CHAPTER IX.

SCIENTIFIC WORK.--ELECTRICAL THEORIES.

Clerk Maxwell’s first electrical paper--that on Faraday’s “Lines of

Force”--was read to the Cambridge Philosophical Society on December

10th, 1855, and Part II. on February 11th, 1856. The author was then a

Bachelor of Arts, only twenty-three years in age, and of less than one

year’s standing from the time of taking his degree.

The opening words of the paper are as follows (Scientific Papers, vol.

i., p. 155):--

“The present state of electrical science seems peculiarly

unfavourable to speculation. The laws of the distribution of

electricity on the surface of conductors have been analytically

deduced from experiment; some parts of the mathematical

theory of magnetism are established, while in other parts the

experimental data are wanting; the theory of the conduction of

galvanism, and that of the mutual attraction of conductors,

have been reduced to mathematical formulæ, but have not

fallen into relation with the other parts of the science. No

electrical theory can now be put forth, unless it shows the

connection, not only between electricity at rest and current

electricity, but between the attractions and inductive effects

of electricity in both states. Such a theory must accurately

satisfy those laws, the mathematical form of which is known,

and must afford the means of calculating the effects in the

limiting cases where the known formulæ are inapplicable.

In order, therefore, to appreciate the requirements of the

science, the student must make himself familiar with a

considerable body of most intricate mathematics, the mere

retention of which in the memory materially interferes with

further progress. The first process, therefore, in the

effectual study of the science, must be one of simplification

and reduction of the results of previous investigation to a

form in which the mind can grasp them. The results of this

simplification may take the form of a purely mathematical

formula or of a physical hypothesis. In the first case we

entirely lose sight of the phenomena to be explained; and

though we may trace out the consequences of given laws, we can

never obtain more extended views of the connections of the

subject. If, on the other hand, we adopt a physical hypothesis,

we see the phenomena only through a medium, and are liable

to that blindness to facts and rashness in assumption which

a partial explanation encourages. We must therefore discover

some method of investigation which allows the mind at every

step to lay hold of a clear physical conception, without being

committed to any theory founded on the physical science from

which that conception is borrowed, so that it is neither drawn

aside from the subject in pursuit of analytical subtleties, nor

carried beyond the truth by a favourite hypothesis.

“In order to obtain physical ideas without adopting a physical

theory we must make ourselves familiar with the existence of

physical analogies. By a physical analogy I mean that partial

similarity between the laws of one science and those of another

which makes each of them illustrate the other. Thus all the

mathematical sciences are founded on relations between physical

laws and laws of numbers, so that the aim of exact science

is to reduce the problems of Nature to the determination of

quantities by operations with members. Passing from the most

universal of all analogies to a very partial one, we find the

same resemblance in mathematical form between two different

phenomena giving rise to a physical theory of light.

“The changes of direction which light undergoes in passing from

one medium to another are identical with the deviations of the

path of a particle in moving through a narrow space in which

intense forces act. This analogy, which extends only to the

direction, and not to the velocity of motion, was long believed

to be the true explanation of the refraction of light; and we

still find it useful in the solution of certain problems, in

which we employ it without danger as an artificial method. The

other analogy, between light and the vibrations of an elastic

medium, extends much farther, but, though its importance and

fruitfulness cannot be over-estimated, we must recollect that

it is founded only on a resemblance _in form_ between the

laws of light and those of vibrations. By stripping it of its

physical dress and reducing it to a theory of ‘transverse

alternations,’ we might obtain a system of truth strictly

founded on observation, but probably deficient both in the

vividness of its conceptions and the fertility of its method.

I have said thus much on the disputed questions of optics, as

a preparation for the discussion of the almost universally

admitted theory of attraction at a distance.

“We have all acquired the mathematical conception of these

attractions. We can reason about them and determine their

appropriate forms or formulæ. These formulæ have a distinct

mathematical significance, and their results are found to be

in accordance with natural phenomena. There is no formula

in applied mathematics more consistent with Nature than the

formula of attractions, and no theory better established in

the minds of men than that of the action of bodies on one

another at a distance. The laws of the conduction of heat in

uniform media appear at first sight among the most different in

their physical relations from those relating to attractions.

The quantities which enter into them are _temperature_, _flow

of heat_, _conductivity_. The word _force_ is foreign to the

subject. Yet we find that the mathematical laws of the uniform

motion of heat in homogeneous media are identical in form

with those of attractions varying inversely as the square of

the distance. We have only to substitute _source of heat_ for

_centre of attraction_, _flow of heat_ for _accelerating effect

of attraction_ at any point, and _temperature_ for _potential_,

and the solution of a problem in attractions is transformed

into that of a problem in heat.

“This analogy between the formulæ of heat and attraction was, I

believe, first pointed out by Professor William Thomson in the

_Cambridge Mathematical Journal_, Vol. III.

“Now the conduction of heat is supposed to proceed by an

action between contiguous parts of a medium, while the force

of attraction is a relation between distant bodies, and yet,

if we knew nothing more than is expressed in the mathematical

formulæ, there would be nothing to distinguish between the one

set of phenomena and the other.

“It is true that, if we introduce other considerations and

observe additional facts, the two subjects will assume very

different aspects, but the mathematical resemblance of some

of their laws will remain, and may still be made useful in

exciting appropriate mathematical ideas.

“It is by the use of analogies of this kind that I have

attempted to bring before the mind, in a convenient and

manageable form, those mathematical ideas which are necessary

to the study of the phenomena of electricity. The methods are

generally those suggested by the processes of reasoning which

are found in the researches of Faraday, and which, though they

have been interpreted mathematically by Professor Thomson and

others, are very generally supposed to be of an indefinite and

unmathematical character, when compared with those employed by

the professed mathematicians. By the method which I adopt, I

hope to render it evident that I am not attempting to establish

any physical theory of a science in which I have hardly made

a single experiment, and that the limit of my design is to

show how, by a strict application of the ideas and methods

of Faraday, the connection of the very different orders of

phenomena which he has discovered may be clearly placed before

the mathematical mind. I shall therefore avoid as much as I can

the introduction of anything which does not serve as a direct

illustration of Faraday’s methods, or of the mathematical

deductions which may be made from them. In treating the simpler

parts of the subject I shall use Faraday’s mathematical methods

as well as his ideas. When the complexity of the subject

requires it, I shall use analytical notation, still confining

myself to the development of ideas originated by the same

philosopher.

“I have in the first place to explain and illustrate the idea

of ‘lines of force.’

“When a body is electrified in any manner, a small body

charged with positive electricity, and placed in any given

position, will experience a force urging it in a certain

direction. If the small body be now negatively electrified, it

will be urged by an equal force in a direction exactly opposite.

“The same relations hold between a magnetic body and the north

or south poles of a small magnet. If the north pole is urged

in one direction, the south pole is urged in the opposite

direction.

“In this way we might find a line passing through any point

of space, such that it represents the direction of the

force acting on a positively electrified particle, or on an

elementary north pole, and the reverse direction of the force

on a negatively electrified particle or an elementary south

pole. Since at every point of space such a direction may be

found, if we commence at any point and draw a line so that,

as we go along it, its direction at any point shall always

coincide with that of the resultant force at that point, this

curve will indicate the direction of that force for every point

through which it passes, and might be called on that account a

_line of force_. We might in the same way draw other lines of

force, till we had filled all space with curves indicating by

their direction that of the force at any assigned point.

“We should thus obtain a geometrical model of the physical

phenomena, which would tell us the _direction_ of the force,

but we should still require some method of indicating the

_intensity_ of the force at any point. If we consider these

curves not as mere lines, but as fine tubes of variable section

carrying an incompressible fluid, then, since the velocity of

the fluid is inversely as the section of the tube, we may make

the velocity vary according to any given law, by regulating the

section of the tube, and in this way we might represent the

intensity of the force as well as its direction by the motion

of the fluid in these tubes. This method of representing the

intensity of a force by the velocity of an imaginary fluid in

a tube is applicable to any conceivable system of forces, but

it is capable of great simplification in the case in which

the forces are such as can be explained by the hypothesis of

attractions varying inversely as the square of the distance,

such as those observed in electrical and magnetic phenomena.

In the case of a perfectly arbitrary system of forces, there

will generally be interstices between the tubes; but in the

case of electric and magnetic forces it is possible to arrange

the tubes so as to leave no interstices. The tubes will then be

mere surfaces, directing the motion of a fluid filling up the

whole space. It has been usual to commence the investigation of

the laws of these forces by at once assuming that the phenomena

are due to attractive or repulsive forces acting between

certain points. We may, however, obtain a different view of the

subject, and one more suited to our more difficult inquiries,

by adopting for the definition of the forces of which we treat,

that they may be represented in magnitude and direction by the

uniform motion of an incompressible fluid.

“I propose, then, first to describe a method by which the

motion of such a fluid can be clearly conceived; secondly

to trace the consequences of assuming certain conditions of

motion, and to point out the application of the method to some

of the less complicated phenomena of electricity, magnetism,

and galvanism; and lastly, to show how by an extension of these

methods, and the introduction of another idea due to Faraday,

the laws of the attractions and inductive actions of magnets

and currents may be clearly conceived, without making any

assumptions as to the physical nature of electricity, or adding

anything to that which has been already proved by experiment.

“By referring everything to the purely geometrical idea of the

motion of an imaginary fluid, I hope to attain generality and

precision, and to avoid the dangers arising from a premature

theory professing to explain the cause of the phenomena.

If the results of mere speculation which I have collected

are found to be of any use to experimental philosophers, in

arranging and interpreting their results, they will have served

their purpose, and a mature theory, in which physical facts

will be physically explained, will be formed by those who by

interrogating Nature herself can obtain the only true solution

of the questions which the mathematical theory suggests.”

The idea was a bold one: for a youth of twenty-three to explain, by

means of the motions of an incompressible fluid, some of the less

complicated phenomena of electricity and magnetism, to show how

the laws of the attractions of magnets and currents may be clearly

conceived without making any assumption as to the physical nature of

electricity, or adding anything to that which has already been proved

by experiment.

It may be useful to review in a very few words the position of

electrical theory[57] in 1855.

Coulomb’s experiments had established the fundamental facts of

electrostatic attraction and repulsion, and Coulomb himself, about

1785, had stated a theory based on these experiments which could “only

be attacked by proving his experimental results to be inaccurate.”[58]

Coulomb supposes the existence of two electric fluids, the theory

developed previously by Franklin, but says--

“Je préviens pour mettre la théorie qui va suivre à l’abri de

toute dispute systématique, que dans la supposition de deux

fluides électriques, je n’ai autre intention que de présenter

avec le moins d’éléments possible les résultats du calcul et

de l’expérience, et non d’indiquer les véritables causes de

l’électricité.”

Cavendish was working in England about the same time as Coulomb, but

he published very little, and the value and importance of his work

was not recognised until the appearance in 1879 of the “Electrical

Researches of Henry Cavendish,” edited by Clerk Maxwell.

Early in the present century the application of mathematical analysis

to electrical problems was begun by Laplace, who investigated the

distribution of electricity on spheroids, and about 1811 Poisson’s

great work on the distribution of electricity on two spheres placed

at any given distance apart was published. Meanwhile the properties

of the electric current were being investigated. Galvani’s discovery

of the muscular contraction in a frog’s leg, caused by the contact of

dissimilar metals, was made in 1790. Volta invented the voltaic pile in

1800, and Oersted in 1820 discovered that an electric current produced

magnetic force in its neighbourhood. On this Ampère laid the foundation

of his theory of electro-dynamics, in which he showed how to calculate

the forces between circuits carrying currents from an assumed law of

force between each pair of elements of the circuits. His experiments

proved that the consequences which follow from this law are consistent

with all the observed facts. They do not prove that Ampère’s law alone

can explain the facts.

Maxwell, writing on this subject in the “Electricity an Magnetism,”

vol. ii., p. 162, says--

“The experimental investigation by which Ampère established the

laws of the mechanical action between electric currents is one

of the most brilliant achievements in science.

“The whole, theory and experiment, seems as if it had leaped

full grown and full armed from the brain of the ‘Newton

of Electricity.’ It is perfect in form and unassailable in

accuracy, and it is summed up in a formula from which all the

phenomena may be deduced, and which must always remain the

cardinal formula of electro-dynamics.

“The method of Ampère, however, though cast into an inductive

form, does not allow us to trace the formation of the ideas

which guided it. We can scarcely believe that Ampère really

discovered the law of action by means of the experiments which

he describes. We are led to suspect, what, indeed, he tells us

himself, that he discovered the law by some process which he

has not shown us, and that when he had afterwards built up a

perfect demonstration, he removed all traces of the scaffolding

by which he had built it.”

The experimental evidence for Ampère’s theory, so far, at least, as

it was possible to obtain it from experiments on closed circuits, was

rendered unimpeachable by W. Weber about 1846, while in the previous

year Grassman and F. E. Neumann both published laws for the attraction

between two elements of current which differ from that of Ampère, but

lead to the same result for closed circuits. In a paper published in

1846 Weber announced his hypothesis connecting together electrostatic

and electro-dynamic action. In this paper he supposed that the force

between two particles of electricity depends on the motion of the

particles as well as on their distance apart. A somewhat similar

theory was proposed by Gauss and published after his death in his

collected works. It has been shown, however, that Gauss’ theory is

inconsistent with the conservation of energy. Weber’s theory avoids

this inconsistency and leads, for closed circuits, to the same results

as Ampère. It has been proved, however, by Von Helmholtz, that, under

certain circumstances, according to it, a body would behave as though

its mass were negative--it would move in a direction opposite to that

of the force.[59]

Since 1846 many other theories have been proposed to explain Ampère’s

laws. Meanwhile, in 1821, Faraday observed that under certain

circumstances a wire carrying a current could be kept in continuous

rotation in a magnetic field by the action between the magnets and

the current. In 1824 Arago observed the motion of a magnet caused by

rotating a copper disc in its neighbourhood, while in 1831 Faraday

began his experimental researches into electro-magnetic induction.

About the same period Joseph Henry, of Washington, was making,

independently of Faraday, experiments of fundamental importance on

electro-magnetic induction, but sufficient attention was not called to

his work until comparatively recent years.

In 1833 Lenz made some important researches, which led him to discover

the connection between the direction of the induced currents and

Ampère’s laws, summed up in his rule that the direction of the induced

current is always such as to oppose by its electro-magnetic action the

motion which induces it.

In 1845 F. E. Neumann developed from this law the mathematical theory

of electro-magnetic induction, and about the same time W. Weber showed

how it might be deduced from his elementary law of electrical action.

The great name of Von Helmholtz first appears in connection with this

subject in 1851, but of his writings we shall have more to say at a

later stage.

Meanwhile, during the same period, various writers, Murphy, Plana,

Charles, Sturm, and Gauss, extended Poisson’s work on electrostatics,

treating the questions which arose as problems in the distribution of

an attracting fluid, attracting or repelling according to Newton’s law,

though here again the greatest advances were made by a self-taught

Nottingham shoemaker, George Green by name, in his paper “On the

Application of Mathematical Analysis to the Theories of Electricity and

Magnetism,” 1828.

Green’s researches, Lord Kelvin writes, “have led to the elementary

proposition which must constitute the legitimate foundation of every

perfect mathematical structure that is to be made from the materials

furnished by the experimental laws of Coulomb.”

Green, it may be remarked, was the inventor of the term Potential.

His essay, however, lay neglected from 1828, until Lord Kelvin called

attention to it in 1845. Meanwhile, some of its most important results

had been re-discovered by Gauss and Charles and Thomson himself.

Until about 1845, the experimental work on which these mathematical

researches in electrostatics were based was that of Coulomb. An

electrified body is supposed to have a charge of some imponderable

fluid “electricity.” Particles of electricity repel each other

according to a certain law, and the fluid distributes itself in

equilibrium over the surface of any charged conductor in accordance

with this law. There are on this theory two opposite kinds of electric

fluid, positive and negative, two charges of the same kind repel, two

charges of opposite kinds attract; the repulsion or attraction is

proportional to the product of the charges, and inversely proportional

to the square of the distance between them.

The action between two charges is action at a distance taking place

across the space which separates the two.

Faraday, in 1837, in the eleventh series of his “Experimental

Researches,” published his first paper on “Electrostatic Induction.”

He showed--as indeed Cavendish had proved long previously, though the

result remained unpublished--that the force between two charged bodies

will depend on the insulating medium which surrounds them, not merely

on their shape and position. Induction, as he expresses it, takes place

along curved lines, and is an action of contiguous particles; these

curved lines he calls the “lines of force.”

Discussing these researches in 1845, Lord Kelvin writes[60]:--

“Mr. Faraday’s researches ... were undertaken with a view to

test an idea which he had long possessed that the forces of

attraction and repulsion exercised by free electricity are not

the resultants of actions exercised at a distance, but are

propagated by means of molecular action among the contiguous

particles of the insulating medium surrounding the electrified

bodies, which he therefore calls the dielectric. By this idea

he has been led to some very remarkable views upon induction,

or, in fact, upon electrical action in general. As it is

impossible that the phenomena observed by Faraday can be

incompatible with the results of experiment which constitute

Coulomb’s theory, it is to be expected that the difference

of his ideas from those of Coulomb must arise solely from a

different method of stating and interpreting physically the

same laws; and further, it may, I think, be shown that either

method of viewing this subject, when carried sufficiently

far, may be made the foundation of a mathematical theory

which would lead to the elementary principles of the other as

consequences. This theory would, accordingly, be the expression

of the ultimate law of the phenomena, independently of any

physical hypothesis we might from other circumstances be led

to adopt. That there are necessarily two distinct elementary

ways of viewing the theory of electricity may be seen from the

following considerations....”

In the pages which follow, Lord Kelvin develops the consequences of an

analogy between the conduction of heat and electrostatic action, which

he had pointed out three years earlier (1842), in his paper on “The

Uniform Motion of Heat in Homogeneous Solid Bodies,” and discusses its

connection with the mathematical theory of electricity.

The problem of distributing sources of heat in a given homogeneous

conductor of heat, so as to produce a definite steady temperature at

each point on the conductor is shewn to be _mathematically_ identical

with that of distributing electricity in equilibrium, so as to produce

at each point an electrical potential having the same value as the

temperature.

Thus the fundamental laws of the conduction of heat may be made the

basis of the mathematical theory of electricity, but the physical

idea which they suggest is that of the propagation of some effect by

means of the mutual action of contiguous particles, rather than that

of material particles attracting or repelling at a distance, which

naturally follows from the statement of Coulomb’s law.

Lord Kelvin continues:--

“All the views which Faraday has brought forward and

illustrated, as demonstrated by experiment, lead to this method

of establishing the mathematical theory, and, as far as the

analysis is concerned, it would in most _general_ propositions

be more simple, if possible, than that of Coulomb. Of course

the analysis of _particular_ problems would be identical in the

two methods. It is thus that Faraday arrives at a knowledge of

some of the most important of the mathematical theorems which

from their nature seemed destined never to be perceived except

as mathematical truths.”

Lord Kelvin’s papers on “The Mathematical Theory of Electricity,”

published from 1848 to 1850, his “Propositions on the Theory of

Attraction” (1842), his “Theory of Electrical Images” (1847), and his

paper on “The Mathematical Theory of Magnetism” (1849), contain a

statement of the most important results achieved in the mathematical

sciences of Electrostatics and Magnetism up to the time of Maxwell’s

first paper.

The opening sentences of that paper have already been quoted. In the

preface to the “Electricity and Magnetism” Maxwell writes thus:--

“Before I began the study of electricity I resolved to read

no mathematics on the subject till I had first read through

‘Experimental Researches on Electricity.’ I was aware that

there was supposed to be a difference between Faraday’s way of

conceiving phenomena and that of the mathematicians, so that

neither he nor they were satisfied with each other’s language.

I had also the conviction that this discrepancy did not arise

from either party being wrong. I was first convinced of this by

Sir William Thomson, to whose advice and assistance, as well as

to his published papers, I owe most of what I have learned on

the subject.

“As I proceeded with the study of Faraday, I perceived that his

method of conceiving the phenomena was also a mathematical

one, though not exhibited in the conventional form of

mathematical symbols. I also found that these methods were

capable of being expressed in the ordinary mathematical forms,

and thus compared with those of the professed mathematicians.

“For instance, Faraday, in his mind’s eye, saw lines of force

traversing all space where the mathematicians saw centres of

force attracting at a distance. Faraday saw a medium where

they saw nothing but distance. Faraday sought the seat of the

phenomena in real actions going on in the medium. They were

satisfied that they had found it in a power of action at a

distance impressed on the electric fluids.”

Now, Maxwell saw an analogy between electrostatics and the steady

motion of an incompressible fluid like water, and it is this analogy

which he develops in the first part of his paper. The water flows along

definite lines; a surface which consists wholly of such lines of flow

will have the property that no water ever crosses it. In any stream

of water we can imagine a number of such surfaces drawn, dividing it

up into a series of tubes; each of these will be a tube of flow, each

of these tubes remain always filled with water. Hence, the quantity

of water which crosses per second any section of a tube of flow

perpendicular to its length is always the same. Thus, from the form of

the tube, we can obtain information as to the direction and strength of

the flow, for where the tube is wide the flow will be proportionately

small, and _vice versâ_.

Again, we can draw in the fluid a number of surfaces, over each of

which the pressure is the same; these surfaces will cut the tubes

of flow at right angles. Let us suppose they are drawn so that the

difference of pressure between any two consecutive surfaces is unity,

then the surfaces will be close together at points at which the

pressure changes rapidly; where the variation of pressure is slow, the

distance between two consecutive surfaces will be considerable.

If, then, in any case of motion, we can draw the pressure surfaces,

and the tubes of flow, we can determine the motion of the fluid

completely. Now, the same mathematical expressions which appear in

the hydro-dynamical theory occur also in the theory of electricity,

the meaning only of the symbols is changed. For velocity of fluid we

have to write electrical force. For difference of fluid pressure we

substitute work done, or difference of electrical potential or pressure.

The surfaces and tubes, drawn as the solution of any hydro-dynamical

problem, give us also the solution of an electrical problem; the

tubes of flow are Faraday’s tubes of force, or tubes of induction,

the surfaces of constant pressure are surfaces of equal electrical

potential. Induction may take place in curved lines just as the tubes

of flow may be bent and curved; the analogy between the two is a

complete one.

But, as Maxwell shows, the analogy reaches further still. An electric

current flowing along a wire had been recognised as having many

properties similar to those of a current of liquid in a tube. When a

steady current is passing through any solid conductor, there are formed

in the conductor tubes of electrical flow and surfaces of constant

pressure. These tubes and surfaces are the same as those formed by the

flow of liquid through a solid whose boundary surface is the same

as that of the conductor, provided the flow of liquid is properly

proportioned to the flow of electricity.

These analogies refer to steady currents in which, therefore, the flow

at any point of the conductor does not depend on the time. In Part

II. of his paper Maxwell deals with Faraday’s electro-tonic state.

Faraday had found that when _changes_ are produced in the magnetic

phenomena surrounding a conductor, an electric current is set up in

the conductor, which continues so long as the magnetic changes are in

progress, but which ceases when the magnetic state becomes steady.

“Considerations of this kind led Professor Faraday to connect

with his discovery of the induction of electric currents the

conception of a state into which all bodies are thrown by the

presence of magnets and currents. This state does not manifest

itself by any known phenomena as long as it is undisturbed,

but any change in this state is indicated by a current or

tendency towards a current. To this state he gave the name of

the ‘Electro-tonic State,’ and although he afterwards succeeded

in explaining the phenomena which suggested it by means of less

hypothetical conceptions, he has on several occasions hinted at

the probability that some phenomena might be discovered which

would render the electro-tonic state an object of legitimate

induction. These speculations, into which Faraday had been

led by the study of laws which he has well established, and

which he abandoned only for want of experimental data for the

direct proof of the unknown state, have not, I think, been

made the subject of mathematical investigation. Perhaps it

may be thought that the quantitative determinations of the

various phenomena are not sufficiently rigorous to be made

the basis of a mathematical theory. Faraday, however, has not

contented himself with simply stating the numerical results

of his experiments and leaving the law to be discovered by

calculation. Where he has perceived a law he has at once stated

it, in terms as unambiguous as those of pure mathematics,

and if the mathematician, receiving this as a physical

truth, deduces from it other laws capable of being tested by

experiment, he has merely assisted the physicist in arranging

his own ideas, which is confessedly a necessary step in

scientific induction.

“In the following investigation, therefore, the laws

established by Faraday will be assumed as true, and it will

be shown that by following out his speculations other and

more general laws can be deduced from them. If it should,

then, appear that these laws, originally devised to include

one set of phenomena, may be generalised so as to extend to

phenomena of a different class, these mathematical connections

may suggest to physicists the means of establishing physical

connections, and thus mere speculation may be turned to account

in experimental science.”

Maxwell shows how to obtain a mathematical expression for Faraday’s

electro-tonic state. In his “Electricity and Magnetism,” this

electro-tonic state receives a new name. It is known as the Vector

Potential,[61] and the paper under consideration contains, though

in an incomplete form, his first statement of those equations of the

electric field which are so indissolubly bound up with Maxwell’s name.

The great advance in theory made in the paper is the distinct

recognition of certain mathematical functions as representing Faraday’s

electrotonic-state, and their use in solving electro-magnetic problems.

The paper contains no new physical theory of electricity, but in a

few years one appeared. In his later writings Maxwell adopted a more

general view of the electro-magnetic field than that contained in his

early papers on “Physical Lines of Force.” It must, therefore, not be

supposed that the somewhat gross conception of cog-wheels and pulleys,

which we are about to describe, were anything more to their author than

a model, which enabled him to realise how the changes, which occur when

a current of electricity passes through a wire, might be represented by

the motion of actual material particles.

The problem before him was to devise a physical theory of electricity,

which would explain the forces exerted on electrified bodies by means

of action between the contiguous parts of the medium in the space

surrounding these bodies, rather than by direct action across the

distance which separates them. A similar question, still unanswered,

had arisen in the case of gravitation. Astronomers have determined the

forces between attracting bodies; they do not know how those forces

arise.

Maxwell’s fondness for models has already been alluded to; it had led

him to construct his top to illustrate the dynamics of a rigid body

rotating about a fixed point, and his model of Saturn’s rings (now in

the Cavendish Laboratory) to illustrate the motion of the satellites

in the rings. He had explained many of the gaseous laws by means of

the impact of molecules, and now his fertile ingenuity was to imagine

a mechanical model of the state of the electro-magnetic field near a

system of conductors carrying currents.

Faraday, as we have seen, looked upon electrostatic and magnetic

induction as taking place along curved lines of force. He pictures

these lines as ropes of molecules starting from a charged conductor, or

a magnet, as the case may be, and acting on other bodies near. These

ropes of molecules tend to shorten, and at the same time to swell

outwards laterally. Thus the charged conductor tends to draw other

bodies to itself, there is a tension along the lines of force, while

at the same time each tube of molecules pushes its neighbours aside; a

pressure at right angles to the lines of force is combined with this

tension. Assuming for a moment this pressure and tension to exist, can

we devise a mechanism to account for it? Maxwell himself has likened

the lines of force to the fibres of a muscle. As the fibres contract,

causing the limb to which they are attached to move, they swell

outwards, and the muscle thickens.

Again, from another point of view, we might consider a line of force

as consisting of a string of small cells of some flexible material

each filled with fluid. If we then suppose this series of cells caused

to rotate rapidly about the direction of the line of force, the cells

will expand laterally and contract longitudinally; there will again be

tension along the lines of force and pressure at right angles to them.

It was this last idea, as we shall see shortly, of which Maxwell made

use--

“I propose now” [he writes (“On Physical Lines of Force,”

_Phil. Mag._, vol. xxi.)] “to examine magnetic phenomena from

a mechanical point of view, and to determine what tensions in,

or motions of, a medium are capable of producing the mechanical

phenomena observed. If by the same hypothesis we can connect

the phenomena of magnetic attraction with electro-magnetic

phenomena, and with those of induced currents, we shall have

found a theory which, if not true, can only be proved to be

erroneous by experiments, which will greatly enlarge our

knowledge of this part of physics.”

Lord Kelvin had in 1847 given a mechanical representation of electric,

magnetic and galvanic forces by means of the displacements of an

elastic solid in a state of strain. The angular displacement at each

point of the solid was taken as proportional to the magnetic force, and

from this the relation between the various other electric quantities

and the motion of the solid was developed. But Lord Kelvin did not

attempt to explain the origin of the observed forces by the effects due

to these strains, but merely made use of the mathematical analogy to

assist the imagination in the study of both.

Maxwell considered magnetic action as existing in the form of pressure

or tension, or more generally, of some stress in some medium. The

existence of a medium capable of exerting force on material bodies and

of withstanding considerable stress, both pressure and tension, is

thus a fundamental hypothesis with him; this medium is to be capable

of motion, and electro-magnetic forces arise from its motion and its

stresses.

Now, Maxwell’s fundamental supposition is that, in a magnetic field,

there is a rotation of the molecules continually in progress about the

lines of magnetic force. Consider now the case of a uniform magnetic

field, whose direction is perpendicular to the paper; we are to look

upon the lines of force as parallel strings of molecules, the axes of

these strings being perpendicular to the paper. Each string is supposed

to be rotating in the same direction about its axis, and the angular

velocity of rotation is a measure of the magnetic force. In consequence

of this rotation there will be differences of pressure in different

directions in the medium; the pressure along the axes of the strings

will be less than it would be if the medium were at rest, that in the

directions at right angles to the axes will be greater, the medium will

behave as though it were under tension along the axes of the molecules

under pressure at right angles to them. Moreover, it can be shown that

the pressure and the tension are both proportional to the square of the

angular velocity--the square, that is, of the magnetic force--and this

result is in accordance with the consequences of experiment.

More elaborate calculation shows that this statement is true generally.

If we draw the lines of force in any magnetic field, and then suppose

the molecules of the medium set in rotation about these lines of force

as axes, with velocities which at each point are proportional to the

magnetic force, the distribution of pressure throughout is that which

we know actually to exist in the magnetic field.

According to this hypothesis, then, a permanent bar magnet has the

power of setting the medium round it into continuous molecular rotation

about the lines of force as axes. The molecules which are set in

rotation we may consider as spherical, or nearly spherical, cells

filled with a fluid, or an elastic solid substance, and surrounded by a

kind of membrane, or sack, holding the contents together.

So far the model does not give any account of electrical actions which

go on in the magnetic field.

The energy is wholly rotational, and the forces wholly magnetic.

Consider, however, any two contiguous strings of molecules. Let them

cut the paper as shown in the two circles in Fig. 1:--

[Illustration: Fig. 1.

Fig. 2.]

Then these cells are both rotating in the same direction, hence at C,

where they touch, their points of contact will be moving in opposite

directions, as shown by the arrow heads, and it is difficult to imagine

how such motion can continue; it would require the surfaces of the

cells to be perfectly smooth, and if this were so they would lose the

power of transmitting action from one cell to the next.

The cells A and B may be compared to two cog-wheels placed close

together, which we wish to turn in the same direction. If the cogs can

interlock, as in Fig. 2, this is impossible: consecutive wheels in the

train must move in opposite directions.

[Illustration: Fig. 3.]

But in many machines the desired end is attained by inserting between

the two wheels A and B a third idle wheel C, as shewn in Fig. 3. This

may be very small, its only function is to transmit the motion of A to

B in such a way that A and B may both turn in the same direction. It is

not necessary that there should be cogs on the wheels; if the surfaces

be perfectly rough, so that no slipping can take place, the same result

follows without the cogs.

Guided by this analogy Maxwell extended his model by supposing each

cell coated with a number of small particles which roll on its surface.

These particles play the part of the idle wheels in the machine, and by

their rolling merely enable the adjacent parts of two cells to move in

opposite directions.

Consider now a number of such cells and their idle wheels lying in a

plane, that of the paper, and suppose each cell is rotating with the

same uniform angular velocity about an axis at right angles to that

plane, each idle wheel will be acted on by two equal and opposite

forces at the ends of the diameter in which it is touched by the

adjacent cells; it will therefore be set in rotation, but there will be

no force tending to drive it onwards; it does not matter whether the

axis on which it rotates is free to move or fixed, in either case the

idle wheel simply rotates. But suppose now the adjacent cells are not

rotating at the same rate. In addition to its rotation the idle wheel

will be urged onward with a velocity which depends on the difference

between the rotations, and, if it can move freely, it will move on from

between the two cells. Imagine now that the interstices between the

cells are fitted with a string of idle wheels. So long as the adjacent

cells move with different velocity there will be a continual stream of

rolling particles or idle wheels between them. Maxwell in the paper

considered these rolling particles to be particles of electricity.

Their motion constitutes an electric current. In a uniform magnetic

field there is no electric current; if the strength of the field

varies, the idle wheels are set in motion and there may be a current.

These particles are very small compared with the magnetic vortices.

The mass of all the particles is inappreciable compared with the mass

of the vortices, and a great many vortices with their surrounding

particles are contained in a molecule of the medium; the particles

roll on the vortices without touching each other, so that so long as

they remain within the same molecule there is no loss of energy by

resistance. When, however, there is a current or general transference

of particles in one direction they must pass from one molecule to

another, and in doing so may experience resistance and generate heat.

Maxwell states that the conception of a particle, having its motion

connected with that of a vortex by perfect rolling contact, may appear

somewhat awkward. “I do not bring it forward,” he writes, “as a mode of

connection existing in Nature, or even as that which I would willingly

assent to as an electrical hypothesis. It is, however, a mode of

connection which is mechanically conceivable and easily investigated,

and it serves to bring out the actual mechanical connections between

the known electro-magnetic phenomena, so that I venture to say that

anyone who understands the provisional and temporary character of this

hypothesis will find himself rather helped than hindered by it in his

search after the true interpretation of the phenomena.”

The first part of the paper deals with the theory of magnetism; in the

second part the hypothesis is applied to the phenomena of electric

currents, and it is shown how the known laws of steady currents and

of electro-magnetic induction can be deduced from it. In Part III.,

published January and February, 1862, the theory of molecular vortices

is applied to statical electricity.

The distinction between a conductor and an insulator or dielectric

is supposed to be that in the former the particles of electricity

can pass with more or less freedom from molecule to molecule. In the

latter such transference is impossible, the particles can only be

displaced within the molecule with which they are connected; the cells

or vortices of the medium are supposed to be elastic, and to resist by

their elasticity the displacement of the particles within them. When

electrical force acts on the medium this displacement of the particles

within each molecule takes place until the stresses due to the elastic

reaction of the vortices balance the electrical force; the medium

behaves like an elastic body yielding to pressure until the pressure is

balanced by the elastic stress. When the electric force is removed the

cells or vortices recover their form, the electricity returns to its

former position.

In a medium such as this waves of periodic displacement could be

set up, and would travel with a velocity depending on its electric

properties. The value for this velocity can be obtained from electrical

observations, and Maxwell showed that this velocity, so found, was,

within the limits of experimental error, the same as that of light.

Moreover, the electrical oscillations take place, like those of light,

in the front of the wave. Hence, he concludes, “the elasticity of the

magnetic medium in air is the same as that of the luminiferous medium,

if these two coexistent, coextensive, and equally elastic media are not

rather one medium.”

The paper thus contains the first germs of the electro-magnetic theory

of light. Moreover, it is shown that the attraction between two small

bodies charged with given quantities of electricity depends on the

medium in which they are placed, while the specific inductive capacity

is found to be proportional to the square of the refractive index.

The fourth and final part of the paper investigates the propagation of

light in a magnetic field.

Faraday had shown that the direction of vibration in a wave of

polarised light travelling parallel to the lines of force in a magnetic

field is rotated by its passage through the field. The numerical laws

of this relation had been investigated by Verdet, and Maxwell showed

how his hypothesis of molecular vortices led to laws which agree in the

main with those found by Verdet.

He points out that the connection between magnetism and electricity

has the same mathematical form as that between certain other pairs

of phenomena, one of which has a _linear_ and the other a _rotatory_

character; and, further, that an analogy may be worked out assuming

either the linear character for magnetism and the rotatory character

for electricity, or the reverse. He alludes to Prof. Challis’ theory,

according to which magnetism is to consist in currents in a fluid

whose directions correspond with the lines of magnetic force, while

electric currents are supposed to be accompanied by, if not dependent

upon, a rotatory motion of the fluid about the axis of the current;

and to Von Helmholtz’s theory of a somewhat similar character. He then

gives his own reasons--agreeing with those of Sir W. Thomson (Lord

Kelvin)--for supposing that there must be a real rotation going on in

a magnetic field in order to account for the rotation of the plane of

polarisation, and, accepting these reasons as valid, he develops the

consequences of his theory with the results stated above.

His own verdict on the theory is given in the “Electricity and

Magnetism” (vol. ii., § 831, first edition, p. 416):--

“A theory of molecular vortices, which I worked out at

considerable length, was published in the _Phil. Mag._ for

March, April, and May, 1861; Jan. and Feb., 1862.

“I think we have good evidence for the opinion that some

phenomenon of rotation is going on in the magnetic field, that

this rotation is performed by a great number of very small

portions of matter, each rotating on its own axis, this axis

being parallel to the direction of the magnetic force, and that

the rotations of these different vortices are made to depend on

one another by means of some kind of mechanism connecting them.

“The attempt which I then made to imagine a working model of

this mechanism must be taken for no more than it really is,

a demonstration that mechanism may be imagined capable of

producing a connection mechanically equivalent to the actual

connection of the parts of the electro-magnetic field. The

problem of determining the mechanism required to establish a

given species of connection between the motions of the parts of

a system always admits of an infinite number of solutions. Of

these, some may be more clumsy or more complex than others, but

all must satisfy the conditions of mechanism in general.

“The following results of the theory, however, are of higher

value:--

“(1) Magnetic force is the effect of the centrifugal force of

the vortices.

“(2) Electro-magnetic induction of currents is the effect of

the forces called into play when the velocity of the vortices

is changing.

“(3) Electromotive force arises from the stress on the

connecting mechanism.

“(4) Electric displacement arises from the elastic yielding of

the connecting mechanism.”

In studying this part of Maxwell’s work, it must clearly be remembered

that he did not look upon the ether as a series of cog-wheels with

idle wheels between, or anything of the kind. He devised a mechanical

model of such cogs and idle wheels, the properties of which would in

some respects closely resemble those of the ether; from this model he

deduced, among other things, the important fact that electric waves

would travel outwards with the velocity of light. Other such models

have been devised since his time to illustrate the same laws. Prof.

Fitzgerald has actually constructed one of wheels connected together by

elastic bands, which shows clearly the kind of processes which Maxwell

supposed to go on in a dielectric when under electric force. Professor

Lodge, in his book, “Modern Views of Electricity,” has very fully

developed a somewhat different arrangement of cog-wheels to attain the

same result.

Maxwell’s predictions as to the propagation of electric waves have

in recent days received their full verification in the brilliant

experiments of Hertz and his followers; it remains for us, before

dealing with these, to trace their final development in his hands.

The papers we have been discussing were perhaps too material to receive

the full attention they deserved; the ether is not a series of cogs,

and electricity is something different from material idle wheels. In

his paper on “The Dynamical Theory of the Electro-magnetic Field,”

_Phil. Trans._, 1864, Maxwell treats the same questions in a more

general manner. On a former occasion he says, “I have attempted to

describe a particular kind of motion and a particular kind of strain

so arranged as to account for the phenomena. In the present paper I

avoid any hypothesis of this kind; and in using such words as electric

momentum and electric elasticity in reference to the known phenomena of

the induction of currents and the polarisation of dielectrics, I wish

merely to direct the mind of the reader to mechanical phenomena, which

will assist him in understanding the electrical ones. All such phrases

in the present paper are to be considered as illustrative and not as

explanatory.” He then continues:--

“In speaking of the energy of the field, however, I wish to

be understood literally. All energy is the same as mechanical

energy, whether it exists in the form of motion or in that of

elasticity, or in any other form.

“The energy in electro-magnetic phenomena is mechanical energy.

The only question is, Where does it reside?

“On the old theories it resides in the electrified bodies,

conducting circuits, and magnets, in the form of an unknown

quality called potential energy, or the power of producing

certain effects at a distance. On our theory it resides in

the electro-magnetic field, in the space surrounding the

electrified and magnetic bodies, as well as in those bodies

themselves, and is in two different forms, which may be

described without hypothesis as magnetic polarisation and

electric polarisation, or, according to a very probable

hypothesis, as the motion and the strain of one and the same

medium.

“The conclusions arrived at in the present paper are

independent of this hypothesis, being deduced from experimental

facts of three kinds:--

“(1) The induction of electric currents by the increase or

diminution of neighbouring currents according to the changes in

the lines of force passing through the circuit.

“(2) The distribution of magnetic intensity according to the

variations of a magnetic potential.

“(3) The induction (or influence) of statical electricity

through dielectrics.

“We may now proceed to demonstrate from these principles the

existence and laws of the mechanical forces, which act upon

electric currents, magnets, and electrified bodies placed in

the electro-magnetic field.”

In his introduction to the paper, he discusses in a general way the

various explanations of electric phenomena which had been given, and

points out that--

“It appears, therefore, that certain phenomena in electricity

and magnetism lead to the same conclusion as those of optics,

namely, that there is an ætherial medium pervading all bodies,

and modified only in degree by their presence; that the parts

of this medium are capable of being set in motion by electric

currents and magnets; that this motion is communicated from

one part of the medium to another by forces arising from the

connection of those parts; that under the action of these

forces there is a certain yielding depending on the elasticity

of these connections; and that, therefore, energy in two

different forms may exist in the medium, the one form being

the actual energy of motion of its parts, and the other being

the potential energy stored up in the connections in virtue of

their elasticity.

“Thus, then, we are led to the conception of a complicated

mechanism capable of a vast variety of motion, but at the

same time so connected that the motion of one part depends,

according to definite relations, on the motion of other parts,

these motions being communicated by forces arising from the

relative displacement of the connected parts, in virtue of

their elasticity. Such a mechanism must be subject to the

general laws of dynamics, and we ought to be able to work out

all the consequences of its motion, provided we know the form

of the relation between the motions of the parts.”

These general laws of dynamics, applicable to the motion of any

connected system, had been developed by Lagrange, and are expressed

in his generalised equations of motion. It is one of Maxwell’s chief

claims to fame that he saw in the electric field a connected system to

which Lagrange’s equations could be applied, and that he was able to

deduce the mechanical and electrical actions which take place by means

of fundamental propositions of dynamics.

The methods of the paper now under discussion were developed further

in the “Treatise on Electricity and Magnetism,” published in 1873; in

endeavouring to give some slight account of Maxwell’s work, we shall

describe it in the form it ultimately took.

The task which Maxwell set himself was a double one; he had first to

express in symbols, in as general a form as possible, the fundamental

laws of electro-magnetism as deduced from experiments, chiefly

the experiments of Faraday, and the relations between the various

quantities involved; when this was done he had to show how these laws

could be deduced from the general dynamical laws applicable to any

system of moving bodies.

There are two classes of phenomena, electric and magnetic, which have

been known from very early times, and which are connected together.

When a piece of sealing-wax is rubbed it is found to attract other

bodies, it is said to exert electric force throughout the space

surrounding it; when two different metals are dipped in slightly

acidulated water and connected by a wire, certain changes take place

in the plates, the water, the wire, and the space round the wire,

electric force is again exerted and a current of electricity is said

to flow in the wire. Again, certain bodies, such as the lodestone, or

pieces of iron and steel which have been treated in a certain manner,

exhibit phenomena of action at a distance: they are said to exert

magnetic force, and it is found that this magnetic force exists in the

neighbourhood of an electric current and is connected with the current.

Again, when electric force is applied to a body, the effects may be in

part electrical, in part mechanical; the electrical state of the body

is in general changed, while in addition, mechanical forces tending to

move the body are set up. Experiment must teach us how the electrical

state depends on the electric force, and what is the connection

between this electric force and the magnetic forces which may, under

certain circumstances, be observed. Now, in specifying the electric

and magnetic conditions of the system, various other quantities, in

addition to the electric force, will have to be introduced; the first

step is to formulate the necessary quantities, and to determine the

relations between them and the electric force.

Consider now a wire connecting the two poles of an electric battery--in

its simplest form, a piece of zinc and a piece of copper in a vessel

of dilute acid--electric force is produced at each point of the wire.

Let us suppose this force known; an electric current depending on the

material and the size of the wire flows along it, its value can be

determined at each point of the wire in terms of the electric force

by Ohm’s law. If we take either this current or the electric force

as known, we can determine by known laws the electric and magnetic

conditions elsewhere. If we suppose the wire to be straight and very

long, then, so long as the current is steady and we neglect the small

effect due to the electrostatic charge on the wire, there is no

electric force outside the wire. There is, however, magnetic force,

and it is found that the lines of magnetic force are circles round the

wire. It is found also that the work done in travelling once completely

round the wire against the magnetic force is measured by the current

flowing through the wire, and is obtained in the system of units

usually adopted by multiplying the current by 4π. This last result then

gives us one of the necessary relations, that between the magnetic

force due to a current and the strength of the current.

Again, consider a steady current flowing in a conductor of any form or

shape, the total flow of current across any section of the conductor

can be measured in various ways, and it is found that at any time this

total flow is the same for each section of the conductor. In this

respect the flow of a current resembles that of an incompressible

fluid through a pipe; where the pipe is narrow the velocity of flow

is greater than it is where the pipe is broad, but the total quantity

crossing each section at any given instant is the same.

Consider now two conducting bodies, two spheres, or two flat plates

placed near together but insulated. Let each conductor be connected

to one of the poles of the battery by a conducting wire. Then, for a

very short interval after the contact is made, it is found that there

is a current in each wire which rapidly dies away to zero. In the

neighbourhood of the balls there is electric force; the balls are said

to be charged with electricity, and the lines of force are curved lines

running from one ball to the other. It is found that the balls slightly

attract each other, and the space between them is now in a different

condition from what it was before the balls were charged. According

to Maxwell, _Electric Displacement_ has been produced in this space,

and the electric displacement at each point is proportional to the

electric force at that point.

Thus, (i) when electric force acts on a conductor, it produces a

current, the current being by Ohm’s law proportional to the force:

(ii) when it acts on an insulator it produces electric displacement,

and the displacement is proportional to the force; while (iii) there

is magnetic force in the neighbourhood of the current, and the work

done in carrying a magnetic pole round any complete circuit linked

with the current is proportional to the current. The first two of

these principles give us two sets of equations connecting together the

electric force and the current in a conductor or the displacement in a

dielectric respectively; the third connects the magnetic force and the

current.

Now let us go back to the variable period when the current is flowing

in the wires; and to make ideas precise, let the two conductors be two

equal large flat plates placed with their faces parallel, and at some

small distance apart. In this case, when the plates are charged, and

the current has ceased, the electric displacement and the force are

confined almost entirely to the space between the plates. During the

variable period the total flow at any instant across each section of

the wire is the same, but in the ordinary sense of the word there is no

flow of electricity across the insulating medium between the plates.

In this space, however, the electric displacement is continuously

changing, rising from zero initially to its final steady value when the

current ceases. It is a fundamental part of Maxwell’s theory that this

variation of electric displacement is equivalent in all respects to a

current. The current at any point in a dielectric is measured by the

rate of change of displacement at that point.

Moreover, it is also an essential point that if we consider any section

of the dielectric between the two plates, the rate of change of the

total displacement across this section is at each moment equal to the

total flow of current across each section of the conducting wire.

Currents of electricity, therefore, including displacement

currents, always flow in closed circuits, and obey the laws of an

incompressible fluid in that the total flow across each section of the

circuit--conducting or dielectric--is at any moment the same.

It should be clearly remembered that this fundamental hypothesis of

Maxwell’s theory is an assumption only to be justified by experiment.

Von Helmholtz, in his paper on “The Equations of Motion of Electricity

for Bodies at Rest,” formed his equations in an entirely different

manner from Maxwell, and arrived at results of a more general

character, which do not require us to suppose that currents flow always

in closed circuits, but permit of the condensation of electricity at

points in the circuit where the conductors end and the non-conducting

part of the circuit begins. We leave for the present the question which

of the two theories, if either, represents the facts.

We have obtained above three fundamental relations--(i) that between

electric force and electric current in a conductor; (ii) that between

electric force and electric displacement in a dielectric; (iii) that

between magnetic force and the current which gives rise to it. And

we have seen that an electric current--_i.e._ in a dielectric the

variation of the strength of an electric field of force--gives rise

to magnetic force. Now, magnetic force acting on a medium produces

“magnetic displacement,” or magnetic induction, as it is called. In

all media except iron, nickel, cobalt, and a few other substances, the

magnetic induction is proportional to the magnetic force, and the ratio

between the magnetic induction produced by a given force and the force

is found to be very nearly the same for all such media. This ratio is

known as the permeability, and is generally denoted by the symbol μ.

A relation reciprocal to that given in (iii) above might be

anticipated, and was, in fact, discovered by Faraday. Changes in a

field of magnetic induction give rise to electric force, and hence to

displacement currents in a dielectric or to conduction currents in a

conductor. In considering the relation between these changes and the

electric force, it is simplest at first not to deal with magnetic

matter such as iron, nickel, or cobalt; and then we may say that (iv)

the work which at any instant would be done in carrying a unit quantity

of electricity round a closed circuit in a magnetic field against the

electric forces due to the field is equal to the rate at which the

total magnetic induction which threads the circuit is being decreased.

This law, summing up Faraday’s experiments on electro-magnetic

induction, gives a fourth principle, leading to a fourth series of

equations connecting together the electric and magnetic quantities

involved.

The equations deduced from the above four principles, together with the

condition implied in the continuity of an electric current, constitute

Maxwell’s equations of the electro-magnetic field.

If we are dealing only with a dielectric medium, the reciprocal

relation between the third and fourth principle may be made more clear

by the following statement:--

(A) The work done at any moment in carrying a unit quantity of

magnetism round a closed circuit in a field in which electric

displacement is varying, is equal to the rate of change of the total

electric displacement through the circuit multiplied by 4 π.[62]

(B) The work done at any moment in carrying a unit quantity of

electricity round a circuit in a field in which the magnetic induction

is varying, is equal to the rate of change of the total magnetic

induction through the circuit.

From these two principles, combined with the laws connecting electric

force and displacement, magnetic force and induction, and with the

condition of continuity, Maxwell obtained his equations of the field.

Faraday’s experiments on electro-magnetic induction afford the proof of

the truth of the fourth principle. It follows from those experiments

that when the number of lines of magnetic induction which are linked

with any closed circuit are made to vary, an induced electromotive

force is brought into play round that circuit. This electromotive force

is, according to Faraday’s results, measured by the rate of decrease

in the number of lines of magnetic induction which thread the circuit.

Maxwell applies this principle to all circuits, whether conducting or

not.

In obtaining equations to express in symbols the results of the fourth

principle just enunciated, Maxwell introduces a new quantity, to which

he gives the name of the “vector potential.” This quantity appears in

his analysis, and its physical meaning is not at first quite clear.

Professor Poynting has, however, put Maxwell’s principles in a slightly

different form, which enables us to see definitely the meaning of the

vector potential, and to deduce Maxwell’s equations more readily from

the fundamental statements.

We are dealing with a circuit with which lines of magnetic induction

are linked, while the number of such lines linked with the circuit is

varying. Now, let us suppose the variation to take place in consequence

of the lines of induction moving outwards or inwards, as the case may

be, so as to cut the circuit. Originally there are none linked with

the circuit. As the magnetic field has grown to its present strength

lines of magnetic induction have moved inwards. Each little element of

the circuit has been cut by some, and the total number linked with the

circuit can be found by adding together those cut by each element. Now,

Professor Poynting’s statement of Maxwell’s fourth principle is that

the electrical force in the direction of any element of the circuit is

found by dividing by the length of the element the number of lines of

magnetic induction which are cut in one second by it.

Moreover, the total number of lines of magnetic induction which have

been cut by an element of unit length is defined as the component

of the vector potential in the direction of the element; hence the

electrical force in any direction is the rate of decrease of the

component of the vector potential in that direction. We have thus a

physical meaning for the vector potential, and shall find that in the

dynamical theory this quantity is of great importance.

Professor Poynting has modified Maxwell’s third principle in a similar

manner; he looks upon the variation in the electric displacement as

due to the motion of tubes of electric induction,[63] and the magnetic

force along any circuit is equal to the number of tubes of electric

induction cutting or cut by unit length of the circuit per second,

multiplied by 4π.

From the equations of the field, as found by Maxwell, it is possible

to derive two sets of symmetrical equations. The one set connects the

rate of change of the electric force with quantities depending on the

magnetic force; the other set connects in a similar manner the rate of

change of the magnetic force with quantities depending on the electric

force. Several writers in recent years adopt these equations as the

fundamental relations of the field, establishing them by the argument

that they lead to consequences which are found to be in accordance with

experiment.

We have endeavoured to give some account of Maxwell’s historical

method, according to which the equations are deduced from the laws of

electric currents and of electro-magnetic induction derived directly

from experiment.

While the manner in which Maxwell obtained his equations is all his

own, he was not alone in stating and discussing general equations

of the electro-magnetic field. The next steps which we are about to

consider are, however, in a special manner due to him. An electrical

or magnetic system is the seat of energy; this energy is partly

electrical, partly magnetic, and various expressions can be found for

it. In Maxwell’s theory it is a fundamental assumption that energy has

position. “The electric and magnetic energies of any electro-magnetic

system,” says Professor Poynting, “reside, therefore, somewhere in the

field.” It follows from this that they are present wherever electric

and magnetic force can be shown to exist. Maxwell showed that all the

electric energy is accounted for by supposing that in the neighbourhood

of a point at which the electric force is R there is an amount of

energy per unit of volume equal to KR²/8π, K being the inductive

capacity of the medium, while in the neighbourhood of a point at which

the magnetic force is H, the magnetic energy per unit of volume is

μH²/8π, μ being the permeability. He supposes, then, that at each point

of an electro-magnetic system energy is stored according to these

laws. It follows, then, that the electro-magnetic field resembles a

dynamical system in which energy is stored. Can we discover more of

the mechanism by which the actions in the field are maintained? Now

the motion of any point of a connected system depends on that of other

points of the system; there are generally, in any machine, a certain

number of points called driving-points, the motion of which controls

the motion of all other parts of the machine; if the motion of the

driving-points be known, that of any other point can be determined.

Thus in a steam engine the motion of a point on the fly-wheel can be

found if the motion of the piston and the connections between the

piston and the wheel be known.

In order to determine the force which is acting on any part of the

machine we must find its momentum, and then calculate the rate at

which this momentum is being changed. This rate of change will give us

the force. The method of calculation which it is necessary to employ

was first given by Lagrange, and afterwards developed, with some

modifications, by Hamilton. It is usually referred to as Hamilton’s

principle; when the equations in the original form are used they are

known as Lagrange’s equations.

Now Maxwell showed how these methods of calculation could be applied

to the electro-magnetic field. The energy of a dynamical system is

partly kinetic, partly potential. Maxwell supposes that the magnetic

energy of the field is kinetic energy, the electric energy potential.

When the kinetic energy of a system is known, the momentum of any

part of the system can be calculated by recognised processes. Thus if

we consider a circuit in an electro-magnetic field we can calculate

the energy of the field, and hence obtain the momentum corresponding

to this circuit. If we deal with a simple case in which the conducting

circuits are fixed in position, and only the current in each circuit is

allowed to vary, the rate of change of momentum corresponding to any

circuit will give the force in that circuit. The momentum in question

is electric momentum, and the force is electric force. Now we have

already seen that the electric force at any point of a conducting

circuit is given by the rate of change of the vector potential in the

direction considered. Hence we are led to identify the vector potential

with the electric momentum of our dynamical system; and, referring to

the original definition of vector potential, we see that the electric

momentum of a circuit is measured by the number of lines of magnetic

induction which are interlinked with it.

Again, the kinetic energy of a dynamical system can be expressed in

terms of the squares and products of the velocities of its several

parts. It can also be expressed by multiplying the velocity of each

driving-point by the momentum corresponding to that driving-point, and

taking half the sum of the products. Suppose, now, we are dealing with

a system consisting of a number of wire circuits in which currents are

running, and let us suppose that we may represent the current in each

wire as the velocity of a driving-point in our dynamical system. We can

also express in terms of these currents the electric momentum of each

wire circuit; let this be done, and let half the sum of the products of

the corresponding velocities and momenta be formed.

In maintaining the currents in the wires energy is needed to supply

the heat which is produced in each wire; but in starting the currents

it is found that more energy is needed than is requisite for the

supply of this heat. This excess of energy can be calculated, and when

the calculation is made it is found that the excess is equal to half

the sum of the products of the currents and corresponding momenta.

Moreover, if this sum be expressed in terms of the magnetic force, it

is found to be equal to μ H²/8 π, which is the magnetic energy of the

field. Now, when a dynamical system is set in motion against known

forces, more energy is supplied than is needed to do the work against

the forces; this excess of energy measures the kinetic energy acquired

by the system.

Hence, Maxwell was justified in taking the magnetic energy of the field

as the kinetic energy of the mechanical system, and if the strengths

of the currents in the wires be taken to represent the velocities of

the driving-points, this energy is measured in terms of the electrical

velocities and momenta in exactly the same way as the energy of a

mechanical system is measured in terms of the velocities and momenta of

its driving-points.

The mechanical system in which, according to Maxwell, the energy is

stored is the ether. A state of motion or of strain is set up in the

ether of the field. The electric forces which drive the currents, and

also the mechanical forces acting on the conductors carrying the

currents, are due to this state of motion, or it may be of strain, in

the ether. It must not be supposed that the term electric displacement

in Maxwell’s mind meant an actual bodily displacement of the particles

of the ether; it is in some way connected with such a material

displacement. In his view, without motion of the ether particles

there would be no electric action, but he does not identify electric

displacement and the displacement of an ether particle.

His mechanical theory, however, does account for the electro-magnetic

forces between conductors carrying currents. The energy of the system

depends on the relative positions of the currents which form part of

it. Now, any conservative mechanical system tends to set itself in

such a position that its potential energy is least, its kinetic energy

greatest. The circuits of the system, then, will tend to set themselves

so that the electro-kinetic energy of the system may be as large as

possible; forces will be needed to hold them in any position in which

this condition is not satisfied.

We have another proof of the correctness of the value found for the

energy of the field in that the forces calculated from this value agree

with those which are determined by direct experiment.

Again, the forces applied at the various driving-points are transmitted

to other points by the connections of the machine; the connections

are thrown into a state of strain; stress exists throughout their

substance. When we see the piston-rod and the shaft of an engine

connected by the crank and the connecting-rod, we recognise that the

work done on the piston is transmitted thus to the shaft. So, too, in

the electro-magnetic field, the ether forms the connection between the

various circuits in the field; the forces with which those circuits

act on each other are transmitted from one circuit to another by the

stresses set up in the ether.

To take another instance, consider the electrostatic attraction between

two charged bodies. Let us suppose the bodies charged by connecting

each to the opposite pole of a battery; a current flows from the

battery setting up electric displacement in the space between the

bodies, and throwing the ether into a state of strain. As the strain

increases the current gets less; the reaction resulting from the strain

tends to stop it, until at last this reaction is so great that the

current is stopped. When this is the case the wires to the battery may

be removed, provided this is done without destroying the insulation of

the bodies; the state of strain will remain and shows itself in the

attraction between the balls.

Looking at the problem in this manner, we are face to face with two

great questions--the one, What is the state of strain in the ether

which will enable it to produce the observed electrostatic attractions

and repulsions between charged bodies? and the other, What is the

mechanical structure of the ether which would give rise to such a state

of strain as will account for the observed forces? Maxwell gives one

answer to the first question; it is not the only answer which could

be given, but it does account for the facts. He failed to answer the

second. He says (“Electricity and Magnetism,” vol. i. p. 132):--

“It must be carefully borne in mind that we have made only

one step in the theory of the action of the medium. We have

supposed it to be in a state of stress, but have not in

any way accounted for this stress, or explained how it is

maintained.... I have not been able to make the next step,

namely, to account by mechanical considerations for these

stresses in the dielectric.”

Faraday had pointed out that the inductive action between two bodies

takes place along the lines of force, which tend to shorten along their

length and to spread outwards in other directions. Maxwell compares

them to the fibres of a muscle, which contracts and at the same time

thickens when exerting force. In the electric field there is, on

Maxwell’s theory, a tension along the lines of electric force and a

pressure at right angles to those lines. Maxwell proved that a tension

K R²/8 π along the lines of force, combined with an equal pressure

in perpendicular directions, would maintain the equilibrium of the

field, and would give rise to the observed attractions or repulsions

between electrified bodies. Other distributions of stress might be

found which would lead to the same result. The one just stated will

always be connected with Maxwell’s name. It will be noticed that the

tension along the lines of force and the pressure at right angles to

them are each numerically equal to the potential energy stored per unit

of volume in the field. The value of each of the three quantities is K

R²/8 π.

In the same way, in a magnetic field, there is a state of stress, and

on Maxwell’s theory this, too, consists of a tension along the lines

of force and an equal pressure at right angles to them, the values of

the tension and the pressure being each equal to that of the magnetic

energy per unit of volume, or μH²/8π.

In a case in which both electric and magnetic force exists, these two

states of stress are superposed. The total energy per unit of volume

is KR²/8π + μH²/8π; the total stress is made up of tensions KR²/8π and

μH²/8π along the lines of electric and magnetic force respectively, and

equal pressures at right angles to these lines.

We see, then, from Maxwell’s theory, that electric force produced at

any given point in space is transmitted from that point by the action

of the ether. The question suggests itself, Does the transmission take

time, and if so, does it proceed with a definite velocity depending on

the nature of the medium through which the change is proceeding?

According to the molecular-vortex theory, we have seen that waves of

electric force are transmitted with a definite velocity. The more

general theory developed in the “Electricity and Magnetism” leads to

the same result. Electric force produced at any point travels outwards

from that point with a velocity given by 1/√(Kμ). At a distant point

the force is zero, until the disturbance reaches it. If the disturbance

last only for a limited interval, its effects will at any future time

be confined to the space within a spherical shell of constant thickness

depending on the interval; the radii of this shell increase with

uniform speed 1/√(Kμ).

If the initial disturbance be periodic, periodic waves of electric

force will travel out from the centre, just as waves of sound travel

out from a bell, or waves of light from a candle flame. A wire carrying

an alternating current may be such a source of periodic disturbance,

and from the wire waves travel outwards into space.

Now, it is known that in a sound wave the displacements of the air

particles take place in the direction in which the wave is travelling;

they lie at right angles to the wave front, and are spoken of as

longitudinal. In light waves, on the other hand, the displacements are,

as Fresnel proved, in the wave front, at right angles, that is, to the

direction of propagation; they are transverse.

Theory shows that in general both these waves may exist in an elastic

solid body, and that they travel with different velocities. Of which

nature are the waves of electric displacement in a dielectric? It

can be shewn to follow as a necessary consequence of Maxwell’s views

as to the closed character of all electric currents, that waves of

electric displacement are transverse. Electric vibrations, like those

of light, are in the wave front and at right angles to the direction

of propagation; they depend on the rigidity or quasi-rigidity of the

medium through which they travel, not on its resistance to compression.

Again, an electric current, whether due to variation of displacement

in a dielectric or to conduction in a conductor, is accompanied by

magnetic force. A wave of periodic electric displacement, then, will be

also a wave of periodic magnetic force travelling at the same rate;

and Maxwell shewed that the direction of this magnetic force also

lies in the wave front, and is always at right angles to the electric

displacement. In the ordinary theory of light the wave of linear

displacement is accompanied by a wave of periodic angular twist about

a direction lying in the wave front and perpendicular to the linear

displacement.

In many respects, then, waves of electric displacement resemble waves

of light, and, indeed, as we proceed we shall find closer connections

still. Hence comes Maxwell’s electro-magnetic theory of light.

It is only in dielectric media that electric force is propagated by

wave motion. In conductors, although the third and fourth of Maxwell’s

principles given on page 185 still are true, the relation between

the electric force and the electric current differs from that which

holds in a dielectric. Hence the equations satisfied by the force are

different. The laws of its propagation resemble those of the conduction

of heat rather than those of the transmission of light.

Again, light travels with different velocities in different transparent

media. The velocity of electric waves, as has been stated, is equal to

1/√(μK); but in making this statement it is assumed that the simple

laws which hold where there is no gross matter--or, rather, where

air is the only dielectric with which we are concerned--hold also in

solid or liquid dielectrics. In a solid or a liquid, as in vacuo, the

waves are propagated by the ether. We assume, as a first step towards

a complete theory, that so far as the electric waves are concerned

the sole effect produced by the matter shews itself in a change of

inductive capacity or of permeability. It is not likely that such a

supposition should be the whole truth, and we may, therefore, expect

results deduced from it to be only approximation to the true result.

Now, electro-magnetic experiments show that, excluding magnetic

substances, the permeability of all bodies is very nearly the same,

and differs very slightly from that of air. The inductive capacity,

however, of different bodies is different, and hence the velocity with

which electro-magnetic waves travel differs in different bodies.

But the refraction of waves of light depends on the fact that light

travels with different velocities in different media; hence we should

expect to have waves of electric displacement reflected and refracted

when they pass from one dielectric, such as air, to another, such as

glass or gutta-percha; moreover, for light the refractive index of

a medium such as glass is the ratio of the velocity in air to the

velocity in the glass.

Thus the electrical refractive index of glass is the ratio of the

velocity of electric waves in air to their velocity in glass.

Now let K₀ be the inductive capacity of air, K₁ that of glass, taking

the permeability of air and glass to be the same, we have the result

that--

Electrical refractive index = √(K₁/K₀).

But the ratio of the inductive capacity of glass to that of air is

known as the specific inductive capacity of glass.

Hence, the specific inductive capacity of any medium is equal to the

square of the electrical refractive index of that medium.

Since Maxwell’s time the mathematical laws of the reflexion and

refraction of electric waves have been investigated by various writers,

and it has been shewn that they agree exactly with those enunciated by

Fresnel for light.

Hitherto we have been discussing the propagation of electric waves

in an isotropic medium, one which has identical properties in all

directions about a point. Let us now consider how these laws are

modified if the dielectric be crystalline in structure.

Maxwell assumes that the crystalline character of the dielectric can

be sufficiently represented by supposing the inductive capacity to

be different in different directions; experiments have since shewn

that this is true for crystals such as Iceland Spar and Aragonite;

he assumes also, and this, too, is justified by experiment, that the

magnetic permeability does not depend on the direction. It follows

from these assumptions that a crystal will produce double refraction

and polarisation of electric waves which fall upon it, and, further,

that the laws of double refraction will be those given by Fresnel for

light waves in a doubly refracting medium. There will be two waves in

the crystal. The disturbance in each of these will be plane polarised;

their velocity and the position of their plane of polarisation can be

found from the direction in which they are travelling by Fresnel’s

construction exactly.

Maxwell’s theory, then, would appear to indicate some close connection

between electric waves and those of light. Faraday’s experiments on

the rotation of the plane of polarisation by magnetic force shew one

phenomenon in which the two are connected, and Maxwell endeavoured to

apply his theory to explain this. Here, however, it became necessary

to introduce an additional hypothesis--there must be some connection

between the motion of the ether to which magnetic force is due and that

which constitutes light. It is impossible to give a mechanical account

of the rotation of the plane of polarisation without some assumption as

to the relation between these two kinds of motion. Maxwell, therefore,

supposes the linear displacements of a point in the ether to be those

which give rise to light, while the components of the magnetic force

are connected with these in the same way as the components of a vortex

in a liquid in vortex motion are connected with the displacements of

the liquid. He further assumes the existence of a term of special form

in the expression for the kinetic energy, and from these assumptions he

deduces the laws of the propagation of polarised light in a magnetic

field. These laws agree in the main with the results of Verdet’s

experiments.