James Clerk Maxwell.

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magnetic forces in aiding the motion of a conductor is equal to the product
of the current in the conductor multiplied by the increment of the electro-
magnetic momentum due to the motion.

Let a short straight conductor of length a move parallel to itself in the
direction of x, with its extremities on two parallel conductors. Then the incre-
ment of the electromagnetic momentum due to the motion of a will be

(dFdx dGdy dH dz\^
\dx ds dx ds dx ds)

That due to the lengthening of the circuit by increasing the length of the
parallel conductors will be

(dFdx dFdy dFdz\^

\dx ds dy ds dz ds)

The total increment is

\ds \dx dyj

ds\dz dx Jj '
which is by the equations of Magnetic Force (B), p. 556,

Let X be the force acting along the direction of x per unit of length of
the conductor, then the work done is XaSx.

Let C be the current in the conductor, and let p, q\ r be its com-
ponents, then

XaZx = Cahx (^ ^y " "^ /^^j .

566 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

or X = iiyq -^l^r^

Similarly, Y=fiar — y.yp \ (J).

These are the equations which determine the mechanical force acting on a
conductor carrying a current. The force is perpendicular to the current and
to the lines of force, and is measured by the area of the parallelogram formed
by lines parallel to the current and lines of force, and proportional to their
intensities.

Mechanical Force on a Magnet.

(J 7) In any part of the field not traversed by electric currents the dis-
tribution of magnetic intensity may be represented by the difterential coefficients
of a function which may be called the magnetic potential. When there are no
currents in the field, this quantity has a single value for each point. When
there are currents, the potential has a series of values at each point, but its
differential coefficients have only one value, namely,

d(f) d4> n ^^

Substituting these values of a, y8, y in the expression (equation 38) for the
intrinsic energy of the field, and integrating by parts, it becomes

^h(-B-'f,-'t)V^-

The expression S (^ + ^ + ^) cZF=S/7icZF (39)

indicates the number of lines of magnetic force which have their origin within
the space F. Now a magnetic pole is known to us only as the origin or
termination of lines of magnetic force, and a unit pole is one which has 47r
lines belonging to it, since it produces unit of magnetic intensity at unit of
distance over a sphere whose surface is 47r.

Hence if m is the amount of free positive magnetism in unit of volume,
the above expression may be written 47r>/i, and the expression for the energy
of the field becomes

E=-X{\^ni)dV (40).

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 567

If there are two magnetic poles 7>i, and vi^ producing potentials ^, and <^,
in the field, then if m, is moved a distance dx, and is urged in that direction
by a force A', then the work done is Xdx, and the decrease of energy in the
field is

and these must be equal by the principle of Conservation of Energy.

Since the distribution <^i is determined by m^, and (f), by vi„ the quantities
^,7?ii and (fi.pn^ will remain constant.

It can be shewn also, as Green has proved (Essay, p. 10), that

m,</>, = mj<^„

so that we get Xdx = d{7n^<f>^),

„ d(f>,
or X = m^-T- = rryx^,

where c^ represents the magnetic intensity due to m^. \ (K).

Similarly, Y= m^,,

Z = m^yi.

So that a magnetic pole is urged in the direction of the lines of magnetic
force with a force equal to the product of the strength of the pole and the
magnetic intensity.

(78) If a single magnetic pole, that is, one pole of a very long magnet,
be placed in the field, the only solution of (f> is

t>—fl (").

where m, is the strength of the pole, and r the distance from it.
The repulsion between two poles of strength m, and w, is

d<f>i m/ni^ f ^ _.

"^^ = 7^5- (^2)-

In air or any medium in which /x = 1 this is simply \ * , but in other

media the force acting between two given magnetic poles is inversely propor-
tional to the coefficient of magnetic induction for the medium. This may be
explained by the magnetization of the medium induced by the action of the
poles.

568 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

Mechanical Force on an Electrified Body.

(79) If there is no motion or change of strength of currents or magnets
in the field, the electromotive force is entirely due to variation of electric
potential, and we shall have (§65)

dx* dy* dz '

Integrating by parts the expression (I) for the energy due to electric
displacement, and remembering that P, Q, R vanish at an infinite distance, it becomes

i^{-(l+|43<^^-

or by the equation of Free Electricity (G), p. 561,

-^t(^e)dV.
By the same demonstration as was used in the case of the mechanical
action on a magnet, it may be shewn that the mechanical force on a small
body containing a quantity e^ of free electricity placed in a field whose
potential arising from other electrified bodies is '^\, has for components

(D).

So that an electrified body is urged in the direction of the electromotive
force with a force equal to the product of the quantity of free electricity and
the electromotive force.

If the electrification of the field arises from the presence of a small
electrified body containing e^ of free electricity, the only solution of ^i is

-.=r.^- (^^)'

where r is the distance from the electrified body.

The repulsion between two electrified bodies e^, e^ is therefore

"'Hf-^T^V (">•

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 569

Measurement of Electrostatic Effects.

(80) The quantities with which we have had to do have been hitherto
expressed in terms of the Electromagnetic System of measurement, which is
founded on the mechanical action between currents. The electrostatic system of
measurement is founded on the mechanical action between electrified bodies,
and is independent of, and incompatible with, the electromagnetic system ; so
that the units of the different kinds of quantity have different values according
to the system we adopt, and to pass from the one system to the other, a
reduction of all the quantities is required.

According to the electrostatic system, the repulsion between two small
bodies charged with quantities 7)^, t], of electricity is

where r is the distance between them.

Let the relation of the tw^o systems be such that one electromagnetic unit
of electricity contains v electrostatic units; then r), = ve, and ri., = ve„ and this
repulsion becomes

v'^^' = A ^J^ by equation (44) (45),

whence h, the coefficient of "electric elasticity" in the medium in which the
experiments are made, i. e. common air, is related to v, the number of electro-
static units in one electromagnetic unit, by the equation

A; = 47ri'' (46).

The quantity v may be determined by experiment in several ways. Ac-
cording to the experiments of MM. Weber and Kohlrausch,
v = 310,740,000 metres per second.

(81) It appears from this investigation, that if we assume that the medium
which constitutes the electromagnetic field is, when dielectric, capable of receiving
in every part of it an electric polarization, in which the opposite sides of every
element into which we may conceive the medium divided are oppositely elec-
trified, and if we also assume that this polarization or electric displacement is
proportional to the electromotive force which produces or maintains it, then we

570 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

can shew that electrified bodies in a dielectric medium will act on one anotKer
with forces obeying the same laws as are established by experiment.

The energy, by the expenditure of which electrical attractions and repul-
sions are produced, we suppose to be stored up in the dielectric medium which
surrounds the electrified bodies, and not on the surface of those bodies them-
selves, which on our theory are merely the bounding surfaces of the air or other
dielectric in which the true springs of action are to be sought.

Note on the Attraction of Gravitation.

(82) After tracing to the action of the surrounding medium both the
magnetic and the electric attractions and repulsions, and finding them to depend
on the inverse square of the distance, we are naturally led to inquire whether
the attraction of gravitation, which follows the same law of the distance, is
not also traceable to the action of a surrounding medium.

Gravitation differs from magnetism and electricity in this ; that the bodies
concerned are all of the same kind, inatead of being of opposite signs, like
magnetic poles and electrified bodies, and that the force between these bodies
is an attraction and not a repulsion, as is the case between like electric and
magnetic bodies.

The lines of gravitating force near two dense bodies are exactly of the
same form as the lines of magnetic force near two poles of the same name ;
but whereas the poles are repelled, the bodies are attracted. Let I^ be the
intrinsic energy of the field surrounding two gravitating bodies M^, M^, and
let E' be the intrinsic energy of the field surrounding two magnetic poles,
ra^, n\ equal in numerical value to iltfj, M^, and let X be the gravitating
force acting during the displacement hx, and X' the magnetic force,

Xhx = hE, X^x = hE';

now X and X are equal in numerical value, but of opposite signs ; so that

hE=-hK,
or E=C-E'

where a, ^, y are the components of magnetic intensity. If R be the resultant

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 571

gravitating force, and li^ the resultant magnetic force at a corresponding part
of the field,

R=-R:, and a' + fi' + '/ = Ii' = R'\
Hence

E=C-tj- ir-dV (47).

The intrinsic energy of the field of gravitation must therefore be less where-
ever there is a resultant gravitating force.

As energy is essentially positive, it is impossible for any part of space to
have negative intrinsic energy. Hence those parts of space in which there is
no resultant force, such as the points of equiUbrium in the space between the
different bodies of a system, and within the substance of each body, must have
an intrinsic energy per unit of volume greater than

where R is the greatest possible value of the intensity of gravitating force in
any part of the universe.

The assumption, therefore, that gravitation arises from the action of the
surrounding medium in the way pointed out, leads to the conclusion that every
part of this medium possesses, when undisturbed, an enormous intrinsic energy,
and that the presence of dense bodies influences the medium so as to diminish
this energy wherever there is a resultant attraction.

As I am unable to understand in what way a medium can possess such
properties, I cannot go any further in this direction in searching for the cause
of gravitation.

572 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

PART V.
THEORY OF CONDENSERS.

Capacity of a Condenser.

(83) The simplest form of condenser consists of a uniform layer of insulating
matter bounded by two conducting surfaces, and its capacity is measured by tbe
quantity of electricity on either surface when the difference of potentials is unity.

Let S be the area of either surface, a the thickness of the dielectric, and
h its coefficient of electric elasticity; then on one side of the condenser the
potential is ^j, and on the other side "^'i + l, and within its substance

di=a = ^f W.

Since ^ and therefore / is zero outside the condenser, the quantity of electricity

on its first surface = - Sf, and on the second + Sf. The capacity of the con-

S
denser is therefore >§/'= -^ in electromagnetic measure.

Specific Capacity of Electric Induction (D).

(84) If the dielectric of the condenser be air, then its capacity in electro-
ns'
static measure is — - (neglecting coiTections arising from the conditions to be

fulfilled at the edges). If the dielectric have a capacity whose ratio to that of

air is D, then the capacity of the condenser will be ~- .

Arra

Hence i) = | (49)^

where k^ is the value of k in air, which is taken for unity.

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 573

Electric Absorption.

(85) When the dielectric of which the condenser is formed is not a perfect
insulator, the phenomena of conduction are combined with those of electric dis-
placement. The condenser, when left charged, gradually loses its charge, and in
some cases, after being discharged completely, it gradually acquires a new charge
of the same sign as the original charge, and this finally disappears. These
phenomena have been described by Professor Faraday {Expenmental Researches,
Series XL) and by Mr F. Jenkin {Report of Committee of Board of Trade on
Submarine Cables), and may be classed under the name of "Electric Absorption."

(86) We shall take the case of a condenser composed of any number of
parallel layers of different materials. If a constant difference of potentials between
its extreme surfaces is kept up for a suflBcient time till a condition of perma-
nent steady flow of electricity is established, then each bounding surface will
have a charge of electricity depending on the nature of the substances on each
side of it. If the extreme surfaces be now discharged, these internal charges
will gradually be dissipated, and a certain charge may reappear on the extreme
surfaces if they are insulated, or, if they are connected by a conductor, a certain
quantity of electricity may be urged through the conductor during the re-
establishment of equilibrium.

Let the thickness of the several layers of the condenser be a^, a„, &c.

Let the values of k for these layers be respectively /:„ k.^, k^, and let

aJc^-\- aJc^-{- kc. =ak (50),

where k is the "electric elasticity" of air, and a is the thickness of an equiva-
lent condenser of air.

Let the resistances of the layers be respectively r^, r,, &c., and let
r, -I- r, -I- &c. = r be the resistance of the whole condenser, to a steady current
through it per unit of surface.

Let the electric displacement in each layer be /, fj, &c.

Let the electric current in each layer be J9„ p^ &c.

Let the potential on the first surface be ^i, and the electricity per unit
of surface e,.

574

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

Let the corresponding quantities at the boundary of the first and second
surface be % and e„ and so on. Then by equations (G) and (H),

de

&c. &c.

But by equations (E) and (F),

&c. &c. &c.

.(51),

(52).

After the electromotive force has been kept up for a sufficient time the
current becomes the same in each layer, and

where ^ is the total difference of potentials between the extreme layers. We
have then

and

t= j- , &c.

r \ajc^ akj'
These are the quantities of electricity on the different surfaces.

(53).

(87) Now let the condenser be discharged by connecting the extreme surfaces
through a perfect conductor so that their potentials are instantly rendered equal,
then the electricity on the extreme surfaces will be altered, but that on the
internal surfaces will not have time to escape. The total difference of potentials
is now

^' = ajc/, + ajc,{e\ + e,) + ajcle\ + e,-\-e^, &c. = (54),

whence if e\ is what e^ becomes at the instant of discharge,

^ r ajc^ ak~ * ak'

(55).

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 575

The instantaneous discharge is therefore —r, or the quantity which would

be discharged by a condenser of air of the equivalent thickness a, and it is
unaffected by the want of perfect insulation.

(88) Now let us suppose the connexion between the extreme surfaces
broken, and the condenser left to itself, and let us consider the gradual dissi-
pation of the internal charges. Let ^ be the difference of potential of the
extreme surfaces at any time t ; then

^' = «A/i + «M + &c (56);

but «i^V/i= -^i^^>

Ml Mi

Hence f^ = Afi'^' , f^^Af' n\ &c. ; and by referring to the values of e\, e^,
&c., we find

■vir ,' -vlr 'I

^ =_ li JL

^ r ajc^ ak

A=t3..^\ ^"*'

' r ajc., ak \

I
&c. J

so that we find for the difference of extreme potentials at any time,

- -{(^S"■■'-(^t>"""-^ <^^'-

(89) It appears from this result that if all the layers are made of the
same substance, "^^ will be zero always. If they are of different substances,
the order in which they are placed is indifferent, and the effect will be the
same whether each substance consists of one layer, or is divided into any number
of thin layers and arranged in any order among thin layers of the other sub-
stances. Any substance, therefore, the parts of which are not mathematically
homogeneous, though they may be apparently so, may exhibit phenomena of
absorption. Also, since the order of magnitude of the coeflicients is the same
as that of the indices, the value of ^' can never change sign, but must start
from zero, become positive, and finally disappear.

576 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

(90) Let us next consider the total amount of electricity which would
pass from the first surface to the second, if the condenser, after being thoroughly-
saturated by the current and then discharged, has its extreme surfaces connected
by a conductor of resistance R. Let p be the current in this conductor; then,
during the discharge,

^'=p^r^-\-jp^r^-\-kc.=pR (59).

Integrating with respect to the time, and calling q„ q^, q the quantities of
electricity which traverse the different conductors,

q^r^ + q^r^-\- kc. = qR (60).

The quantities of electricity on the several surfaces will be

e. + qi-qt.

&c. ;
and since at last all these quantities vanish, we find
qi = e\-q,
q, = e\ + e,-q',

a quantity essentially positive; so that, when the primary electrification is in
one direction, the secondary discharge is always in the same direction as the
primary discharge *.

♦ Since this paper was communicated to the Royal Society, I have seen a paper by M. Gaugain
in the Annates de Chimie for 1864, in which he has deduced the phenomena of electric absorption and
secondary discharge from the theory of compound condensers.

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 577

PART VI.

ELECTROMAUNETIC THEORY OF LIGHT.

(91) At the commencement of this paper we made uae of the optical
hypothesis of an elastic medium through which the vibrations of light are
propagated, in order to shew that we have warrantable grounds for seeking,
in the same medium, the cause of other phenomena as well as those of light.
We then examined electromagnetic phenomena, seeking for their explanation in
the properties of the field which surrounds the electrified or magnetic bodies.
In this way we arrived at certain equations expressing certain properties of
the electromagnetic field. We now proceed to investigate whether these pro-
perties of that which constitutes the electromagnetic field, deduced from electro-
magnetic phenomena alone, are sufficient to explain the propagation of light
through the same substance.

(92) Let us suppose that a plane wave whose direction cosines are Z, m, n
is propagated through the field with a velocity V. Then all the electro-
ma^etic functions will be functions of

w = Ix + my-{-7iz— Vt.

The equations of Magnetic Force (B), p. 556, will become

dH dG

iia = m — i n -j — ,

'^ dw dw

'^ dw dw '

^^ ~ dw dw '

If we multiply these equations respectively by /. m, n, and add, we find

lfj.a-\-7nixj3 + nny = (62),

which shews that the direction of the magnetization must be in the plane of
the wave.

578

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

(93) If we combine the equations of Magnetic Force (B) with those of
Electric Currents (C), and put for brevity

dF dG dH J _ ,
+ —-jf.-r-=J, and
dx dy dz

daf'^df^dz^''^

,.,p' = ^^V^F

4irfiq'

^-VG

dJ

-V'H

(63).

(64).

If the medium in the field is a perfect dielectric there is no true conduction,
and the currents p, q, r are only variations in the electric displacement, or,
by the equations of Total Currents (A),

i'-f' «-4- ^-=§ <«^)-

But these electric displacements are caused by electromotive forces, and by the
equations of Electric Elasticity (E),

P = hf, Q = kg, R = Tck (66).

These electromotive forces are due to the variations either of the electro-
magnetic or the electrostatic functions, as there is no motion of conductors in
the field; so that the equations of electromotive force (D) are

dF^_d^'
dt dx

P =

^ ~ dt dy

R=-

dH d^

(67).

dt dz
(94) Combining these equations, we obtain the following:-

.(^_,..)..„(«.«)..

\dy

dm

df

"*■ dzdt)

(68).

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 579

If we differentiate the third of these equations with respect to y, and

the second with respect to z, and subtract, J and "^ disappear, and by remem-
bering the equations (B) of magnetic force, the results may be written

de

(69).

(95) If we assume that a, fi, y are functions of Ix + my + 7iz — Vt = ic, the
first equation becomes

*''S=^VF'S (70).

^=±^/5 (")■

The other equations give the same value for V, so that the wave is propa-
gated in either direction with a velocity V.

This wave consists entirely of magnetic disturbances, the direction of mag-
netization being in the plane of the wave. No magnetic disturbance whose
direction of magnetization is not in the plane of the wave can be propagated
as a plane wave at all.

Hence magnetic disturbances propagated through the electromagnetic field
agree with light in this, that the disturbance at any point is transverse to
the direction of propagation, and such waves may have all the properties of
polarized light.

(96) The only medium in which experiments have been made to determine
the value of ^ is air, in which /u,= l, and therefore, by equation (46),

V=v (72).

By the electromagnetic experiments of MM. Weber and Kohlrausch *,
t> = 310,740,000 metres per second

• Leipzig Transactions, Vol. v. (1857), p. 260, or PoggendorflTs AnnaUn, Aug. 1856, p. 10.

73—2

580 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

is the number of electrostatic units in one electromagnetic unit of electricity,
and this, according to our result, should be equal to the velocity of light in
air or vacuum.

The velocity of light in air, by M. Fizeau's * experiments, is

F= 314,858,000;

according to the more accurate experiments of M. Foucatilt t,

F= 298,000,000.

The velocity of light in the space surrounding the earth, deduced from
the coefficient of aberration and the received value of the radius of the earth's
orbit, is

F= 308,000,000.

(97) Hence the velocity of light deduced from experiment agrees sufficiently
well with the value of v deduced from the only set of experiments we as yet
possess. The value of v was determined by measuring the electromotive force
with which a condenser of known capacity was charged, and then discharging
the condenser through a galvanometer, so as to measure the quantity of electricity
in it in electromagnetic measure. The only use made of light in the experiment
was to see the instruments. The value of V found by M. Foucault was
obtained by determining the angle through which a revolving mirror turned,
while the light reflected from it went and returned along a measured course.
No use whatever was made of electricity or magnetism.

The agreement of the results seems to shew that light and magnetism
are affections of the same substance, and that light is an electromagnetic dis-
turbance propagated through the field according to electromagnetic laws.

(98) Let us now go back upon the equations in (94), in which the
quantities J and ^ occur, to see whether any other kind of disturbance can
be propagated through the medium depending on these quantities which disappeared
from the final equations.

* Comptes Re-ndus, Vol. xxix. (1849), p. 90.
t Ibid. Vol. LV. (1862), pp. 501, 792.

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

If we determine x ^^om the equation

Vx=g-|-§=-^ (")•

and F', G', //' from the equations

581

r=F-

dx

G' = G-

dx

then

dx' dy

dF' dG' ^ dlT^^
dx dy dz

ir=H-

dx
dz'

•(74),
(75),

and the equations in (94) become of the form

.V.^ = 4..{^f'.^(..|)} (76).

Differentiating the three equations with respect to x, y, and z, and adding, we
find that

^=-:^+^(^' y^ ^)-

■{77),

and that

(78).

dt

Hence the disturbances indicated by F', G', H' are propagated with the velocity

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