to find the forms of R" and 1%, one for the inner, the other for the outer conductor. If this be imagined to be done, and we put
R" = L0{d/dt) + R,' + R2y the equation (756) becomes
e - dF/dx = R"C=L0C + R”C+ R%C, (776)
wherein R" is known. The complete scheme will therefore be,
e-dVjdx = R"C, q = SF, - dCidx = dqjdt = SF, (786)
which should be compared with (716) and (726). As for the equations of F and of C\ they may be obtained by elimination, but it is unnecessary to write them at present.
We have supposed R” and R" to be known. The question is, then, how to find them. We know that in steady-flow they must be R1 and R2y the steady resistances of the conductors. We know, further, that they are Rl + L^d/dt) and R2 + L2(d/dt), when only the first derivative C of the current is allowed for. Now, we know that, under all ordinary circumstances, the length of a wire must be a very large multiple of its diameter before the influence of the electric charge becomes sensible. When it does become sensible, the current is of a different strength in different parts of the line during the setting up of a steady current. But in a section of the line which, though long compared with the diameter of the wire, is short compared with its length, the current changes insensibly, even when the change is very great between the current-strength in that section and in another, which, by contrast, may be called distant from the first.
It is, clear, therefore, that we shall come exceedingly near the truth if, in the investigation of the function R!( we altogether disregard the change in strength of the current in passing along the line. This amounts to ignoring the small radial component of the current in the conductors, and making the current quite longitudinal. This is only done for purposes of simplification, and does not involve any physical assumption in contradiction of the continuity of the current; for we join on the dielectric current to that in the conductors, by means of the equation of continuity, the third of (786).
The determination of R" and R" is thus made a magnetic problem, of which I have already given the solution. See equation (506), Section xxx., where the first big fraction represents R" for the inner conductor, and the second R" for the outer. The separation of these into even and odd differential coefficients, thus,
nz-m+up,
is of principal utility in the periodic applications. It may, perhaps, be as well pointed out that the first equation (78b) should, in strictness, be cleared of fractions to obtain the rational differential equation. But the advantages of the form (78b) are too great to be lightly sacrificed to formal accuracy.
We have now the means of fully investigating the transmission of disturbances along the line, including the retardation to inward transmission from the dielectric into the conductors as well as the effects of the electrostatic charge. The system is a practical working one; for, the electrical variables being Vand C, we are enabled to submit the line to any terminal conditions arising from the attachment of apparatus, the effect of which is fully determinable, because the differential equation of the apparatus itself is one between V and C. Both the ratio of V to C and their product are important quantities. The first is, in steady-flow, a mere resistance. In variable states it becomes a complex operator of great importance in the theoretical treatment. The second, VC, is the energy-current, concerning which more in the next Section.
In the meantime I will briefly indicate the nature of the changes made when we go further towards a complete representation of Maxwell’s electric and magnetic connections. First, as regards the small radial component of current in the conductors. The quantity s that appears in the expression for R" is given by
-Sj2 = 4TT/Z1&LP,
Hj being the inductivity and kx the conductivity of the inner conductor, whilst p is, when we are dealing with a normal system of subsidence, a constant; thus, ept is the time-factor showing how it subsides, p being always negative in an electromagnetic problem, and also always negative in an electrostatic problem, whilst in a combined electrostatic and magnetic case it is either negative and real, or negative with an imaginary part, when its term must be paired with a companion to make a real oscillatorily subsiding system. Now the simplest form of terminal condition possible is V=0 at both ends of the line, i.e., short- circuits. Then
V=A sin (jiTx/l),
where j is any integer, represents a V system, satisfying the condition of vanishing at both ends. Let the factor of x, which is jirjl, be denoted by m. Only the first few/s are of much importance, 1, 2, 3, etc. Now, if we change the connection between and p above-given to
Si = - l-rrnfap + ml,
we shall be able to take the radial component of current in the conductors into account; but the change made is usually very insignificant. There are four other cases in which we can work similarly— viz., when the line is insulated at both ends, or C=0; when it is insulated at either end and short-circuited at the other—two cases; and when the line is closed upon itself, each conductor making a closed circuit without interposed resistances, etc. In all except the last case, when the line has no ends, the quantity VC vanishes at both ends of the line, either V or C being zero at these places, so that no energy can enter or leave the line (dielectric and two conductors). Nor can this happen in the last case. But if we join on terminal apparatus, thus making VC finite at one or both ends, the system breaks down, and we require to fall back upon the preceding.
But if we keep to the five cases mentioned, we may make a further refinement, by taking the longitudinal current in the dielectric into account, which we have previously considered negligible in comparison with the current C. We cannot do this in terms of V, which is inadequate to express the electric energy. But we may do it in terms of the electric and magnetic forces, and then obtain a full representation of Maxwell’s connections, instead of an approximate. But even in this it is assumed that there is no magnetic disturbance outside the outer conducting tube or inside the inner, which there must really be, for we must have continuity of the tangential electric force, which necessitates electric force, and therefore also electric displacement and current and magnetic force, outside the outer tube and inside the inner, having some minute disturbing effect on the current in the conductors.
We may, however, leave these refinements to take care of themselves, and return to the V and C system of representation. The advantage of dealing with concentric tubes is due to the circularity of the lines of magnetic force, which produces considerable mathematical simplifications, as well as physical. Suppose, however, the tubes are not concentric, although the dielectric is still shut in by them. Here, clearly, to a first approximation, we have merely to give changed values to the constants S and L, whilst B is unchanged. But to go further, the determination of B" and B" will present great difficulties. This, however, is clear: that the full L' will have for its minimum value, approached with very rapid oscillations, Z0, such that SL0 = v~2, where v is the speed of propagation of undissipated disturbances through the dielectric. This follows by regarding the conductors as infinitely conducting, so that there is no waste in them, when the equation of V becomes
dW dW _ nT d2V /7QM
~IVW OT (79J)
showing wave propagation with velocity v.
But if the two conductors be parallel solid wires or tubes (not concentric), and be placed at a sufficient distance from one another, the lines of magnetic force in and close round the conductors will be very nearly circles, so that we may regard B" and B% as known by the preceding ; and we can therefore go beyond the approximate method of representation founded upon B, S, and L only. Even if we bring the conductors so close that there is considerable disturbance from the assumed state, we should still, in reckoning B" and B" in the same way, go a long distance in the direction required, especially in the case of iron wires, in which, by reason of the high inductivity, the magnetic retardation is so great.
The effect of leakage has not been allowed for in the preceding. The making of the necessary changes is, however, quite an elementary matter in comparison with those connected with magnetic retardation. We require to change the form of the equation of continuity. If there be a leakage-fault on an otherwise perfectly insulated line, we have the line divided into two sections, in each of which the former equations hold good ; whilst at the place of the leak there is continuity of V and discontinuity of C, the current arriving at the leak on the one side exceeding that leaving it on the other by the current in the leak itself, which is the quotient of V by the resistance of the leak, if it be representable as a resistance merely. But when the leakage is widely distributed it must be allowred for in the line-equations. Even in the case of leakage over the surface of the insulators of a suspended wire, the proper and rational course is to substitute continuously distributed leakage for the large number of separate leaks; which amounts to the same thing as substituting a continuous curve for a large number of short straight lines joined together so as to closely resemble the curve. The equation of continuity becomes
-dC/dx = KF+SF, (806)
where the fresh quantity K is the conductance, or reciprocal of the insulation resistance, per unit length of line. That is, the true current leaving the line is the sum of the former SV\ the condenser-current, and of KV\ the leakage-current, both of which co-operate to make the current in the line vary along its length, although in the steady state it is the leakage alone that thus operates. But as regards retardation, their effects are opposed. The setting up of the permanent state is greatly facilitated by leakage, as is most easily seen by considering the converse, viz., the subsidence of the previously set-up steady state to zero when the impressed force is removed. If, then, we wish to increase the clearness of definition of current-changes at the distant end of a line on which electrostatic retardation is important, we can do it by lowering the insulation-resistance as far as is practicable.
SECTION XXXV. THE TRANSFER OF ENERGY AND ITS APPLICATION TO WIRES.
ENERGY-CURRENT.
When the sage sits down to write an elementary work he naturally devotes Chapter I. to his views concerning the very foundation of things, as they present themselves to his matured intellect. It may be questioned whether this is to the advantage of the learner, who may be well advised to “ skip the Latin,” as the old dame used to say to her pupils when they came to a polysyllable, and begin at Chapter II. If this be done, Prof. Tait’s “ Properties of Matter ” is such an excellent scientific work as might be expected from its author. But Chapter I. is metaphysics. There are only two Things going, Matter and Energy.
Nothing else is a thing at all; all the rest are Moonshine, considered as Things.
However this be, the transfer of energy is a fact well known to all, even when we put the statement in such a form that the energy seems to lose its thinginess, by calling it the transfer of the power of doing work. Thus, after transfer of energy from the sun ages ago, followed by long storage underground and convection to the stove or furnace, we set free the imprisoned energy, to be generally diffused by the most varied paths. The transfer from place to place can be, in great measure, traced so far as quantity and time are concerned; but it does not seem possible to definitely follow the motion of an atom of energy, so to speak, or to give a fixed individuality to any definite quantity of energy.
Whenever the dynamical connections are known, the transfer of energy can be found, subject to a certain reservation. In the elementary case of a force, F> acting on a particle of mass m and velocity v, F is measured by the rate of acceleration of momentum, or F=mv; and, to obtain the equation of activity, we merely multiply this equation by the velocity, getting Fv = mvv = the rate of increase of the kinetic energy T', or ^mv2, which is the amount of work the particle can do against resistance in coming to rest. Where the energy came from is here left unspecified. In a case of impact, we may clearly understand that the transfer of kinetic energy is from one of the colliding bodies to the other through the forces of elasticity brought into play, thus making potential energy an intermediary, though what the potential energy may be, and whether it is not itself kinetic, or partly kinetic, we are not able to decide.
It is much more difficult in the case of gravity. As the stone falls to the ground, it acquires kinetic energy truly; and if energy moves continuously, as its indestructibility seems to imply, it must receive its energy from the surrounding medium; or the energy of gravitation must be in space generally, wholly or in part, and be transferred through space by definite paths through stresses in the medium, by which means Maxwell endeavoured to account for gravitation. In general, we have only to frame the equations of motion of a continuous system of forces, and it stands to reason that the transfer of energy is to be got by forming the equation of activity, not of the system as a whole, but of a unit volume.
Now, in the admirable electromagnetic scheme framed by Maxwell, continuous action through space is involved, and the kinetic and potential energies (or magnetic and electric) are definitely located, as well as the seat and amount of dissipation of energy. We therefore need only form the equation of activity to find the transfer-of-energy vector. Of course impressed forces are subject to the energy definition. No other is possible in a dynamical system.
But if we take Maxwell’s equations and endeavour to immediately form the equation of activity (like Fv=T from F=mv), it will be found to be impossible. They will not work in the manner proposed. But we may consider the energy, electric and magnetic, entering and leaving a given space, and that dissipated within it, and by laborious transformations evolve the expression for the vector transfer. This was first done by Prof. Poynting for a homogeneous isotropic medium (Phil. Trans., 1884). In my independent investigation of this matter,
I also followed this method in the first place (The Electrician, June 21, 1884) [p. 377, vol. i.J in the case of conductors. But the roundabout nature of the process to obtain what ought to follow immediately from the equations of motion, led me to remodel Maxwell’s equations in some important particulars, as in the commencing Sections of this Article (Jan., 1885) with the result of producing important simplifications, and bringing to immediate view useful analogies which are in Maxwell’s equations hidden from sight by the intervention of his vector-potential. This done, the equation of activity is at once derivable from the two cross-connections of electric force and magnetic current, magnetic force and electric current, in a manner analogous to Fv = t, without roundabout work, and applicable without change to heterogeneous and heterotropic media, with distinct exhibition of what are to be regarded as impressed forces, electric and magnetic. (Electrician, Feb. 21, 1885) [p. 449, vol. I.]
Knowing the electric field and the magnetic field everywhere, the transfer of energy becomes known. The vector transfer at any place is perpendicular to both the electric and the magnetic forces there, not counting impressed forces. Its amount per unit area equals the product of the intensities of the two forces and the sine of their included angle.
But I mentioned that there is a reservation to be made. It is like this. If a person is in a room at one moment, and the door is open, and we find that he is- gone the next moment, the irresistible conclusion is that he has left the room by the door. But he might have got under the table. If you look there you can make sure. But if you are prevented from looking there, then there is clearly a doubt whether the person left the room by the door or got under the table hurriedly. There is a similar doubt in the electromagnetic case in question, and in other cases. Thus, we can unhesitatingly conclude from the properties of the magnetic field of magnets that the mechanical force on a complete closed circuit supporting a current is the sum of the electromagnetic forces per unit volume (vector-product of current and induction), but it does not follow strictly that the so-called electromagnetic force is the force really acting per unit volume, for any system of forces might be superadded which cancel when summed up round a closed circuit.
So, in the transfer-of-energy case, there may be any amount of circulation of energy in closed paths going on (as pointed out in another manner by Prof. J. J. Thomson), besides the obviously suggested transfer, provided this superposed closed circulation is without dissipation of energy. Or, if W be the vector energy-current density, according to the above-mentioned rule, we may add to it another vector, say w, provided w have no convergence anywhere. The existence of w is possible, but there does not appear to be any present means of finding whether it is real, and how it is to be expressed.
Its consideration may seem quite useless, in fact. But it is forced upon us in quite another way, by the fact that, when w = 0, we are sometimes led to the circuital flux of energy. Let, for instance, a magnet be placed in the field of an electrified body; or, more simply, let a magnet be itself electrified. There is no waste of energy ; hence the flux of energy caused by the coexistence of the two fields, electric and magnetic, is entirely circuital. E.g., in the case of a spherical uniformly magnetised body, uniformly superficially electrified, it takes place in circles in parallel planes perpendicular to the axis of magnetisation, the circles being centred on this axis. This circuital flux is entirely through the air or other dielectric. What is the use of it? On the other hand, what harm does it do ? And if the medium is really strained by coexistent electric and magnetic stresses, why should there not be this circuital flux ? But, if we like, we maj- cancel it by introducing the auxiliary w.
There is yet another kind of closed circulation, according to W alone, not existing by itself, but set going by impressed forces causing a useful transfer of energy, and ceasing when the useful transfer ceases. If, for instance, we close a conductive circuit containing a battery, we set up a useful transfer from the battery to all parts of the wire, through the dielectric usually. Suppose there is also impressed electric force in the dielectric, or electrification, or any stationary electric field.
If the battery does not work there is no transfer of energy. But when it does, there is, besides the regular first-mentioned transfer from the battery to the wire, a closed circulation due to the coexistence of the stationary electric field and the magnetic field of the wire-current, the resultant transfer being got by superposing the regular flux and the closed circulation. Here again, by introducing w, we may reduce it to the regular undisturbed transfer. It is clear, then, in considering the nature of the transfer in a useful problem, that it is of advantage to entirely ignore the useless transfer, and confine our attention to the undisturbed.
A general description of the transfer along a straight wire was given in Section II. [vol. I., p. 434]. It takes place, in the vicinity of the wire, very nearly parallel to it, with a slight slope towards the wire, as there described. Prof. Poynting, on the other hand (Royal Society, Transactions, February 12, 1885), holds a different view, representing the transfer as nearly perpendicular to a wire, i.e., with a slight departure from the vertical. This difference of a quadrant can, I think, only arise from what seems to be a misconception on his part as to the nature of the electric field in the vicinity of a wire supporting electric current.
The lines of electric force are nearly perpendicular to the wire. The departure from perpendicularity is usually so small that I have sometimes spoken of them as being perpendicular to it, as they practically are, before I recognised the great physical importance of the slight departure. It causes the convergence of energy into the wire. To estimate the amount of departure, we may compare the normal and tangential components of electric force. Let there be a steady current in a straight wire, and the fall of potential from beginning to end be V0 - V1; the tangential component is then (VQ - V\) I, if I be the length of wire. On the other hand, the fall of potential from the wire to its return—of no resistance, for simplicity—at any distance from the beginning of the line, is V, which is V0 at one end and at the other. It is clear at once that the tangential is an exceedingly small fraction of the normal component of electric force, if the wire be long, and that it is only under quite exceptional circumstances anything but a small fraction. Prof. Poynting should therefore, I think, make his tubes of displacement stick nearly straight up as they travel along the wire, instead of having them nearly horizontal, unless 1 have greatly misunderstood him.
But if we distribute the impressed force uniformly throughout the circuit, so that there shall be, in the steady state, no difference of potential and no transfer of energy, owing to the impressed force at any place being just sufficient to support the current there then, on starting the impressed force, the transfer of energy will be perpendicular to the wire outward, ceasing when the steady state is reached; and, on the other hand, on stopping the impressed force the transfer will be perpendicular to the wire inward, the magnetic energy travelling back again (assisted by temporary longitudinal electric force, which has no existence in the steady state) to be dissipated in the wire. But this case, though imaginable, is not practically realisable.
In the vicinity of the wire the radial electric force varies inversely as the distance, and so does the intensity of magnetic force. The density of the energy-current therefore varies inversely as the square of the distance approximately. This does not continue indefinitely. Thus, if the return be a parallel wire the middle distance is the place of minimum density of the energy-current, in the plane of the two wires. As regards the total energy-current, this is VC, the product of the fall of potential from one wire to the other into the current in each. One factor, V, is the line-integral of the electric force across the dielectric.
The other, C, is the line-integral (-i- far) of the magnetic force round either wire.
In the figure, AB and CD are the two wires, enormously shortened in length compared with their distance apart, joined through terminal terminal arrangement entered, to be wasted in frictional heat-genera- tion RXC2 therein, or otherwise disposed of. The curved lines and arrows perpendicular to them show lines of electric force and the direction of the energy-flux at a certain place, the inclination of the lines of force to the perpendicular being greatly exaggerated, as well as that of the lines of flux of energy to the horizontal, in order to show the convergence of energy upon the wires, there to be wasted. Its further transfer belongs to another science.
The rate of decrease of VC as we travel along the line is the waste per unit length. Thus, <8,J)
R1 and R2 being the resistances of the wires per unit length. This is in steady-flow, with no leakage. But if there be leakage, we have the equation of continuity
- (dCjdx) = KV,
making - A( VC) = KV2 + (R1 + R2)C2, (826)
where KV2 is the waste-heat per second due to the leakage-resistance. But when the state is not steady, we have the equation of continuity
M=RV+SV, (806) bis.
U/X
and the equation of electric force
dV
-aŁ. = L0C + E-F, (73b) bis.
so that - J-(FC) = KV2+jMV1) + i-UAC2) + EG-FC. (836)
Here we account for the leakage-heat, for the increase of electric energy, and for the increase of magnetic energy in the dielectric by the first, second, and third terms on the right side. EC, the fourth term, represents the energy entering the first wire per second, E being the tangential electric force; and — FC, the last term, represents the energy entering the second wire per second, F being the tangential electric force at its boundary reckoned the same way as E. The energy-flux is now perpendicular to the current, i.e., after entering the wires, ceasing when the axes are reached. And,
EC=Q1 + fv -FC=Q2 + T2, (846)
if Qv Q2, are the dissipativities, Tx and T2 the magnetic energies in the two wires, per unit length of line.
If the impressed force is a S.H. function of the time, so is the current, etc., everywhere, and
E = R[C+L[C, -F=R'2C+L'2C, (856)
where R{, i2Ł, L[, and L'2 are constants depending upon the frequency, reducing to the steady resistances and inductances when the frequency is infinitely low. In this S.H. case
<3i=wfil Q2=WCS, Twines, T,=mci,
are the mean dissipativities and magnetic energies in the wires, CQ being the amplitude of the current; the halving arising from the mean value of the square of a sinusoidal function being half the square of its amplitude. But in no other case is there anything of the nature of a definite resistance, although, if the magnetic retardation to inward transmission is small, we may ignore it altogether, and drop the accents in (856).