Chapter IX: Scientific Work.--Electrical Theories (1)

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:--

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.

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--