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lines must be repeated.

Thus in the figure of five points, each point is the meeting place of four
lines. The forces in these lines may be represented by five gauche quadrilaterals
(that is, quadrilaterals not in one plane) ; and one of these being chosen, the
other four may be applied to its sides and to each other so as to form five
sides of a gauche hexahedron. The sixth side, that opposite the original quad-
rilateral, will be a parallelogram, the opposite sides of which are repetitions of
the same line.

We have thus a complete but redundant diagram of forces consisting of
eight points joined by twelve lines, two pairs of the lines being repetitions.
This is a more convenient though less elegant construction of a diagram of
forces, and it never becomes geometrically impossible as long as the problem is
mechanically possible, however complicated the original figure may be.



[From the Royal Society Transactions, Vol. CLV.]
XXV. A Dynamical Theory of tJie Electromagnetic Field.



(Received October 27,— Read December 8, 1864.)



PAET I.

INTRODUCTORY.

(1) The most obvious mechanical phenomenon in electrical and magnetical
experiments is the mutual action by which bodies in certain states set each
other in motion while still at a sensible distance from each other. The first
step, therefore, in reducing these phenomena into scientific form, is to ascertain
the magnitude and direction of the force acting between the bodies, and when
it is found that this force depends in a certain way upon the relative position
of the bodies and on their electric or magnetic condition, it seems at first sight
natural to explain the facts by assuming the existence of something either at
rest or in motion in each body, constituting its electric or magnetic state, and
capable of acting at a distance according to mathematical laws.

In this way mathematical theories of statical electricity, of magnetism, of
the mechanical action between conductors carrying currents, and of the induction
of currents have been formed. In these theories the force acting between the
two bodies is treated with reference only to the condition of the bodies and
their relative position, and without any express consideration of the surrounding
medium.

These theories assume, more or less explicitly, the existence of substances
the particles of which have the property of acting on one another at a distance
by attraction or repulsion. The most complete development of a theory of this



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 527

kind is tliat of M. W. Weber*, who has made the same theory include elec-
trostatic and electromagnetic phenomena.

In doing so, however, he has found it necessary to assume that the force
between two electric particles depends on their relative velocity, as well as on
their distance.

This theor}% as developed by MM. W. Weber and C. Neumann t, is ex-
ceedingly ingenious, and wonderfully comprehensive in its application to the
phenomena of statical electricity, electromagnetic attractions, induction of currents
and diamagnetic phenomena ; and it comes to us with the more authority, as
it has serv-ed to guide the speculations of one who has made so great an
advance in the practical part of electric science, both by introducing a consistent
system of units in electrical measurement, and by actually determining electrical
quantities with an accuracy hitherto unknown.

(2) The mechanical difficulties, however, which are involved in the assump-
tion of particles acting at a distance with forces which depend on their velocities
are such as to prevent me from considering this theory as an ultimate one.
though it may have been, and may yet be useful in leading to the coordina-
tion of phenomena,

I have therefore preferred to seek an explanation of the fact in another
direction, by supposing them to be produced by actions which go on in the
surrounding medium as well as in the excited bodies, and endeavourino- to
explain the action between distant bodies without assuming the existence of
forces capable of acting directly at sensible distances.

(3) The theory I propose may therefore be called a theory of the EJectro-
mcignetic Field, because it has to do with the space in the neighbourhood of
the electric or magnetic bodies, and it may be called a Dynamical Theory,
because it assumes that in that space there is matter in motion, by which
the observed electromagnetic phenomena are produced.

(4) The electromagnetic field is that part of space which contains and
surrounds bodies in electric or magnetic conditions.

* " Electrodynamische Maassbestimmungen." Leipzic Trans. Vol. i. 1849, and Taylor'.s Scieudjir
Memoirs, Vol. v. art. xiv.

t JUxplirnre tentntur quomodo fiat ut lucis planum polarizationis per vires electricas vel matfVJ'tints
declinetur. — Halis Saxonum, 1858.



528 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

It may be filled with any kind of matter, or we may endeavour to render
it empty of all gross matter, as in the case of Geissler's tubes and other po-
called vacua.

There is always, however, enough of matter left to receive and transmit
the undulations of light and heat, and it is because the transmission of these
radiations is not greatly altered when transparent bodies of measurable density
are substituted for the so-called vacuum, that we are obliged to admit that the
undulations are those of an sethereal substance, and not of the gross matter,
the presence of which merely modifies in some way the motion of the sether.

"We have therefore some reason to beheve, from the phenomena of light
and heat, that there is an ^ethereal medium filling space and permeating bodies,
capable of being set in motion and of transmitting that motion from one part
to another, and of communicating that motion to gross matter so as to heat
it and afiect it in various ways.

(5) Now the energy communicated to the body in heating it must have
formerly existed in the moving medium, for the undulations had left the source
of heat some time before they reached the body, and during that time the
energy must have been half in the form of motion of the medium and half in
the form of elastic resilience. From these considerations Professor W. Thomson
has argued'", that the medium must have a density capable of comparison with
that of gross matter, and has even assigned an inferior limit to that density.

(6) We may therefore receive, as a datum derived from a branch of science
independent of that with which we have to deal, the existence of a pervading
medium, of small but real density, capable of being set in motion, and of trans-
mitting motion from one part to another with great, but not infinite, velocity.

Hence the parts of this medium must be so connected that the motion of
one pai-t depends in some way on the motion of the rest; and at the same
time these connexions must be capable of a certain kind of elastic yielding,
since the communication of motion is not instantaneous, but occupies time.

The medium is therefore capable of receiving and storing up two kinds of
energy, namely, the "actual" energy depending on the motions of its parts, and
"potential" energy, consisting of the work which the medium will do in recover-
ing from displacement in virtue of its elasticity.

* "On the Possible Density of the Lnminiferous Medium, and on the Mechanical Value of a
Cubic Mile of Sunlight," Trarisactwis of the Royal Society of Edinburgh (1854), p. 57.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 520

The propagation of undulations consists in the continual transformation of
one of these forms of energy into the other alternately, and at any instant
the amount of energy in the wliole medium is equally divided, so that half
is energy of motion, and half is elastic resilience.

(7) A medium having such a constitution may be capable of other kinds
of motion and displacement than those which produce the phenomena of light
and heat, and some of these may be of such a kind that they may be
evidenced to our senses by the phenomena they produce.

(8) Now we know that the luminiferous medium is in certain cases acted
on by magnetism ; for Faraday* discovered that when a plane polarized ray
traverses a transparent diamagnetic medium in the direction of the lines of
magnetic force produced by magnets or currents in the neighbourhood, the plane
of polarization is caused to rotate.

This rotation is always in the direction in which positive electricity must
be carried round the diamagnetic body in order to produce the actual mag-
netization of the field.

M. Verdetf has since discovered that if a paramagnetic body, such as
solution of perchloride of iron in ether, be substituted for the diamagnetic body,
the rotation is in the opposite direction.

Now Professor W. Thomson J has pointed out that no distribution of forces
actincr between the parts of a medium whose only motion is that of the lumi-
nous vibrations, is sufficient to account for the phenomena, but that we must
admit the existence of a motion in the medium depending on the magnetization,
in addition to the vibratory motion which constitutes light.

It is true that the rotation by magnetism of the plane of polarization has
been observ^ed only in media of considerable density; but the properties of the
magnetic field are not so much altered by the substitution of one medium for
another, or for a vacuum, as to allow us to suppose that the dense medium
does anything more than merely modify the motion of the ether. We have
therefore warrantable grounds for inquiring whether there may not be a motion
of the ethereal medium going on wherever magnetic effects are observed, and

* ExperimenUil Researches, Series xix.

t Cainptes Reudus (185G, second half year, p. 529, ami 1857, first half year, p. 1209).

% Proceedings of the Royal ISociety, June 185G and June ISGl.



530 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

we have some reason to suppose that this motion is one of rotation, having
the direction of the magnetic force as its axis.

(9) We may now consider another phenomenon observed in the electro-
magnetic field. When a body is moved across the lines of magnetic force it
experiences what is called an electromotive force ; the two extremities of the
body tend to become oppositely electrified, and an electric current tends to flow
through the body. When the electromotive force is sufficiently powerful, and is
made to act on certain compound bodies, it decomposes them, and causes one
of their components to pass towards one extremity of the body, and the other
in the opposite direction.

Here we have evidence of a force causing an electric current in spite of
resistance ; electrifying the extremities of a body in opposite ways, a condition
which is sustained only by the action of the electromotive force, and which, as
soon as that force is removed, tends, with an equal and opposite force, to
produce a counter current through the body and to restore the original electrical
state of the body ; and finally, if strong enough, tearing to pieces chemical
compounds and carrying their components in opposite directions, while their
natural tendency is to combine, and to combine with a force which can generate
an electromotive force in the reverse direction.

This, then, is a force acting on a body caused by its motion through the
electromagnetic field, or by changes occurring in that field itself; and the effect
of the force is either to produce a current and heat the body, or to decompose
the body, or, when it can do neither, to put the body in a state of electric
polarization, — a state of constraint in which opposite extremities are oppositely
electrified, and from which the body tends to relieve itself as soon as the
disturbing force is removed.

(10) According to the theory which I propose to explain, this "electro-
motive force " is the force called into play during the communication of motion
from one part of the medium to another, and it is by means of this force
that the motion of one part causes motion in another part. When electromotive
force acts on a conducting circuit, it produces a current, which, as it meets
with resistance, occasions a continual transformation of electrical energy into
heat, which is incapable of being restored again to the form of electrical energy
by any reversal of the process.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 531

(11) But when electromotive force acts on a dielectric it produces a state
of polarization of its parts similar in distribution to the polarity of the parts
of a mass of iron under the influence of a magnet, and like the magnetic
polarization, capable of being described as a state in which every^ particle has
its opposite poles in opposite conditions'".

In a dielectric under the action of electromotive force, we may conceive
that the electricity in each molecule is so displaced that one side is rendered
positively and the other negatively electrical, but that the electricity remains
entirely connected with the molecule, and does not pass from one molecule to
another. The effect of this action on the whole dielectric mass is to produce
a general displacement of electiicity in a certain direction. This displacement
does not amount to a current, because when it has attained to a certain value
it remains constant, but it is the commencement of a current, and its varia-
tions constitute currents in the positive or the negative direction according as
the displacement is increasing or decreasing. In the interior of the dielectric
there is no indication of electrification, because the electrification of the surface
of any molecule is neutralized by the opposite electrification of the surface of
the molecules in contact with it ; but at the bounding surface of the dielectric,
where the electrification is not neutralized, we find the phenomena which indicate
positive or negative electrification.

The relation between the electromotive force and the amount of electric
displacement it produces depends on the nature of the dielectric, the same
electromotive force producing generally a greater electric displacement in solid
dielectrics, such as glass or sulphur, than in air.

(12) Here, then, we perceive another effect of electromotive force, namely,
electric displacement, which according to our theory is a kind of elastic yielding
to the action of the force, similar to that which takes place in structures and
machines owing to the want of perfect rigidity of the connexions.

(13) The practical investigation of the inductive capacity of dielectrics is
rendered difficult on account of two disturbing phenomena. The first is the
conductivity of the dielectric, which, though in many cases exceedingly small,
is not altogether insensible. The second is the phenomenon called electric absorp-

* Faraday, Experimental Researches, Series xi. ; Mossotti, Mem. delta Soc. Italiana (Modena),
Vol. XXIV. Part 2, p. 49.

67—2



532 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

tion*, in virtue of which, when the dielectric is exposed to electromotive force,
the electric displacement gradually increases, and when the electromotive force
is removed, the dielectric does not instantly return to its primitive state, but
only discharges a portion of its electrification, and when left to itself gradually
acquires electrification on its surface, as the interior gradually becomes depolarized.
Almost all solid dielectrics exhibit this phenomenon, which gives rise to the
residual charge in the Leyden jar, and to several phenomena of electric cables
described by Mr F. Jenkint.

(14) We have here two other kinds of yielding besides the yielding of
the perfect dielectric, which we have compared to a perfectly elastic body. The
yielding due to conductivity may be compared to that of a viscous fluid (that
is to say, a fluid having great internal friction), or a soft solid on which the
smallest force produces a permanent alteration of figure increasing with the
time during which the force acts. The yielding due to electric absorption may
be compared to that of a cellular elastic body containing a thick fluid in its
cavities. Such a body, when subjected to pressure, is compressed by degrees
on account of the gradual yielding of the thick fluid ; and when the pressure
is removed it does not at once recover its figure, because the elasticity of the
substance of the body has gradually to overcome the tenacity of the fluid before
it can regain complete equilibrium.

Several solid bodies in which no such structure as we have supposed can
be found, seem to possess a mechanical property of this kind J ; and it seems
probable that the same substances, if dielectrics, may possess the analogous
electrical property, and if magnetic, may have corresponding properties relating
to the acquisition, retention, and loss of magnetic polarity.

(15) It appears therefore that certain phenomena in electricity and mag-
netism lead to the same conclusion as those of optics, namely, that there is
an Ebthereal 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

* Fai-aday, Experwiental Researches, 1233 — 1250.

t Reports of British Association, 1859, p. 248; and Report of Committee of Board of Trade on
Submarine Cables, pp. 136 & 464.

J As, for instance, the composition of glue, treacle, <fec., of wliich small plastic figures are made,
which after being distorted gradually recover theii- shape.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 533

part of the medium to another by forces arising from the connexions of those
pai-ts ; that under the action of these forces there is a certain yielding depending
on the elasticity of these connexions ; 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 con-
nexions, in virtue of their elasticity.

(IG) 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.

(17) We know that when an electric current is established in a conducting
circuit, the neighbouring part of the field is characterized by certain magnetic
properties, and that if two circuits are in the field, the magnetic properties of
the field due to the two currents are combined. Thus each part of the field
is in connexion with both currents, and the two currents are put in connexion
with each other in virtue of their connexion with the magnetization of the field.
The first result of this connexion that I propose to examine, is the induction of
one current by another, and by the motion of conductors in the field.

The second result, which is deduced fi:om this, is the mechanical action
between conductors carrying currents. The phenomenon of the induction of
currents has been deduced from their mechanical action by Helmholtz * and
Thomson t. I have followed the reverse order, and deduced the mechanical action
from the laws of induction. I have then described experimental methods of
determining the quantities L, M, N, on which these phenomena depend.

(18) I then apply the phenomena of induction and attraction of cuiTents
to the exploration of the electromagnetic field, and the laying down systems
of lines of magnetic force which indicate its magnetic properties. By exploring

* "Conservation of Force," Physical Society of Berlin, 1847; and Taylor's Scieniijic Memoirs, 1853,
p. lU.

t Reports of live British Association, 1848; Philosophical Magazine, Dec. 1851.



534 A DYNAMICAL THEORY OF THE JJLECTROMAGNETIC FIELD.

the same field with a magnet, I shew the distribution of its equipotential
magnetic surfaces, cutting the hnes of force at right angles.

In order to bring these results within the power of symbolical calculation,
I then express them in the form of the General Equations of the Electro-
magnetic Field. These equations express —

(A) The relation between electric displacement, true conduction, and the
total current, compounded of both.

(B) The relation between the lines of magnetic force and the inductive
coefficients of a circuit, as already deduced from the laws of induction.

(C) The relation between the strength of a current and its magnetic effects,
according to the electromagnetic system of measurement.

(D) The value of the electromotive force in a body, as arising from the
motion of the body in the field, the alteration of the field itself, and
the variation of electric potential from one part of the field to
another.

(E) The relation between electric displacement, and the electromotive force
which produces it.

(F) The relation between an electric current, and the electromotive force
which produces it.

(G) The relation between the amount of free electricity at any point, and
the electric displacements in the neighbourhood.

(H) The relation between the increase or diminution of free electricity and

the electric currents in the neighbourhood.
There are twenty of these equations in all, involving twenty variable
quantities.

(19) I then express in terms of these quantities the intrinsic energy of
the Electromagnetic Field as depending partly on its magnetic and partly on
its electric polarization at every point.

From this I determine the mechanical force acting, 1st, on a moveable con-
ductor carrying an electric current; 2ndly, on a magnetic pole; 3rdly, on an
electrified body.

The last result, namely, the mechanical force acting on an electrified body,
gives rise to an independent method of electrical measurement founded on its



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 535

electrostatic effects. The relation between the units employed in the two methods
is shewn to depend on what I have called the "electric elasticity" of the medium,
and to be a velocity, which has been experimentally determined by MM. Weber
and Kohlrausch.

I then shew how to calculate the electrostatic capacity of a condenser, and
the specific inductive capacity of a dielectric.

The case of a condenser composed of parallel layers of substances of different
electric resistances and inductive capacities is next examined, and it is shewn
that the phenomenon called electric absorption will generally occur, that is, the
condenser, when suddenly discharged, will after a short time shew signs of a
residual charge.

(20) The general equations are next applied to the case of a magnetic
disturbance propagated through a non-conducting field, and it is shewn that
the only disturbances which can be so propagated are those which are transverse
to the direction of propagation, and that the velocity of propagation is the
velocity v, found from experiments such as those of Weber, which expresses
the number of electrostatic units of electricity which are contained in one electro-
magnetic unit.

This velocity is so nearly that of light, that it seems we have strong
reason to conclude that light itself (including radiant heat, and other radiations
if any) is an electromagnetic disturbance in the form of waves propagated through
the electromagnetic field according to electromagnetic laws. If so, the agree-
ment between the elasticity of the medium as calculated from the rapid alterna-
tions of luminous vibrations, and as found by the slow processes of electrical
experiments, shews how perfect and regular the elastic properties of the medium
must be when not encumbered with any matter denser than air. If the same
character of the elasticity is retained in dense transparent bodies, it appears
that the square of the index of refraction is equal to the product of the specific
dielectric capacity and the specific magnetic capacity. Conducting media are
shewn to absorb such radiations rapidly, and therefore to be generally opaque.

The conception of the propagation of transverse magnetic disturbances to
the exclusion of normal ones is distinctly set forth by Professor Faraday* in
liis " Thoughts on Ray Vibrations." The electromagnetic theory^ of light, as



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