Heaviside Electromagnetic Induction And Its Propagation Sec I 2 X

SECTION I. ROUGH SKETCH OF MAXWELL’S THEORY.

Conductivity, Capacity, and Permeability.

In the electromagnetic scheme of Maxwell there are recognized to be three distinct properties of a body considered with reference to electric force and magnetic force, viz., conductivity, electrostatic capacity, and magnetic permeability. The body may support a conduction current, it may support electric displacement, and it may support magnetic induction. These three phenomena may, and in general do, coexist at any one point. Quantitatively considered, they are all vector magnitudes, having definite directions as well as strengths, which are reckoned per unit area perpendicular to their directions in terms of chosen units.

The facility of supporting the three states of conduction current, electric displacement and magnetic induction varying with the nature of the medium for equal amounts of energy concerned, brings in three coefficients, the electric conductivity k, the electric capacity c, and the magnetic permeability $\mu$. At first sight it might appear as if three other vector magnitudes related to the former by these coefficients were involved; but in reality there are but two, the electric force and the magnetic force, the former being connected with both the conduction current and the displacement. First we have Ohm’s law. C being the conduction current-density, E the electric force, and k the specific conductivity,

$ C = kE $

(Conduction current) (1)

Far more is known about conductivity than about capacity or permeability. In an unstrained isotropic metal, k appears to depend on the temperature only, and not to vary rapidly with it. That is, k is practically a constant, which simplicity is of great utility. Within wide limits k is independent of the current or the electric force. The range of conductivity in different media is very great. From the conductivity of copper to that of cold glass is such an enormous range as to compare with astronomical ratios, and it speaks well for electrical science that it can compare definitely such widely differing magnitudes.

Dry air in its ordinary state appears to have no conductivity. But it is a vacuum that is the perfect non-conductor in Maxwell’s theory. Where there is no matter, in the ordinary sense, there is no dissipation of energy; and ether, whatever it be, is perfectly conservative and non-dissipative, dynamically considered. Dissipation of energy is a necessary accompaniment of a conduction current, so far as is known ; though of course a perfect conductor can be imagined in which a continuous current developed no heat. But ether cannot be this perfect conductor consistently with the propagation of magnetic disturbances, for none can be propagated in a perfect conductor. Grant that they are propagated in pure ether (space from which all "matter" has been removed) without loss of energy in the medium, and it follows that ether is the perfect non-conductor. This however, somewhat anticipates electric displacement and magnetic induction.

Equation (1) is a vector equation. In an isotropic medium k is a scalar constant. We may symbolise E, C, or other physical vector magnitudes by geometrical vectors, lines drawn of the proper lengths and in the proper directions. Thus E is one vector, C is another, and when, as ordinarily, k is a scalar constant, (1) says simply that C and E are parallel, and that C is k times as long as E. Vector quantities are compounded like velocities; in a vector equation containing n vectors, separated by + or — signs, the n vectors form the n sides of a polygon. But two straight lines cannot enclose a space, so, in equation (1), C and kE are parallel and equal.

But in a body eolotropic as regards conductivity, C and E are only exceptionally parallel. Using the same equation (1) to represent the relation between them, k, from being a scalar constant, becomes a linear operator ; kE must be regarded as a single symbol, being E operated upon by k in a certain manner, turning it into a new vector kE. The operation is a little complex when expressed in Cartesian co-ordinates referred to any axes, so it is better to define once for all the meaning to be attached to k when eolotropy is to be included, and then use equation (1), rather than be repeating the Cartesian operations over and over again. The following defines the operation k, and the same will serve for c and $\mu$ later. First let there be no rotatory power. Then, in three directions, mutually perpendicular, fixed in a body at the point considered, depending on its structure there, Ohm’s law, as ordinarily considered, is obeyed. That is to say, if electric forces $E_{1}$, $E_{2}$, $E_{3}$ act successively parallel to the above mentioned directions of the principal axes of conductivity, and $C_{1}$, $C_{2}$, $C_{3}$ be the corresponding currents, $C_{1}$ will be parallel to $E_{1}$, $C_{2}$ to $E_{2}$, and $C_{3}$ to $E_{3}$, and we shall have

$ C_{1} = k_{1}E_{1}, C_{2} = k_{2}E_{2}, C_{3} = k_{3}E_{3} $

and

$ C_{1} = k_{1}E_{1}, C_{2} = k_{2}E_{2}, C_{3} = k_{3}E_{3} $

(2)

where $k_{1}$, $k_{2}$, $k_{3}$ are scalar constants, being the principal conductivities, and $C_{1}$ is the tensor or magnitude of $C_{1}$, $C_{2}$ of $C_{2}$, etc. From these we may find the current when the force acts in any other direction than parallel to one of the principal axes. For if E be the force, let its components parallel to the axes be $E_{1}$, $E_{2}$, $E_{3}$; the components of the current will then be $C_{1}$ $C_{2}$, $C_{3}$, as defined by (2). Compounding them, we get C. Thus the relation of C to E requires a knowledge of the principal conductivities and the directions of the principal axes.

But should the body possess rotatory power, the above process is incomplete. Let $\epsilon$ be a vector, directed parallel to the conductivity axis of rotation, and of length properly chosen; then, to the current as found by the above process must be added another current expressed by

$ V \epsilon E $

(Vector product) (3)

which stands for a vector whose direction is perpendicular to the plane containing $\epsilon$ and E, and whose length equals the product of the length of $\epsilon$ into that of E, into the sine of the angle between their directions. This also defines the prefix V before two vectors. The + direction is defined thus. Let $\epsilon$ and E be the short and the long hands of a watch. Let $\epsilon$ point to XII. and E anywhere else. The angle between $\epsilon$ and E is measured positive in the usual direction of motion of the hands, and the direction of V$\epsilon$E when positive is from the face to the back.

It is possible, consistent with the linear principle, for $k_{1}$, $k_{2}$, $k_{3}$ to be all zero, and $\epsilon$ not zero. Then

$ C = V \epsilon E $

simply; the current is always perpendicular to the force, of maximum strength when $\epsilon$ and E are perpendicular, and vanishing when they are parallel.

Returning to equation (1), multiply it by E. Then

$ EC = EkE = Q, say. $

(Dissipativity) (4)

Q is the dissipativity per unit volume. It is, in the first place, the rate of working of the force E, and next, by the experimental law of Joule, the rate of generation of heat per unit volume. (4) is a scalar equation. All our equations will be either wholly scalar or wholly vector. In case of isotropy, with k a scalar constant, we may write

$ Q = kE^{2} $

or, since

$ E = k^{-1}C $

$ Q = k^{-1}C^{2} $

where $k_{-1}$ is the specific resistance, a more familiar form of Joule’s law. But in general, when k is a linear operator, we must not take $EkE = kE^{2}$, unless E act parallel to one of the principal axes, when we may do so, with the appropriate value of k for that axis. When E and C are not parallel, the product EC means the strength of E multiplied by that of C, and by the cosine of the angle between their directions ; which of course includes the common algebraic meaning of EC, since when E and C are parallel, cos 0° = l. Referred to three rectangular axes, it $E_{1}$, $E_{2}$, $E_{3}$ are the scalar components of E, and $C_{1}$, $C_{2}$, $C_{3}$ those of C, then

$ EC = E_{1}C_{1} + E_{2}C_{2} + E_{3}C_{3} $

(Scalar product) (5)

which is an equivalent definition of EC.

Coming next to specific capacity, although there are media, as air, which appear to have no conductivity, yet, by the continuity of the electric current, they can support current; not steady, but transient, and stopped elastically. By an obvious mechanical analogy the integral current is termed the electric displacement. Let this be D, and let E, as before, be the electric force. We have then

$ D = cE/4 \pi $

(Electric displacement) (6)

The excrescence 4$\pi$ is a mere question of units, and need not be discussed here. The $4/\pi$’s are particularly obnoxious and misleading in the theory of magnetism. Privately I use units which get rid of them completely, and then, for publication, liberally season with $4/\pi$’s to suit the taste of B.A. unit-fed readers. Of course, if it comes to numerical comparisons we should have to consider the ratios of units in the ordinary to what I may call the rational system. Sometimes it is $\sqrt{4\pi}$, sometimes $(4\pi)^{-\frac{1}{2}}$, sometimes $4/\pi$, sometimes unity, but in the mere algebra it is simply a matter of putting in $4/\pi$’s here and there in translating from rational to ordinary units. [See pp. 199, 262.]

In a dielectric medium, the force and its displacement are simultaneous, like the force and the current in a conductor. Time does not appear in the equations. In an isotropic dielectric, c is simply a scalar constant; in an eolotropic dielectric it is, as described above for k, a linear vector operator, with this difference, however, that there is no rotatory vector $\epsilon$, so that the relation of D to E is settled by the values of the principal capacities, and their axes.

Multiply (6) by $\frac{1}{2}$E; then

$ \frac{1}{2}ED = EcE/8\pi = U, say. $

(Electric energy) (7)

U is the electric energy per unit volume, the work done by the force on the displacement as they rise from 0 to their final values, or the final displacement multiplied by the mean force which produced it. This energy is stored, and is recoverable in work like the energy of a perfectly elastic strained spring. It is unnecessary to assume that there is any real displacement of anything in the direction of the electric displacement. All the electric and magnetic quantities are more or less abstractions, measurable abstractions, whose real signification is as yet unknown.

Far less is known of c than of k, and it is not so agreeably definite as k. Solid dielectrics appear to have imperfect electric elasticity, as they have imperfect mechanical elasticity. The bent spring, with the applied force removed, and brought quietly to rest, is not exactly in its equilibrium position. A small part of the displacement remains, and slowly disappears. This is easily shown when not visible to the eye by using a microphonic contact; though, by the way, the variability of the contact itself makes it a bad method. Most likely there is no such thing as a perfect return even with small displacements; we cannot draw a hard and fast line to mark the limit of perfect elasticity.

All non-conductors are dielectrics. Bad conductors are also dielectrics. Good conductors, even the best, may be dielectrics as well, so that with a force E we shall have a conduction current kE and a displacement $(4\pi)^{-1}cE$ co-existing. But in such case, as well as in the case of known dielectric power of bad conductors, kE is not the complete or true current, unless the displacement remains steady. The time-variation of the displacement is itself an electric current, and the true current is the sum of the conduction current and of the rate of increase of the displacement. Let $\tau$ be the true current; we then have, in a conducting dielectric, or dielectric conductor,

$ C = kE, D = cE/4\pi $
$ \tau = C + D = kE + cE/4\pi $	(True current) (8)

Put c = 0 in a pure conductor, and k = 0 in a pure dielectric. It is the true current that is "the current" when we come to induction and variable states.

In the equation $\tau = C + D$ we have three vectors. They form the three sides of a triangle, unless D should be parallel to C. But D may not be parallel to C, nor need it be parallel to D. If we charge a condenser formed of two large flat opposed conductors very close together, the displacement current, when setting up the displacement, is, by general reasoning, parallel to the displacement—at least away from the edges. But this is not invariable. When charged conductors are discharged, the displacement current does not in general follow the tubes of displacement. To do so would require instantaneous propagation of the disturbances to infinite distances. The displacement current may be perpendicular to the displacement—viz., when the displacement at a certain place changes its direction without changing in amount.

Multiply (8) by E; then

$ E\tau = EkE + Ec\dot{E}/4\pi = Q + \dot{U} $

(9)

The rate of working of the force is accounted for partly in heating (Q per second), and partly in the increase in the energy U of the displacement. (Equations (4) and (7)). The first is lost from the system, the latter is stored.

Whilst conductivity depends on the presence of matter, the existence of capacity is independent of matter, though modified in amount by its presence. That is, capacity is a function of the ether, which is the standard dielectric medium of least capacity. Ether is a very wonderful thing. It may exist only in the imaginations of the wise, being invented and endowed with properties to suit their hypotheses; but we cannot do without it. How is energy to be transmitted through space without a medium? Yet, on the other hand, gravity appears to be independent of time. Perhaps this is an illusion. But admitting the ether to propagate gravity instantaneously, it must have wonderful properties, unlike anything we know.

Coming next to permeability, all bodies sustain magnetic induction, and most of them to nearly the same degree. H being the magnetic force, B the induction, and $\mu$ the permeability,

$ B = \mu H $

(Magnetic induction) (10)

$\mu$ is taken as unity in ether (in the "electromagnetic" system of units), and is either a little greater or a little less in most bodies. But in some bodies it, very singularly, runs up to large numbers. Iron is the principal offender; then come nickel and cobalt, minor magnetics, but far removed from the crowd of almost unmagnetisable substances. Fe = 56, Ni and Co about 58.5. What can it be?

The linear connection between H and B is very unsatisfactory. Not merely does $\mu$ vary with the temperature, and enormously from one piece of iron to another, being, with moderate strength of magnetic force, largest in the softest iron and smallest in hard steel, but it varies with the magnetic force, first increasing with the force, and then, more importantly, decreasing greatly; how far down is unknown. To make matters worse, part of the induction produced by applied magnetising force becomes fixed, for the time, remaining after the removal of the force. Thus the linear connection between H and B must be taken with salt. But within moderate limits, and excluding permanent magnetisation, which requires separate consideration, $\mu$ in equation (10) may be taken to be, like k and c before, a scalar constant in case of isotropy, and a linear vector operator in eolotropic media, being then, like c, self-conjugate, or without the rotatory power.

$\mu$ in soft iron is said to run up to 5,000 or 10,000 (Rowland’s experiments. I forget the exact figures). But in general it is very far lower than these tremendous figures. From experiments on the retardation of coils made some years ago, including straight solenoids, I concluded that $\mu$ = from 50 to 200 was safe, [for small forces].

Not B, but $B/4\pi$ should be the magnetic induction to compare with D, the electric induction, or displacement. So, dividing (11) by $4\pi$, and then multiplying by $\frac{1}{2}H$, we have

$ \frac{1}{2}HB/4\pi = H\muH/8\pi = T, say. $

(Magnetic energy) (11)

T is the energy of the magnetic induction per unit volume, when wholly induced, and acting conservatively, [within the elastic limits].

SECTION II. On THE TRANSMISSION OF ENERGY THROUGH WIRES BY THE ELECTRIC CURRENT.

Consider the electric current, how it flows. From London to Manchester, Edinburgh, Glasgow, and hundreds of other places, day and night, are sent with great velocity, in rapid succession, backwards and forwards, electric currents, to effect mechanical motions at a distance, and thus serve the material interests of man.

By the way, is there such a thing as an electric current? Not that it is intended to cast any doubt upon the existence of a phenomenon so called ; but is it a current—that is, something moving through a wire? Now, although nothing but very careful inculcation at a tender age, continued unremittingly up to maturity, of the doctrine of the materiality of electricity, and its motion from place to place, would have made me believe it, still, there is so much in electric phenomena to support the idea of electricity being a distinct entity, and the force of habit is so great, that it is not easy to get rid of the idea when once it has been formed. In the historical development of the science, static phenomena came first. In them the apparent individuality of electricity, in the form of charges upon conductors, is most distinctly indicated. The fluids may be childish notions, appropriate to the infancy of science ; but still electric charges are easily imaginable to be quantities of a something, though not matter, which can be carried about from place to place. In the most natural manner possible, when dynamic electricity came under investigation, the static ideas were transferred to the electric current, which became the actual motion of electricity through a wire. This has reached its fullest development in the hands of the German philosophers, from Weber to Clausius, resulting in ingenious explanations of electric phenomena based upon forces acting at a distance between moving or fixed individual elements of electricity. It so happened that my first acquaintance with electricity was with the dynamic phenomena, and after I had read with absorbed interest that instructive book, Tyndall’s "Heat as a Mode of Motion." This may explain why, when it came later to book-learning regarding electricity, I had the greatest possible repugnance to all the explanations, and could not accept the electric current to be the motion of electricity (static) through a wire, but thought it something quite different. I simply did not believe, except so far as mere statements of experimental facts were concerned. This had its disadvantages; one can get on faster if one has sufficient faith—which we know moves mountains—to accept a certain hypothesis unhesitatingly as a fact, and work out its consequences undoubtingly, regardless of the danger of fixing one’s ideas prematurely.

As Maxwell remarked, we know nothing about the velocity of electricity; it may be an inch in a year or a million miles in a second. Following this up, it may be nothing at all. In fact, it is only on the hypothesis that the electric current is something moving, a definite quantity in a given space, that we can entertain the idea of its possessing velocity. Then, the product of its hypothetical density into its velocity is the measure of the current; but, being a mere hypothesis, unless we chose to accept it, to talk of the velocity of electricity in the electric current becomes meaningless. On the other hand, when we apply the ideas of abstract dynamics to electricity, and compare the electric current to a velocity, it is not the above supposititious velocity of electricity that is referred to in any way. It has no meaning now. It is the supposed velocity of electricity in the electric current; whereas, in the dynamical theory, it is the electric current itself that is a velocity, in the generalized sense, with the electromotive force as the generalized force ; so that force x velocity = activity. In only one sense do I think we can speak of the velocity of electricity, consistent with Maxwell’s theory, viz., by the hypothesis that the electric current in a wire is the continuous discharge of contiguous charged molecules, when plainly we can call the velocity of motion of a molecule the velocity of the charge it carries. As between the molecules we have the electric medium the ether, this view of the conduction current ultimately resolves itself into "displacement" currents in a dielectric.

But is there not the fact that we can send a current into a long circuit, and that it plainly travels along the wire, taking some time to arrive at the other end ? Does that not show that electricity travels through the wire? To this I should have answered formerly, when filled with "Heat as a Mode of Motion," that it is a fact that there is a transformation of energy in the battery, and that this energy is transmitted through the wire, there suffering another transformation, viz., into heat; that when the current is set up steadily, the heat is generated uniformly; that the electric current in the wire is therefore some kind of stationary motion of the particles of the wire, not exactly like heat, but having some peculiarity of a directional nature making the difference between a positive and a negative current; but that there was no evidence in the closed circuit of any motion of electricity through the wire, but only of a transfer of energy through the wire.

However, leaving personal details of no importance to anyone but myself, let us consider the transmission of energy through a wire. To fix ideas, let our circuit be an insulated suspended wire from London to Edinburgh, and that we transmit energy to Edinburgh from a battery in London, the circuit being completed through the earth. Let the current be kept on. In the first place the phenomenon is steady. It does not change with the time. Next we find that the magnetic force about the wire is the same everywhere at the same distance, or the wire is in the same condition as regards the magnetic induction outside it, and when we apply our knowledge to the interior of the wire, regarded as a bundle of smaller wires, we find that the magnetic force in the wire does not vary along its length. Again, heat is being generated within the wire at a uniform rate (a part of the steadiness above mentioned), and next, this phenomenon is also the same all along the wire. Heat is undoubtedly a kinetic phenomenon, hence the electric current is also, at least in part, a kinetic phenomenon. The electric current is not itself heat; but as its existence in the wire involves the continued production of heat, we conclude that some kind of motion is necessarily involved in the electric current apart from the heat produced, and from the uniformity of effect in different parts of the wire, that it is a kind of stationary motion. Again, the electric force is the same all through the wire. There seems no difference between one part and another. Outside the wire, in the dielectric, however, there is a difference, for the electric force varies not only at different distances from the wire but also at the same distance outside different parts of the wire. (We disregard here all irregularities due to other conductors and currents.)

Passing to the battery, the complexity of conditions makes it more difficult to follow, though the state of electric force and magnetic force and heat generation is reducible to the same, and may be made identically the same as in the wire by properly choosing its shape, etc. But in the battery there is a very remarkable thing taking place, namely, the loss of chemical energy at a steady rate; and in the system generally, a still more remarkable thing, an exactly equivalent steady gain of heat. Heat that might have been produced on the spot by the chemical action, otherwise conducted, appears all over the circuit. How does it get there? The natural answer is, through the wire.

But to get to the further parts of the wire it must go through the nearer, hence there must be what we may call an energy-current, which, in the wire, at a given place, would be the rate of transfer of energy through a cross section there. Now, which way is the energy-current directed? It would seem only fair to let it go both ways equally from the battery. Let it be so first. Then there is an energy-current entering the wire, equal to one-half the dissipativity, which falls in strength regularly up to the middle of the wire, where it is zero. It falls in strength on account of the heat generation. Similarly the other energy-current goes through the earth to Edinburgh almost unabated in strength, and is then directed from Edinburgh to the middle of the wire, where its strength also falls to nothing. This seems absurd. Then let the energy-current be directed one way only, say with the positive current. If the positive pole of the battery is to line, we have an energy-current in one direction all round the circuit, London to Edinburgh, and back through earth. If of maximum strength at the battery it falls nearly to nothing at the distant end, and quite to nothing through the earth up to the other pole of the battery. But we have no data whatever to fix whereabouts the place of maximum energy-current is. It requires a second assumption. The reader may similarly consider the effect of reversing the battery, or of making the energy-current be directed with the negative current. There is no getting at anything definite, except that the energy-current must vary very widely, though regularly, in strength, whilst there is nothing to fix which way it is directed, or where the maximum strength is. Again, the energy-current is a kinetic phenomenon, and as it varies so widely in different parts, we might expect different parts of the wire itself to be in different electrical states, which is exactly what we do not do; for though its potential varies, yet potential is not a physical state, but a mere scientific concept.

Had we not better give up the idea that energy is transmitted through the wire altogether? That is the plain course. The energy from the battery neither goes through the wire one way nor the other. Nor is it standing still. The transmission takes place entirely through the dielectric. What, then, is the wire? It is the sink into which the energy is poured from the dielectric and there wasted, passing from the electrical system altogether. All [the above mentioned] difficulties now disappear.

That the energy of the battery passes into heat immediately would require its instantaneous transmission to all parts of the wire, which cannot be entertained. There must be an intermediate state or states, after leaving the battery and before becoming heat. And there must be a definite amount of energy in transit at a given moment; in the steady state this must be of constant amount, just as the total rate of transmission is of constant amount. We must not, however, individualize particular elements of energy, and follow their motions, but regard the matter quantitatively only. The energy in transit may be compared to the energy of a machine which is transmitting motion ; if done at a steady rate, it remains constant and definite, and the rate of transmission is definite.

Now, in Maxwell’s theory there is the potential energy of the displacement produced in the dielectric parts by the electric force, and there is the kinetic or magnetic energy of the magnetic induction due to the magnetic force in all parts of the field, including the conducting parts. They are supposed to be set up by the current in the wire. We reverse this; the current in the wire is set up by the energy transmitted through the medium around it. The sum of the electric and magnetic energies is the energy of the electric machinery which is transmitting energy from the battery to the wire. It is definite in amount, and the rate of transmission of energy (total) is also definite in amount.

It becomes important to find the paths along which the energy is being transmitted. First define the energy-current at a point to be the amount of energy transferred in unit time across unit area perpendicular to the direction of transmission. As the present section is argumentative and descriptive only, we cannot enter into mathematical details further than to say that if H be the vector magnetic force, and E the vector electric force, not counting impressed forces, the energy-current, as above defined, is $VEH/4\pi$ (see equation (3) for definition of V). This is true universally, irrespective of the nature of the medium as to conductivity, capacity, and permeability, or as to eolotropy or isotropy, and true in transient as well as in steady states. A line of energy-current is perpendicular to the electric and the magnetic force, and is a line of pressure. We here give a few general notions.

Return to our wire from London to Edinburgh with a steady current from the battery in London. The energy is poured out of the battery sideways into the dielectric at a steady rate. Divide into tubes bounded by lines of energy-current. They pursue in general solenoidal paths in the dielectric, and terminate in the conductor. The amount of energy entering a given length of the conductor is the same wherever that length may be situated. The lines of energy-current are the intersections of the magnetic and electric equipotential surfaces. Most of the energy is transmitted parallel to the wire nearly, with a slight slant towards the wire in the direction of propagation; thus the lines of energy-current meet the wire very obliquely. But some of the outer tubes go out into space to an immense distance, especially those which terminate on the further end of the wire. Others pass between the wire and the earth, but none in the earth itself from London to Edinburgh, or vice versa, although there is a small amount of energy entering the earth straight downwards, especially at the earth "plates." If there is an instrument in circuit at Edinburgh, it is worked by energy that has travelled wholly through the dielectric, then finding its way into the instrument, where it enters the coil and is there dissipated, or else used up by the visible motions it effects in moving parts of the instrument; which, however, is a different kind of affair from dissipation, as it involves impressed force.

Now, go into the line-wire. A tube of energy-current arriving at the surface of the wire by a long slant, at once turns round and goes straight to the axis. In passing from the battery to the wire through the dielectric the energy-current is continuous, the state being steady (or the ether machinery frictionless) ; but directly it reaches the conducting matter of the wire dissipation commences and the current begins to fall in strength, and on reaching the axis has fallen to nothing. Not a fraction of an erg is transmitted along the wire. Some small part of the energy leaving the battery may enter it again, but most of the dissipation in the battery itself is accounted for by the weakening of strength in tubes which are on their way to leave the battery.

Put the battery in the middle of the line; earth at both ends. Now, one half of the energy-current tubes leaving the battery sideways turn round to one section of the line, the other half to the other section. Otherwise the case is similar to the last.

When we have a double wire looped without earth, and battery at one end, most of the energy is transmitted between the wires.

In a circular circuit, with the battery at one end of a diameter, its other end is the neutral point; the lines of energy-current are distributed symmetrically with respect to the diameter.

On closing the battery circuit there is an immediate rush of energy into the dielectric, and, at the first moment, into all bodies in the neighbourhood of the battery, and wasted there in induced currents according to their conductivity. In the variable state the tubes of energy-current are themselves in motion. It takes some time to set the electric machinery going steadily. Also the energy-current is not continuous in the dielectric, for the potential energy of displacement and the magnetic energy have to be supplied at every place. But, in the end, the energy-current becomes continuous in the dielectric, goes round an external conductor instead of entering it, as it would do in the transient state, and finally reaches the conductor to which the battery is connected, penetrating which it terminates.

If we neglect the magnetic energy, as in Sir W. Thomson's original telegraph theory, against the energy of electric displacement, we can easily get a general idea of the setting up of the permanent state in a long suspended wire; a submarine cable is more complex on account of the sheath. The energy reaches the beginning of the wire first, and only reaches the end, save insignificantly, later on. But the theory indicates instantaneous setting up of current at the far end, though not in recognisable amount. This result follows from the neglect of the magnetic energy. In a dielectric medium the velocity of undisturbed propagation is $(c\mu)^{-i}$; where c is the capacity, and $\mu$ the permeability; that the magnetic energy = 0 is equivalent to assuming $\mu$ = 0 everywhere, whence instantaneous transmission. The "retardation," however, arises from the setting up of the potential energy of displacement. But, strictly speaking, we must not neglect $\mu$. It is, then, not so easy to follow the transient state without simplifications. There is an oscillatory phenomenon in the dielectric, a to-and-fro transmission of energy and pressure parallel to the wire all round it with a velocity whose possible maximum is that of undisturbed transmission. This is modified as it progresses by dissipation in the wire, and so gets wiped out. This usually occurs so rapidly that the waves are of importance only at the battery end of a long wire. The electric machinery must have mass, as well as elasticity, by reason of this phenomenon, since there is reason to believe (from Maxwell’s theory of light) that it is not the air, but something between the air molecules that is the electromagnetic medium, the air merely modifying the phenomena somewhat.

In the state of steady current through a submarine cable, with an iron sheath outside the dielectric, the energy is transmitted wholly through the gutta percha or other suitable insulator (neglecting the small amount going to earth), thus going nearly parallel to the wire, practically quite parallel, except as regards the lines near the wire itself, as they all eventually meet the wire. There is no transmission in the sheath lengthwise, though there is dissipation there if it should contain, as it does sometimes, part of the return current. In the transient state there is, of course, always dissipation in the sheath more or less, besides the loss of energy to magnetise it.

Now to speak more generally. In the steady state of current due to any impressed forces, the tubes of energy-current start sideways from the places of impressed force, where energy is supplied to the electric system, and travel through definite paths, without loss in dielectric, with loss in conducting parts, to terminate finally in conducting matter; or else they may go from one place of impressed force to another with, or without dissipation on the way when the current is with the impressed force at one source, and against it at the other But with special arrangements (solenoidal) of impressed force, there is no transmission of energy in the steady state.

Since on starting a current the energy reaches the wire from the medium without, it may be expected that the electric current in the wire is first set up in the outer part, and takes time to penetrate to the middle. This I have verified by investigating some special cases.

Increase the conductivity of a wire enormously, still keeping it finite, however. Let it, for instance, take minutes to set up current at the axis. Then ordinary rapid signalling "through the wire" would be accompanied by a surface-current only, penetrating to but a small depth. The disturbance is then propagated parallel to the wire in the manner of waves, with reflection at the end, and hardly any tailing off. With infinite conductivity there can be no current set up in the wire at all. There is no dissipation; wave propagation in the medium is perfect. The wire-current is wholly superficial—an abstraction—yet it is nearly the same with very high conductivity. This illustrates the impenetrability of a perfect conductor to magnetic induction (and similarly to electric current), applied by Maxwell to the molecular theory of magnetism. Whatever state of magnetic induction and of current there may be in a perfect conductor is a fixture. If we move the conductor about in a magnetic field, superficial currents are instantaneously induced, whose only function is to ward off external induction and keep the interior state unchanged.

In a thermo-electric circuit of two metals, with one junction a little hotter than the other, there is a transmission of energy from one junction to the other through the dielectric, with a trifling amount of loss in the circuit generally. Here the source of the electric energy is heat, and the final result is heat. One junction is cooled, the other is heated, reversibly. Now, heat is the energy of molecular agitation, and at first sight the only difference is that the agitation is a little more brisk at one junction than at the other. Again, all parts of the circuit are agitating the ether. It would appear, then, that the ordinary molecular agitations set up no electric manifestations on account of their irregularity; although the electric machinery may be influenced vigorously, yet it must be done in some regularly symmetrical manner to constitute an impressed electric force. At the junctions there is a change of material, the molecules are different, and at their contact some directed quality is given to the agitations. This is very vague, no doubt, but is merely to point out that the impressed force is a symmetrical kind of radiation.

After these general remarks the temporarily interrupted mathematical treatment will be resumed.

SECTION III. RESUMPTION OF ROUGH SKETCH. EXTENSIONS.

Real transient, and suggested dissipative Magnetic Current. As the rate of increase of the displacement in a non-conducting dielectric is the electric current, so the rate of increase of Bjiir may be called the magnetic current. Let it be G. Then

G = B/47t = /aH/47t. (Magnetic current) (12)

Like electric displacement currents, magnetic currents are transient only, i.e., they cannot continue indefinitely in one direction, like an electric conduction current. Also, like electric currents in a dielectric, they are unaccompanied by heat generation. In ether, the electric current and the magnetic current are of equal significance.

There is probably no such thing as a magnetic conduction current, with dissipation of energy. If there be such, analogous to an electric conduction current, then let G = f/H + //H/47T (13)

Here gH is the magnetic conduction current, which, added to the undoubted magnetic current as in (12), gives G the true magnetic current. g may be scalar, or similar to /r, with rotatory c. Multiply (13) by H. Then, using (11),

HG = H^H + T. (14)

Here H^H is the rate of dissipation. Compare with (9).

Effect of g in a Closed Iron Iling.

The permanency of state of a steel magnet makes it improbable that g has any existence at all, so that the conduction magnetic current is quite imaginary. But we may inquire what would happen in a closed ring of iron under magnetising force, on the supposition that g exists. Let the ring be uniformly lapped with wire, through which we pass a current from a voltaic battery.

If the radius of the ring be large compared with its section, the core may be treated as straight, and the manner in which the current would rise in the coil and the accompanying core phenomena may be easily worked out by a slight modification of the corresponding case with <7^0 [Alt. xxvii., § 29, Example 2, p. 394]. Let a be the radius of the core, also of the coil of negligible depth surrounding it, having windings per unit length of core. Let k and /z be the conductivity and permeability of the core, and H (parallel to the axis) the magnetic force at distance r from the axis. The differential equation of TI will be

f -(‘'ft**♦*«*/**!

whence J0(nr)emt is a normal system of magnetic force, if n2 Airn

Thus the effect of g is to increase the reciprocal of the time-constant of every normal system by the same quantity ■iTrg/fx; in this respect resembling the effect of uniform leakage along a telegraph line, and having a similar result, viz., to accelerate the establishment of the permanent state. When this is reached, we do not have uniform strength of magnetic force in the core; but, if H0 is the strength at the axis, that at distance r therefrom is

• =".(<+^+PP+->

where x — (i~)-gk. This is accompanied by core-currents parallel to the coil-current, of density The coil-current will be a little less strong than if g = 0; for the work of the battery is spent not merely in supporting the coil-current, but in heating the core, both by reason of the weak electric current in the core and the supposed weak magnetic current gH. The back E.M.F. in the coil will be of strength f *i+-+-r Tj 1 + _ ^ i.

where E.M.F. of battery, II resistance of coil-circuit, and L its inductance without the core—i.e., with air replacing it. Or, since g is to be small,

- F(Y+Rji-Lg)-\

If LjPi= ‘01 second, g = \jiiv would make the back force = 1/101 of the battery force, so g, if existent, must be very small.

In the following, g = 0, so that equation (12) is the equation of the magnetic current.

First Cross Connection of Magnetic and Electric Force.

In the foregoing we have been dealing with the direct connection of the electric force and its consequences, electric conduction current and displacement, and of the magnetic force and magnetic induction. We have also brought in the displacement current in a dielectric, and the true current in a conducting dielectric. Also, to balance the displacement current, we have introduced the magnetic current. But, so far, we have no relations whatever between the electric and the magnetic quantities, which we must have, in order to make a consistent system.

The first cross connection is expressed by curlH = 47rr, (15)

H being the magnetic force and T the true current. Here “curl” is, like sin and cos, the symbol of an operation. It is so recurrent in electromagnetism that it might be termed the electromagnetic operator. It may be defined with reference to Cartesian coordinates thus: If Hv h9 h3, are the three rectangular components of H, those of curl H are dJL, _ ilU, tlH^ _ ilH.. dll.^ _ dU1 (h/ dz ’ dz d.r 1 dx dij

But the most useful definition is that which is virtually contained in the fundamental Theorem of VersionThe line-integral of a vector H round any closed curve or circuit (or the “circulation” of H) equals the surface- integral of another vector, viz., curl H. over any surface bounded by the circuit. Apply this to small squares in planes perpendicular to x, y, and z successively, and the three expressions given in (1(3) for the components of curl H follow at once. Apply the theorem to suitably chosen infinitely small areas in any system of coordinates and we obtain the proper expressions in, usually, a fir simpler manner than by laborious transformations of differential coefficients. Whilst the expressions for the components vary according to the system of coordinates chosen as most suitable for a special problem, the theorem, on the other hand, is universal, and gives us the inner meaning of the operation. It is far the best in general investigations not to employ any system of coordinates, but to emancipate one’s self from their complexity by employing symbols which only relate to the intrinsic meaning of the operations; besides which, there is a great gain in the ease of manipulation. In the present paper the meanings of all forms of expression likely to be unfamiliar are briefly stated, and we shall avoid occupying valuable space by lengthy formulae

The operator “ curl ” is connected with rotation thus: if H be tlie instantaneous velocity at a point in a moving fluid, curl H is a vector whose direction is that of the axis of instantaneous rotation of the fluid surrounding the point, and whose length equals twice the angular velocity of rotation.

Notice that (15) contains 110 physical constants. It is therefore, in a sense, a purely geometrical equation. Given a system of magnetic force H, mentally represented by lines or tubes of force mapping out space in one way, by the operator “curl ” we find another system of lines or tubes mapping out space in another way, viz., the lines and tubes of current. Whether H be wholly continuous or not, the derived F is necessarily continuous [that is, circuital]. The curl of a vector can have no divergence anywhere, which we express by

div r = 0; or, fiEl + ^ + S. = 0, (17)

a.c tly clz

which defines “ divergencewith reference to Cartesian coordinates. The divergence of T is the amount of T leaving a point, reckoned per unit volume. When T, as here, signifies electric current, it is continuous ; as much current leaves as enters any volume, or the integral amount leaving it, reckoning that entering it as negative, is zero. That (17) is involved in (15) is tested by differentiating the three components in (16) to x, y, and Ł respectively and adding them, when (17) results.

Given H, we have T, by (15), perfectly definite. But given T (necessarily continuous), H is not definitely fixed by (15). For, on finding one function H satisfying (15) with F given, we may add to H any function I such that curl I = 0, without disturbing the relation (15). The nature of I is given by

1= -VI2; or, Ix = -dQ/d.r, I2= -dQ/dy, Ig= -dŁl/dz, (18)

where 12 is a scalar function of position, a scalar potential in fact. We require some other condition than (15) to find H completely when T is given; this is, that the magnetic induction B = (equation (10),) is continuous, or divB = 0. H is now perfectly definite. If /u. = constant, or all space is equally magnetisable isotropically, then B is the same multiple of H everywhere, hence div H = 0, so that the proper solution of (15) is that function H satisfying (15) which is continuous, like T. But H is not continuous when [x varies from one part of the field to another.

Having now defined “curl,” “divergence,”and V applied to a scalar function, consider (15) from a less abstract point of view, in the light of the Version Theorem. Let there be any closed circuit in space,— whether passing through conducting or dielectric matter is immaterial. The amount of current passing through the circuit in the positive direction (that passing the other way being counted negativefy) equals the circulation of H round the circuit 4- Air. The actual distribution of I' is got by taking the circuit infinitely small and applying it to all parts of the field. Let us, whilst considering a finite circuit, yet take it sufficiently small to make the current pass all one way through it. Then, setting up current through the circuit, we set up magnetic force round it.

But there is another way of setting up magnetic force round the circuit, viz., by motion of the circuit itself in a previously undisturbed electric field. Thus, let there be a steady field of electric force, say in air, with therefore steady electric displacement, and no electric current. Let the closed circuit be a thin wire. When at rest in the field there is no current through it, and no magnetic force round it. But if we move the circuit so that the amount of electric displacement through it varies, there is electric current through the circuit, to be measured by the rate of increase of the amount of displacement through it at any moment; or, in another form, by the number of tubes of displacement added to the circuit per second by the motion of the circuit across them. Hence there will be magnetic force round the circuit, and if it be a thin iron wire, it will become magnetised by the motion in the electric field. In general, the motion of matter in an electric field sets up magnetic force.

As an example, fix a thin circular iron ring in air. Call the line through its centre perpendicular to its plane the axis. Let there be no current or magnetic force in the first place. Now shoot a small bullet, having an electrical charge, through the ring, along its axis. The electric displacement due to the charge will be continually charging; thus, there is a system of electric current in the air accompanying the motion of the bullet. The velocity of propagation of disturbances in air is so great that, unless the velocity of the bullet be not a very small fraction of the velocity of propagation, we may neglect the disturbance in the field of force due to the latter velocity not being infinite, and suppose that the bullet carries with it in its motion its normal field of force (radiating straight lines) unchanged. The distribution of displacement current about the moving bullet is then the same as that of the lines of magnetic force that would come from it if it were uniformly magnetised parallel to the axis, or line of actual motion in the real case, and the lines of magnetic force accompanj'ing the displacement currents are circles centred upon the axis, in planes perpendicular thereto, the strength of magnetic force in the air being inversely proportional to the square of the distance from the centre of the bullet, and directly proportional to the cosine of the latitude; the equator being the circle on the bullet’s surface in the plane perpendicular to the axis passing through the centre of the bullet. (With very high velocity this distribution of displacement current and magnetic force is departed from.) The fixed ring coincides with the lines of magnetic force during the whole motion of the bullet, and is therefore solenoidally magnetised thereby, most strongly when the magnetic force is strongest there, i.e., when the bullet has just reached the centre of the ring, and the current through the ring is a maximum. The current through the ring may be measured either by the displacement current through a surface bounded by the ring, or by the rate at which the ring cuts the lines of electric force (supposed undisturbed) of the bullet.

Next, fix the charged bullet and move the ring instead, so that their relative motion shall be as before. There is exactly the same amount of electric displacement through the circuit added per second as before, in corresponding positions of the bullet and ring, with, therefore, the same magnetic force in the ring and the same magnetisation. Otherwise, however, there is a great difference in the two experiments. In the first case, changing electric displacement or electric current all through the dielectric, the greatest strength of current being at the poles of the bullet: whilst in the latter case the field is practically undisturbed except near the moving ring itself. Compare with the induction of electric force in a ring in a magnetic field, first when the field is moving, and next when the ring is moved in the field.

The induced magnetic force per unit length in a wire moved perpendicularly across the lines of force in an electric field equals the amount x 47r of electric displacement of the field crossed by the unit length of wire per second, and is perpendicular to the electric displacement and to the direction of motion. In general,

h = VDvx47r, (18it)

where D is the displacement of the field, v the velocity, h the induced magnetic force, and V is as in equation (3). There are, of course, corrections due to the reactions set up, due to the wire not being infinitely thin, and to finite length.

In electromagnetic units, c in air=(r1)-2, if t\ = velocity of propagation =3 x 1010cm. per sec. Therefore, in the case of motion of a thin wire perpendicularly across the lines of force in a uniform electric field of strength E,

h = Er(t\)~2 = El'/(9 x 1(J-U).

Let E= 1012c.g.s., or 104 volts per cm., which is less than the disruption force in air in its ordinary state, then

h =vj(9 x 108).

To get magnetic force of strength 10_5c.g.s., v must equal 90 metres or 300 feet per second. Magnetic Ewrgt) of Moving Charged Spheres.

In passing, I may remark that J. J. Thomson {Phil. Mag., April, 1881) found the magnetic energy 5 T due to a sphere of radius a with an electric charge q moving with velocity v in a medium of permeability ju to be

I find that the fs should be Also, he found the mutual magnetic energy — T10 of two infinitely small spheres at distance r with charges qx and <].„ moving with velocities defined by the rectangular components uv «2, us, and rp r2, vs, with ux and r, the velocities parallel to the line joining the spheres, to be

- Tu = (P7i72/3r) OVj + n.p, + »3r3), against which I find it to be

2 Tv> = (Mi'kl-'0 (2^1 + u2v2 + usvs).

I do not know what corrections, if any, have been published, and should be glad to receive information on the point, whether in corroboration of my results or otherwise.

SECTION IV. COMPLETION OF ROUGH SKETCH.

Second Connection between Electric Force and Magnetic Force.

The equation (15), curlH = 47rI\ expressing a relation, independent of physical constants, between the magnetic force and the electric current, is an extension of Ampere’s results for linear circuits. By T must be understood Maxwell’s tiue current—that is, the sum of the conduction current and the displacement current if the body considered be both a dielectric and a conductor, or the conduction current alone or the displacement current alone if the body have 110 dielectric capacity or no conductivity respectively. All bodies are either conducting or dielectric, or both, and ether is dielectric, so that electric current may exist everywhere. Putting Y in terms of E by the equation of true current (S), [p. 443], we get

curl H = 47TZ.’E + cE (19)

which is one connection between E and H.

The second connection may be obtained by translating Faraday’s law of induced electric force in a linear circuit into a mathematical form. It is remarkable that the ideas of Faraday, who was no mathematician, should admit of immediate translation into mathematical language; a fact due to his dispensing with the direct action-at-a- clistance hypothesis, and employing the intermediate mechanism of lines or tubes of force. In popular language, the total E.M.F. of induction round a linear circuit is measured by the number of lines of force taken out of the circuit per second. Here the conventional connection between the assumed positive direction of translation through a circuit, and the assumed positive direction of motion in the circuit must be remembered. Selecting either direction through a circuit as the positive direction of translation, look through the circuit in this direction. Then the positive direction of rotation is right- handed, or with the hands of a watch whose front faces the spectator. Thus, increasing the number of lines of force through a circuit sets up negative E.M.F. round it.

So far in a medium of unit permeability. But when we make allowances for differences of magnetic permeability, it is not the variation of the magnetic force H, but of the magnetic induction B=/xH, which determines the induced E.M.F. The amended statement is that the total E.M.F. of induction round a circuit equals tlie rate of decrease of the amount of magnetic induction through the circuit. Now, since we have here a line-integral, viz., of the electric force of induction round a circuit, and a surface-integral, viz., of -/xH or - B over any surface bounded by the circuit, we may at once apply the Version Theorem before referred to [p. 443] and deduce

curl E = - B = - //H, (20)

which is one form of tlie second relation between E and H.

The following method is also instructive. Since the rate of increase of the magnetic induction at a point equals ATTG, where G is the magnetic current, as defined by equation (12), we may state the law of induced electric force thus:—The total E.M.F. of induction round a circuit in the negative direction equals Att times the total magnetic current through the circuit in the positive direction. Now compare this statement with the statement regarding equation (15) [p. 443], viz., that the total magnetic force round a circuit equals 47r times the total electric current through the circuit, and change this so as to produce the statement in the last sentence. We must change magnetic force to electric force taken negatively, and electric current to magnetic current. Hence

curl H = 4LTTT (15) bis

becomes - curl E = 47rGr, (21)

which is equivalent to (20).

We have, in order to simplify the establishment of (20) or .(21), avoided mentioning the E.M.F. induced in a linear circuit by its motion in the field, which may or may not be varying independently. The amount of induction added to or taken out of a circuit from this cause may be obviously represented by a line-integral, as it depends upon the rate at which the different elements of the circuit cross the lines of induction. If the induction were of the same strength at all the moving parts of the circuit, and they all moved at right angles to their lengths and also perpendicularly across the lines of induction in the same sense, the total E.M.F. would be of strength =B x rate of increase of area of circuit. But when B varies, and likewise the velocity of the different elements across the lines of B, each element must be considered separately. The amount contributed to the total E.M.F. by an element of unit length equals the component parallel to its length of

YvB, (21a)

if v be the vector velocity. But, if there be current induced, this brings in working mechanical forces, and should therefore be separately considered. At present we return to the case of r, k, and n constant with respect to the time, and no parts moveable.

In equation (21), E is the electric force of induction only, not the actual electric force. There may be in addition electrostatic force, and also impressed electric force. But the electrostatic force is polar; it is derived from a scalar potential. If this be P, the force is - VP. But curl VP = 0, as was before remarked [p. 444] with reference to 12, consequently the polar force may be included in E in equation (21). Similarly any polar force may be included in H in the previous equation (15). Now in all our equations, from (1) up to (14), not containing any relations between E and H, those symbols mean the actual resultant electric and magnetic force from all causes. Hence, in order that the two equations (15) and (21) may harmonise with the preliminary equations (1) to (14), not only in space where there is no impressed force, but at the places where such exist as well, we must, whilst still using E and H to denote the actual forces, deduct from them the impressed forces in using the relations (15) and (21). So,

let e be the impressed electric force, and h the impressed magnetic force. Our two connections between E and H are then

curl (H - h) = 4TTF = 47rZ-E + cE, (22)

curl(e-E) = 4jrG = 4jr0H + /iH, (23)

where the coefficient g of magnetic conductivity is introduced to show the symmetry, and may be put =0. We have now a dynamically complete system.

The subject of impressed force will be considered in a following section more fully, especially as regards impressed magnetic force, and its interpretation in terms of magnetisation. In the meantime we may define impressed electric force thus. If e be the impressed electric force at a point, and T the electric current there, er of energy is taken into the electromagnetic system there per unit volume per second. Similarly, we may define the impressed magnetic force h at a point, by saying that if there be a magnetic current G there, hG of energy is taken in per second per unit volume by the electromagnetic system there. In general, er and hG are scalar products [see equation (5)], having the ordinary signification when e is parallel to T, or h to G; in other cases to be multiplied by the cosine of the angle between e and r or between h and G.

The Equation of Energy and its Transfer.

We must find the rate of working of the impressed forces, and compare with the dissipativity and with the changes taking place in the energy of displacement and the magnetic energy. Multiply (22) by (e - E), and (23) by (h - H), and add the results. We get

47r{(e - E)r + (h - H)G} = (e - E) curl (H - h) + (h - H) curl (e - E).

Or

er + hG = Er + HG + (H - h) curl(E - e) - (E - e) curl(H - h)/4tt, (24)

by rearrangement. Er and HG here occurring have been already expressed in terms of the dissipativity Q, the electric energy of displacement U, and the magnetic energy T; see equations (9) and (14). Thus

ET + -EG = Q+U+t. (25)

On the left side we have the rate of working per unit volume of the actual forces E and H on the currents T and G; on the right side the dissipativity, or rate at which energy is being lost from the system irreversibty, producing heat according to Joule’s law, and the rate of increase of the electric and magnetic energies, all per unit volume.

Now looking at (24), the left side expresses the rate at which energy is being taking in (reversibly) per unit volume, in virtue of the impressed forces e and h. Therefore the excess of (el1 + hG) over (Er + HG) must be the energy leaving the unit volume per second through its sides. Now, X and Y being any two vectors,

Y curl X - X curl Y = div VXY ; (26)

11. e p.—vol. 1. 2 F

L 450 ELECTRICAL PAl’EliS. or in full, by (5), (16), (17), and (3),

Y^dXJdy - dXJiM + Y.2(dXJch - dXJdjr) + Y.JdXJdx - dXJdy) - X^dYJdy - dYJdz) - XJjlYJdz - dYJdx) - Xs(dYJdx - dYJdy) = (d/d,)(X,Y3 - X,YJ + {d!dy){X, Yl - X1 Ya) + (d!dz)(XxY2 - Y1X2);

using the numbers 1, 2, 3, to denote the x, y, and z components. Let then

W = V(E-e)(H-h)/47r> (27)

then by (26), with X = E-e, and Y = H-h, equation (24) becomes

er + hG= Er + HG + divW\ = Q+U+T+ divWj '

the equation of energy put in its most significant form. Summing up through all space, W goes out; or the total work per second of the impressed forces equals the total dissipativit}' plus the rate of increase of the total electric and magnetic energy.

W is the vector rate of transfer of energy, or what we before [p. 438] termed the energy-current, a vector whose direction is that of the transfer of energy, and whose magnitude equals the amount transferred per second across unit area of a plane perpendicular to that of transfer. Note [p. 438] that impressed forces were said to be not counted; hence as E and H are the actual forces now, the impressed forces are deducted, as shown in (27). The magnitude of W is the product of the strengths of the two forces and the sine of the angle between their directions, and the direction of W is perpendicular to both forces, with the beforestated convention regarding positive directions.

The general nature of the energy-current was described in Section II. “ On the transmission of energy through wires by the electric current ” [p. 434], where, however, only impressed electric force was considered. The same general results apply to impressed magnetic force; energy proceeding from places where such exists, to be dissipated as heat in conducting matter, or to increase the electric and magnetic energies, or to go to other places of impressed magnetic force. But there are great practical differences between impressed electric and magnetic force, owing to the transient nature of magnetic currents and other causes.

Differential Equations of E and H.

By eliminating E or H between (22) and (23) we obtain the characteristic equation of E or of H. Put /7 = 0, and eliminate H. Then,

curl /A-1 curl (e - E) = curl h + 47T&E + HB, (29)

which is the equation of E. Here e and h, being impressed, must be supposed to be given, /a-1 is the operator inverse to y., that is, in the general case of eolotropy /x_1 is defined by the three principal axes and the values 1//^, l//z;J, along them, as was explained [p. 430] in speaking of k. .Similar remarks apply to k~l and c_1 should they occur.

In space where there is no impressed electric force, and no, or else constant, impressed magnetic force,

curl /x”1 curl E + 4-/.E + cE = 0 (30)

In a non-dielectric conductor, curl /x_1 curl E + iirkE = 0,) ,,. and curl Zr1 curl H + ATT/JLK = 0. J

Here propagation of E and of H is by diffusion. And in a non-conducting dielectric,

curl fjr1 curl E + cE = 0,) , „ 0,

curl c~l curl H + /xH = 0. J

Here propagation is by waves, i.e., propagation of E or H, not of energy. c and /x self-conjugate ; k 'not necessarily so.

We should note that

f= (rf/^)(H/xH/S7r) = H/xH( 8tt + H/xH 8tt, whilst HG = H^H/4tt ;

so for HG to equal T we require

H/xH = H/xH,

i.e., /x must be self-conjugate, or contain no rotatory e [equation (3), p. 431]. Similarly, for U to equal ED, c must be self-conjugate. But there is no such limitation thrown upon k the electric conductivity operator, nor would there be upon g the magnetic conductivity operator, did such exist. There are other proofs of these conclusions, but the above are very short. There is, however, an objection to be raised against the rotatory conductivity vector e, which want of space does not permit to be mentioned at present.

SECTION V. IMPRESSED MAGNETIC FORCE. INTRINSIC MAGNETIC ACTION.

The energy definition of impressed electric force, due originally, it not explicitly, at least substantially, to Sir W. Thomson, has long been well recognised by most writers on electrical subjects, especially since the practical introduction of dynamo machines, accumulators, etc., which raised the energy transformations concerned in electrical phenomena from being matters of almost purely scientific interest to matters of the extremest practical commercial importance.

But in our last we gave an energy definition of impressed magnetic force, precisely similar to that of impressed electric force. Thus, if h be the impressed magnetic force at a point, and G the magnetic current there, the rate of working is hG per unit volume, and this amount of energy is taken in per second by the electromagnetic system at the place, and is employed in increasing the energy of electric displacement and the magnetic energy, or wasted in the heat of conduction currents. Also it may be used in effecting bodily motions when there is yielding to the mechanical forces, or in chemical work, etc. Should there be no impressed force, electric or magnetic, except an impressed force h in a single unit volume, we have the summation extending through all space, where Q is the dissipativity, U the energy of displacement, and T the magnetic energy, all per unit volume. In order to identify the quantity thus defined, and show the relation it bears to the quantity termed intensity of magnetisation, let there be no electric force in the field, and its state be steady. The second equation of induction (23), goes out, since G = 0, and the first, (22), is reduced to

curl (H - h) = 0 (33)

Integrating once, we have

H = h + F, where curlF = 0, or F = -Vfi (34)

Thus the actual magnetic force H differs from the impressed force h by a polar force F, a force which, when analysed, is found to be made up by the superposition of radial forces proceeding from points, i2 is the potential, a scalar, variable from point to point.

But, so far, there is nothing to settle what particular distribution of polar force F must be. A second condition is wanted. Now we know from equation (23) that the magnetic current, like the electric current, is always circuital, i.e.,

divG = 0, therefore divB = 0, (35)

and, if we take the time-integral, we find

div B =/(#, y, z),

any scalar function of position, independent of the time. If, then, the magnetic induction B were not also circuital, its divergence would continue unaltered at any place, however the field might otherwise vary. It could only be altered by convection, shifting the arrangement of matter. It would then, by a suitable arrangement of matter, be possible to have a unipolar magnet, a quantity of matter round which the magnetic force was everywhere directed outward, or everywhere inward. This being contradicted by universal experience, we must conclude that

divB = 0, as well as divG = 0 (36)

The second equation of induction (23), if we use the full expression for G there given, is too general, requiring the limitation # = 0, from the absence of magnetic conductivity. The now added limitation div B = 0, as it does not contradict the second equation of induction, must be considered as an auxiliary condition. Though not necessarily dependent upon the first limitation g = 0, it is yet intimately connected with it. Our two equations are, therefore, (33) and (36), or,

curl (H - h) = 0, and div /xH = 0, (37)

and the question is, given /x the permeability, and h the impressed force, find H the real force. Or, using (34), we have

div /x(h + F) = 0, curlF = 0; (38)

and now, with the same data, we have to find F. By the second of (38), F is restricted to be a polar force; by the first it is still further restricted to have, when multiplied by /x, a given amount of divergence, thus,

div fjiF = - div /xh.

The two conditions together make F perfectly definite, and therefore H and B definite through all space. Of this a variety of proofs may be given (the first was given by Sir W. Thomson in 1848, relating to similar equations, /x being isotropic). The following perhaps puts the matter in as simple a form as it can be put, and is best adapted to the present circumstances.

In the first place there cannot be two solutions of (38) for F. For if (38) are satisfied by F and also by F + f, we have, by subtraction,

div/xf=0, curlf=0, (39)

as the equations that f must satisfy. But consider the quantity ^ f/xf integrated through all space. It cannot be negative, for every element of it is positive or else zero. Thus

f/xf = fxj* + /x2f22 + fxj./,

if /Ltj, /x2, be the principal permeabilities, and f1? f2, f3 the corresponding components of f. We suppose the permeability always positive (to deny this would lead to absurdities), hence f/xf is always positive, or else zero, viz., when f=0. But, by the second of (39), f = - Vp, if p is the potential of f, and therefore, by a potential property,

2 f/xf = 'Zp div /xf, (40)

and therefore vanishes, by the first of (39). Hence f = 0, making the supposed two solutions identical. Next, to show that there is one solution, consider 3 H/xH through all space. This is also positive, or zero, whatever H may be, by the same reasoning, so has necessarily a minimum value. If H be quite arbitrary, the minimum is zero, when H = 0 everywhere. But H = h + F, and here h is constant. Let F vary. The corresponding variation in

2 H/xH is

8 2 H/xH = 2 8F/x(h + F) + 2 (h + F)8/xF = 2 2SF/x(h + F) (41)

Now subject F to either of the two equations (38), say the second, so that F = - Then, similarly to (40),

8 2 KJJ.1L = 2 25ft div /x(h + F) (42)

Hence, to make 2H/xK a minimum requires

div/x(h + F) = 0,

which is the first equation (38). Thus, when F satisfies both equations, it makes ^ H/<H a minimum. But this quantity has a minimum, therefore there is a solution of the equations (38), or F is a definite vector for every point of space.

If we assume that /x contains a rotatory vector e, so that

^H = /x0H + VeH,'l /,o\

let /x'H = /x0H - VcH, I v ' then // is conjugate to /JL, and

|(/x + /x')H = /x0H,

where //0 is self-conjugate or non-rotatory. Instead of (41) we shall have

S^H/xH = 2 26F//0(h + F), (44)

and the minimum is given by

div /x0( h + F) = 0,

which is not the proper condition.

There is a similar failure in the mathematically analogous problem of conduction current kept up by impressed electric force, when the conductivity k is rotatory.

In connection with the above, we may notice two special solutions of (37). First, if curlh = 0, then curl H — 0, which, with div/xH = 0, requires H = 0, and therefore B = 0. That is, if the impressed force be wholly polar, there is no induction. The simplest example is a closed magnetic shell of uniform strength, and any thickness, an assemblage of magnets put together side by side in such a manner that there is no induction anywhere.

Secondly, if div/xh = 0, then div/xF = 0, which, with curl F = 0, requires F = 0, therefore H = h, and B = /di. Here the impressed force everywhere produces the full induction, and there is no polar force.

Comparing our equations with those occurring in the problem of magnetisation, we find that, if I be the intensity of intrinsic magnetisation, it is related to h thus :—

I = /xh/47r (45)

h may therefore be called the intrinsic magnetic force, if we like. The real magnetisation is the sum of the intrinsic and the “ induced,” which we shall call i, and the ordinary form of the magnetic induction equation is equivalent to

B = F + 47r(I + i), (46)

where F is the polar force due to both the intrinsic and the induced magnetisation. It is the same as F above. And, to identify (46) with the equation B = jxH we use always, we have first

i = #cF,

giving the induced magnetisation in terms of the polar force, K being the coefficient of induced magnetisation, next the equation (45), and lastly,

1 + 4ttk ■

so that (46) becomes

B = F + ywh + 4TTKF = /x(h + F j — /txH (47)

It will be seen that this separation of magnetisation into intrinsic and induced is a roundabout way of treating the subject, and that there is considerable simplification obtained bv always using the equation B = /xH, both in an mtiinsic magnet and without it, evei employing the two ideas of magnetic force and magnetic induction, and letting magnetisations alone. The quantity B is the magnetic induction as ordinarily understood ; H is also the magnetic force as ordinarily understood everywhere, except where there is intrinsic magnetisation. There, however, we add the intrinsic or impressed force h, making

H = h + F

in general, just as in a conduction current system we add the impressed force, where there is any, to the polar electric force and that of induction to obtain the resultant efficient electric force. H is thus always the resultant magnetic force from all causes, and although in the above we have considered no electric currents to exist, yet we may add that should there be any,

H = h + F + F1}

where F1 is the magnetic force of the current; and B = /xH always. The term “intrinsic,” as applied to magnetisation, is used by Sir W. Thomson, but not by Maxwell, though he gives the same theory of induced magnetisation as the former. It is not always clear in Maxwell’s treatise whether by magnetisation he refers to the intrinsic only or to the actual, including the induced. Maxwell’s equations of disturbances, also, break down when they are applied to the interior of an intrinsic magnet, owing to his use of the equation

curl H — 47TT,

as the relation between the magnetic force and the current, whereas it must be—equation (32)— curl (H - h) = 4TTF, where there is intrinsic magnetisation, if we are to obtain consistent results.

Although, in identifying I with /xh/4-, we have, by means of the two equations of induction, (32) and (33), and the equation of energy deduced therefrom, obtained a justification of the energy definition of h, similar to that for impressed electric force, yet, as the consequences are of some importance in variable states, they may be advantageously followed up from the impressed force point of view.

Magnetic Energy. Double work of Magnet.

Let there be a distribution of impressed magnetic force h in a medium of given variable permeability /x—for example, a r~ ^net in air containing soft iron and any other substances. Imagine the impressed forces to be put on suddenly. We know, by the above, that a certain definite distribution of magnetic induction is set up, which is steady when the arrangement of matter is fixed. During the transient state there is magnetic current everywhere unless //. = 0 somewhere, which we must believe to be impossible, since /x is very little less than unity for any known substance. The magnetic currents are wholly closed. They are accompanied by electric currents, also closed, in all space, in general. When they have ceased, there is no electric force anywhere, and the magnetic induction at any point is the time-integral of the magnetic current x 4TT. Compare the work done by the impressed forces with that done by the actual forces through all space. The latter is, per unit volume,

JhgJ = Jh^H/4tt. dt = = H/xH/8tt,

where, in the last expression, H is the final value of the magnetic force. In the integrals, H is the variable value at time t. Let T be the whole work thus done in all space, then

r= 2 HJU-H/STT (48)

On the other hand, the work done by the impressed force per unit volume is

jhGtft = Jh^tH/47r. dt — h/xH/47r •

so, if Tx is the whole work done by the impressed forces,

T1 = ^ h/^H/47r, (49)

where the summation may also extend through all space, since where there is no h nothing is contributed to the sum. Now H =h + F, and

2F/*H = 2fldiv/*H = 0, (50)

because F is polar and fiH. circuital. (Similar to (40).) So in (49) we may add F to h, making

Tx = 2 (h + F)/u.H/4tt = 2 H/xH/4tt = 2 T, (51)

by (48). The impressed forces therefore do double the work of the actual magnetic forces during the transient state. The excess is done by the electric forces. For, integrating (28), to the time, with e = 0, and also through all space, to get rid of Z/7,

2 jhGrft = sjEr^ + 2jHG(^ j

and, since finally U= 0, we have Tx = ^Qdt+T. (52)

Hence, by (51), T=Ji

the total heat in conductors arising from induced currents.

One half the work done by the impressed forces is wasted in heat of induced currents, the other half is the magnetic energy set up, expressed by (48). Now, suddenly remove the impressed forces j there will be a similar inverse transient state, during which, as the magnetic induction subsides, the whole of the energy T will find its way to the conducting parts to be there wasted as heat. The intrinsic magnet itself, it should be remembered, would come in for a large share of this in general, or, in special cases, the whole. T is thus the whole amount of work that can be got out of the magnetic system by removing the intrinsic forces, which is why we term it the magnetic energy. [In the above investigation it is assumed that there is a steady state. When there cannot be, there are exceptional peculiarities, treated of in later papers.]

It may be objected that the above is unrealisable, that we cannot put on, or suddenly remove, the retentiveness of a magnet, and so obtain the whole magnetic energy existing in a given configuration in work. Admitting this, it may be remarked that we can do something equivalent, or at least approximate thereto, in the following manner. By means of a properly chosen distribution of impressed electric force we may set up electric current that shall exactly neutralise the field of the magnet, producing a state of no induction. Now suddenly cut oft' the impressed electric forces that kept up the currents. We then start with 110 induction, and terminate with the proper distribution of induction due to the magnet, and the impressed electric forces do 110 work, being cut out. Hence the effect is the same as suddenly putting on the impressed magnetic forces, doing 2T of work, one half magnetically, the other half being expended in the heat of induced currents. Some simple examples to illustrate this will be worked out later.

This extreme case will serve to illustrate the meaning of the distinction between the work done by h and by H, also the meaning to be attached to magnetic energy. There are other ways, of course, of using up the energy T above said to be wasted in heat. Thus, if we alter the configuration of the matter 111 our magnetic system, we usually alter T. Let it be increased from T to T+&T. Then the impressed forces h will do 28T of work, one half magnetically, in increasing J\ the other half mechanically, by the mutual stresses assisting the motion, and this latter half will be partly wasted in heat at once by the induced current accompanying the motion, and all may be ultimately thus wasted.

This naturally brings us to the subject of the mechanical forces.

SECTION VI. THE MECHANICAL FORCES AND THEIR POTENTIAL ENERGY.

The expression for the magnetic energy T may be conveniently put into another remarkable form, thus : By (50),

2F/*h = - 2 F/xF;

then, by (48) and (51), we obtain

T= 2 h/xh/87r - 2 F/xF/8tt, = T0-M, say (53)

Here the magnetic energy T is expressed as the difference of two quantities T0 and M, of which the first is constant, being the maximum value of T. Short-circuit the magnet by an infinitely permeable skin; there will be then no induction outside the skin, and T will be the greatest possible consistent with not altering the interior permeability /x and impressed force h. But it will not in general be as great as T0. This requires there to be no polar force, so that h at any place can produce the full induction /xh. But if we imagine every element of volume surrounded by an infinite^ permeable skin, T becomes T0. Should the magnet be uniformly magnetised, and of uniform permeability, it is sufficient to coat the outer surface with an infinitely permeable skin to increase T to T0. In some other cases no skin is needed.

For the same reason as before given for T, T0 and M are both necessarily positive. The quantity M is the same as what Sir AV. Thomson calls the “ mechanical value ” of the magnetic system, or the amount of work that would have to be done against repulsions to build up the intrinsic magnet if it were given in the state of infinitely slender filaments, magnetised parallel to their lengths, placed infinitely widely apart. Or, reversing the operations, imagine the intrinsic magr.et to be divided into infinitely slender filaments parallel to the lines of intrinsic magnetisation, and the filaments cut up into short straight pieces (though infinitely long compared with their diameters). Then if the elementary parts thus defined be infinitely widely separated from one another, and from all matter susceptible to induced magnetisation, the work done during the separation by the polar forces would amount to M. But I am unable to verify the statement that M may be either positive or negative (“ Electrostatics and Magnetism,” Art. 731, end of p. 565). The form 2F/xF/87r (not given in the paper quoted) shows that it must be always positive. The following are the principal forms :—

M= T0-T=? h/xh/87r - 2 H/xH/Stt = 2 F,xF/8tt ) , .

- Ł2 ftp, = Ł2 niP = 2 FFj/87T = - 12 IF, = - 12 I,F; 1 ( '

to understand which, it is necessary to say that I, = /xh/47r is the intensity of intrinsic magnetisation, and I the actual, the sum of the intrinsic and the induced; F, the polar force of the -intrinsic, and F the actual polar force ; pl the density of free intrinsic magnetism, and p the actual density; ft, the potential of F,, and ft of F; and, lastly, H = h + F. For the intrinsic magnetisation, we have

F, = - Yft,, brp1 = div F, = -47r div Ir

Similarly, if the number 2 refer to the induced magnetisation, we have

F. j = - Vft2, 47rp„ = div F., = - 4TT div I.,.

Lastly, F = F, + F2, ft = ft, + ft2, P ~ P\~Y- Poi I = I, +10.

The connection between the induced and intrinsic magnetisations is I2 = (/x - 1)(F, +Fo)/4?r = (/X - 1)F/4TT, which makes the induction B be

B — F + 471-1, + 47rl2 - /x(F + 47TI,//X) = /x(F + h) = /xH.

The various forms in (54) are got by application of the elementary potential property

2f/*f=2_pdiv/*f

through all space, f being any “ polar” force, whose potential is p, with the assistance of the various relations following (54). The forms in the first line of (54), in terms of forces, are the most important.

Now return to the subject from the impressed force point of view. (Our language may be suggestive of our believing magnetic induction to be a purely static state, but such a conclusion is not meant to be con- veyed). We suppose there to be impressed magnetic force in the intrinsic magnet, of strength h at any place, which is always present. The impressed forces try to do as much work as they can. They have in any configuration of the system (referring to the external arrangement of magnetisable matter) already done the amount T magnetically, in setting up the state of induction B, which =- jxH everywhere, H being the magnetic force as ordinarily understood outside the intrinsic magnet, including inductively magnetised matter, and the same with the impressed force h added, where there is h, that is, in the intrinsic magnet, which is of course inductively magnetised as well, unless its permeability should be unity. The impressed forces take advantage of all displacements of the system to do more work, if possible. If parts of the system be free to move, move they will, in such a manner as to let the impressed forces do more work, and increase T. The generalised “force,” assisting a displacement d.r, is expressed by

dT/dx, or, -d.Mjdx;

since, T0 being constant, any increase in T is accompanied by an equal decrease in M.

Any increase of permeability increases the induction and T, unless there be counteracting decrease elsewhere. A sphere of soft iron has no tendency to move anyway when placed in a perfectly uniform field of magnetic force. T is the same for any position of the sphere. But if the field be not uniform it will move so as to increase T. Any small piece of matter inductively magnetised will move in the direction of fastest increase in the square of the force of the field if its permeability be greater than that of the medium in which it moves; and in the direction of fastest decrease when its permeability is less. This is irrespective of the direction of the force. Thus iron moves to, and bismuth from, either magnet pole, and in certain positions they may move straight across the lines of force. This also happens when a wire conveying an electric current attracts iron, the motion being across the lines of force. (This is not a case of intrinsic magnetic force, but the principle is the same.)

Imagine a uniformly intrinsically magnetised magnet to be wholly surrounded by imaginary impermeable matter to begin with. There is no induction anywhere, and T= 0. Let outside the magnet there be matter of all degrees of permeability with no retentiveness, and divisible as much as we please, all floating in the standard medium of unit permeability. If we remove some of the impermeable matter from the surface of the magnet, the impressed forces immediately act, and some induction comes out into the surrounding space, and with it there are mechanical stresses set up, which, if yielded to by the matter, assisted by suitable guidance if required, will have the effect of bringing the most permeable matter to and driving the least permeable matter away from the magnet. All the while, the impressed forces h are working, increasing T tlie magnetic energy, and equivalently reducing M the “ mechanical value,” which was 7’0, its greatest value, at starting; all the while doing an equal amount of work mechanically, viz., on the matter set in motion, which we may conveniently dispose of by frictional resistance. In the end, supposing we have infinitely permeable matter to surround the magnet with, the whole work done by the impressed forces will be 2T0y half magnetically, half mechanically. The final magnetic energy is T0, the mechanical value or potential energy, nil. The distribution of T as it rises in value from nil to T0 becomes ultimately confined to the magnet alone. For in the final state, the magnet is short-circuited by the infinitely permeable skin. It is only necessary for the ends of the lines of impressed force to be connected by infinitely permeable matter, and this is most simply done by the skin. To obtain the magnetic energy T0 that is left locked up in the magnet, tlie impressed force must be removed; then, the equations of induction show that T0 of heat is generated by the induced currents accompanying the subsidence of the induction.

Regarding the before-given definition of the mechanical value, notice that the more slender a filament (longitudinally magnetised) is, the less important is the effect of the polar force 011 the induction inside, which differs little from /xh, except near the ends, /xh being the maximum induction h can produce itself. Thus by slitting up the magnet into filaments as described, and separating them infinitely, we have a final state in which the impressed forces have done infinitely nearly the full amount of work they can do, the same amount as if the magnet, without any slitting and separation, were short-circuited, if it be uniformly magnetised.

When work is done by external agency against the mechanical forces, as in drawing soft iron away from a magnet, we reduce T by the same amount. There is, during the motion, magnetic current in the magnet opposed to the impressed force, and the work done against h is twice the decrease in T. Half of this is accounted for by the magnetic energy returned to the magnet (becoming latent, as it were), the other half by the work done mechanically in drawing away the soft iron.

271/ might be called the potential energy of the impressed forces in any configuration, being at a maximum '2T0 when the forces are prevented from working by an impermeable skin, and zero when short-cir- cuited (with necessary modifications for irregular distributions of h). There are so many senses in which the energy of a magnet may be understood that it is necessary to be precise in stating one’s meaning. Therefore, I repeat that by the magnetic energy I always mean the quantity T, which has the value H/xH/87r (or I force x induction /Air) per unit volume, both in the magnet and without (and also when there are electric currents, only then it will not be the magnetic energy of the magnet alone), the intrinsic force, where there is any, to be included in the reckoning of H, the magnetic force. We are thereby enabled t< make use of electromagnetic ideas without bringing in the hypothetical Amperean currents. The inclusion of h in the magnetic force, making B = /xH always, is specially useful in simplifying both ideas and formula1.

without loss of generality. The theory of magnetism is quite difficult enough already (owing to imperfect retentiveness and variable permeability) without additional gratuitous difficulties.

To return to the mechanical forces set up by a magnet. There is a very important reservation to be made when considering the forces and the variation in the value of T as they are yielded to. The motion must be sufficiently slow to not appreciably alter the force by electric currents set up. That is, the variation of T the magnetic energy (of the magnet) does not ever give -exactly the value of the generalised force, and in rapid motions it may be something very different.

For distinctness, consider a round bar magnet (intrinsic) and a round cylinder of soft iron moving in a line with its axis. As the soft iron moves, the induction increases or decreases, both in the magnet and the soft iron, according as it approaches or recedes from the magnet. From the symmetry, the lines of induced E.M.F. are circles about their common axis, and as they are conductors there are currents set up in both (there are also similar circular currents in the dielectric, but not involving waste of energy); their directions are such as to retard the increase of induction on approach, and retard the decrease on recession ; hence the attraction of the magnet and soft iron is reduced as they approach, and increased as they recede from one another. More work must be done externally against the attraction in drawing away the soft iron than is done by the magnet in the reverse motion. The difference is accounted for fully, according to the laws of induction, by the heat of the induced currents. This will be greatest in the magnet itself, less in the soft iron, and a very little in surrounding conductors. This effect has nothing to do with any lagging or retardation of magnetisation in the soft iron, which, if there be any, requires separate reckoning, but is of the same nature as the resistance to the motion of a (practically) unmag- netisable conductor in the magnetic field. Substitute a cylinder of copper for the soft iron; there is no appreciable force now when the cylinder is very slowly moved, as there was no appreciable departure from the normal attraction of the magnet for the soft iron when it was very slowly moved; in both cases the currents set up by rapid motions waste energy, as heat in the magnet and in the soft iron or copper respectively, and this waste must be externally accounted for. Or, supposing the magnet to draw the soft iron from rest at a certain distance, the kinetic energ}’ communicated to the soft iron mass after it has moved a certain length will be less than the increase of T during the motion, the deficit being wasted by the heat of induced currents.

The following brings into a strong light the connection between intrinsic force and intrinsic magnetisation. Suppose we double the permeability in every part of a magnetic system, how will it affect the magnetic energy % That depends on whether we keep the intrinsic force constant or the intrinsic magnetisation. If we keep the intrinsic force constant, we double the magnetic energy, since we keep the actual force unchanged as well, whilst we double the induction. On the other hand, if we keep the intrinsic magnetisation constant, we halve the magnetic energy; for we halve the force everywhere, whilst keeping the induction unchanged.

Similarly, any increase of permeability outside a magnet increases T and the induction. Also any increase of permeability inside the magnet increases the induction and T provided the intrinsic force at the place is unaltered. T0 is also increased. But if, whilst increasing the permeability at a place inside a magnet, we keep the intrinsic magnetisation constant, we reduce T.

The conduction current analogue will make this plain. In any system of conduction cnrrent kept up by impressed E.M.F., any increase of conductivity outside the seat of impressed force will increase the current, and also the heat generation ; the same is true if we increase the conductivity at a place where there is impressed force, if we do not alter the impressed force. But if, whilst, say, doubling the conductivity at a certain place where there is impressed force, we halve the strength of the latter, we decrease the current and the heat generation.

SECTION VII. WORK DOSE BY IMPRESSED FORCES DURING TRANSIENT STATES.

When we charge a condenser by means of a voltaic battery a transient current is set up in the circuit, which is quickly stopped by the elastic reaction of the electric displacement in the dielectric. There is then a certain amount of electrostatic energy set up in the condenser, say U, and, during the charge, a certain amount of heat was generated in the conductor. That its value is also U (expressed as energy, to save the perfectly useless introduction of the mechanical equivalent of heat), may be seen at once on remembering that when we discharge the condenser through the same resistance (without impressed force), the current passes through the same series of values at corresponding times as during the charge, and must therefore generate the same heat, which, being now derived from the potential energy of the condenser, must amount to U. And we further see that whether the discharge circuit has or has not the same resistance as the charge circuit, the heat during the charge and discharge are equal, namely U. Thus, in charging the condenser, the battery does 2Z7of work, half of which is accounted for by the Joule-heat during the charge, and the other half by the energy of displacement in the condenser.

This property is wholly irrespective of the manner in which the charge takes place, if no other work be finally done than in heating and in setting up electrostatic energy. Thus a coil may be inserted in the circuit, which may materially alter the manner of the charge, and rer ler it oscillatory; still, the heat will amount to U as before. And 11 we put another coil near the first, so that there is a current during the charge in it as well as in the main circuit, the heat will still be exactly U, provided we include the heat in the secondary coil as well.

Similarly, in charging a submarine cable, the distant end being insulated or only connected to earth inductively through condensers, so that the final state is one of no current (practicallj'), the total heat during the charge exactly equals in value that of the electrostatic energy set up in the dielectric of the cable, condensers, etc., when we count the Joule-heat in the conductor, sheath, and wherever else there may be conduction current during the charge.

The general law, of which the above are examples, is as follows :—If, in any arrangement of matter, conducting (metallically), or dielectric, or both, originally uncharged and free from current, we cause any stead}'' impressed forces to suddenly commence to act, and we keep them on, whose distribution is such that the final state is one of no current, the Joule-heat generated in the conducting parts during the transient state will exactly equal in amount the value of the final electrostatic energy set up. The impressed forces may be either in the dielectric or the conducting matter. If in the former, it does not matter how they are distributed, for the final state will be one of no current; but if in conducting matter the distribution of impressed force ceases to be permissibly arbitrary. In a linear conducting circuit, for example, their sum must be zero round the circuit. If partly conductive, partly inductive, this ceases to be necessary, but the impressed forces must act equally over the whole cross-section of the linear conductor, otherwise the final state will not be one of no current.

Suppose, however, other things being the same, the distribution of impressed force is left perfectly arbitrary in the conductors as well as in the dielectrics, with the result that the final state is a certain distribution of steady electric current in conductors, of electrostatic energy in dielectrics, and of magnetic energy in both, all to be definitely known from the given data, the distribution of impressed forces, of conductivity, capacity, and permeability. This we may conveniently divide into three cases; first, the final magnetic energy negligible in comparison with the electrostatic; next, the electrostatic energy negligible in comparison with the magnetic; and last, the real case, both being counted.

In the first case we have a transient state, during which the actual electric force anywhere is that due to the impressed force on the spot and the changing electrostatic force, (though “static” is rather misapplied), with electric current, both in conductors and dielectrics, leading to a final state in which the current is confined to conductors. Now, if there had been no electrostatic capacity, the final state of current would have been set up instantaneously, the activity of the impressed forces would be 2eC0, e being the impressed force anywhere and C0 the current, the summation to include all places where e exists. This activity would have existed from the first moment, so that at the time t after putting on the impressed forces, the whole work done by them would have been 2 eC0Ł, wholly accounted for in Joule-heating. In reality, when there is electrostatic energy set up as well, the whole work done by the impressed forces up to the time t, to include the transient state, exceeds the amount 2eC0tf, which would have been done had there been no electrostatic energy to set up, by the amount 2 U, if U is the final energy of electric displacement. Besides doing an additional amount of work U in setting up the energy of electric displacement, the battery does an equal additional amount, which is accounted for as extra Joule-heating.

Thus, in charging a shunted condenser, or a cable Avhose further end is to earth (if we do not count the electromagnetic induction), U being the value of the final electrostatic energy, the battery will do 2 V more work than if the electrostatic capacity were nil, half in extra heating, half in setting up the electrostatic energy.

In the second case we have a. great difference. Here electromagnetic induction is predominant. The law is now (other things being the same) that the impressed forces do ‘2T ks,S work than they would have done had there been no magnetic energy to set up, T being the value of the final magnetic energy. In the electrostatic case the work done was

2 eC,/ + 2 U; it is now 3 eC0i — 2T,

up to any time t including the transient state. Hence the Joule-heat (instead of being U more) is now ‘ST less than if no magnetic work had been done.

In both cases, U and 1\ the electric and magnetic energy, are recoverable, appearing as Jonle-heat in the conductors when the impressed forces are removed; but the doing of electrostatic work makes the impressed forces work faster, and of magnetic work slower. The one is potential energy, the other kinetic. The one is connected with elasticity, the other with inertia.

Lastly, coming to what is more usually the case, both electric and magnetic energy set up. During the transient state the coexistence of the two inductions causes a singularly complex state of affairs, by no means the mere resultant of the tvvo taken separately. Yet the law, which we might guess from the preceding, is that the additional work done by the impressed forces above the amount ~ eC(/, that they would have done had there been no electric and magnetic energy to set up, amounts to 2 Ł7- 2T, being 2 U more on account of the electric energy U, and 2T less on account of the magnetic energy T; whilst the Joule- heat is increased by U-'ST.

This includes, of course, all the preceding special cases. Thus, in the case of charging a condenser, the final current 6'0 = 0, and T=0. The additional work 2(U- T) may of course be either positive or negative, according to the values of U and T.

The following proof covers the whole, the impressed forces being arbitrarily distributed, and the matter having any conductivity, capacity, and permeability. Also, eolotropy in these three respects is included. Let e be the steady impressed force at any place, put on at the time / = 0, and kept on. Let E, H, T, G, be the electric force, magr Jc force, electric current, and magnetic current, at the time t after the commencement, and E0, H0, ro, their final values, G0 being zero.

The activity of e is eT at any moment, and the total activity is - el1 through all space, or wherever e exists. Let F be any “polar” electric force, then

^Fr = 0, (55)

because F lias no curl, and T no divergence; a well-known theorem that is made visibly true by considering that the tubes of T are closed, whilst the line-integral of F in any circuit is zero. Now. in the transient state, the equation of induction is

curl (e - E) = 47rG; * (56)

which becomes, in the final state,

curl (e - E0) = 0;

whence E0 = e + F0, (57)

where F() is polar, or F0 = — P being a scalar potential, the electric potential. Choose then F = F0; then, by (55) and (57),

2er = 2er + 2Fur = 2E(,r; (58)

or the activity is the same 011 putting the final real electric force for tlie impressed force.

Now r = C + i), (50)

C being the conduction and J) the displacement current, D being the displacement (elastic). Therefore, by (58) and (59),

2er = 2E1(C + 2EuD,

= 2 EQ/L’E + 2 E0D, because C = /.E ;

= 2 C0E + 2 E0D, (60)

k being the conductivity, and C0 = ŁE0 the final conduction current, = T0. But curl H0 = 4~C0; (61) therefore

2 CnE = 2 -^°-= curl E = - H0 (curl e k - G),

curl

by (61) and (56). Therefore

2 C0E = 2 e curl H0/477 -2 H0G = 2 eC0 - 2 H0G.

Putting this in (60), we get

2 er = 2 eC0 + 2 E0D - 2 H0G (62)

This is true at every moment. Now integrate (62) to the time, from 0 to t, to include the transient state (t must mathematically be infinity), and we get

2 e jTdt = 2 eCy + 2 E0D0 - 2 H0B0/47r, (63)

D0 being the final displacement, B(J the final magnetic induction. But

i/=2JE0D0, T= 2iH0B0/47r

are the values of the final electric and magnetic energies. So (63) becomes

2e/r(// = 2eCy + 2f/-27; (64)

which is the required result, showing that the work done by the impressed forces is increased by 21 on account of the electric energy, n.rc.r.—vol. i. 2g and reduced by 2T on account of the magnetic energy, the Joule-heat being increased by U — 2>T.

I have written out the above rather fully. Without the explanations, it goes simply thus, 2er = S(e + P0)r = 2Eor = 2E0C + SE0D - 2 C0E + 2 E0i) = SH0 (curl e/47r - G) + 2 E0D = 2eC0 + 2EuD-2H0G, whose time-integral is

2efr,/t = ?eC0t + 2U-2T.

This shows how much may be put in a small compass.

We should remark that it is F the true current that is circuital, in general, not C or I) separately, except in the final state, when the current is wholly conductive. Also, that we twice make use of the theorem

- A curl B = 2 B curl A (65)

through all space, A and B being any vector functions; of this (55) above is a special case. Giving proofs of all the potential properties made use of is out of the question. I entered fully into these matters in former articles. It is customary in mathematical investigations in electromagnetism to virtually prove this and similar theorems over anti over again in the course of working out results, instead of merely quoting them; like proving a proposition in “Euclid” ab initio, from the axioms and definitions. It would, however, be very desirable to have special names for the various useful vector theorems connected with the V operator. This is sometimes done by quoting a man’s name, and leads to confusion, if two theorems are called, for instance, Laplace’s theorem. I think the three fundamental theorems of Slope, Version, and Divergence would be recognisable by these names by anyone acquainted Avitli the theorems, though not previously Avith these names for them. From them folloAv a number of others of the greatest utility, of which (65) is an example.

SECTION VIII. ELECTRIC ENERGY. CIRCUITAL DISPLACEMENT.

In the theory of electrostatics a tube of displacement has a beginning and an end, at its beginning there being positive, at its end negative electrification. The terminations of the tubes are usually upon C' n- ducting surfaces; there may, hoAvever, be interior electrification ir .lie dielectric, if so, it has got there by convection, or by disruption. Impressed force in the dielectric is not considered in the theory of electrostatics. But should there be any, there will usually be closed tubes of displacement without electrification, as Avell as terminated tubes, due to the presence of conductors; and should there be no conductors, the displacement set up by impressed force is Avholly circuital. This Avill be briefly considered later. At present Ave take the case of circuital displacement in a dielectric arising from electromagnetic induction. Conductors are temporarily excluded for simplicity. Let there he any state of electric displacement D and magnetic induction B in an infinitely extended dielectric, without impressed forces. A possible state of induction and displacement is meant, of course. For instance, set up any state of displacement by impressed force, which then remove, leaving the system to itself. E and H being the electric and magnetic forces at any moment, we have

B = /xH, D = cE4—; (66)

H being the permeability and c the specific capacity. The tubes of B are always closed. Those of D are also closed, if there be no bodily electrification. AVe cannot, therefore, express the electric energy in terms of the scalar electric potential and the electrification. The appropriate form is in terms of the magnetic current and its vector- potential.

Let Z be such that

-curlZ = cE, (67)

which is possible because this is the general integral of divcE = 0, expressing that the displacement is closed.* Z is the vector-potential of the magnetic current. It is given by

ry ^ curl t'E 1 v (E /ŁQ\

-Z = 2, , or = curl 2 -—, (68)

■iTry 47rr

by potential properties. Here r is the distance from the point where Z is reckoned, of the element of the quantity summed up through all space. To Z, as given by (68), any polar term may be added without affecting (67); Z, after (68), being circuital, like B and D. Since the second equation of induction is

— curl E = /xH = 4—G, (^9)

G being the magnetic current, the equation of Z, by (67) and (69), is

curl c_1 curl Z = 4-G; (70)

from which we see that when c is constant, we have

Z = cSG/r, (71)

verifying that Z is the vector-potential of the magnetic current. If U be the electric energy (energy of electric displacement), we have

Z7= 2 JED = 2 I-?-curl E = 2 ^ZG, (72)

“curl

by potential properties, and (67) and (69).

These may be instructively compared with the corresponding magnetic equations. If A be Maxwell’s vcctor-potential of the electric current, we have

curl A = B =/xH, (67a)

[The first use (not then, but now) known to me of the function Z in a dielectric, to give the displacement by curling, ia in Professor Fitzgeralds paper “On the Electromagnetic Theory of the Reflection and Refraction of Light, ’ Pin!. 7 ran-*., 1SS0.J

resulting from integrating div B = 0. And the first equation of induction is

curlH=<E = ITTV, (09(0

r being the electric current. So, T being the magnetic energy, we have

T = 2 iHB/477 = 2 J--B-curl H/4tt = 3 \AV (72a.)

curl

by (07a) and (69c?)- And, if /a be constant, we have

A = /^I>. (71rt)

These four equations marked a correspond to the former equations with the same numbers. We have also

E = - A -VP, (73r0

and, to correspond thereto,

H = - Z - Vfl, (73)

P being the scalar single-valued electric potential, and Ł2 the scalar single-valued magnetic potential. (73(0 is Maxwell’s equation. To prove (73), we may merely remark that by (69(0 ^ie magnetic force round a closed curve equals 47t times the electric current through the curve; and by (67) the same relation holds between - Z and in times the displacement. Or, differentiate (67) to the time, and compare with (69«). So far as the energy expressions in (72) and (72(0 gf>, it does not matter whether Ł1 and P are counted or not, though they usually exist, especially if there are variations of permeability or capacity from place to place. Other forms of U and T:—When A ami Z are wholly circuital,

U=^ ArA/Sir-21 P<T, T=? Z//Z/S- - 2 IQp,

if - \-rrp = div [J.Z and - 47ro- = div cA;

(73) and (73«) holding good. Here <r is imaginary electrification, and p imaginary magnetic matter, the first being where c and the second where /x varies.

Simple Example of Closed Displacement.

In this example there is a conductor in the field, but as, from symmetry, it will be obvious that the displacement is wholly closed, it will not matter. If an intrinsic magnet be at rest in a dielectric, there is I ^ electric force, but merely a state of magnetic force. But if it be set in motion there is immediately a field of electric force set up as well, and of displacement and electric current due to changing displacement. Whether the displacement does not or does cause electrification will depend upon whether it is, at the surface of a conductor, wholly tangential or not, for it is the normal component that introduces surface electrification. Now, if a straight bar magnet of circular section be

' o o

carried through the air parallel to its length, the lines of electric force are clearly circles about the line of motion, so that the displacement in the air is wholly circuital. Or, let a uniformly intrinsically magnetised sphere move through a dielectric in the direction of its axis of magnetisation with constant velocity r small compared with that of the propagation of light, so that we may regard the sphere’s field of magnetic force to be rigidly attached to it and move with it without change. A certain state of electric force in circles about tlie line of motion, the continuation both ways of the sphere’s axis, is set up, changing at any fixed point continuously. But if we travel through the air with the sphere, the electric field is stationary. We may thus regard the magnet as carrying with it, in rigid connection, a certain constant electric field as well as its magnetic field. The first approximation to the solution, by far the most important part, is readily found.

Let M be the magnetic moment of the sphere, of radius « ; z distance measured along the line of motion, from a fixed origin, of the centre of the sphere at time t; r the distance of any point, P, from the centre of the sphere, and 0 the angle r makes with z. The magnetic potential at P is fl = (J/r-)cos 6.

Let v be the velocity of the sphere, then t being the time and r:, constant.

This gives dSljdt = - c dil/dz ;

so that the magnetic current at the point P is

G = /iH/Ijt = — V& 47T — (V/47r)V(dŁ2 d:.); or, using the above value of Q,

G = (zJ//4tt)V{(1 - 3 cos-0)/**}, (74)

the differentiation V being conducted at P.

Calculate the total of G through the circle r, 6. This is, if w = cos0,

rMf f c, 7, 7 d 1-3 cos2tf 3<-J/ i „v

4irJJ th' — = '2r- - “■>

The circle r, 0 is a line of electric force E. So, by the relation (G9),

— 2TTI E sin 0 — 67rrJ/(o(l — w-)//■-, or E — — 4f cMr~s sin 20.

This could be got more simply, but we wanted an expression for G, and having it, made use of the relation (69) applied to a closed curve, or the Theorem of Version.

The last equation gives the strength of electric force at the point r, 0 referred to the sphere’s centre and axis of motion. The potential energy U of the displacement is

U= 2 rE2 Sir = Jjjg - (3^/yt,^(l - io-)f-dr do> d<f>.

The limits for r are a and co • for </>, 0 and 2TT; for o, - 1 and + 1. This wiv^s

U= cv-M-jW = r?M'2/5r?« >, (70)

if r, is the v elocity of light, the electromagnetic value of r being thr reciprocal of r,2.

T=^ fiir2}87r = ^ is easily shown to be T=3M2 + 2),

if p. be the permeability of the sphere, the outside value being unity. Thus, if }i= 1 in the sphere also,

U/T=\vVvI

Travelling with the sphere, we have a steady electric field, and no current. But at a fixed point, the current, or rate of increase of displacement, is

cEjiir = - (cvj4:7T)((IEjdrS) = 2>crM sin (9(1 - 3 COS20)/47T)4V{.

As before remarked, the solution is the first approximation. In a complete theory there would be no discontinuity in the electric force at the surface of the magnet, as the above supposes, and there would be electric current in the magnet, with waste of energy by the Joule heating, thus requiring a continuously applied mechanical force to keep up the motion. Whilst, therefore, we should have disturbance inside the magnet, the solution outside would be not exactly that given. In fact, we see that the calculated dielectric current itself has its magnetic field, thus slightly altering the assumed magnetic field, that of the magnet at rest; and to the motion of this new magnetic field (very weak) there corresponds a new electric field, and so on. However insignificant these corrections may be in point of magnitude, they are yet required to make up a complete sjstem satisfying the laws of induction. Taking, however, the above field of electric displacement by itself, we may close the magnetic currents in the appropriate, manner on the surface of tiie magnet itself. The surface value of E is, by (75),

- fiiIvti~3 sin 26, (77)

and it is tangential. Hence the same expression divided by 4?-, and taken positively, is, by the surface interpretation of (69), the strength of the complementary surface magnetic current, directed at right angles to the electric force—that is, along the meridional lines, if the poles be those points of the sphere cut by the line of motion through the centre. This system, with the former, makes a closed system of magnetic current, whose vector-potential may be taken to be given by

Z=cMrr~2 cos 6, parallel to z (78)

For this satisfies (67). This is literally the vector-potential of a surface magnetic current of strength o-v, parallel to z, a being the surface-density of free magnetism; but it is unnecessary to calculate the part due to the complementary current required to close it.

W e can now check the value of Ł/by the formula (72).

U= 2 iZG = I ,3 ~ TL*(l - <o-)a-du d<f> (79)

4/. a- ii- JJ

Here we have to integrate the scalar product of Z and G through all space, Z being given by (7S) and G by (71) outside the magnet, and by

(77) (when divided by — 47t) on its surface. But the volume-integral outside the magnet vanishes, because Z contains Qx and G contains as a factor (zonal harmonics). Hence there is simply the surface integral left, expressed by the right side of (79), the .-.-component of the surface G alone counting. It gives the value (76) again.

The reason why (78) suffices for use in the formula 2-?rZG--Ł' is because G is circuital. If, on the other hand, we employed the full formula for Z, we would not need to close G. A surface-current crv parallel to z would then suifice. Thus, if Z-Zl + Z., and G = Gx + G.„ wherein Z and G are both circuital, whilst the parts Z„ and G„ are polar, we have

2 ZG = 2(Zj + Z2)GX - 2 Zi(Gj + G,), since 2 ZG2 = 0 and 2 Z2G = 0.

SECTION IX. IMPRESSED ELECTRIC FORCE IN DIELECTRICS.

A comparison is often made between distributions of magnetic induction and of electric current. There is, however, a far more satisfactory analogy between magnetic induction and electric displacement in a dielectric, which may be pushed much further before correspondence ceases. So far as mere distributions in space go, of the three phenomena of conduction current, electric displacement in a dielectric, and magnetic induction, we may conveniently compare them simultaneously.

First, let there be a distribution of impressed electric force e15 in a conductor of conductivity k (infinitely extended in the general case, with the conductivity different at different places), setting up a steady state of electric force E1} and conduction current C. We have the three conditions

C = mit div C = 0, curl (e1 - Ex) = 0 (80a)

Secondly, in a non-conducting unelectrified dielectric of capacity r, in which a distribution of impressed electric force e2, sets up a steady state of electric force E2, and displacement D, we have D = cE0/47r, div D = 0, curl (e2 - E2) = 0 (801)

Thirdly, in a medium of permeability /j., in which a distribution of impressed magnetic force h, sets up a steady state of magnetic force H, and induction B, we have

B = //H, div B = 0, curl (H - h) = 0 (80)

These three sets of conditions are exactly similar. We have in each case a “force” and a “flux.” The first condition is the linear relation between the force and the flux, i.e., Ohm’s law, etc. The second condition is that of continuity of the flux, asserting that its divergence or convergence is zero everywhere, or that the flux is circuital. The third is the force equation, what the equation of induction becomes when the state is steady ; the third conditions in (80(/) and (806) being examples of

curl (e - E) — fj.il,

with H = 0, and that in (80c) arising from

curl (H - h) = l7rr,

with T = 0. The difference between the actual and the impressed force, or the natural force of the field itself, has no rotation, or its line-integral round any closed line whatever is zero. Or,

Ej = ex + Fj, Eo = eJ + F2, H = h + F3,

where F1? F2, Fs are “polar” forces entirely, derived from single-valued scalar potentials, whose space-variations give the polar forces; thus

Fx = - vPj, f 2=-vr,, F3=-vfl,

where 1\ and 1\ are electric potentials and i] magnetic potential.

The three conditions serve to determine unambiguously the complete solution, so far as the force and the flux are concerned, when the impressed force and the distribution of conductivity, etc., are given. To the impressed force we require to add a polar force to make up a complete system of force satisfying the continuity of the flux. At the poles, or places where the polar force converges, or diverges, we may, if we like, put imaginary matter, electric or magnetic, as the case may be, repelling according to the inverse-square law, and regard the potentials as the potentials of the matter. Each distribution of impressed force requires a particular polar force to supplement it; except when the impressed force is so distributed that it can by itself satisfy the continuity of the flux, and the linear relation between the flux and the force. Thus, when

div /,el = 0, or divce2 = 0, or div/xh = 0,

no polar force is needed, and there is none, or the potential does not vary. We have then

C = Łev D = te2/4:7r, and B^/xh

respectively. On the other hand, should the impressed force be itself polar in its distribution, there is no flux produced. We then have

e1= - Fj, etc., and Ej = 0, etc.

Now, if in the above three problems, the distributions of impressed force—electric or magnetic, as the case may be—are identical, and also the distributions of conductivity, etc., in space, then also the three fluxes have identical distributions. Practically, as neither the permeability nor the capacity (in non-conductors, at any rate) can vanish, we must not let the conductivity be zero anywhere in the conduction current problem; i.e., all space must be conducting, more or less, to tret identical distributions of current to those of induction and dis- placement respectively. We cannot confine either of the last to definite closed channels, as we do electric currents, by arrangement of matter, although we can do so by proper distributions of the impressed force, viz., in the above mentioned cases of no polar force.

The distributions of Joule-heat per second (or dissipativity), of electric energy (or energy of displacement), and of magnetic energy are also similar, being E1/i'E1, E2rE.,/S-n-, and H//.H '877 respectively per unit volume; and if their totals through all space be Q, U, and T respectively, we have

(J = 2 E,/,-E1 = 2 e^-E, = 2 e1/i-e1 - 2 F^F,, (81.)

87TU= 2 E2cE2 - 2 ey'Eo - 2 e.ce, - 2 F2rF,, (816)

STTT = ?KIXK = 2h,ja = 2h,Ji -2F3/*F3 (81r)

In the first form of expression, as 2 E^E-p the quantity summed is the actual amount in the unit volume. In the second form the summation extends only where there is impressed force, being of the scalar product of the impressed force and the actual flux.

In the third form, there are two summations, both necessarily of positive amounts, whose difference gives the dissipativity, etc. The first extends only where there is impressed force, being the greatest value of (J, etc. The second extends over all space where there is polar force, and vanishes when there is none. The parts depending on the polar forces may also be expressed in terms of the potential and of

imaginary matter. Thus—

^Fj/iFj = 2 l\pv if = conv Lev (82 a)

2 F2(;F2/87r = 2 lP2p2, if p., = conv cej-iir, (82ft)

2 F3/*F3/8tt = 2 if cr = conv

Note here that the distribution of imiiginary matter is not the same (in general) as that before mentioned, measured by the divergence of the polar force. Here the matters are distributed where the impressed force, multiplied by the conductivity, etc., has convergence. In (S'Jc), o■ is the density of imaginary magnetic matter 011 the ends of a magnet. But in (82.) and (826), pl and p2 are not distributions of electrification, for there is none in either case. The electrification is measured by the divergence of the displacement, which is zero under the stated conditions.

We may also employ vector-potentials in all three cases. Thus

<2=2 E^ = 211^, (83.)

Ł/= 2 AEJ) =2iZG2> (836)

7=2 J HB/4TT = 2 ’ AF (83c)

The relations of these new quantities are

curl H1 = 4;rC, curl ea = ITTG^ (84.)

curl Z =cE.,, curl e.2 = 4TTG.,, (846)

curl A curlh=47rF (84c )

In the conduction current case (83.) and (84.), HL is the magnetic force of the current, and Gt is an imaginary distribution of magnetic current, viz., where the impressed force has rotation, or varies laterally. In (836) and (846), G., is also an imaginary magnetic current, similarly related to the impressed force, whilst in the magnetic case (83c) and (84r), F is similarly related to the impressed magnetic force, and is the well-known imaginary electric current which would (if it were a reai current) correspond to the same state of magnetic induction as the impressed force h sets up. The summations do not extend over the whole region of impressed force, but to portions only, perhaps round a single line. This we will illustrate in the conduction current case. Tlie formula (J = 2 H1G] has some suggestiveness in connection with the transfer of energy, but turns out to have no very important application. As we remarked before, if the impressed force be polar, there is no fiux. For there to be any flux at all, the impressed force must have curl somewhere, and Q - HjG-j shows the exact dependence of the activity on the situation and amount of this curl. Take a simple voltaic circuit, copper and zinc in acid, and the copper continued by a copper wire to the zinc outside the liquid. Suppose the impressed forces are entirely confined to the metal-acid surfaces, and are of uniform strength over each metal. Then the places of summation are the two wind-and-water lines. One place gives the amount of energy leaving the zinc per second, the other the amount arriving at the copper, and their difference is the amount of J oule-heat in the circuit. Next suppose that the zinc-air force equals the zinc-acid force, and that the copper-air force equals the copper-acid force. There is now only one place of curl of impressed force, viz., the air-boundary of the copper-zinc junction, and the summation round that line only gives the value of Q. But not only may wo thus shift the places of summation outside the battery, but we may locate them altogether away from the circuit if we like by suitable dispositions of impressed force outside the circuit, which will not in any way disturb the state of magnetic force and current provided we keep the same impressed forces in the battery as before, although they will alter the paths of the transfer of energy. In problems (b) and (<■) there is, of course, no transfer of energy at all after the steady states, which are alone considered, have been set up.

In problems (/>) and (r) we may also compare the mechanical stresses. They are such as to increase U or T respectively when allowed to act, by letting the impressed forces do more work. Two bar magnets repel with like poles, and attract with unlike poles approached, whilst either pole of either will attract soft iron and repel bismuth. Similarly, take two bars of a dielectric and put in them impressed electric force parallel to their lengths. Like ends will repel and unlike attract one another, whilst either pole will attract a piece of solid dielectric of greater capacity than air (in which all are immersed), in which there is 110 impressed force, or repel it if its capacity be less than that of the air, or other surrounding medium. The same will happen if there be intrinsic displacement in the solid bars, such as arises from so-called “absorption.” (If in the conduction current problem there were a corresponding tendency for (J to increase, then a copper ball in mercury conve3ring a current would be attracted by cither electrode, where the current has greater density.)

Dismissing now the problems (a) and (b), consider something quite peculiar to (/■), that of dielectric displacement. Both the conduction current and the magnetic induction are continuous, as expressed by tlie second condition in each case. We have also written down the same condition for the electric displacement. No arrangement of impressed electric force can create any discontinuity in the displacement, provided there be no conduction anywhere. Should there be any discontinuity, its proper measure is the divergence of the displacement. That is, p the volume-density of the electrification—for it is nothing else than discontinuity in the displacement—is given b}r

p = div D.

Supposing there to be any such discontinuity, it will remain a fixture, or only be varied in distribution by motion of matter carrying the electrification with it—of matter, because it appears most probable that there cannot be any electrification without the presence of matter ; or, the displacement in the electromagnetic medium itself, the ether, is always continuous. To get rid of the electrification there must be conduction. Conversely, to create electrification there must be conduction. The two statements go together.

Now the essential property of a conduction current is dissipation of energy by the heating effect produced. Destruction of electrification by disruption of the dielectric is therefore of the nature of a conduction current, at least partly, although the simple metallic conduction law is not followed, and, in fact, it is not known definitely what is the exact course of events, even when considered merely electrically. But we should never, in attempting to explain something, go from the complex and ill-understood to the comparatively simple and mathematically expressible, but pursue the other course. We may say then, with tolerable certainty, that to create electrification there must be (1), the presence of matter; (’2), impressed electric force, i.e., a cause in operation which tends to produce an electric current (if we like we may say which tends to produce magnetic force, and regard the current as an affection of the magnetic force), which is neither the polar electric force nor the electric force of induction, these being the only two naturally belonging to the electromagnetic medium ; (3), conduction, with dissipation of energy.

Nothing is easier than to create electrification experimentally or involuntarily by the contact or friction of bodies. Imaginative explanations may be readily made up, and are likely to be of very little value. It is, then, necessarv to be somewhat general, or vague, in

f J «/ O * O 3

order to keep on the straight and narrow path. We may create electrification by the contact of different conductors, either by means of the small known contact-force of thermal origin, or of the much larger air-surface contact-forces of chemical origin, and. so far, it is easy to recognise the presence of the matter, of the impressed force, and of the conduction. Similarly, by the contact of a conductor ami a dielectric, if there be similar impressed forces present. No friction is absolutely necessary. But if we set two dielectrics in contact, without any connection with conducting matter, no impressed electric forces can, without a change of conditions, set up any electiiHcation. Some friction is needed, and Avitli it there is conduction, or equivalently disruption. The actual original impressed electric forces need not be great, and are probably small. It is the act of separating mechanically the opposite electrifications produced by the friction, at first extremely close together, that is the main cause of the high difference of potential observed after separation, together with the similar separation taking place during the friction. The electric field is, for the most part, set up during the separation, and derives its energy in the main from the mechanical work then done.

Now, having got electrification—p per unit volume, say— the iiekl due to it is settled by the three conditions,

D = rE/47r, divD = /.>, curlE = 0 (85)

differing from the former, equations (806), iu the second and third, in the absence of impressed force and the existence of p. The field is definitely determined by (85); the force is polar completely, and the energy is

2 EfE, 8 =

if P be the potential.

►Should there be also impressed forces, the actual field will be the sum, in the vector sense, of the two fields, due separately to the electrification and to the impressed forces. Not only that, but the total energy of displacement will be the sum of the amounts in the separate fields; or, the mutual energy of the two fields is zero. This is true because the displacement in the field of the impressed force is everywhere continuous, and the force of the other field wholly polar.

Electrification has no magnetic analogue, the magnetic induction being always continuous. It is important that the distinction between electrification, as above considered, and imaginary free electricity (as when the force is discontinuous, though not the displacement) should be clearly recognised. The latter is the analogue of free magnetism.

Similarly, there is no electrification in a conductor supporting a current, provided it be not dielectric as well. In the latter case it is the true current, the sum of the conduction and displacement current, that is continuous, making them sometimes separately discontinuous,

thus - div C = + div i) = /J; div V = 0 ;

C being the conduction, D the displacement current, ami their sum F. It is usually only at the surface of a conductor that there is electrification ; should, however, the specific capacity of the conducting matter itself be variable from place to place, there will generally be interior electrification as well, during the existence of a conduction current.

SECTION X. DIELECTRIC DISPLACEMENT AND ABSORPTION.

The most remarkable and distinguishing feature of Maxwell’s theory of electromagnetism is his dielectric current, whose introduction into the theory gives us a dynamically complete system, with propagation of disturbances in time through the medium surrounding and between conductors, doing away with the mathematically expressible but practically unimaginable, instantaneous actions of currents upon one another at a distance, and similar, though more simply expressible, instantaneous forces between imaginary accumulations of the “ electric fluid.'*' Some might say that the distinguishing feature is his dielectric displacement. But it is scarcely that, strict^. For that a dielectric medium was put into a state of polarisation by electric force was Faraday’s idea, and this polarisation is only another name for electric displacement. Again, Sir YY . Thomson had given mathematical expression to the idea of polarisation in a dielectric; the statement D==f‘E/’47r, expressing the displacement or polarisation in terms of the electric force, had been given in a somewhat similar form by him. Whether we call it displacement or polarisation does not matter; the important step made by Maxwell was the recognition that changes in the displacement constitute a real electric current (though without dissipation of energy in Joule-heating), and that the electric current, whether conductive or not, is always continuous. Accumulations are done away with altogether, and with their abolition the fluid or fluids become meaningless.

As regards the term displacement, though it may be objected to as misleading, suggesting a real displacement in a certain direction, whereas the phenomenon, though undoubtedly having some directional peculiarity, is probabl}' not of the nature of a simple displacement, just as the conductive current is not the motion of a fluid through a wire, yet on the other hand, it is to be remarked that the term current is firmly fixed in use, and, to accompany it, there could be no better term than displacement. The time-integral of the current, from the zero configuration, in a dielectric, is the displacement; converse!}7, the current is the tiine-variation of the displacement. It would therefore be a pity to abolish the term displacement, unless we simultaneously abolish current ; and it would be hard to find two words which fit together so well.

If we define the displacement as the time-integral of the current anywhere, not merely in a non-conducting dielectric, the current is the „i me-variation of the displacement anywhere. The displacement, reckoned from a proper zero at a certain time (any time when there was no dielectric displacement) is then, like Maxwell's true current, a vector magnitude of no convergence at every moment and everywhere. In some special investigations this is useful. We may then speak of conductive displacement and elastic displacement, but I believe in general it is best to confine the term displacement to elastic displacement (in a dielectric) 011I3', unless we are careful to qualify the word by a preliminary adjective. In the absence of any qualification displacement in a dielectric is meant, and, should it be also conducting, the conductive displacement must be separately reckoned.

The difference between Maxwell’s and older views regarding electricity and the electric current is instructively brought into prominence by making a small change in Maxwell's system, or rather 111 taking an imaginably possible, though really untrue, special case of the same. Put c — 0 everywhere. That is, stop all elastic displacement. Make 110 other change. The electric current is left continuous, and is now neccesarily confined to conductors only, and at their surfaces must always be tangential. But there will still be magnetic force and induction, mutual forces between circuits, induction of one circuit on another, and there will still be transfer of energy through the medium from sources to sinks of energy. We have simply done away with the “ elastic yielding of the connecting mechanism.” The velocity of propagation of waves, as the specific capacity is imagined to be reduced to nothing, becomes infinitely great, instead of being only that of light. The theory of the induction of linear circuits becomes that given by Maxwell, wherein dielectric displacement is ignored. No effect is produced upon the distribution of steady current set up by steady impressed forces, but in the first transient states there are great changes. In a linear circuit, for example, the current is now absolutely constrained to keep iu the conductor, so that there can be 110 surface charges or static retardation. Though there is 110 electrification, there is still electric potential. But we cannot charge a condenser. Therein lies the difference from reality. To be able to do this, without employing Maxwell’s dielectric current, we are necessitated to suppose that the current is not continuous, but that it is the real motion of something that can accumulate in places. We are also obliged to change the relation between magnetic force andcurrent, as it implies continuity of the current everywhere. Many other changes are also required to make a consistent system, for one change necessitates another, and we shall ultimately come to something extremely different from Maxwell’s system.

In view of the extreme relative simplicity of Maxwell’s views, and their completeness without any artificial contrivances to save appearances, and in their modernness, referring to modern views regarding action at a distance, one is almost constrained to believe that the dielectric current, the really essential part of Maxwell’s theory, is not merely an invention but a reality, and that Maxwell’s theory, or something very like it, is the theory of electricity, all others being makeshifts, and that it is the basis upon which all future additions will have to rest, if they are to have any claims to permanency.

Electric displacement is primarily a phenomenon of the ether. Ether is perfectly elastic. It must be so if there be 110 absorption of the energy of radiation during transmission through space. This conclusion is of course independent of Maxwell’s view of light being itself an electromagnetic disturbance. If the energy of displacement be potential energy, the displacement, whatever it really be, is of a perfectly elastic character, in the absence of ordinary matter.

But when electric displacement occurs in a solid dielectric, it there be, as there must be, mutual influence between the ether and matter, we may expect the elastic properties of the matter to be communicated, apparently, to the ether. Thus, no solid is perfectly elastic, and, consequently, electric displacement in a solid dielectiic is not perfectly elastic, as it is assumed to be in the formula D = cE 4-, with c invariable at any place, the linear relation between the displacement and the electric force.

A solid, under the influence of externally-applied force, is strained. If the strain be under a certain magnitude it is assumed to be perfectly elastic, so that whilst the stress remains steady the strain does not change; and on the removal of all stress the strain entirely ceases. But in reality, if a certain applied force deform an elastic solid, and the force be kept 011, the deformation slowly increases, as if the elasticity slowly decreased. And on the removal of the applied force the original zero configuration is not immediately recovered. The residual deformation will then slowly subside. If, after the first approximate return to the original state, the body be, by constraint, prevented from changing shape, the solid, which at first did not react against the constraint, will gradually do so, so that on the removal of the constraint, a second return, in a lump, towards the original configuration, will take place.

Bemembering that a dielectric under electric stress is in a state of strain, we may expect there to be a corresponding electrical phenomenon. The displacement produced by a given constant impressed force should be, at first, appreciably quite elastic, but should thereafter slowly increase. On the removal of the impressed force the original displacement should at once subside, leaving a small residual displacement which should subside very slowly, as it came on. Or, if' the residual displacement be fixed, so that it cannot subside, it should appear to gi adually come back into existence, as elastic displacement referred to the original zero, so that on the removal of the constraint a second sudden subsidence should take place.

This is what happens in the phenomenon of electric absorption in a perfectly insulating dielectric. Whether it be really true or not that a part of the electric displacement becomes intrinsic by reason of some 5w«si-rigid connection with a similar phenomenon taking place in the strained solid, there is 110 doubt that there is a remarkable resemblance in the details of the two cases. That the presence of the matter causes the displacement to be increased from what it would be with the same electric force in ether is a separate matter, as this refers to elastic displacement (c >1 in all dielectrics, if unity in vacuo).

An elastic spring is therefore the most correct analogue to a condenser when we wish to make up a mechanical illustration of the electro-elastic properties of a solid dielectric. We may, for instance, take a flat spring, clamp it firmly at one end, and apply pressure or pull to the free end in a direction perpendicular to its flat sides. Consider the applied force to represent the E.M.F. of a battery joined to a condenser, and the displacement of the free end of the spring to represent the electric displacement in the condenser. The displacement will be proportional to the force, in both eases, approximately; and if by any suitable means we magnify, mechanically or optically, the motions of the spring under varied circumstances, we shall see a corresponding set of phenomena to those occurring under similarly varied circumstances in the case of the electric displacement.

But an illustration, which, though less exact, is more easily followed by the mind's eye, when we cannot render visible the absorption properties of a spring, is that which occurred to me when first making acquaintance with Maxwells mechanical illustration, the parallel vertical tubes, containing water. This being rather complex, I substituted the following :—Electric displacement 111 a condenser is represented by the actual displacement of a piston in a cylinder from its position of natural equilibrium in the middle thereof, when the cylinder is perfectly airtight and contains an equal amount of air 011 each side. To give motion to the piston, rods may be attached to it, passing through holes in the closed ends of the cylinder. Call the ends of these rods, outside the cylinder, a and b, and the two corresponding air spaces in the cylinder A and B, a and A being to the left, b and B to the right. If pressure be applied to a or a pull to b, or both, the piston will be displaced from left to right, and small displacements will be proportional to the corresponding applied forces.

The displacement of the piston corresponds to the total electric displacement in a condenser; the applied force to the E.M.F, of the battery 011 the condenser; the back pressure of the rod u to the difference of potential of the condenser plates ; the displacement of a inward to the positive charge, and of b outward to the equal negative charge. Insulation of the condenser is represented by fixing a or b so that the displacement cannot change.

If we like to carry the illustration further, we may cause the rods a and b to meet with frictional resistance when moving, proportional to the speed of their motion. This speed will correspond to the strength of current, and the coefficient of friction to the resistance of the conductor joining the poles of the condenser. We may go further, and suppose the mass of the piston and rods to represent the inductance of the electric circuit, thus obtaining an illustration of the oscillatory discharge which occurs with suitable values of the resistance, capacity and inductance. Although this analogy, which is well known in one form or another, is very close, and therefore educationally valuable, it should be remembered that it suggests that the momentum of an electric current is that of matter moving with the current, or of the current itself, if it be the motion of matter, having therefore necessarily momentum. So far it is apt to mislead ; for electricity has no momentum itself, or kinetic energy. The momentum is that of the magnetic induction, or is proportional thereto, and it, and the (nominal) energy of the current, exist wherever there is magnetic induction, not merely in the wire.

To imitate absorption, make the piston very slightly leaky. Then if a be pushed in by a steady pressure, the first displacement of the piston is elastic with reference to the proper zero, the middle of the cylinder. But air then leaks slowly from B to A 011 account of the increased pressure in B, which causes the back pressure of a to decrease, and allows the same applied force to slowly increase the displacement. (This corresponds to the slow continuous increase of electric displacement in a condenser when a constant battery is kept on). Or, if a be fixed, the back pressure will slowly fall whilst the displacement remains constant. (If the condenser be insulated, its difference of potential will fall whilst its charge remains constant.) Unfix a; the first return equals the first displacement, approximately, but there is left a small displacement which will slowly subside if the piston-rods be free; or, if a be fixed, a pressure at a will be gradually brought on by the slow leakage from A to B, so that a second sudden motion of the piston back to its proper zero can be got on unfixing a. (Remove the battery, but close the circuit still; the first discharge will approximately equal the first charge, but there is left a small charge which will slowly subside if the circuit remain closed; or, if it be insulated, a difference of potential of the same kind as before will gradually come on, so that a second discharge can be got on again closing the circuit.)

To instantly remove the displacement of the piston after leakage has occurred, we must apply an opposite force, a pressure at b or pull at n of the right amount to bring the piston to the middle. If we then fix a and leave the thing to itself the pressure of the rod at b will gradually subside. Similarly if we, after absorption has occurred in a condenser, and the first discharge has been taken, charge it oppositely by a reverse E.M.F. of the right amount to make the real displacement zero, and leave the condenser insulated, the apparent opposite charge will gradually disappear.

Since the first discharge equals the first charge we may regard the capacity of the condenser as being constant, and the displacement at any moment to consist of two parts, first that due to the battery E.M.F., which can be got rid of at any moment, and next a temporary intrinsic displacement which is kept up b}7 impressed electric force in the dielectric itself arising from its altered structure, or the changed zero of its elastic deformation. The displacement due to a certain total E.M.F. in the circuit is the same however it be distributed, whether in the conductor or dielectric, if, over any cross-section, it be evenly distributed. But only the part in the conductor causes difference of potential in a steady state between the ends of the conductor, so that the intrinsic displacement due to internal impressed force gives no external sign of its existence until, by the removal of the impressed force, difference of potential is developed between the terminals of the condenser.

In the case of the piston in the cylinder, the intrinsic force keeping up displacement of the piston after the rods have been set free is the pressure of air in A against the piston from left to right, which, had it not been for the leakage, would have been in B, and have pressed the piston the other way.

Another effect may be mentioned. If the dielectric of the condenser be of a such a nature that its capacity increases or decreases with the temperature, then, on suddenly charging it, there will be a cooling or a heating effect produced in the dielectric. This has also its parallels in the metal spring and “ spring of the air ” illustrations.

Intrinsic displacement in a dielectric, without conducting matter surrounding it, to render the displacement wholly latent., has also some interesting features, which will be considered later.