On the Transfer of Energy in the Electromagnetic Field

On the Transfer of Energy in the Electromagnetic Field

By J. H. Poynting, M.A., late Fellow of Trinity College, Cambridge, Professor of Physics, Mason College, Birmingham. Communicated by Lord Rayleigh, M.A., D.C.L., F.R.S. Received December 17, 1883 — Read January 10, 1884. Philosophical Transactions of the Royal Society of London, volume 175, pages 343 to 361.

Editorial note: the paper is carried here in Poynting's own words. His displayed equations are given as named results in words — the symbols and their definitions are his — and the six figures, which are line drawings of equipotential surfaces, are described. The complete original, algebra and figures included, is in the linked transcription of the scanned volume.

Introduction

A space containing electric currents may be regarded as a field where energy is transformed at certain points into the electric and magnetic kinds by means of batteries, dynamos, thermoelectric actions, and so on, while in other parts of the field this energy is again transformed into heat, work done by electromagnetic forces, or any form of energy yielded by currents. Formerly a current was regarded as something travelling along a conductor, attention being chiefly directed to the conductor, and the energy which appeared at any part of the circuit, if considered at all, was supposed to be conveyed thither through the conductor by the current. But the existence of induced currents and of electromagnetic actions at a distance from a primary circuit from which they draw their energy has led us, under the guidance of Faraday and Maxwell, to look upon the medium surrounding the conductor as playing a very important part in the development of the phenomena. If we believe in the continuity of the motion of energy — that is, if we believe that when it disappears at one point and reappears at another it must have passed through the intervening space — we are forced to conclude that the surrounding medium contains at least a part of the energy, and that it is capable of transferring it from point to point.

Upon this basis Maxwell has investigated what energy is contained in the medium, and he has given expressions which assign to each part of the field a quantity of energy depending on the electromotive and magnetic intensities and on the nature of the matter at that part in regard to its specific inductive capacity and magnetic permeability. These expressions account, as far as we know, for the whole energy. According to Maxwell's theory, currents consist essentially in a certain distribution of energy in and around a conductor, accompanied by transformation and consequent movement of energy through the field.

Starting with Maxwell's theory, we are naturally led to consider the problem: how does the energy about an electric current pass from point to point — that is, by what paths and according to what law does it travel from the part of the circuit where it is first recognisable as electric and magnetic to the parts where it is changed into heat or other forms?

The aim of this paper is to prove that there is a general law for the transfer of energy, according to which it moves at any point perpendicularly to the plane containing the lines of electric force and magnetic force, and that the amount crossing unit of area per second of this plane is equal to the product of the intensities of the two forces, multiplied by the sine of the angle between them divided by four pi, while the direction of flow of energy is that in which a right-handed screw would move if turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity. After the investigation of the general law several applications will be given to show how the energy moves in the neighbourhood of various current-bearing circuits.

The general account of the method

Denote the electromotive intensity at a point — that is, the force per unit of positive electrification which would act upon a small charged body placed at the point — by E, and the specific inductive capacity of the medium at that point by K; denote the magnetic intensity — the force per unit pole which would act on a small north-seeking pole placed at the point — by H, and the magnetic permeability by mu. Maxwell's expression for the electric and magnetic energies per unit volume of the field is then the sum of two terms: the specific inductive capacity multiplied by the square of the electromotive intensity and divided by eight pi, added to the permeability multiplied by the square of the magnetic intensity and divided by eight pi. If any change is going on in the supply or distribution of energy, the change in this quantity per second is the corresponding sum of the two products of each intensity with its own rate of change, each divided by four pi rather than eight.

According to Maxwell the true electric current is in general made up of two parts: the conduction current, and a part due to change of electric displacement in the dielectric, this latter being called the displacement current. Now the displacement is proportional to the electromotive intensity, being that intensity multiplied by the specific inductive capacity and divided by four pi; so when change of displacement takes place, due to change in the electromotive intensity, the rate of change — that is, the displacement current — is the specific inductive capacity multiplied by the rate of change of the electromotive intensity and divided by four pi, and this is equal to the difference between the true current and the conduction current.

Multiply that difference by the electromotive intensity. The first term of the change per second in the energy then becomes the electromotive intensity multiplied by the true current, less the electromotive intensity multiplied by the conduction current. The first of these may be transformed by substituting for the components of the total current their values in terms of the components of the magnetic intensity; the second, by Ohm's law — which states that the conduction current is the specific conductivity multiplied by the electromotive intensity — becomes the square of the conduction current divided by the conductivity, and this is the energy appearing as heat in the circuit per unit volume according to Joule's law.

If we sum up that quantity for the whole space within a closed surface, the integral of the first term can be integrated by parts, and we find that it consists of two terms: one an expression depending on the surface alone, to which each part of the surface contributes a share depending on the values of the electromotive and magnetic intensities at that part; the other being the change per second in the magnetic energy, with a negative sign. The integral of the second term is the total amount of heat developed in the conductors within the surface per second. We have then the following result.

The change per second in the electric energy within a surface is equal to a quantity depending on the surface, less the change per second in the magnetic energy, less the heat developed in the circuit. Or, rearranging: the change per second in the sum of the electric and magnetic energies within a surface, together with the heat developed by currents, is equal to a quantity to which each element of the surface contributes a share depending on the values of the electric and magnetic intensities at the element. That is, the total change in the energy is accounted for by supposing that the energy passes in through the surface according to the law given by this expression.

On interpreting the expression it is found that it implies that the energy flows as stated before — perpendicularly to the plane containing the lines of electric and magnetic force — and that the amount crossing unit area per second of this plane is equal to the electromotive intensity multiplied by the magnetic intensity multiplied by the sine of the included angle, all divided by four pi, while the direction of flow is given by the three quantities — electromotive intensity, magnetic intensity, flow of energy — being in right-handed order.

It follows at once that the energy flows perpendicularly to the lines of electric force, and so along the equipotential surfaces where these exist. It also flows perpendicularly to the lines of magnetic force, and so along the magnetic equipotential surfaces where these exist. If both sets of surfaces exist, their lines of intersection are the lines of flow of energy.

The proof

The energy of the field may be expressed, following Maxwell's Electricity and Magnetism, volume ii, second edition, page 253, as the sum of two volume integrals — the electrostatic energy, written with the displacement components, and the electromagnetic energy, written with the induction and magnetic force components. Substituting the relations between displacement and electromotive force, and between induction and magnetic force, turns these into the specific inductive capacity multiplied by the integral of the square of the electromotive force over the volume and divided by eight pi, added to the permeability multiplied by the integral of the square of the magnetic force and divided by eight pi.

Consider the space within any fixed closed surface. If changes are taking place, the rate of increase of energy of the electric and magnetic kinds per second is the corresponding pair of integrals, each intensity multiplied by its own rate of change, and each divided by four pi.

Now Maxwell's equations for the components of the true current give each component as the sum of a conduction-current component and the rate of change of a displacement component; and each displacement rate may be replaced by the specific inductive capacity multiplied by the rate of change of the corresponding electromotive-force component and divided by four pi. Substituting into the first of the two integrals gives the integral of the electromotive force multiplied by the true current, less the integral of the electromotive force multiplied by the conduction current.

The equations for the components of electromotive force — Maxwell, volume ii, page 222 — write each component as a part depending on the velocities of the moving matter, together with a part not containing the velocities, which comes from the rates of change of the vector potential components and from the gradient of the scalar function. Sorting the products accordingly, the electromotive force multiplied by the true current becomes the work done per second by the components of the electromagnetic force per unit of volume, taken with the reverse sign, together with the products of the velocity-free parts of the electromotive force with the current components.

Putting for the current components their values in terms of the magnetic force — Maxwell, volume ii, page 233 — and transposing, the remaining volume integral may be integrated by parts term by term. It splits into a surface integral, with the direction cosines of the outward normal appearing, and a further volume integral. The identities satisfied by the velocity-free parts of the electromotive force — their curl being the rate of change of the magnetic induction, Maxwell, volume ii, page 216 — turn that further volume integral into the rate of change of the magnetic energy with a negative sign, and it is transposed to the other side.

What stands at the end is this. The gain per second in the electric energy within the closed surface, added to the gain per second in the magnetic energy, added to the work done per second by the electromagnetic forces — that is, the energy transformed by the motion of the matter in which the currents exist — added to the energy the conductors transform into heat, chemical energy and so on, is equal to a single integral over the bounding surface. The left side is the total gain in energy per second within the surface, and the equation asserts that this energy comes through the bounding surface, each element contributing the amount expressed by the right side.

This may be put in another form. Writing the velocity-free electromotive intensity and the magnetic intensity as two vectors with an angle between them, the surface integral becomes their product with the sine of that angle, divided by four pi, and resolved along the normal to the element. An element drawn to coincide with the plane containing the two intensities therefore contributes the greatest amount of energy to the space; in other words, the energy flows perpendicularly to the plane containing them, at the rate per unit area already named. Taking the electromotive intensity along one axis and the magnetic intensity along a second, and the normal along the third, shows that the energy moves in the direction in which a screw would move if its head were turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity. If the surface be taken where the matter has no velocity, the velocity-free intensity becomes the full electromotive intensity, and the amount of energy crossing unit area perpendicular to the flow per second is the electromotive intensity multiplied by the magnetic intensity multiplied by the sine of the included angle, divided by four pi.

Since the surface may be drawn anywhere we please, then wherever there is both magnetic and electromotive intensity there is flow of energy. Since the energy flows perpendicularly to the plane containing the two intensities, it must flow along the electric and magnetic level surfaces when these exist, so that the lines of flow are the intersections of the two surfaces.

Applications of the law of transfer of energy

(1) A straight wire conveying a current

In this case very near the wire, and within it, the lines of magnetic force are circles round the axis of the wire. The lines of electric force are along the wire, if we take it as proved that the flow across equal areas of the cross section is the same at all parts of the section. If the current runs from one end of the wire to the other, then a tangent plane to the surface at any point contains the directions of both the electromotive and the magnetic intensity, and energy is therefore flowing in perpendicularly through the surface — that is, along the radius towards the axis. Take a portion of the wire bounded by two plane sections perpendicular to the axis. Across the ends no energy is flowing, for they contain no component of the electromotive intensity. The whole of the energy then enters in through the external surface of the wire, and by the general theorem the amount entering in must just account for the heat developed owing to the resistance, since if the current is steady there is no other alteration of energy.

It is worth showing independently that the energy moving in will just account for the heat developed. Let the wire have a given radius and carry a given current, with a stated magnetic intensity at its surface, a stated electromotive intensity within it, and a stated difference of potential between the two ends. The energy entering from the outside per second is the area of the length considered, multiplied by the electromotive intensity and by the magnetic intensity, divided by four pi. The line integral of the magnetic intensity round the wire is four pi multiplied by the current through it, and the electromotive intensity multiplied by the length is the difference of potential. The expression therefore reduces to the current multiplied by the difference of potential; and by Ohm's law that is the current squared multiplied by the resistance, which is the heat developed according to Joule's law.

It seems then that none of the energy of a current travels along the wire, but that it comes in from the non-conducting medium surrounding the wire; that as soon as it enters it begins to be transformed into heat, the amount crossing successive layers of the wire decreasing till by the time the centre is reached — where there is no magnetic force, and therefore no energy passing — it has all been transformed into heat. A conduction current then may be said to consist of this inward flow of energy with its accompanying magnetic and electromotive forces, and the transformation of the energy into heat within the conductor.

We have now to inquire how the energy travels through the medium on its way to the wire.

(2) Discharge of a condenser through a wire

Consider first the slow discharge of a simple condenser consisting of two charged parallel plates connected by a wire of very great resistance, since in this case we can form an approximate idea of the actual path of the energy.

Let the two plates be charged positively and negatively. Before discharge the sections of the equipotential surfaces run between and around them; the chief part of the energy resides in the part of the dielectric between the two plates, but there will be some energy wherever there is electromotive intensity, and between the plates the electromotive intensity runs from the positive to the negative plate, everywhere perpendicular to the level surfaces. Now connect the plates by a fine wire of very great resistance, following a line of force, with the resistance so adjusted that it is the same for the same fall of potential throughout — the arrangement is chosen so that the level surfaces shall not be disturbed by the flow of the current — and the wire so fine that the discharge takes place very slowly.

While the discharge goes on a current flows round the wire, and there is also an equal displacement current from the negative to the positive plate due to the yielding of the displacement there. The current will be encircled by lines of magnetic force forming closed curves about the circuit, running one way round the wire and the other way round the space between the plates. The electromotive intensity is always from the higher level surfaces to the lower, both near the wire and in the space between the plates.

Since the energy always moves perpendicularly to the lines of electromotive intensity it must travel along the equipotential surfaces; and since it also moves perpendicularly to the lines of magnetic intensity it moves, as in case (1), inwards on all sides to the wire, where it is all converted into heat. But between the plates the electromotive intensity is opposed to the current, while the magnetic intensity bears the same relation to the current as in the wire; and remembering that the electromotive intensity, the magnetic intensity and the direction of flow of energy are connected by the right-handed screw relation, we see that the energy moves outwards from the space between the plates. As the strain of the dielectric between them is gradually released by what we call a discharge current along the wire, the energy thus given up travels outwards through the dielectric, following always the equipotential surfaces, and gradually converges once more on the circuit where the surfaces are cut by the wire. There the energy is transformed into heat. If the current may be considered steady, the energy moves along at the same level throughout.

(3) A circuit containing a voltaic cell

When a circuit contains a voltaic cell we do not know with certainty what is the distribution of potential, but most probably it is as follows. Suppose we have a simple copper, zinc and acid cell producing a steady current. There is probably a considerable sudden rise in passing from the zinc to the acid — the place where the chemical energy is given up — a fall through the acid depending on the resistance, a sudden fall on passing from the acid to the copper, where some energy is absorbed with evolution of hydrogen, and then a gradual fall through the wire of the circuit round to the zinc again. The equipotential surfaces will then all start from where the acid comes in contact with the zinc: some of the highest potential passing through the acid, others passing between the acid and copper and crowding in there, the rest lower than these cutting the circuit at right angles at intervals representing equal falls of potential.

If this be the actual arrangement, then the current, which travels round the circuit from zinc through acid to copper, is in opposition to the electromotive intensity between the zinc and the acid, while the magnetic intensity is related to the current in the ordinary way. The energy will therefore pass outwards from there along the level surfaces. In fact the medium between the zinc and the acid behaves like the medium between the plates of the condenser in case (2), and it seems possible that the chemical action produces continually fresh electric displacement from acid towards zinc which yields as rapidly as it is formed, the energy of the displacement moving out sideways.

Some of this energy, travelling along the highest level surfaces, will converge on the acid and there be, at any rate ultimately, converted into heat. Some of it will move along those surfaces which crowd in between the acid and the copper, and there converge to supply the energy taken up by the escaping hydrogen. The rest spreads out to converge at last at different parts of the circuit, and there to be transformed into heat according to Joule's law.

It may be noticed that if the level surfaces be drawn with equal differences of potential, equal amounts of energy travel out per second between successive pairs of surfaces. For the amount transformed in the circuit in a length having a given difference of potential between its ends will be that difference multiplied by the current; and since the current and the field are steady, the energy transformed will be equal to the energy moving out from the cell between the same surfaces — the energy never crossing level surfaces.

This result has a consequence which, though already well known, is worth mentioning. Let the difference of potential between the zinc and the acid be the first quantity, and that between the acid and the copper the second. The first multiplied by the current is the total energy travelling out per second from the zinc surface; of this, the second multiplied by the current is absorbed at the copper surface, and the remainder is transformed in the circuit. The fraction of the whole energy sent out which is transformed in the circuit is therefore the difference of the two potential differences divided by the first of them — a result analogous to the expression for the amount of heat which can be transformed into work in a reversible heat-engine.

Two illustrations may be mentioned here. Suppose that we are sending a current through a submarine cable by a battery with, say, the zinc to earth, and suppose the sheath is everywhere at zero potential. Then the wire will everywhere be at higher potential than the sheath, and the level surfaces will pass from the battery through the insulating material to the points where they cut the wire. The energy then which maintains the current, and which works the needle at the further end, travels through the insulating material, the core serving as a means to allow the energy to get in motion. Again, when the only effect in a circuit is the generation of heat, we have energy moving in upon the wire, there undergoing some sort of transformation, and then moving out again as heat or light. If Maxwell's theory of light be true, it moves out again still as electric and magnetic energy, but with a definite velocity and intermittent in type. We have in the electric light, for instance, the curious result that energy moves in upon the arc or filament from the surrounding medium, there to be converted into a form which is sent out again, and which, though still the same in kind, is now able to affect our senses.

(4) Thermoelectric circuits

Take first a circuit composed of two metals, neither of which has any Thomson effect, and suppose the current at the hot junction flows from the first metal to the second. According to Professor Tait's theory it would appear that the electromotive intensity at the hot junction is to that at the cold as the absolute temperature at the hot junction is to that at the cold. If the current is steady there is probably a sudden rise in potential at the hot junction, a gradual fall along the second metal, a sudden fall at the cold junction — less, however, than the rise at the other — and a gradual fall along the first. The level surfaces will then all start from the hot junction, the higher ones cutting the circuit at successive points along the second metal, several converging at the cold junction, and the rest cutting the circuit at successive points along the first. The heat at the hot junction is converted into electric and magnetic energy, which here moves outwards, since the current is against the electromotive intensity. Some of this energy converges upon the two metals, to be converted into heat according to Joule's law, and some on the cold junction, there producing the Peltier heating effect.

Now suppose a circuit of the same two metals, all at the same temperature, but with a battery interposed which sends a current in the same direction as before. We know that the current will tend to cool the junction which was hot and to heat the one which was cold. The heat developed in the parts of the circuit near the first junction will thus be partly supplied from that junction, where the current is against the electromotive intensity; the energy therefore moves out thence, giving a cooling effect.

The Thomson effect may be considered in somewhat the same way. Suppose a metal of the iron type, with temperature falling along it, forms part of a circuit between two neutral metals of the lead type, and suppose these are each at the neutral temperatures with respect to it, so that there is no electromotive intensity at the junction. If we drive a current through by means of some external electromotive intensity elsewhere in the circuit, the potential will tend to fall along the way the current goes. But a current in iron from hot to cold cools the metal — that is, the electromotive intensity appears to be in opposition to the current, so that the energy moves outwards. The potential therefore tends to rise along the unequally heated piece, and actually will do so if its resistance is negligible compared with that of the rest of the circuit. The energy moving outwards will come into the circuit again at the parts near the junctions, where it will be transformed once more into heat. If the resistance of the heated piece be gradually increased, the fall of potential according to Ohm's law will tend to lessen the rise, and fewer surfaces will cut it. It would seem possible so to adjust matters that the two exactly neutralised each other, so that no energy either entered or left it; in that case we should only have lines of magnetic force round it, and no other characteristic of a current in that part of the circuit.

If this is the true account of the Thomson effect, it would appear that it should be described not as an absorption of heat or development of heat by the current, but rather as a movement of energy outwards or inwards, according as the electromotive intensity in the unequally heated metal opposes or agrees with the direction of the current.

(5) A circuit containing a motor

This case closely resembles the third, the motor playing a part analogous to that of the surface of contact of the acid with the copper. Suppose for simplicity that the motor has no internal resistance. When it has no velocity all the level surfaces cut the circuit, and the energy leaving the dynamo or battery is all transformed into heat due to resistance. But if the motor is being worked the current diminishes, the level surfaces begin to converge on the motor, and fewer cut the circuit. Some of the energy therefore passes into the motor, and is there transformed into work. As the velocity increases the number cutting the rest of the circuit decreases, for the current diminishes and therefore, by Ohm's law, the fall of potential along the circuit is less; and ultimately, when the velocity of the motor becomes very great, the current becomes very small. In the limit no level surface cuts the circuit, all converging on the motor. That is, all the energy passes into the motor, where it is transformed into work, and the efficiency of the arrangement is perfect, though the rate of doing work is infinitely slow.

(6) Induced currents

It is not so easy to form a mental picture of the movement of energy which takes place when the field is changing and induced currents are created. But we can see in a general way how these currents are accounted for. When there is a steady current in a field there is corresponding to it a definite distribution of energy. If there is a secondary circuit present, so long as the primary current is constant there is no electromotive intensity in the secondary circuit, for it is all at the same potential. The energy neither moves into nor out of it, but streams round it somewhat as a current of liquid would stream round a solid obstacle. But if the primary current changes there is a redistribution of the energy in the field. While this takes place there will be a temporary electromotive intensity set up in the conducting matter of the secondary circuit, energy will move through it, and some of the energy will there be transformed into heat or work — that is, a current will be induced in the secondary circuit.

(7) The electromagnetic theory of light

The velocity of plane waves of polarised light on the electromagnetic theory may be deduced from the consideration of the flow of energy. If the waves pass on unchanged in form with uniform velocity, the energy in any part of the system due to the disturbance also passes on unchanged in amount with the same velocity. If this velocity be some quantity, then the energy contained in a unit volume of cubical form with one face in a wave front will all pass out through that face in one over that velocity of a second. Suppose the direction of propagation is straightforward while the displacements are up and down; then the magnetic intensity will be right and left. The energy within the cube is the usual sum of the electric and magnetic energy densities, and the rate at which energy crosses the face in the wave front is the product of the two intensities divided by four pi. Equating the one to the other, divided by the velocity, is the first relation.

Taking a face of the cube perpendicular to the direction of displacement, and therefore containing the magnetic intensity, the line integral of the magnetic intensity round this face is four pi multiplied by the current through the face; that current, being an alteration of displacement, is the specific inductive capacity multiplied by the rate of change of the electromotive intensity and divided by four pi. Since the displacement is propagated unchanged with the velocity, the rate of change in time is the velocity multiplied by the rate of change in space, with the sign reversed. Combining these gives the magnetic intensity as the specific inductive capacity multiplied by the velocity and by the electromotive intensity; and taking the line integral of the electromotive intensity round a face perpendicular to the magnetic intensity, and equating it to the decrease of magnetic induction through that face, gives similarly the electromotive intensity as the permeability multiplied by the velocity and by the magnetic intensity.

The product of these two relations at once gives the value of the velocity, for dividing out the two intensities leaves unity equal to the permeability multiplied by the specific inductive capacity and by the square of the velocity — that is, the velocity is one divided by the square root of the permeability multiplied by the specific inductive capacity. Using either relation alone in the energy equation gives the same result. It also gives the magnetic energy equal to the electric energy.

It may be noted that this velocity is the greatest with which the two energies can be propagated together, and that they must be equal when travelling with this velocity. For if the velocity of propagation and the angle between the two intensities are left general, the velocity is twice the sine of that angle divided by the sum of two terms, one carrying the specific inductive capacity and the ratio of the electromotive to the magnetic intensity, the other the permeability and the reciprocal ratio. The greatest value of the numerator is two, when the angle is a right angle, and the least value of the denominator is twice the square root of the permeability multiplied by the specific inductive capacity, when the two terms are equal to each other. The maximum value of the velocity therefore occurs when the intensities are at right angles and the two energy densities are equal.

Concluding remarks

The preceding examples will suffice to show that it is easy to arrange some of the known experimental facts in accordance with the general law of the flow of energy. I am not sure that there has hitherto been any distinct theory of the way in which the energy developed in various parts of the circuit has found its way thither, but there is, I believe, a prevailing and somewhat vague opinion that in some way it has been carried along the conductor by the current. Probably Maxwell's use of the term "displacement" to describe one of the factors of the electric energy of the medium has tended to support this notion. It is very difficult to keep clearly in mind that this "displacement" is, as far as we are yet warranted in describing it, merely a something with direction which has some of the properties of an actual displacement in incompressible fluids or solids. When we learn that the "displacement" in a conductor having a current in it increases continually with the time, it is almost impossible to avoid picturing something moving along the conductor, and it then seems only natural to endow this something with energy-carrying power. Of course it may turn out that there is an actual displacement along the lines of electromotive intensity. But it is quite as likely that the electric "displacement" is only a function of the true displacement, and it is conceivable that many theories may be formed in which this is the case, while they may all account for the observed facts. Mr. Glazebrook has already worked out one such theory. It seems to me then that our use of the term is somewhat unfortunate, as suggesting to our minds so much that is unverified or false, while it is so difficult to bear in mind how little it really means.

I have therefore given several cases in considerable detail of the application of the mode of transfer of energy in current-bearing circuits according to the law given above, as I think it is necessary that we should realise thoroughly that if we accept Maxwell's theory of energy residing in the medium, we must no longer consider a current as something conveying energy along the conductor. A current in a conductor is rather to be regarded as consisting essentially of a convergence of electric and magnetic energy from the medium upon the conductor and its transformation there into other forms. The current through a seat of so-called electromotive force consists essentially of a divergence of energy from the conductor into the medium. The magnetic lines of force are related to the circuit in the same way throughout, while the lines of electric force are in opposite directions in the two parts of the circuit — with the so-called current in the conductor, against it in the seat of electromotive force. It follows that the total electromotive intensity round the circuit with a steady current is zero, or the work done in carrying a unit of positive electricity round the circuit with the current is zero. For work is required to move it against the electromotive intensity in the seat of energy, this work sending energy out into the medium, while an equal amount of energy comes in in the rest of the circuit where it is moving with the electromotive intensity. This mode of regarding the relations of the various parts of the circuit is, I am aware, very different from that usually given, but it seems to me to give us a better account of the known facts.

It may seem at first sight that we ought to have new experimental indications of this sort of movement of energy, if it really takes place. We should look for proofs at points where the energy is transformed into other modifications, that is, in conductors. Now in a conductor, when the field is in a steady state, there is no electromotive intensity, and therefore no motion and no transformation of energy. The energy merely streams round the outside of the conductor, if in motion at all in its neighbourhood. If the field is changing, energy can pass into the conductor, as there may be temporary electromotive intensity set up within it, and there will be transformation. But we already know the nature of this transformation, for it constitutes the induced current. Indeed, the fundamental equation describing the motion of energy is only a deduction from Maxwell's equations, which are formed so as to express the experimental facts as far as yet known. Among these are the laws of induction in secondary circuits, and they must therefore agree with the law of transfer. We can hardly hope, then, for any further proof of the law beyond its agreement with the experiments already known, until some method is discovered of testing what goes on in the dielectric independently of the secondary circuit.