A Treatise on Electricity and Magnetism/Part IV/Chapter VI

by James Clerk Maxwell

CHAPTER VI.

DYNAMICAL THEORY OF ELECTROMAGNETISM.

568.] We have shewn, in Art. 552, that, when an electric current exists in a conducting circuit, it has a capacity for doing a certain amount of mechanical work, and this independently of any external electromotive force maintaining the current. Now capacity for performing work is nothing else than energy, in whatever way it arises, and all energy is the same in kind, however it may differ in form. The energy of an electric current is either of that form which consists in the actual motion of matter, or of that which consists in the capacity for being set in motion, arising from forces acting between bodies placed in certain positions relative to each other.

The first kind of energy, that of motion, is called Kinetic energy, and when once understood it appears so fundamental a fact of nature that we can hardly conceive the possibility of resolving it into anything else. The second kind of energy, that depending on position, is called Potential energy, and is due to the action of what we call forces, that is to say, tendencies towards change of relative position. With respect to these forces, though we may accept their existence as a demonstrated fact, yet we always feel that every explanation of the mechanism by which bodies are set in motion forms a real addition to our knowledge.

569.] The electric current cannot be conceived except as a kinetic phenomenon. Even Faraday, who constantly endeavoured to emancipate his mind from the influence of those suggestions which the words electric current and electric fluid are too apt to carry with them, speaks of the electric current as 'something progressive, and not a mere arrangement'^[1].

The effects of the current, such as electrolysis, and the transfer of electrification from one body to another, are all progressive actions which require time for their accomplishment, and are therefore of the nature of motions.

As to the velocity of the current, we have shewn that we know nothing about it, it may be the tenth of an inch in an hour, or a hundred thousand miles in a second^[2]. So far are we from knowing its absolute value in any case, that we do not even know whether what we call the positive direction is the actual direction of the motion or the reverse.

But all that we assume here is that the electric current involves motion of some kind. That which is the cause of electric currents has been called Electromotive Force. This name has long been used with great advantage, and has never led to any inconsistency in the language of science. Electromotive force is always to be understood to act on electricity only, not on the bodies in which the electricity resides. It is never to be confounded with ordinary mechanical force, which acts on bodies only, not on the electricity in them. If we ever come to know the formal relation between electricity and ordinary matter, we shall probably also know the relation between electromotive force and ordinary force.

570.] When ordinary force acts on a body, and when the body yields to the force, the work done by the force is measured by the product of the force into the amount by which the body yields. Thus, in the case of water forced through a pipe, the work done at any section is measured by the fluid pressure at the section multiplied into the quantity of water which crosses the section.

In the same way the work done by an electromotive force is measured by the product of the electromotive force into the quantity of electricity which crosses a section of the conductor under the action of the electromotive force.

The work done by an electromotive force is of exactly the same kind as the work done by an ordinary force, and both are measured by the same standards or units.

Part of the work done by an electromotive force acting on a conducting circuit is spent in overcoming the resistance of the circuit, and this part of the work is thereby converted into heat. Another part of the work is spent in producing the electromagnetic phenomena observed by Ampere, in which conductors are made to move by electromagnetic forces. The rest of the work is spent in increasing the kinetic energy of the current, and the effects of this part of the action are shewn in the phenomena of the induction of currents observed by Faraday.

We therefore know enough about electric currents to recognise, in a system of material conductors carrying currents, a dynamical system which is the seat of energy, part of which may be kinetic and part potential.

The nature of the connexions of the parts of this system is unknown to us, but as we have dynamical methods of investigation which do not require a knowledge of the mechanism of the system, we shall apply them to this case.

We shall first examine the consequences of assuming the most general form for the function which expresses the kinetic energy of the system.

571.] Let the system consist of a number of conducting circuits, the form and position of which are determined by the values of a system of variables x₁, x₂, &c., the number of which is equal to the number of degrees of freedom of the system.

If the whole kinetic energy of the system were that due to the motion of these conductors, it would be expressed in the form

$T={\frac {1}{2}}(x_{1}x_{1}){\dot {x}}_{1}^{2}\,+\And c.+(x_{1}x_{2}){\dot {x}}_{1}{\dot {x}}_{2}\,+\And c.$

where the symbols (x₁, x₁, &c.) denote the quantities which we have called moments of inertia, and (x₁, x₂, &c., ) denote the products of inertia.

If X' is the impressed force, tending to increase the coordinate x, which is required to produce the actual motion, then, by Lagrange's equation,

${\frac {d}{dt}}{\frac {dT}{d{\dot {x}}}}-{\frac {dT}{dx}}=0.$

When T denotes the energy due to the visible motion only, we shall indicate it by the suffix _m, thus, T_m.

But in a system of conductors carrying electric currents, part of the kinetic energy is due to the existence of these currents. Let the motion of the electricity, and of anything whose motion is governed by that of the electricity, be determined by another set of coordinates y₁, y₂, &c., then T will be a homogeneous function of squares and products of all the velocities of the two sets of coordinates. We may therefore divide T into three portions, in the first of which, T_m the velocities of the coordinates x only occur, while in the second, T_e , the velocities of the coordinates y only occur, and in the third, T_me, each term contains the product of the velocities of two coordinates of which one is x and the other y.

We have therefore

$T=T_{m}+T_{e}+T_{m}e,\,$

where

${\begin{aligned}T_{m}&={\frac {1}{2}}(x_{1}x_{1}){\dot {x}}_{1}^{2}\,+\And c.+(x_{1}x_{2}){\dot {x}}_{1}{\dot {x}}_{2}\,+\And c.,\\T_{e}&={\frac {1}{2}}(y_{1}y_{1}){\dot {y}}_{1}^{2}\,+\And c.+(y_{1}y_{2}){\dot {y}}_{1}{\dot {y}}_{2}\,+\And c.,\\T_{me}&=(x_{1}y_{1}){\dot {x}}_{1}{\dot {y}}_{1}\,+\And c.\end{aligned}}$

572.] In the general dynamical theory, the coefficients of every term may be functions of all the coordinates, both x and y. In the case of electric currents, however, it is easy to see that the coordinates of the class y do not enter into the coefficients.

For, if all the electric currents are maintained constant, and the conductors at rest, the whole state of the field will remain constant. But in this case the coordinates y are variable, though the velocities ${\dot {y}}$ are constant. Hence the coordinates y cannot enter into the expression for T, or into any other expression of what actually takes place.

Besides this, in virtue of the equation of continuity, if the conductors are of the nature of linear circuits, only one variable is required to express the strength of the current in each conductor. Let the velocities ${\dot {y}}_{1}$, ${\dot {y}}_{2}$, &c. represent the strengths of the currents in the several conductors.

All this would be true, if, instead of electric currents, we had currents of an incompressible fluid running in flexible tubes. In this case the velocities of these currents would enter into the expression for T, but the coefficients would depend only on the variables x, which determine the form and position of the tubes.

In the case of the fluid, the motion of the fluid in one tube does not directly affect that of any other tube, or of the fluid in it. Hence, in the value of T_e, only the squares of the velocities ${\dot {y}}$, and not their products, occur, and in T_me any velocity ${\dot {y}}$ is associated only with those velocities of the form ${\dot {x}}$ which belong to its own tube.

In the case of electrical currents we know that this restriction does not hold, for the currents in different circuits act on each other. Hence we must admit the existence of terms involving products of the form ${\dot {y}}_{1}{\dot {y}}_{2}$, and this involves the existence of something in motion, whose motion depends on the strength of both electric currents ${\dot {y}}_{1}$ and ${\dot {y}}_{2}$. This moving matter, whatever it is, is not confined to the interior of the conductors carrying the two currents, but probably extends throughout the whole space surrounding them.

573.] Let us next consider the form which Lagrange's equations of motion assume in this case. Let X' be the impressed force corresponding to the coordinate x, one of those which determine the form and position of the conducting circuits. This is a force in the ordinary sense, a tendency towards change of position. It is given by the equation

$X'={\frac {d}{dt}}{\frac {dT}{d{\dot {x}}}}-{\frac {dT}{dx}}.$

We may consider this force as the sum of three parts, corresponding to the three parts into which we divided the kinetic energy of the system, and we may distinguish them by the same suffixes. Thus

$X'=X'_{m}+X'_{e}+X'_{me}.\,$

The part X'_m is that which depends on ordinary dynamical considerations, and we need not attend to it.

Since T_e does not contain ${\dot {x}}$, the first term of the expression for X'_e is zero, and its value is reduced to

$X'_{e}=-{\frac {dT_{e}}{dx}}.$

This is the expression for the mechanical force which must be applied to a conductor to balance the electromagnetic force, and it asserts that it is measured by the rate of diminution of the purely electrokinetic energy due to the variation of the coordinate x. The electromagnetic force, X_e, which brings this external mechanical force into play, is equal and opposite to it, and is therefore measured by the rate of increase of the electrokinetic energy corresponding to an increase of the coordinate x. The value of X_e, since it depends on squares and products of the currents, remains the same if we reverse the directions of all the currents.

The third part of X' is

$X'_{me}={\frac {d}{dt}}{\frac {dT_{me}}{d{\dot {x}}}}-{\frac {dT_{me}}{dx}}.$

The quantity T_me contains only products of the form ${\dot {x}}{\dot {y}}$, so that ${\frac {dT_{me}}{d{\dot {x}}}}$ is a linear function of the strengths of the currents ${\dot {y}}$. The first term, therefore, depends on the rate of variation of the strengths of the currents, and indicates a mechanical force on the conductor, which is zero when the currents are constant, and which is positive or negative according as the currents are increasing or decreasing in strength.

The second term depends, not on the variation of the currents, but on their actual strength. As it is a linear function with respect to these currents, it changes sign when the currents change sign. Since every term involves a velocity ${\dot {x}}$, it is zero when the conductors are at rest.

We may therefore investigate these terms separately. If the conductors are at rest, we have only the first term to deal with. If the currents are constant, we have only the second.

574.] As it is of great importance to determine whether any part of the kinetic energy is of the form T_me, consisting of products of ordinary velocities and strengths of electric currents, it is desirable that experiments should be made on this subject with great care.

The determination of the forces acting on bodies in rapid motion is difficult. Let us therefore attend to the first term, which depends on the variation of the strength of the current.

Fig. 33. If any part of the kinetic energy depends on the product of an ordinary velocity and the strength of a current, it will probably be most easily observed when the velocity and the current are in the same or in opposite directions. We therefore take a circular coil of a great many windings, and suspend it by a fine vertical wire, so that its windings are horizontal, and the coil is capable of rotating about a vertical axis, either in the same direction as the current in the coil, or in the opposite direction.

We shall suppose the current to be conveyed into the coil by means of the suspending wire, and, after passing round the windings, to complete its circuit by passing downwards through a wire in the same line with the suspending wire and dipping into a cup of mercury.

Since the action of the horizontal component of terrestrial magnetism would tend to turn this coil round a horizontal axis when the current flows through it, we shall suppose that the horizontal component of terrestrial magnetism is exactly neutralized by means of fixed magnets, or that the experiment is made at the magnetic pole. A vertical mirror is attached to the coil to detect any motion in azimuth.

Now let a current be made to pass through the coil in the direction N.E.S.W. If electricity were a fluid like water, flowing along the wire, then, at the moment of starting the current, and as long as its velocity is increasing, a force would require to be supplied to produce the angular momentum of the fluid in passing round the coil, and as this must be supplied by the elasticity of the suspending wire, the coil would at first rotate in the opposite direction or W.S.E.N., and this would be detected by means of the mirror. On stopping the current there would be another movement of the mirror, this time in the same direction as that of the current.

No phenomenon of this kind has yet been observed. Such an action, if it existed, might be easily distinguished from the already known actions of the current by the following peculiarities.

(1) It would occur only when the strength of the current varies, as when contact is made or broken, and not when the current is constant.

All the known mechanical actions of the current depend on the strength of the currents, and not on the rate of variation. The electromotive action in the case of induced currents cannot be confounded with this electromagnetic action.

(2) The direction of this action would be reversed when that of all the currents in the field is reversed.

All the known mechanical actions of the current remain the same when all the currents are reversed, since they depend on squares and products of these currents.

If any action of this kind were discovered, we should be able to regard one of the so-called kinds of electricity, either the positive or the negative kind, as a real substance, and we should be able to describe the electric current as a true motion of this substance in a particular direction. In fact, if electrical motions were in any way comparable with the motions of ordinary matter, terms of the form T_me would exist, and their existence would be manifested by the mechanical force X_me.

According to Fechner's hypothesis, that an electric current consists of two equal currents of positive and negative electricity, flowing in opposite directions through the same conductor, the terms of the second class T_me would vanish, each term belonging to the positive current being accompanied by an equal term of opposite sign belonging to the negative current, and the phenomena depending on these terms would have no existence.

It appears to me, however, that while we derive great advantage from the recognition of the many analogies between the electric current and a current of a material fluid, we must carefully avoid making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as measured in feet per second.

A knowledge of these things would amount to at least the beginnings of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mechanism and details of the motion, are taken as the objects of study.

575.] The experimental investigation of the second term of X_me, namely ${\frac {dT_{me}}{dx}}$, is more difficult, as it involves the observation of the effect of forces on a body in rapid motion.

Fig. 34.

The apparatus shewn in Fig. 34, which I had constructed in 1861, is intended to test the existence of a force of this kind.

The electromagnet A is capable of rotating about the horizontal axis BB', within a ring which itself revolves about a vertical axis.

Let A, B, C be the moments of inertia of the electromagnet about the axis of the coil, the horizontal axis BB', and a third axis CC' respectively.

Let θ be the angle which CC' makes with the vertical, φ azimuth of the axis BB', and ψ a variable on which the motion of electricity in the coil depends.

Then the kinetic energy of the electromagnet may be written

$2T=A{\dot {\phi }}^{2}\sin ^{2}\theta +B{\dot {\theta }}^{2}+C{\dot {\phi }}^{2}\cos ^{2}\theta +E({\dot {\phi }}\sin \theta +{\dot {\psi }})^{2},$

where E is a quantity which may be called the moment of inertia of the electricity in the coil.

If Θ is the moment of the impressed force tending to increase θ, we have, by the equations of dynamics,

$\Theta =B{\frac {d^{2}\theta }{dt^{2}}}-\{(A-C){\dot {\phi }}^{2}\sin \theta \cos \theta +E{\dot {\phi }}\cos \theta ({\dot {\phi }}\sin \theta +{\dot {\psi }})\}.$

By making Ψ, the impressed force tending to increase ψ, equal to zero, we obtain

${\dot {\phi }}\sin \theta =\gamma ,\,$

a constant, which we may consider as representing the strength of the current in the coil.

If C is somewhat greater than A, Θ will be zero, and the equilibrium about the axis BB' will be stable when

$\sin \theta ={\frac {E\gamma }{(C-A){\dot {\phi }}}}.$

This value of θ depends on that of γ, the electric current, and is positive or negative according to the direction of the current.

The current is passed through the coil by its bearings at B and B', which are connected with the battery by means of springs rubbing on metal rings placed on the vertical axis.

To determine the value of θ, a disk of paper is placed at C, divided by a diameter parallel to BB' into two parts, one of which is painted red and the other green.

When the instrument is in motion a red circle is seen at C when θ is positive, the radius of which indicates roughly the value of θ. When is negative, a green circle is seen at C.

By means of nuts working on screws attached to the electromagnet, the axis CC' is adjusted to be a principal axis having its moment of inertia just exceeding that round the axis A, so as to make the instrument very sensible to the action of the force if it exists.

The chief difficulty in the experiments arose from the disturbing action of the earth s magnetic force, which caused the electromagnet to act like a dip-needle. The results obtained were on this account very rough, but no evidence of any change in θ could be obtained even when an iron core was inserted in the coil, so as to make it a powerful electromagnet.

If, therefore, a magnet contains matter in rapid rotation, the angular momentum of this rotation must be very small compared with any quantities which we can measure, and we have as yet no evidence of the existence of the terms T_me derived from their mechanical action.

576.] Let us next consider the forces acting on the currents of electricity, that is, the electromotive forces.

Let Y be the effective electromotive force due to induction, the electromotive force which must act on the circuit from without to balance it is Y' = -Y, and, by Lagrange s equation,

$Y=-Y'=-{\frac {d}{dt}}{\frac {dT}{d{\dot {y}}}}+{\frac {dT}{dy}}.$

Since there are no terms in T involving the coordinate y, the second term is zero, and Y is reduced to its first term. Hence, electromotive force cannot exist in a system at rest, and with constant currents.

Again, if we divide Y into three parts, Y_m, Y_e, Y_me, corresponding to the three parts of T, we find that, since T_m does not contain ${\dot {y}}$, Y_m = 0.

We also find

$Y_{e}=-{\frac {d}{dt}}{\frac {dT_{e}}{d{\dot {y}}}}.$

Here ${\frac {dT_{e}}{d{\dot {y}}}}$ is a linear function of the currents, and this part of the electromotive force is equal to the rate of change of this function. This is the electromotive force of induction discovered by Faraday. We shall consider it more at length afterwards.

577.] From the part of T, depending on velocities multiplied by currents, we find

$Y_{me}=-{\frac {d}{dt}}{\frac {dT_{me}}{d{\dot {y}}}}.$

Now ${\frac {dT_{me}}{d{\dot {y}}}}$ is a linear function of the velocities of the conductors. If, therefore, any terms of T_me have an actual existence, it would be possible to produce an electromotive force independently of all existing currents by simply altering the velocities of the conductors. For instance, in the case of the suspended coil at Art. 559, if, when the coil is at rest, we suddenly set it in rotation about the vertical axis, an electromotive force would be called into action proportional to the acceleration of this motion. It would vanish when the motion became uniform, and be reversed when the motion was retarded.

Now few scientific observations can be made with greater precision than that which determines the existence or non-existence of a current by means of a galvanometer. The delicacy of this method far exceeds that of most of the arrangements for measuring the mechanical force acting on a body. If, therefore, any currents could be produced in this way they would be detected, even if they were very feeble. They would be distinguished from ordinary currents of induction by the following characteristics.

(1) They would depend entirely on the motions of the conductors, and in no degree on the strength of currents or magnetic forces already in the field.

(2) They would depend not on the absolute velocities of the con ductors, but on their accelerations, and on squares and products of velocities, and they would change sign when the acceleration becomes a retardation, though the absolute velocity is the same.

Now in all the cases actually observed, the induced currents depend altogether on the strength and the variation of currents in the field, and cannot be excited in a field devoid of magnetic force and of currents. In so far as they depend on the motion of conductors, they depend on the absolute velocity, and not on the change of velocity of these motions.

We have thus three methods of detecting the existence of the terms of the form T_me, none of which have hitherto led to any positive result. I have pointed them out with the greater care because it appears to me important that we should attain the greatest amount of certitude within our reach on a point bearing so strongly on the true theory of electricity.

Since, however, no evidence has yet been obtained of such terms, I shall now proceed on the assumption that they do not exist, or at least that they produce no sensible effect, an assumption which will considerably simplify our dynamical theory. We shall have occasion, however, in discussing the relation of magnetism to light, to shew that the motion which constitutes light may enter as a factor into terms involving the motion which constitutes magnetism.^[3]

1. Exp. Res., 283.
2. Exp. Res., 1648.
3. The Lorentz force, the forca acting on a charged particle in electric and magnetic fields, has been for the first time correctly described by Hendryk Lorentz in 1892