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Full text of "A Dynamical Theory of the Electromagnetic Field"

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VIII. A Dynamical Theory of the Electromagnetic Field. By J. Clerk Maxwell, F.B.S. 

Received October 27, — Read December 8, 1864. 



PART 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 some- 
thing 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 mecha- 
nical 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 parti- 
cles 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 kind is that of % 
M. W. Weber*, who has made the same theory include electrostatic 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 theory, as developed by MM. W. Weber and C. Neumann f, is exceedingly 
ingenious, and wonderfully comprehensive in its application to the phenomena of 
statical electricity, electromagnetic attractions, induction of currents apd diamagnetic 
phenomena ; and it comes to us with the more authority, as it has served 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. 

* Electrodynamische Maassbestimnmngen. Leipzic Trans, vol. i. 1849, and Taylor's Scientific Memoirs, vol.r. 
art. xiv. . 

f " Explicare tentatur quomodo fiat ut lucis planum polarizationis per vires electricas vel magneticas decli- 
netur."— Halis Saxomim, 1858. 

MDCCCLXV. 3 E 



460 PEOEESSOE CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 

(2) The mechanical difficulties, however, which are involved in the assumption 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 coordination 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 endeavouring 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 Electromagnetic 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. 

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 Geisslbr's tubes and other so-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 eethereal 
substance, and not of the gross matter, the presence of which merely modifies in some 
way the motion of the aether. 

We have therefore some reason to believe, from the phenomena of light and heat, 
that there is an sethereal medium filling space and permeating bodies, capable of being 
set in motion and of transmitting that motion from one part to another, and of com- 
municating that motion to gross matter so as to heat it and affect 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 infe- 
rior limit to that density. 

(6) We may therefore receive, as a datum derived from a branch of science inde- 
pendent 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 transmitting 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 part 

* " On the Possible Density of the Luminiferous Medium, and on the Mechanical Yalue of a Cubic Mile of 
Sunlight/' Transactions of the Royal Society of Edinburgh (1854), p. 57. 



PEOFESSOE CLEKK MAXWELL ON THE ELECTROMAGNETIC FIELD. 461 

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 recovering from displace- 
ment in virtue of its elasticity. . 

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 whole 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 pheno- 
mena they produce. 

(8) Now we know that the luminiferous medium is in certain cases acted on by 
magnetism; for Faraday f discovered that when a plane polarized ray traverses a trans- 
parent 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 magnetization of the field. 

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

Now Professor W. Thomson^ has pointed out that no distribution of forces acting 
between the parts of a medium whose only motion is that of the luminous 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 
observed 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 elects 
are observed, and 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 electromagnetic 

■* Experimental Besearches, Series 19. 

f Comptes Bendus (1856, second half year, p. 529, and 1857, first half year, p. 1209). 

% Proceedings of the Boyal Society, June 1856 and June 1861. 

3b2 



462 PEOEESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 

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 oppo- 
sitely 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 resist- 
ance; 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 oppo- 
site 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 electro- 
magnetic 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 "electromotive 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 pro- 
duces a current, w 7 hich, 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. 

(11) But when electromotive force acts on a dielectric it produces a state of polari- 
zation 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 con- 
ditions *. 

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 electricity in a cer- 
tain 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 variations constitute currents in the positive or the negative direction according 

* Faraday, Exp. Ees. Series XI. ; Mossom, Mem. della Soc. Italiana (Modena), vol. xxiy. part 2. p. 49. 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 463 

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 pro- 
ducing 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 absorption *, 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. Jenkin f. 

(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 com- 
plete 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 

* Faraday, Exp. Ees. 1233-1250. 

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

$ As, for instance, the composition of glue, treacle, &c, of which small plastic figures are made, which after 
being distorted gradually recover their shape. 



464 PBOFESSOE CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD, 

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 magnetism lead 
to the same conclusion as those of optics, namely, that there is an sethereal medium 
pervading "all bodies, and modified only in degree by their presence; that the parts of 
this medium are capable of being set in motion by electric currents and magnets ; that 
this motion is communicated from one part of the medium to another by forces arising 
from the connexions of those parts; 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 
connexions, in virtue of their elasticity. . 

(16) 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 con- 
nexion 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 from this, is the mechanical action between con- 
ductors carrying currents. The phenomenon of the induction of currents has been 
deduced from their mechanical action by Helmholtz* and Thomson f . 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 currents to the 
exploration of the electromagnetic field, and the laying down systems of lines of mag- 
netic force which indicate its magnetic properties. By exploring the same field with a 
magnet, I show the distribution of its equipotential magnetic surfaces, cutting the lines 
of force at right angles. 

* "Conservation of Force," Physical Society of Berlin, 1847; and Taylok's Scientific Memoirs, 1853, 

p. 114. 

f Beports of the British Association, 1848; Philosophical Magazine, Dec. 1851. 



PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 465 

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 Electromagnetic 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. 

(G) 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 electrio 
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 pro- 

duces 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 elec- 
tric 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 Electro- 
magnetic Field as depending partly on its magnetic and partly on its electric polariza- 
tion at every point. 

From this I determine the mechanical force acting, 1st, on a moveable conductor 
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 electrostatic effects. 
The relation between the units employed in the two methods is shown 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 show 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 shown that the pheno- 
menon called electric absorption will generally occur, that is, the condenser, when 
suddenly discharged, will after a short time show 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 shown that the only disturbances 
which can be so propagated are those which are transverse to the direction of propaga- 
tion, and that the velocity of propagation is the velocity v, found from experiments such 



466 PBOFESSOE CLEEK MAXWELL ON THE ELECTROMAGNETIC FIELD. 

as those of Weber, which expresses the number of electrostatic units of electricity 
which are contained in one electromagnetic unit. 

This velocity is so nearly that of light, that it seems we have strong reason to con- 
clude that light itself (including radiant heat, and other radiations if any) is an electro- 
magnetic disturbance in the form of waves propagated through the electromagnetic field 
according to electromagnetic laws. If so, the agreement between the elasticity of the 
medium as calculated from the rapid alternations of luminous vibrations, and as found 
by the slow processes of electrical experiments, shows 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 
shown to absorb such radiations rapidly, and therefore to be generally opaque. 

The conception of the propagation of transverse magnetic disturbances to the exclu- 
sion of normal ones is distinctly set forth by Professor Faraday* in his "Thoughts on 
Eay Vibrations/' The electromagnetic theory of light, as proposed by him, is the same 
in substance as that which I have begun to develope in this paper, except that in 1846 
there were no data to calculate the velocity of propagation. 

(21) The general equations are then applied to the calculation of the coefficients of 
mutual induction of two circular currents and the coefficient of self-induction in a coil. 
The want of uniformity of the current in the different parts of the section of a wire at 
the commencement of the current is investigated, I believe for the first time, and the 
consequent correction of the coefficient of self-induction is found. 

These results are applied to the calculation of the self-induction of the coil used in 
the experiments of the Committee of the British Association on Standards of Electric 
Eesistance, and the value compared with that deduced from the experiments. 

PART II.— ON ELECTROMAGNETIC INDUCTION. 

Electromagnetic Momentum of a Current. 

(22) We may begin by considering the state of the field in the neighbourhood of an 
electric current. We know that magnetic forces are excited in the field, their direction 
and magnitude depending according to known laws upon the form of the conductor 
carrying the current. When the strength of the current is increased, all the magnetic 
effects are increased in the same proportion. Now, if the magnetic state of the field 
depends on motions of the medium, a certain force must be exerted in order to increase 
or diminish these motions, and when the motions are excited they continue, so that the 
effect of the connexion between the current and the electromagnetic field surrounding 
it, is to endow the current with a kind of momentum, just as the connexion between 
the driving-point of a machine and a fly-wheel endows the driving-point with an addi- 

* Philosophical Magazine, May 1846, or Experimental Researches, iii. p. 447. 



PB0EES30B CLEEK MAXWELL ON THE ELECTEOMAaKETIC EIELD. 467 

tional momentum, which may be called the momentum of the fly-wheel reduced to 
the driving-point. The unbalanced force acting on the driving-point increases this 
momentum, and is measured by the rate of its increase. 

In the case of electric currents, the resistance to sudden increase or diminution of 
strength produces effects exactly like those of momentum, but the amount of this mo- 
mentum depends on the shape of the conductor and the relative position of its different 
parts. 

Mutual Action of two Currents. 

(23) If there are two electric currents in the field, the magnetic force at any point is 
that compounded of the forces due to each current separately, and since the two currents 
are in connexion with every point of the field, they will be in connexion with each other, 
so that any increase or diminution of the one will produce a force acting with or con- 
trary to the other. 

Dynamical Illustration of Beduced Momentum. 

(24) As a dynamical illustration, let us suppose a body C so connected with two 
independent driving-points A and B that its velocity is p times that of A together with 
q times that of B. Let u be the velocity of A, v that of B, and w that of C, and let &f, 
hy, iz be their simultaneous displacements, then by the general equation of dynamics*, 

where X and Y are the forces acting at A and B. 
But 

dw du dv 

and 

}>z=.phw-{-qly. 

Substituting, and remembering that Ix and ly are independent, 



Y= 5 (Qp2W+C2H 



(1) 



We may call Cp*u+Opqv the momentum of C referred to A, and Cpqu+Ctfv its 
momentum referred to B ; then we may say that the effect of the force X is to increase the 
momentum of C referred to A, and that of Y to increase its momentum referred to B. 

If there are many bodies connected with A and B in a similar way but with different 
values of p and q, we may treat the question in the same way by assuming 

L=2(qp»), M=2(CJp2), and N=2(C 2 2 ), 

* Lagkakge, Mec. Anal. ii. 2. § 5. 
MDCCCLXV. 3 S 



468 PEOFESSOK CLEEK MAXWELL ON THE ELECTEOMAGKETIC FIELD. 

where the summation is extended to all the bodies with their proper values of C, p, and q. 
Then the momentum of the system referred to A is 

\m +M#, 
and referred to B, 

and we shall have 



(2) 



X=j i (Lu+Mv), 

Y = |(M„ +N ,), 

where X and Y are the external forces acting on A and B. 

(25) To make the illustration more complete we have only to suppose that the 
motion of A is resisted by a force proportional to its velocity, which we may call Rw, 
and that of B by a similar force, which we may call St;, R, and S being coefficients of 
resistance. Then if f and n are the forces on A and B 






(3) 



If the velocity of A be increased at the rate —^ then in order to prevent B from moving 

(a/v 

d 

a force, *7=-^(Mw) must be applied to it. 

This effect on B, due to an increase of the velocity of A, corresponds to the electro- 
motive force on one circuit arising from an increase in the strength of a neighbouring 
circuit. 

This dynamical illustration is to be considered merely as assisting the reader to under- 
stand what is meant in mechanics by Reduced Momentum. The facts of the induction 
of currents as depending on the variations of the quantity called Electromagnetic Mo- 
mentum, or Electrotonic State, rest on the experiments of Faraday*, FELicif, &c. 

Coefficients of Induction for Two Circuits. 

(26) In the electromagnetic field the values of L, M, N depend on the distribution 
of the magnetic effects due to the two circuits, and this distribution depends only on 
the form and relative position of the circuits. Hence L, M, N are quantities depending 
on the form and relative position of the circuits, and are subject to variation with the 
motion of the conductors. It will be presently seen that L, M, N are geometrical 
quantities of the nature of lines, that is, of one dimension in space ; L depends on the 
form of the first conductor, which we shall call A, N on that of the second, which we 
shall call B, and M on the relative position of A and B. 

(27) Let | be the electromotive force acting on A, x the strength of the current, and 

* Experimental Researches, Series I., IX. f Annales de Chimie, ser. 3. xxxiv. (1852) p. 64. 



PEOEESSOB CLEKK MAXWELL ON THE ELECTROMAGNETIC FIELD. 469 

E the resistance, then Hx will be the resisting force. In steady currents the electro- 
motive force just balances the resisting force, but in variable currents the resultant 
force |=Eo? is expended in increasing the " electromagnetic momentum," using the 
word momentum merely to express that which is generated by a force acting during a 
time, that is, a velocity existing in a body. 

In the case of electric currents, the force in action is not ordinary mechanical force, at 
least we are not as yet able to measure it as common force, but we call it electromotive 
force, and the body moved is not merely the electricity in the conductor, but something 
outside the conductor, and capable of being affected by other conductors in the neighbour- 
hood carrying currents. In this it resembles rather the reduced momentum of a drivings- 
point of a machine as influenced by its mechanical connexions, than that of a simple 
moving body like a cannon ball, or water in a tube. 

Electromagnetic Relations of two Conducting Circuits. 

(28.) In the case of two conducting circuits, A and B, we shall assume that the 
electromagnetic momentum belonging to A is 

La? +My, 
and that belonging to B, 

. Mar + %, 

where L, M, N correspond to the same quantities in the dynamical illustration, except 
that they are supposed to be capable of variation when the conductors A or B are 
moved. 

Then the equation of the current x in A will be 

g=Ea?+^(La?+My), . . (4) 

and that of y in B 

d 
^=Sj/ + ^(M#+%), (5) 

where I and q are the electromotive forces, x and y the currents, and E and S the 
resistances in A and B respectively, 

Induction of one Current by another. 

(29) Case 1st. Let there be no electromotive force on B, except that which arises 
from the action of A, and let the current of A increase from to the value x, then 

d 

S^+^(M^+%)=0, 
whence 

Y= j ydt=—-gX) 

that is, a quantity of electricity Y, being the total induced current, will flow through B 
when x rises from to x. This is induction by variation of the current in the primary 

3s2 



470 PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC HELD. 

conductor. When M is positive, the induced current due to increase of the primary 
current is negative. 

Induction by Motion of Conductor. 

(30) Case 2nd. Let x remain constant, and let M change from M to M', then 

M'-M mm 

X — y, X y 

so that if M is increased, which it will be by the primary and secondary circuits 
approaching each other, there will be a negative induced current, the total quantity of 
electricity passed through B being Y. 

This is induction by the relative motion of the primary and secondary conductors. 

Equation of Work and Energy. 

(31) To form the equation between work done and energy produced, multiply (1) by 
x and (2) by y, and add 

^+^=B^+% 2 +^(I^ + My)+y^(Mo;+%) (8) 

Here %x is the work done in unit of time by the electromotive force § acting on the 
current x and maintaining it, and ny is the work done by the electromotive force q. 
Hence the left-hand side of the equation represents the work done by the electromotive 
forces in unit of time. 

Heat produced by the Current. 

(32) On the other side of the equation we have, first, 

E^+S^=H, ........... (9) 

which represents the work done in overcoming the resistance of the circuits in unit of 
time. This is converted into heat. The remaining terms represent work not converted 
into heat. They may be written 

Intrinsic Energy of the Currents. 

(33) If L, M, N are constant, the whole work of the electromotive forces which is 
not spent against resistance will be devoted to the development of the currents. The 
whole intrinsic energy of the currents is therefore 

|I^ 2 +M^+i% 2 =E (10) 

This energy exists in a form imperceptible to our senses, probably as actual motion, the 
seat of this motion being not merely the conducting circuits, but the space surrounding 
them. 



PKOEESSOR CLEEK MAXWELL ON THE ELECTROMAGNETIC EIELD, 471 

Mechanical Action between Conductors. 
(34) The remaining terms, 

2 (it X + ^ ^+2 ^ y — yy ....... . (1.1 J 

represent the work done in unit of time arising from the variations of L, M, and N, or, 
what is the same thing, alterations in the form and position of the conducting circuits 
A and B. 

Now if work is done when a body is moved, it must arise from ordinary mechanical 
force acting on the body while it is moved. Hence this part of the expression shows 
that there is a mechanical force urging every part of the conductors themselves in that 
direction in which L, M, and N will be most increased. 

The existence of the electromagnetic force between conductors carrying currents is 
therefore a direct consequence of the joint and independent action of each current on 
the electromagnetic field. If A and B are allowed to approach a distance ds> so as to 
increase M from M to M f while the currents are x and y, then the work done will be 

(M'— M)xy, 

and the force in the direction of ds will be 

and this will be an attraction if x and y are of the same sign, and if M is increased as 
A and B approach. 

It appears, therefore, that if we admit that the unresisted part of electromotive force 
goes on as long as it acts, generating a self-persistent state of the current, which 
we may call (from mechanical analogy) its electromagnetic momentum, and that this 
momentum depends on circumstances external to the conductor, then both induction of 
currents and electromagnetic attractions may be proved by mechanical reasoning. 

What I have called electromagnetic momentum is the same quantity which is called 
by Faraday* the electrotonic state of the circuit, every change of which involves the 
action of an electromotive force, just as change of momentum involves the action of 
mechanical force. 

If, therefore, the phenomena described by Faraday in the Ninth Series of his Expe- 
rimental Eesearches were the only known facts about electric currents, the laws of 
Ampere relating to the attraction of conductors carrying currents, as well as those 
of Faraday about the mutual induction of currents, might be deduced by mechanical 
reasoning. 

In order to bring these results within the range of experimental verification, I shall 
next investigate the case of a single current, of two currents, and of the six currents 
in the electric balance, so as to enable the experimenter to determine the values of 
L, M, N. 

* Experimental Researches, Series I. 60, &c. 



472 PROEESSOE -CLERK MAXWELL OF THE ELECTEOMAGNBTIO FIELD. 

Case of a single Circuit. 

(35) The equation of the current x in a circuit whose resistance is R, and whose 
coefficient of self-induction is L, acted on by an external electromotive force fj, is 

d 

£5 ' " XY^V T~" J± J-it.t' » * a « * • • e » • IJLOJ 

When 5 is constant, the solution is of the form 



^-t 



x=b-\-(a—b)e L 

where a is the value of the current at the commencement, and b is its final value. 
The total quantity of electricity which passes in time £, where t is great, is 






ocdt^=^bt J r(a—b)- n . .......... (14) 

JLV 



The value of the integral of x 2 with respect to the time is 

^dt=m+{a-b)h^^j (15) 

The actual current changes gradually from the initial value a to the final value #, but 
the values of the integrals of x and x 2 are the same as if a steady current of intensity 

\(a+b) were to flow for a time 2 — , and were then succeeded by the steady current 6. 

The time 2 ~ is generally so minute a fraction of a second, that the effects on the galvano- 

meter and dynamometer may be calculated as if the impulse were instantaneous. 

If the circuit consists of a battery and a coil, then, when the circuit is first completed, 
the effects are the same as if the current had only half its final strength during the time 

2 ~. This diminution of the current, due to induction, is sometimes called the counter- 
R 

current. 

(36) If an additional resistance r is suddenly thrown into the circuit, as by breaking 

contact, so as to force the current to pass through a thin wire of resistance r, then the 

original current is #==J>5 and the final current is b= 



R R + r 

The current of induction is then ^%^ T ^-^. and continues for a time 2^- — -. This 

2 R(R-fr)' R + r 

current is greater than that which the battery can maintain in the two wires R and r, 
and may be sufficient to ignite the thin wire r. 

When contact is broken by separating the wires in air, this additional resistance is 
given by the interposed air, and since the electromotive force across the new resistance 
is very great, a spark will be forced across. 



PBOEESSOE. CLEEK MAXWELL ON THE ELEOTEOMAGNBTIO FIELD. 478 

If the electromotive force is of the form E sinj?£, as in the case of a coil revolving in 
a magnetic field, then 

ar=- — sin (jp#— -a), 

where g 2 =R 2 +I/p 9 , and tana=-j£' 

Case of two Circuits. 

(37) Let It be the primary circuit and S the secondary circuit, then we have a case 
similar to that of the induction coil. 

The equations of currents are those marked A and B, and we may here assume 
L, M, N as constant because there is no motion of the conductors. The equations 
then become 

E *+ L S+M J=& 



Sy+Mg+Nt=o. 



(13*) 



To find the total quantity of electricity which passes, we have only to integrate these 
equations with respect to t; then if o? , y be the strengths of the currents at time 0, 
and x x , y x at time t, and if X, Y be the quantities of electricity passed through each 
circuit during time #, 

X=: s {g*4-L(^o— ^i)4-M(y — yj}, 



Y=^{M(d? -o? 1 )+N(y -y 1 )}. 



(14*) 



When the circuit E is completed, then the total currents up to time t, when t is 
great, are found by making 

#* =:U, 0C l :=: -^^, ^o^U, y l zzz\) \ 

then 

JSl. — (h j I v wmmm "zr I , X — ""-■"* "7™" 0u i . . . • • • • • y J- ^ y 
\ K/ o 

The value of the total counter-current in E is therefore independent of the secondary 
circuit, and the induction current in the secondary circuit depends only on M, the 
coefficient of induction between the coils, S the resistance of the secondary coil, and 
x x the final strength of the current in R. 

When the electromotive force g ceases to act, there is an extra current in the pri- 
mary circuit, and a positive induced current in the secondary circuit, whose values are 
equal and opposite to those produced on making contact. 

(38) All questions relating to the total quantity of transient currents, as measured 
by the impulse given to the magnet of the galvanometer, may be solved in this way 
without the necessity of a complete solution of the equations. The heating effect of 



474 PKOFESSOB CLEEK MAXWELL OH THE ELECTBOMAQKETIC EIELD. 

the current, and the impulse it gives to the suspended coil of Weber's dynamometer, 
depend on the square of the current at every instant during the short time it lasts. 
Hence we must obtain the solution of the equations, and from the solution we may find 
the effects both on the galvanometer and dynamometer ; and we may then make use of 
the method of Weber for estimating the intensity and duration of a current uniform 
while it lasts which would produce the same effects, 
(39) Let n 19 n 2 be the roots of the equation 

(LN~-M> 2 +(EN+LS>+ES=0 5 (16) 

and let the primary coil be acted on by a constant electromotive force Ke, so that c is 
the constant current it could maintain ; then the complete solution of the equations for 
making contact is 

^L-^ 2 {(l+ N )^-(|+ N )^+ s? ^ • • • • m 

y=T^^-^> as) 

From these we obtain for calculating the impulse on the dynamometer, 



£*»=(* 



o L M 2 
V a R"*RN+LSf' ( 19 ) 



Wdt-c% — — (20) 

iy 2 S(RN+LS) l ' 

The effects of the current in the secondary coil on the galvanometer and dynamometer 
are the same as those of a uniform current 

i MR 

~* C RN+LS 
for a time 

9 /L N' 

(40) The equation between work and energy may be easily verified. The work done 
by the electromotive force is 

%§xdt=c 2 (Rt— L). 

Work done in overcoming resistance and producing heat, 

TL$tfdt+&Sfdt=<?(Bt-$L). 
Energy remaining in the system, 



— — Q.lJ JLJt 



(41) If the circuit E is suddenly and completely interrupted while carrying a current 
c, then the equation of the current in the secondary coil would be 

M -It 

N 



y=c-e « 



This current begins with a value c -^ , and gradually disappears. 



PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 475 

The total quantity of electricity is c~^ , and the value of §y*dt is ^ 2 o§m* 

The effects on the galvanometer and dynamometer are equal to those of a uniform 

current A c — for a time 2 -^ • 

2 N S 

The heating effect is therefore greater than that of the current on making contact. 

(42) If an electromotive force of the form |=E cos pt acts on the circuit R, then if 
the circuit S is removed, the value of $ will be 

a?= j sin (jpt— a), 

where 

A 2 =R 2 +Ly, 

and 

Jjp 
tan a=-jp» 

The effect of the presence of the circuit S in the neighbourhood is to alter the value 
of A and a, to that which they would be if E become 

and L became 

T 2 MN 

Hence the effect of the presence of the circuit S is to increase the apparent resistance and 
diminish the apparent self-induction of the circuit R. 

On the Determination of Coefficients of Induction by the Electric Balance. 

(43) The electric balance consists of six con- 
ductors joining four points, ACDE, two and two. 
One pair, A C, of these points is connected through 
the battery B. The opposite pair, D E, is connected 
through the galvanometer G. Then if the resistances 
of the four remaining conductors are represented by 
P, Q, R, S, and the currents in them by #, x—z, y, 
and y-\-z, the current through G will be z. Let the 

potentials at the four points be A, C, D, E. Then the conditions of steady currents may 
be found from the equations 

P#=A-D Q(#-s)=D-C, 

%=A-E S(y+s)=E— C, 

Gs=D-E B(#+y)=-A+C+F. 

Solving these equations for -z 9 we find 

4p + a + s4+ B (F+!)(<3+^ ( 22 > 

MDCCCLXV. 3 T 




>. 



(21) 



476 



PBOFESSOE CEEKK MAXWELL ON THE ELECTEOMAaNETIC FIELD. 



In this expression F is the electromotive force of the battery, z the current through 
the galvanometer when it has become steady. P, Q, E, S the resistances in the four 
arms. B that of the battery and electrodes, and G that of the galvanometer. 

(44) If PS=QR, then £=0, and there will be no steady current, but a transient 
current through the galvanometer may be produced on making or breaking circuit on 
account of induction, and the indications of the galvanometer may be used to determine 
the coefficients of induction, provided we understand the actions which take place. 

We shall suppose PS=QR, so that the current z vanishes when sufficient time is 
allowed, and 

x(P+Q)=y(B, + S)= (p + Q)( ft + S ) +B (P + cj)(R + sy 

Let the induction coefficients between P, Q, E S, be 
given by the following Table, the coefficient of induction 
of P on itself being p, between P and Q, A, and so on. 

Let g be the coefficient of induction of the galvanometer 
on itself, and let it be out of the reach of the inductive 
influence of P, Q, R, S (as it must be in order to avoid 
direct action of P, Q, R, S on the needle). Let X, Y, Z be the integrals of #, y, z 
with respect to t. At making contact #, y, z are zero. After a time z disappears, and 
s and y reach constant values. The equations for each conductor will therefore be 

PX +(p+h )x+{k +1 )y=SAdt-$T)dt, ) 

Q(X-Z)+(h+q )x+(m+n)y=§Ddt~lCdt, 
RY +(k+m)x+(r +o)y=]Adt-]Edt, 

S(Y-f Z) +(l +n )w+(o +s)y=$Edt-§Cdt, 
GZ=$Dtd-~$Edt. 





P 


Q 


E 


S 


p 


P 


h 


& 


I 


Q 


h 


9. 


m 


n 


K 


Jc 


m 


r 





S 


I 


n 





s 



(24) 



1 



Solving these equations for Z, we find 



F 



PS 



p-Q-i+l+ A (p-a) +*(i-p) +*(i+i) - m (p+|) 



■ (25) 



+ w (i-s)+°(i-!t)} 



(45) Now let the deflection of the galvanometer by the instantaneous current whose 
intensity is Z be «. 

Let the permanent deflection produced by making the ratio of PS to QR, ? instead of 
unity, be 6, 

Also let the time of vibration of the galvanometer needle from rest to rest be T. 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 477 

Then calling the quantity 



we find 



Z 2sml*T 



9 



z tau 9 7T 1 — g 



,. ( Z i ) 



In determining r by experiment, it is best to make the alteration of resistance in one 
of the arms by means of the arrangement described by Mr. Jenkin in the Report of the 
British Association for 1863, by which any value off from 1 to 1*01 can be accurately 
measured. 

We observe (a) the greatest deflection due to the impulse of induction when the 
galvanometer is in circuit, when the connexions are made, and when the resistances are 
so adjusted as to give no permanent current. 

We then observe (j3) the greatest deflection produced by the permanent current when 
the resistance of one of the arms is increased in the ratio of 1 to g , the galvanometer 
not being in circuit till a little while after the connexion is made with the battery. 

In order to eliminate the effects of resistance of the air, it is best to vary g till /3 = 2& 
nearly; then 2sini* /oon 

7r^ *' tan |/3 

If all the arms of the balance except P consist of resistance coils of very fine wire of 
no great length and doubled before being coiled, the induction coefficients belonging to 

these coils will be insensible, and r will be reduced to £ . The electric balance there- 

fore affords the means of measuring the self-induction of any circuit whose resistance is 
known. 

(46) It may also be used to determine the coefficient of induction between two 
circuits, as for instance, that between P and S which we have called m ; but it would be 
more convenient to measure this by directly measuring the current, as in (37), without 

using the balance. We may also ascertain the equality of ^ and £ by there being no 

current of induction, and thus, when we know the value of p, we may determine that of 
q by a more perfect method than the comparison of deflections. 

Exploration of the Electromagnetic Field. 

(47) Let us now suppose the primary circuit A to be of invariable form, and let us 
explore the electromagnetic field by means of the secondary circuit B, which we shall 
suppose to be variable in form and position. 

We may begin by supposing B to consist of a short straight conductor with its extre- 
mities sliding on two parallel conducting rails, which are put in connexion at some 
distance from the sliding-piece. 

3t2 



478 PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 

Then, if sliding the moveable conductor in a given direction increases the value of M, 
a negative electromotive force will act in the circuit B, tending to produce a negative 
current in B during the motion of the sliding-piece. 

If a current be kept up in the circuit B, then the sliding-piece will itself tend to 
move in that direction, which causes M to increase. At every point of the field there 
will always be a certain direction such that a conductor moved in that direction does 
not experience any electromotive force in whatever direction its extremities are turned. 
A conductor carrying a current will experience no mechanical force urging it in that 
direction or the opposite. 

This direction is called the direction of the line of magnetic force through that point. 

Motion of a conductor across such a line produces electromotive force in a direction 
perpendicular to the line and to the direction of motion, and a conductor carrying a 
current is urged in a direction perpendicular to the line and to the direction of the 
current. 

(48) We may next suppose B to consist of a very small plane circuit capable of being 
placed in any position and of having its plane turned in any direction. The value of M 
will be greatest when the plane of the circuit is perpendicular to the line of magnetic 
force. Hence if a current is maintained in B it will tend to set itself in this position, 
and will of itself indicate, like a magnet, the direction of the magnetic force. 

On Lines of Magnetic Force. 

(49) Let any surface be drawn, cutting the lines of magnetic force, and on this sur- 
face let any system of lines be drawn at small intervals, so as to lie side by side without 
cutting each other. Next, let any line be drawn on the surface cutting all these lines, 
and let a second line be drawn near it, its distance from the first being such that the 
value of M for each of the small spaces enclosed between these two lines and the lines 
of the first system is equal to unity. 

In this way let more lines be drawn so as to form a second system, so that the value of 
M for every reticulation formed by the intersection of the two systems of lines is unity. 

Finally, from every point of intersection of these reticulations let a line be drawn 
through the field, always coinciding in direction with the direction of magnetic force. 

(50) In this way the whole field will be filled with lines of magnetic force at regular 
intervals, and the properties of the electromagnetic field will be completely expressed 
by them. 

For, 1st, If any closed curve be drawn in the field, the value of M for that curve will 
be expressed by the number of lines of force which pass through that closed curve. 

2ndly. If this curve be a conducting circuit and be moved through the field, an 
electromotive force will act in it, represented by the rate of decrease of the number of 
lines passing through the curve. 

3rdly. If a current be maintained in the circuit, the conductor will be acted on by 
forces tending to move it so as to increase the number of lines passing through it, and 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 479 

the amount of work done by these forces is equal to the current in the circuit multi- 
plied by the number of additional lines. 

4thly. If a small plane circuit be placed in the field, and be free to turn, it will place 
its plane perpendicular to the lines of force. A small magnet will place itself with its 
axis in the direction of the lines of force. 

5thly. If a long uniformly magnetized bar is placed in the field, each pole will be 
acted on by a force in the direction of the lines of force. The number of lines of force 
passing through unit of area is equal to the force acting on a unit pole multiplied by a 
coefficient depending on the magnetic nature of the medium, and called the coefficient 
of magnetic induction. 

In fluids and isotropic solids the value of this coefficient p is the same in whatever 
direction the lines of force pass through the substance, but in crystallized, strained, and 
organized solids the value of p may depend on the direction of the lines of force with 
respect to the axes of crystallization, strain, or growth. 

In all bodies \h is affected by temperature, and in iron it appears to diminish as the 
intensity of the magnetization increases. 

On Magnetic Equipotential Surfaces. 

(51) If we explore the field with a uniformly magnetized bar, so long that one of its 
poles is in a very weak part of the magnetic field, then the magnetic forces will perform 
work on the other pole as it moves about the field. 

If we start from a given point, and move this pole from it to any other point, the 
work performed will be independent of the path of the pole between the two points ; 
provided that no electric current passes between the different paths pursued by the pole. 

Hence, when there are no electric currents but only magnets in the field, we may 
draw a series of surfaces such that the work done in passing from one to another shall 
be constant whatever be the path pursued between them. Such surfaces are called 
Equipotential Surfaces, and in ordinary cases are perpendicular to the Lines of mag- 
netic force. 

If these surfaces are so drawn that, when a unit pole passes from any one to the 
next in order, unity of work is done, then the work done in any motion of a magnetic 
pole will be measured by the strength of the pole multiplied by the number of surfaces 
which it has passed through in the positive direction. 

(52) If there are circuits carrying electric currents in the field, then there will still 
be equipotential surfaces in the parts of the field external to the conductors carrying the 
currents, but the work done on a unit pole in passing from one to another will depend 
on the number of times which the path of the pole circulates round any of these 
currents. Hence the potential in each surface will have a series of values in arith- 
metical progression, differing by *the work done in passing completely round one of the 
currents in the field. 

The equipotential surfaces will not be continuous closed surfaces, but some of them 



480 PROFESSOR CLEEK MAXWELL ON THE ELECTROMAGNETIC FIELD. 

will be limited sheets, terminating in the electric circuit as their common edge or 
boundary. The number of these will be equal to the amount of work done on a unit 
pole in going round the current, and this by the ordinary measurement = 4sry, where y 
is the value of the current. 

These surfaces, therefore, are connected with the electric current as soap-bubbles are 
connected with a ring in M. Plateau's experiments. Every current y has 4<ry surfaces 
attached to it. These surfaces have the current for their common edge, and meet it at 
equal angles. The form of the surfaces in other parts depends on the presence of other 
currents and magnets, as well as on the shape of the circuit to which they belong. 

PART III.— GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD. 

(53.) Let us assume three rectangular directions in space as the axes of x, y, and 3, 

and let all quantities having direction be expressed by their components in these three 

directions. 

Electrical Currents (p, q, r). 

(54) An electrical current consists in the transmission of electricity from one part of 
a body to another. Let the quantity of electricity transmitted in unit of time across 
unit of area perpendicular to the axis of x be called jp, then p is the component of the 
current at that place in the direction of x. 

We shall use the letters p, q, r to denote the components of the current per unit of 
area in the directions of #, y, z. 

Electrical Displacements (f, g, h). 

(55) Electrical displacement consists in the opposite electrification of the sides of a 
molecule or particle of a body which may or may not be accompanied with transmission 
through the body. Let the quantity of electricity which would appear on the faces 
dy.dz of an element dx, dy, dz cut from the body be f .dy.dz, then /is the component 
of electric displacement parallel to x. We shall use /, g, h to denote the electric 
displacements parallel to x, y, z respectively. 

The variations of the electrical displacement must be added to the currents p, q, r to 
get the total motion of electricity, which we may call/, #', r\ so that 






1 , dh 



• e 



(A) 



Electromotive Force (P, Q, E). 
(56) Let P, Q, R represent the components of the electromotive force at any point. 
Then P represents the difference of potential per unit of length in a conductor 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 481 

placed in the direction of x at the given point. We may suppose an indefinitely short 
wire placed parallel to x at a given point and touched, during the action of the force P, 
by two small conductors, which are then insulated and removed from the influence of 
the electromotive force. The value of P might then be ascertained by measuring the 
charge of the conductors. 

Thus if I be the length of the wire, the difference of potential at its ends will be PZ, 
and if C be the capacity of each of the small conductors the charge on each will be 
|CPZ. Since the capacities of moderately large conductors, measured on the electro- 
magnetic system, are exceedingly small, ordinary electromotive forces arising from 
electromagnetic actions could hardly be measured in this way. In practice such measure- 
ments are always made with long conductors, forming closed or nearly closed circuits. 

Electromagnetic Momentum (F, G, H). 

(57) Let F, G, II represent the components of electromagnetic momentum at any 
point of the field, due to any system of magnets or currents. 

Then F is the total impulse of the electromotive force in the direction of x that would 
be generated by the removal of these magnets or currents from the field, that is, if P 
be the electromotive force at any instant during the removal of the system 

F=ftdt. 

Hence the part of the electromotive force which depends on the motion of magnets or 
currents in the field, or their alteration of intensity, is 

P=-^, Q=-A K=-^. ...... (29) 

dt at at 

Electromagnetic Momentum of a Circuit, 

(58) Let s be the length of the circuit, then if we integrate 

j"(*i+G5+ni)* ■ ■ (») 

round the circuit, we shall get the total electromagnetic momentum of the circuit, or the 
number of lines of magnetic force which pass through it, the variations of which measure 
the total electromotive force in the circuit. This electromagnetic momentum is the 
same thing to which Professor Faraday has applied the name of the Electrotonic State. 
If the circuit be the boundary of the elementary area dy dz, then its electromagnetic 
momentum is 






and this is the number of lines of magnetic force which pass through the area dydz. 

Magnetic Force {a, |3, y). 

(59) Let a, |8, y represent the force acting on a unit magnetic pole placed at the 
given point resolved in the directions of w 9 y, and z. 



482 PKOFESSOK CLEBK MAXWELL ON THE ELECTBOMAGNETIC FIELD. 

Coefficient of Magnetic Induction (^). 

(60) Let p be the ratio of the magnetic induction in a given medium to that in air 
under an equal magnetizing force, then the number of lines of force in unit of area 
perpendicular to x will be pa, (p is a quantity depending on the nature of tHe medium, 
its temperature, the amount of magnetization already produced, and in crystalline bodies 
varying with the direction). 

(61) Expressing the electric momentum of small circuits perpendicular to the three 
axes in this notation, we obtain the following 



Equations of Magnetic Force. 



(JbU 



W 3 



W= 



dR dG ] 



dy 


dz' 


d¥ 


dK 


dz 


dx' 


dG 


d¥ 


dx 


d/ii 



. (B) 



Equations of Currents. 

(62) It is known from experiment that the motion of a magnetic pole in the electro- 
magnetic field in a closed circuit cannot generate work unless the circuit which the pole 
^escribes passes round an electric current. Hence, except in the space occupied by the 

electric currents, 

udx+fidy-\-ydz=d<p (31) 

a complete differential of <p, the magnetic potential. 

The quantity <p may be susceptible of an indefinite number of distinct values, according 
to the number of times that the exploring point passes round electric currents in its 
course, the difference between successive values of <p corresponding to a passage com- 
pletely round a current of strength c being inc. 

Hence if there is no electric current, 



but if there is a current jp', 



Similarly, 



_dp 
dy dz 



0; 



dy 
dy 

da 
dz 



dj$ 
dz 



- T =±vp'. 



dy 
dx 



**§[ > 



dB dx . , 

We may call these the Equations of Currents. 



» * • * »• » * « 



. (C) 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 483 

Electromotive Force in a Circuit. 
(63) Let | be the electromotive force acting round the circuit A, then 

5=J(*£+Q|+b£)*» ■ .......... (32) 

where ds is the element of length, and the integration is performed round the circuit. 

Let the forces in the field be those due to the circuits A and B, then the electro- 
magnetic momentum of A is 

J(F§+at+H£)&=I*+M«, (33) 

where u and v are the currents in A and B, and 

%=-§ t (Lu+M.v) (34) 

Hence, if there is no motion of the circuit A, 



dt dx 
dG dV 



dt ~~ dy' ' 



_ dR dW 

'™~ MO " dt dz ' 



(35) 



where T is a function of #, y^ z 9 and £, which is indeterminate as far as regards the 
solution of the above equations, because the terms depending on it will disappear on 
integrating round the circuit. The quantity *¥* can always, however, be determined in 
any particular case when we know the actual conditions of the question. The physical 
interpretation of **¥ is, that it represents the electric potential at each point of space. 

Electromotive Force on a Moving Conductor. 
(64) Let a short straight conductor of length &, parallel to the axis of #, move with 
a velocity whose components are -^, -^, ~, and let its extremities slide along two 

7 

parallel conductors with a velocity ~ . Let us find the alteration of the electro- 

tit 

magnetic momentum of the circuit of which this arrangement forms a part. 

In unit of time the moving conductor has travelled distances ~, -^, ~ along the 

O/Z ttz az 

directions of the three axes, and at the same time the lengths of the parallel conductors 

ds 
included in the circuit have each been increased by ^- 



Hence the quantity 



foJ+Gj+H*)*. 



MDCCCLXV. 3 U 



485 PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC MELD. 

will be increased by the following increments, 

fd¥ dx , d¥ dy , d¥ dz\ -, , , • n i , 

® t- -7-+-,- -r+T- -r h due to motion ot conductor, 

\dx dt dy dt dz dt/ 
ds fd¥ dx . dG dy , dH dz\ i , i ,*, £ . . , 

—a-r-l -r -T-+-T- -r-H — r- -r )? due to lengthening ot circuit. 

dt \dx ds dx ds dx ds/ ° ° 

The total increment will therefore be 

/d¥ dGyly /dR d¥\dz # 

\ % dx J dt \ dx dz J dt ' 

or, by the equations of Magnetic Force (8), 

/ dy n dz' 

If P is the electromotive force in the moving conductor parallel to x referred to unit 
of length, then the actual electromotive force is Ya ; and since this is measured by the 
decrement of the electromagnetic momentum of the circuit, the electromotive force due 
to motion will be 

(65) The complete equations of electromotive force on a moving conductor may now 
be written as follows : — 

Equations of Electromotive Force. 



^ / dy n dz\ dF d"^ 

v =t A {y^-Pdt)-w-'te' 

„ ( dz dx\ dG d^t 



E 



dt r dt J dt dy 

dx dy\ dft^dW 

dt dz 



(D) 



/ n dx dy \ 

(*{PTt- a dt) 



The first term on the right-hand side of each equation represents the electromotive 
force arising from the motion of the conductor itself. This electromotive force is per- 
pendicular to the direction of motion and to the lines of magnetic force ; and if a 
parallelogram be drawn whose sides represent in direction and magnitude the velocity 
of the conductor and the magnetic induction at that point of the field, then the area of 
the parallelogram will represent the electromotive force due to the motion of the con- 
ductor, and the direction of the force is perpendicular to the plane of the parallelogram. 

The second term in each equation indicates the effect of changes in the position or 
strength of magnets or currents in the field. 

The third term shows the effect of the electric potential Y. It has no effect in 
causing a circulating current in a closed circuit. It indicates the existence of a force 
urging the electricity to or from certain definite points in the field. 



PBOFESSOB CLEEK MAXWELL ON THE ELECTEOMAGNETIC EIELD. 485 

Electric Elasticity. 

(66) When an electromotive force acts on a dielectric, it puts every part of the 
dielectric into a polarized condition, in which its opposite sides are oppositely electri- 
fied. The amount of this electrification depends on the electromotive force and on the 
nature of the substance, and, in solids having a structure defined by axes, on the direc- 
tion of the electromotive force with respect to these axes. In isotropic substances, if k 
is the ratio of the electromotive force to the electric displacement, we may write the 

Equations of Electric Elasticity, 

Q=h/, V (E) 



E = Mi. 

Electric Resistance. 

(67) When an electromotive force acts on a conductor it produces a current of elec- 
tricity through it. This effect is additional to the electric displacement already con- 
sidered. In solids of complex structure, the relation between the electromotive force 
and the current depends on their direction through the solid. In isotropic substances, 
which alone we shall here consider, if £ is the specific resistance referred to unit of 
volume, we may write the 

Equations of Electric Resistance, 

• •••(F) 




Electric Quantity. 

(68) Let e represent the quantity of free positive electricity contained in unit of 
volume at any part of the field, then, since this arises from the electrification of the 
different parts of the field not neutralizing each other, we may write the 

Equation of Free Electricity, 

(69) If the medium conducts electricity, then we shall have another condition, which 
may be called, as in hydrodynamics, the 

Equation of Continuity, 
^4.^ + ^4 *==() 

dt dx dy dz 

(70) In these equations of the electromagnetic field we have assumed twenty variable 

3 u2 



486 



PEOPESSOE CLEEK MAXWELL OK THE ELECTROMAGNETIC MELD. 



quantities, namely, 

For Electromagnetic Momentum 

Magnetic Intensity . . 

Electromotive Force 

Current due to true conduction 

Electric Displacement 

Total Current (including variation of displacement) 

Quantity of free Electricity . 

Electric Potential 



¥ 


G 


H 


a, 


a 


7 


P 


Q 


R 


P 


1 


r 


f 


9 


h 


p' 


4 


/yJ 


e 







"\Tf 



Between these twenty quantities we have found twenty equations, viz. 

Three equations of Magnetic Force (B) 

Electric Currents (C) 

Electromotive Force (D) 

Electric Elasticity (E) 

Electric Resistance (F) 

Total Currents ......... (A) 

One equation of Free Electricity (G) 

„ Continuity (H) 

These equations are therefore sufficient to determine all the quantities which occur 
in them, provided we know the conditions of the problem. In many questions, how- 
ever, only a few of the equations are required. 

Intrinsic Energy of the Electromagnetic Field. 

(71) We have seen (33) that the intrinsic energy of any system of currents is found 
by multiplying half the current in each circuit into its electromagnetic momentum. 
This is equivalent to finding the integral 

E=i2(Fp'+G#'+Hr')dV (37) 

over all the space occupied by currents, where p, %, r are the components of currents, 
and F, G, H the components of electromagnetic momentum. 

Substituting the values of p\ q\ r f from the equations of Currents (C), this becomes 

1 



8?r 



Kg-f)+«(£-£)+ H (l4;)h 



Integrating by parts, and remembering that a, /3, y vanish at an infinite distance, the 
expression becomes 

i x r /<m dG\ , „/<zf <m\ . /dG d¥\] 7Tr 

where the integration is to be extended over all space. Referring to the equations of 
Magnetic Force (B), p. 482, this becomes 






(38) 



PEOFESSOK CLERK MAXWELL ON THE ELECTROMAGNETIC MELD. 487 

where a, )3, y are the components of magnetic intensity or the force on a unit magnetic 
pole, and ^a, ^/3, py are the components of the quantity of magnetic induction, or the 
number of lines of force in unit of area. 

In isotropic media the value of p is the same in all directions, and we may express 
the result more simply by saying that the intrinsic energy of any part of the magnetic 
field arising from its magnetization is 

" r 



8tt 



per unit of volume, where I is the magnetic intensity. 

(72) Energy may be stored up in the field in a different way, namely, by the action 
of electromotive force in producing electric displacement. The work done by a variable 
electromotive force, P, in producing a variable displacement, f, is got by integrating 

ft*/ 

from P = to the given value of P. 

Since 7=kf^ equation (E), this quantity becomes 

Hence the intrinsic energy of any part of the field, as existing in the form of electric 
displacement, is P (P/ + Q^+EA)^. 

The total energy existing in the field is therefore 



E=2- 



l(^ a+ ^/3+y^)+i(P/+Q^+M)MV. ... . (I) 



The first term of this expression depends on the magnetization of the field, and is 
explained on our theory by actual motion of some kind. The second term depends on 
the electric polarization of the field, and is explained on our theory by strain of some 
kind in an elastic medium. 

(73) I have on a former occasion* 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 induc- 
tion of currents and the polarization of dielectrics, I wish merely to direct the mind of 
the reader to mechanical phenomena which will assist him in understanding the elec- 
trical ones. All such phrases in the present paper are to be considered as illustrative, 
not as explanatory. 

(74) 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 electromagnetic phenomena is 
mechanical energy. The only question is, Where does it reside % On the old theories 

* " On Physical Lines of Force," Philosophical Magazine, 1861-62. 



488 PEOFESSOE CLEEK MAXWELL ON THE ELE0TEOMAG2STBTIC FIELD. 

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 electromagnetic 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 polarization 
and electric polarization, or, according to a very probable hypothesis, as the motion and 
the strain of one and the same medium. 

(75) The conclusions arrived at in the present paper are independent of this hypo- 
thesis, 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 electromagnetic field. 

PAET IV.— MECHANICAL ACTIONS IN THE FIELD. 

Mechanical Force on a Moveable Conductor. 

(76) We have shown (§§ 34 & 35) that the work done by the electromagnetic forces 
in aiding the motion of a conductor is equal to the product of the current in the con- 
ductor multiplied by the increment of the electromagnetic momentum due to the 
motion. 

Let a short straight conductor of length a move parallel to itself in the direction of 
#, with its extremities on two parallel conductors. Then the increment of the electro- 
magnetic momentum due to the motion of a will be. 



/d¥ dx dG dy d\l dz\ v 
' I dx ds ' dx ds ' dx ds J 



That due to the lengthening of the circuit by increasing the length of the parallel con- 
ductors will be 



The total increment is 



/d¥ dx ( d¥ dy dF dz\ . ^ 
I dx ds ' dy ds ' ds ds J 



dhx< 



\dyfdG d¥\ dz/d'F dll 



It Ub 



ds I dx dy J ds \ dz 

which is by the equations of Magnetic Force (B), p. 482, 



fl^(s^y-S^)- 



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



PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 489 

Let C be the current in the conductor, and letj?', q', r' be its components, then 

Xa*=Cato(Jpy-|p/3), 
or 'X=[Ayq f —pfir\ 



Similarly, Y=t»»> -prf, 



(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. 

(77) In any part of the field not traversed by electric currents the distribution of 
magnetic intensity may be represented by the differential 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, 

££— „ ^_« **—„ 

Substituting these values of «, /3, y in the expression (equation 38) for the intrinsic 
energy of the field, and integrating by parts, it becomes 



\j 8t \ doc ' dy ' dz 

The expression 



•dY. 



indicates the number of lines of magnetic force which have their origin within the 
space V. 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 4<zr lines belonging to it, since 
it produces unit of magnetic intensity at unit of distance over a sphere whose surface 

is 4-r. 

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

T£=—%(%<pm)dV. (40) 

If there are two magnetic poles m t and m 2 producing potentials (p l and p 2 in the field , 
then if m a is moved a distance dx, and is urged in that direction by a force X, then the 
work done is Xefcr, and the decrease of energy in the field is 

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



490 PEOFESSOE CLEEK MAXWELL OJST THE ELECTEOMAQKETIC FIELD. 

Since the distribution <p x is determined by m 15 and <p 2 by m 2 , the quantities ^ 1 m 1 and 
p 2 m 2 will remain constant. 

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

so that we get 

X.dw=d(7n 2 <p l )^ 
or 

L « (K) 



where a x represents the magnetic intensity due to m x . 
Similarly, Y=m 2 fi 1 , 

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 <p is 

a 1== -!!!il, . . (41) 

/x r 

where m x is the strength of the pole and r the distance from it. 
The repulsion between two poles of strength m x and m 2 is 

dp = rn^ (42) 

dr pr* ' 



In air or any medium in which ^=1 this is simply — L?> but in other media the force 

acting between two given magnetic poles is inversely proportional 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. 

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) 

v dW dW ^ dV 

x== — '-r" 5 \4= — -y-9 lt= r~" 

ax dy dz 

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



*s{*(i+i+s)h, 



or by the equation of Free Electricity (G), p. 485, 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 491 

By the same demonstration as was used in the case of the mechanical action on a magnet, 
it may be shown that the mechanical force on a small body containing a quantity e 2 of 
free electricity placed in a field whose potential arising from other electrified bodies 
is lf x , has for components 



dx 






P) 



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 x of free electrity, the only solution of If x is 

*>=s£' (43) 

where r is the distance from the electrified body. 

The repulsion between two electrified bodies e 19 e 2 is therefore 

«**!=*!$. (44) 

Measurement of Electrical Phenomena by 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 mecha- 
nical action between currents. The electrostatic system of measurement is founded on 
the mechanical action between electrified bodies, and is independent of, and incom- 
patible 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 q x , tj 2 of electricity is 

where r is the distance between them. 

Let the relation of the two systems be such that one electromagnetic unit of elec- 
tricity contains v electrostatic units ; then ^=1;^ and q 2 =ve 2 , and this repulsion becomes 

ff "£A = ^!A by equation (44), ....... (45) 

whence #, the coefficient of "electric elasticity " in the medium in which the experi- 
ments are made, i. e. common air, is related to v, the number of electrostatic units in one 

electromagnetic unit, by the equation 

k=4:vv 2 . ........... (46) 

mdccclxv. 3 x 



492 PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 

The quantity v may be determined by experiment in several ways. According) to the 
experiments of MM. Weber and Kohlrausch, 

#=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 electrified, and if we also 
assume that this polarization or electric displacement is proportional to the electro- 
motive force which produces or maintains it, then we can show that electrified bodies 
in a dielectric medium will act on one another with forces obeying the same laws as are 
established by experiment. 

The energy, by the expenditure of which electrical attractions and repulsions are pro- 
duced, we suppose to be stored up in the dielectric medium which surrounds the electri- 
fied bodies, and not on the surface of those bodies themselves, 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 npt 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, instead 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 E be the intrinsic energy of the field surrounding 
two gravitating bodies Mj , M 2 , and let E' be the intrinsic energy of the field surrounding 
two magnetic poles m,, m 2 , equal in numerical value to M 2 , M 2 , and let X be the gravi- 
tating force acting during the displacement he, and X' the magnetic force, 

3&F=.aE, X&r=SE',; 
now X and X' are equal in numerical value, but of opposite signs; so that 

XE=-&E', 
or 

E=C-E' 



PE0FESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 493 

where «, j3, y are the components of magnetic intensity. If R be the resultant gravi- 
tating force, and R! the resultant magnetic force at a corresponding part of the field, 

R=~R', and a 2 +/3 2 +y 2 =R 2 =R' 2 . 

Hence 

Iij=z:C— 2rT~ R d v (4») 

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

As energy is essentially positive, it is impossible for any part of space to have nega- 
tive intrinsic energy. Hence those parts of space in which there is no resultant force, 
such as the points of equilibrium 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 

JLpa 

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. 



PART Y.—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 the 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 k its coeffi- 
cient of electric elasticity; then on one side of the condenser the potential is ^„ and on 
the other side ^,+1, and within its substance 

-~^=2~=Jcf. (48) 

Since -^ and therefore /is zero outside the condenser, the quantity of electricity on its 
first surface = — Sf, and on the second +S/. The capacity of the condenser is there- 
fore Sf = jjjjj in electromagnetic measure. 

3x2 



494 PEOEESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 

Specific Capacity of Electric Induction (D). 

(84) If the dielectric of the condenser be air, then its capacity in electrostatic mea- 

S 
sure is - — (neglecting corrections 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 capa- 

DS 
city of the condenser will be — -• 

Hence D=^, .. (49) 

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

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 displacement. 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 
Faeaday (Experimental Researches, Series XI.) and by Mr. F. Jenkin (Report of Com- 
mittee 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 sufficient time till a condition of permanent 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 con- 
nected by a conductor, a certain quantity of electricity may be urged through the con- 
ductor during the reestablishment of equilibrium. 

Let the thickness of the several layers of the condenser be a x , # 2 , &c. 
Let the values of k for these layers be respectively k„ Tc^ # 3 , and let 

a l fc 2 -\-a 2 k 2 -{-8tc.=ak, . (50) 

where Jc is the " electric elasticity" of air, and a is the thickness of an equivalent con- 
denser of air. 

Let the resistances of the layers be respectively r 19 r 2 , &c, and let f\+r 2 + &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 f„ f< 2 , &c. 

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

Let the potential on the first surface be ^„ and the electricity per unit of surface e x . 

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



PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC EIELD. 



495 



e, 



■/» 



e *=fi-f» 



de x 
~dt 

a\ 
dt 



"JP 



13 



-Pl-P» 



r 



. . (51) 



&c. 
But by equations (E) and (F), 

x 2 x 3 

&C. 



&C. 



•a l kf l =—r l p l9 

&C. &C. 



(52) 



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



\p 



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



and 



"'' r ajc^ 






/. 






&,=: — ( — f . ), &C. 



• ( 53 ) 



These are the quantities of electricity on the different surfaces. 

(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 

f =4^1+42(4+^+43(^+^2+4 &c « = ' ..... (54) 
whence if d x is what e x becomes at the instant of discharge, 



e\ 



>Jr 



\jr 



r a x k t ak 



1 ak 



(55) 



yfr 



The instantaneous discharge is therefore —-, or the quantity which would be dis- 

dK 

charged by a condenser of air of the equivalent thickness #, 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 dissipation of the internal charges. 
Let *¥"' be the difference of potential of the extreme surfaces at any time t ; then 

^=ajc l f l +ajc 2 f 2 +&cc.; ........ (56) 

but a i Jc ifi" =z ~ r ^~i: 



^2^2/2 — ~~ r * dt 



496 PEOFESSOE CEBRK MAXWELL OF THE ELECTEOMAGNETIC FIELD. 



Hence f 1 =A 1 e"~ : ^ , f 2 =A 2 e~ r * , &c. ; and by referring to the values of d l9 e 2 , &c.. 



we find 



1 r a x k x ak 



r a 2 k q ak 



M* r ^p ► • • (^ ' ) 



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

*=»{(5-^)«-*'+(?-*)^ , + *-} (58) 

(89) It appears from this result that if all the layers are made of the same sub-, 
stance, H 1 will be zero always. If they are of different substances, the order in which 
the} 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 substances. 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 coefficients 
is the same as that of the indices, the value of V can never change sign, but must start 
from zero, become positive, and finally disappear. 

(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, 

y l ==p l r 1 +p 2 r 2 +&c.=p'R. ........ (59) 

Integrating with respect to the time, and calling q l , q 2 , q the quantities of electricity 
which traverse the different conductors, 

q i r l +q 2 rt+&c.=giR (60) 

The quantities of electricity on the several surfaces will be 

^S + ft — ft, 

&c. ; 
and since at last all these quantities vanish, we find 

q, =e f 1 ~q, 

q 2 =£+61—4; 

whence ? R = _(^ + ^ + & c /)-^, 



or 



?=^{«A^(^-^) 2 +«A^ 8 (-^-^r)V&c.}, . . . (61) 



PEOFESSOE CLEEK MAXWELL OW THE ELECTEOMAGNEKC FIELD. 19(7 

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

PAET VL— ELECTROMAGNETIC THEORY OF LIGHT. 

(91) At the commencement of this paper we made use of the optical hypothesis df 
an elastic medium through which the vibrations of light are propagated, in order to^ 
show 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 pheno- 
mena, 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 properties 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 propa- 
gated through the field with a velocity V. Then all the electromagnetic functions will 
be functions of 7 . , v . 

The equations of Magnetic Force (B), p. 482, will become 

dR dG 

dF 7 dn 

7 dQ dF 

ri duo aw 

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

Zfi»a+wiffcjS+W|Bi»y==0, ......... (§2) 

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

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

d* + dj+lF~ J > md a^dy*+d^-- V > ..... [bd) 



dJ 



(64) 



* Since this paper was communicated to the Eoyal Society, I have seen a paper by M. Gatoain in the Annales 
de Chimie for 1864, in which he has deduced the phenomena of electric absorp tion and secondary discharge 
from the theory of compound condensers. 



498 



PEOEESSO& CLEEK MAXWELL ON THE ELECTROMAGNETIC EIELD, 



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



r>'- df 



2 



dff 

dt 



m 



dh 
dt' 



••••••• I vU I 



But these electric displacements are caused by electromotive forces, and by the equations 

of Electric Elasticity (E), 

P=Af, Q=%, TH=kh. (66) 

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



at ax 
dG dW 



Q= — 



dt dy 



, y • ' • : • • • • » • • • l^'y 



dt dz 



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

dJ 






d*F d*¥ 
dt 3 ~r ' dxdt 

d^G d*P 

dt % ~^dydt 

dm . d*v 



)=o, 
)=o, 



dt % ^~dzdt 



=0. 



. . . . (68) 



If we differentiate the third of these equations with respect to y, and the second with 
respect to z, and subtract, J and "V disappear, and by remembering the equations (B) of 
magnetic force, the results may be written 



d* 

k V (*u = 45Tj& -jp (Ace, 

AV»3= 4*^6 ^j/^3, 

W>y = 4^^^y. 



. . . (69) 



(95) If we assume that a, /3^ y are functions of lw-}-my-{-nZ'—Yt=w, the first equa- 
tion becomes 



flfw 2 



or 



V 



= ±\/ 



4*tjx 



• •' 



(71) 



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



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC HELD- 499 

This wave consists entirely of magnetic disturbances, the direction of magnetization 
being in the plane of the wave. No magnetic disturbance whose direction of magneti- 
zation 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 propaga- 
tion, 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 k is air, in which ^=1, and therefore, by equation (46), 

V=v. (72) 

By the electromagnetic experiments of MM. Webee and Kohlrausch *, 

v= 310,740,000 metres per second 

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'sf experiments, is 

V=314,858,000; 

according to the more accurate experiments of M. Foucault J, 

V=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 

V= 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 galvano- 
meter, 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 show that light and magnetism are affections 
of the same substance, and that light is an electromagnetic disturbance 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. 

* Leipzig Transactions, vol. v. (1857), p. 260, or Poggendobff's c Annalen/ Aug. 1856, p. 10. 
t Comptes Eendus, vol. xxix. (1849), p. 90. % Ibid. vol. lv. (1862), pp. 501, 792. 

MDCCCLXY. 3 Y 



500 PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC EIELD. 

If we determine % from the equation 

VV-^4-^4-^-J (73) 

v ^~ due 2 + % 2 ^ ^ 2 "^ ^ ' 

and F, G', H' from the equations 

F'=F-^, cy=G— $£, H'=H-^, .... (74) 

then 

<flF' d& dlV _ r5) 

dx ' dy ' /fe ' ' ^ ' 

and the equations in (94) become of the form 

a 2 F' , d /, TJ . , ^ 

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

*=-f +#*.**). (77) 



*V^==W^+^(¥+g)) (76) 



and that £V 2 F' = 4t^ ^ , 

dm 1 



jfcV*H'=4*7* 



(78) 



r2 



df j 

Hence the disturbances indicated by F, G', H' are propagated with the velocity 
V = \ / — through the field ; and since 

V 47Tju, 

doc * dy ~* doc ~ ' 

the resultant of these disturbances is in the plane of the wave. 

(99) The remaining part of the total disturbances F, G, H being the part depending 
on %, is subject to no condition except that expressed in the equation 

dt ^ df "" u * 

If we perform the operation V 2 on this equation, it becomes 

dJ 
ke=-T7— #V 2 p(#, y, z) (79) 

Since the medium is a perfect insulator, #, the free electricity, is immoveable, and 

dJ . 
therefore -j, is a function of x, y, 2, and the value of J is either constant or zero, or 

uniformly increasing or diminishing with the time ; so that no disturbance depending 
on J can be propagated as a wave. 

(100) The equations of the electromagnetic field, deduced from purely experimental 
evidence, show that transversal vibrations only can be propagated. If we were to go 
beyond our experimental knowledge and to assign a definite density to a substance which 



PE0FE8S0E CLEEK MAXWELL ON THE ELECTEOMAGNETIC HELD. 501 

we should call the electric fluid, and select either vitreous or resinous electricity as the 
representative of that fluid, then we might have normal vibrations propagated with a 
velocity depending on this density. We have, however, no evidence as to the density of 
electricity, as we do not even know whether to consider vitreous electricity as a sub- 
stance or as the absence of a substance. 

Hence electromagnetic science leads to exactly the same conclusions as optical science 
with respect to the direction of the disturbances which can be propagated through the 
field; both affirm the propagation of transverse vibrations, and both give the same velocity 
of propagation. On the other hand, both sciences are at a loss when called on to affirm 
or deny the existence of normal vibrations. 

Relation between the Index of Refraction and the Electromagnetic Character of the 

substance, 

(101) The velocity of light in a medium, according to the Undulatory Theory, is 

-V 

i v °' 

where i is the index of refraction and V is the velocity in vacuum. The velocity, 
according to the Electromagnetic Theory, is 



/z 

V 47TU 



k 

47rp 
I 

where, by equations (49) and (71), k=^k , and & =:4:7rY 2 . 

Hence D=-, ........... (80) 

or the Specific Inductive Capacity is equal to the square of the index of refraction 
divided by the coefficient of magnetic induction. 

Propagation of Electromagnetic Disturbances in a Crystallized Medium. 

(102,) Let us now calculate the conditions of propagation of a plane wave in a 
medium for which the values of h and ^ are different in different directions. As we 
do not propose to give a complete investigation of the question in the present imperfect 
state of the theory as extended to disturbances of short period, we shall assume that the 
axes of magnetic induction coincide in direction with those of electric elasticity. 

(103) Let the values of the magnetic coefficient for the three axes be X, ^, v, then 
the equations of magnetic force (B) become 

dR dG 

Ace =~7~ — ~j— 3 
ay dz 

~ dF drL 

^P dz dx ' 

_dG_dF 
' dx dy \ 

3 y2 



... 



. . . (81) 



502 



PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 



The equations of electric currents (C) remain as before. 
The equations of electric elasticity (E) will be 

P = w/; 

Q=4?r% ? I . . . . . 

K=WA, 

where 47m 2 , farb 2 , and Awe* are the values of k for the axes of x, y, z. 

Combining these equations with (A) and (D), we get equations of the form 

dW 



(82) 



(iv \ da? 'P dy* ' dz 2 ) pv dx\ dx '> dy ~*~ 



dz 



1 /d*F d*V 

d'X dp ~T~dxdt 



(83) 



(104) If £, m, n are the direction-cosines of the wave, and V its velocity, and if 



lx-\-my-\-nz — Vt=w 9 



(84) 



then F, G, H, and Y will be functions of w ; and if we put F, G', H', "¥' for the second 
differentials of these quantities with respect to w, the equations will be 



V 2 -^(^+?))G'+^H'+^F-mV^=0, 



V 



If we now put 



e+3) 



c q nl 



c q nm 



H'+^F+^fff-nV^O. 



P 



v. 



. . 



(85) 






Z*Ji( J*/» + &v) + m>(^» + « 2 A) + n*v(a\ + £» 



+ 



cfib^c 2 /P . m 2 m 



we shall find 



Aftv V« 2 '6 3 



5+ « + 3 (a+m>+^)= u > 



(86) 



FV 2 U-Z^'VU=0, . . 

with two similar equations for G' and H'. Hence either 

V=0, ....... 

u=o 



(87) 



or 



VF=W, VG'=m^' and YU.'=nV. 



(88) 
(89) 

(90) 



The third supposition indicates that the resultant of F, G', H' is in the direction 
normal to the plane of the wave ; but the equations do not indicate that such a disturb- 
ance, if possible, could be propagated, as we have no other relation between ty ! and 
F', G', H'. 

The solution V = refers to a case in which there is no propagation. 

The solution U=0 gives two values for V 2 corresponding to values of F', G', H', which 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 



503 



are given by the equations 



at -f- , % (j -J- o M U, 



(91) 



c%v 



(105) The velocities along the axes are as follows:- 



Direction of propagation . 



, . (92) 



» • 



Direction of the electric displacements < 





# 


y 


2 






a 2 


« 2 


# 




• — 


— 






V 


f* 




6 2 




6 2 


y 


V 




A 




c 2 


c 2 




s 


— 


— 






f* 


A 





Now we know that in each principal plane of a crystal the ray polarized in that 
plane obeys the ordinary law of refraction, and therefore its velocity is the same in 
whatever direction in that plane it is propagated. 

If polarized light consists of electromagnetic disturbances in which the electric dis- 
placement is in the plane of polarization, then 

If, on the contrary, the electric displacements are perpendicular to the plane of pola- 
rization, 

•k=p=v. , # . (94) 

We know, from the magnetic experiments of Faraday, Plucker, &c, that in many 
crystals X, ^, v are unequal. 

The experiments of Knoblauch * on electric induction through crystals seem to show 
that a, b and e, may be different. 

The inequality, however, of X, ^, v is so small that great magnetic forces are required 
to indicate their difference, and the differences do not seem of sufficient magnitude to 
account for the double refraction of the crystals. 

On the other hand, experiments on electric induction are liable to error on account 
of minute flaws, or portions of conducting matter in the crystal. 

Further experiments on the magnetic and dielectric properties of crystals are required 
before we can decide whether the relation of these bodies to magnetic and electric 
forces is the same, when these forces are permanent as when they are alternating with 
the rapidity of the vibrations of light. 

* Philosophical Magazine, 1852, 



504 PKOEE8SOB CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 

Relation between Electric Resistance and Transparency. 

(106) If the medium, instead of being a perfect insulator, is a conductor whose resist- 
ance per unit of volume is g>, then there will bid not only electric displacements, but true 
currents of conduction in which electrical energy is transformed into heat, and the undu- 
lation is thereby weakened. To determine the coefficient of absorption, let us investi- 
gate the propagation along the axis of x of the transverse disturbance G. 

By the former equations 

= - 4 *? 6 (f+2) b y( A )> 

rf 2 Cr /I fl^Cr 1 flCr\ 

^=+M^4^) by(E)and(F) ' • • < 95) 

If G is of the form 

0=6'^ cos (qx+nt), (96) 

we find that 

« = ^2=?^V (97) 

g q g i K J 

where V is the velocity of light in air, and i is the index of refraction. The proportion 
of incident light transmitted through the thickness x is 

e~^ x . . . . (98) 

Let R be the resistance in electromagnetic measure of a plate of the substance whose 
thickness is x, breadth #, and length 7, then 

OX 

2px=^l±.- (99) 

(107) Most transparent solid bodies are good insulators, whereas all good conductors 
are very opaque. 

Electrolytes allow a current to pass easily and yet are often very transparent. We 
may suppose, however, that in the rapidly alternating vibrations of light, the electro- 
motive forces act for so short a time that they are unable to effect a complete separation 
between the particles in combination, so that when the force is reversed the particles 
oscillate into their former position without loss of energy. 

Gold, silver, and platinum are good conductors, and yet when reduced to sufficiently 
thin plates they allow light to pass through them. If the resistance of gold is the same 
for electromotive forces of short period as for those with which we make experiments, 
the amount of light which passes through a piece of gold-leaf, of which the resistance 
was determined by Mr. C. Hockin, would be only 10~~ 50 of the incident light, a totally 
imperceptible quantity. I find that between ^^ and y<f -o of green light gets through 



PBOFESSOB CLEEK MAXWELL ON THE ELECTBOMAGNETIC MELD. 505 

such gold-leaf. Much of this is transmitted through holes and cracks;. there is enough, 
however, transmitted through the gold itself to give a strong green hue to the 
transmitted light. This result cannot be reconciled with the electromagnetic theory 
of light, unless we suppose that there is less loss of energy when the electromotive forces 
are reversed with the rapidity of the vibrations of light than when they act for sensible 
times, as in our experiments. 

Absolute Values of the Electromotive and Magnetic Forces called into play in the 

Propagation of Light. 

(108) If the equation of propagation of light is 

F=Acos~(s— W), 
the electromotive force will be 

P=— Ay Vsin~(2: — Vt); 

and the energy per unit of volume will be 

P 2 

8^V 2 ' 

where P represents the greatest value of the electromotive force. Half of this consists 
of magnetic and half of electric energy. 

The energy passing through a unit of area is 



8?raV 

so that 

P = X /8^VW, 

where V is the velocity of light, and W is the energy communicated to unit of area by 
the light in a second. 

According to Pouillet's data, as calculated by Professor W. Thomson*, the mecha- 
nical value of direct sunlight at the Earth is 

83*4 foot-pounds per second per square foot. 

This gives the maximum value of P in direct sunlight at the Earth's distance from the Sun, 

P=60,000,000, 

or about 600 Daniell's cells per metre. 

At the Sun's surface the value of P would be about 

13,000 Daniell's cells per metre. 

At the Earth the maximum magnetic force would be *193f. 

At the Sun it would be 4*13. 

These electromotive and magnetic forces must be conceived to be reversed twice in 
every vibration of light; that is, more than a thousand million million times in a second. 

* Transactions of the Eoyal Society of Edinburgh, 1854 (" Mechanical Energies of the Solar System"), 
f The horizontal magnetic force at Kew is about 1-76 in metrical units. 



506 PEOEESSOK CLEEK MAXWELL OJST THE ELECTEOMAGNETIC FIELD. 

PAET VIL— CALCULATION OP THE COEFFICIENTS OP ELECTEOMAGNETIC INDUCTION. 

General Methods. 

(109) The electromagnetic relations between two conducting circuits, A and B, 
depend upon a function M of their form and relative position, as has been already 
shown. 

M may be calculated in several different ways, which must of course all lead to the 
same result. 

First Method. M is the electromagnetic momentum of the circuit B when A carries 
a unit current, or n , , , , v 

m=J(f5+gJ+hS)^ 

where F, G, H are the components of electromagnetic momentnm due to a unit current 
in A, and ds 1 is an element of length of B, and the integration is performed round the 
circuit of B. 

To find F, G, H, we observe that by (B) and (C) 

d*F , d*F . d q F , 

with corresponding equations for G and H, p ! , g ; , and ^ being the components of the 
current in A. 

Now if we consider only a single element ds of A, we shall have 

p'= Js ds, d=j s ds, r'= Js ds, 
and the solution of the equation gives 

g as ■ g ds ' g ds ' 

where g is the distance of any point from ds. Hence 



CCfifdx dx dy dy dz dz\ 
J J 9 \ ds ds' + dsds'+ds ds l ) asaS 

= l i - cos 6dsds\ 



where 6 is the angle between the directions of the two elements ds, ds ! , and g is the 
distance between them, and the integration is performed round both circuits. 

In this method we confine our attention during integration to the two linear circuits 
alone. 

(110) Second Method. M is the number of lines of magnetic force which pass 
through the circuit B when A carries a unit current, or 

where pec, p(3, py are the components of magnetic induction due to unit current in A, 



PEOEESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC HELD. 507 

S' is a surface bounded by the current B, and Z, m, n are the direction- cosines of the 

normal to the surface, the integration being extended over the surface. 

We may express this in the form 

1 
M=^2~2 sin 6 sin (I sin <pdS f ds, 

where dS f is an element of the surface bounded by B, ds is an element of the circuit A, 
q is the distance between them, 6 and 8 are the angles between g and ds and between 
g and the normal to dS 1 respectively, and <p is the angle between the planes in which 
6 and & are measured. The integration is performed round the circuit A and over the 
surface bounded by B. 

This method is most convenient in the case of circuits lying in one plane, in which 
case sin 0=1, and sin<p=l. 

111. Third Method. M is that part of the intrinsic magnetic energy of the whole 
field which depends on the product of the currents in the two circuits, each current 
being unity. 

Let a, /3, y be the components of magnetic intensity at any point due to the first 
circuit, a f , /3', y' the same for the second circuit; then the intrinsic energy of the 
element of volume dV of the field is 

The part which depends on the product of the currents is 

Hence if we know the magnetic intensities I and I' due to unit current in each circuit, 
we may obtain M by integrating 

•t2u.IV cos 4dV 

4tt r 

over all space, where 6 is the angle between the directions of I and I'. 

Application to a Coil. 

(112) To find the coefficient (M) of mutual induction between two circular linear 
conductors in parallel planes, the distance between the curves being everywhere the same, 
and small compared with the radius of either. 

If r be the distance between the curves, and a the radius of either, then when r is 
very small compared with «, we find by the second method, as a first approximation, 



M=4^Aog e ^-2V 



To approximate more closely to the value of M, let a and a x be the radii of the circles, 
and b the distance between their planes ; then 

r 2 =(a— a^+b 2 . 
mdccclxv. 3 z 



508 PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC MELD. 

We obtain M by considering the following conditions : — 
1st. M must fulfil the differential equation 

(PM d*M IdM__ 
da 2, * db 2 * a da 

This equation being true for any magnetic field symmetrical with respect to the common 
axis of the circles, cannot of itself lead to the determination of M as a function of a, a„ 
and b. We therefore make use of other conditions. 

2ndly. The value of M must remain the same when a and a Y are exchanged. 

3rdly. The first two terms of M must be the same as those given above. 

M may thus be expanded in the following series : — 



M: 



b r \ ' 2 a l 16 d l 32 a s j 



_4 ffg /2+ lg - fl i+ 1 b*-3(a-a*)__ 1 (6&M«-*i) a )(«-«i) + &c l 

( 2 a ' 16 a 2 48 a s j 

(113) We may apply this result to find the coefficient of self-induction (L) of a circular 
coil of wire whose section is small compared with the radius of the circle. 

Let the section of the coil be a rectangle, the breadth in the plane of the circle being 
c, and the depth perpendicular to the plane of the circle being b. 

Let the mean radius of the coil be a, and the number of windings n; then we find, 

by integrating, ^ „„„„ 

L=to| I j i M(#y x'y ! )dx dy dx dy\ 

where M(xyx ! y') means the value of M for the two windings whose coordinates are xy 
and #y respectively ; and the integration is performed first with respect to x and y over 
the rectangular section, and then with respect to x ! and y' over the same space. 

L==47mVJlog e ™+ 

, 13 sin 4 , . .) , 

Here a~ mean radius of the coil. 
„ ft=z diagonal of the rectangular section ~sfl?A r &. 
„ 0= angle between r and the plane of the circle. 
„ n= number of windings. 
The logarithms are Napierian, and the angles are in circular measure. 

In the experiments made by the Committee of the British Association for deter- 
mining a standard of Electrical Resistance, a double coil was used, consisting of two 
nearly equal coils of rectangular section, placed parallel to each other, with a small 
interval between them. 



PEOHBSSOB CLEEK MAXWELL ON THE ELECTBOMAGNETIC FIELD. 509 

The value of L for this coil was found in the following way. 

The value of L was calculated by the preceding formula for six different cases, in 

which the rectangular section considered has always the same breadth, while the depth 

was 

A, B, C, A+B 5 B+C, A+B+C, 

and n=l in each case. 

Calling the results 

L(A), ' L(B), L(C), &c, 

we calculate the coefficient of mutual induction M(AC) of the two coils thus, 
2ACM(AC)=(A+B+C) 2 L(A+ 

Then if n Y is the number of windings in the coil A and n 2 in the coil B, the coefficient 
of self-induction of the two coils together is 

L = nfL( A) + 2n l n 2 L( AC) + wJL(B) . 

(114) These values of L are calculated on the supposition that the windings of the 
wire are evenly distributed so as to fill up exactly the whole section. This, however, is 
not the case, as the wire is generally circular and, covered with insulating material. 
Hence the current in the wire is more concentrated than it would have been if it had 
been distributed uniformly over the section, and the currents in the neighbouring wires 
do not act on it exactly as such a uniform current would do. 

The corrections arising from these considerations may be expressed as numerical 
quantities, by which we must multiply the length of the wire, and they are the same 
whatever be the form of the coil. 

Let the distance between each wire and the next, on the supposition that they are 
arranged in square order, be D, and let the diameter of the wire be d, then the correc- 
tion for diameter of wire is 



log 7 + 3lo g 2 + 3--g-j. 



The correction for the eight nearest wires is 

+0-0236. 
For the sixteen in the next row 

+0-00083. 

These corrections being multiplied by the length of wire and added to the former 
result, give the true value of L, considered as the measure of the potential of the coil 
on itself for unit current in the wire when that current has been established for some 
time, and is uniformly distributed through the section of the wire. 

(115) But at the commencement of a current and during its variation the current is 
not uniform throughout the section of the wire, because the inductive action between 
different portions of the current tends to make the current stronger at one part of the 
section than at another. When a uniform electromotive force P arising from any cause 

3z2 



510 PEOFESSOK CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD, 

acts on a cylindrical wire of specific resistance g 9 we have 

-r. d¥ 

where F is got from the equation 

d 2 F 1 d¥_ 

l^ + ^W — ix W> 

r being the distance from the axis of the cylinder. 

Let one term of the value of F be of the form IV", where T is a function of the time, 
then the term of p which produced it is of the form 



Hence if we write 






PZ-\ l+ dt) e dt* r 



dl 



2 



l 2 2 2 dfi ^ ' 



2 1 d 8 T 



/ — — ^£,\_, # 



1 2 .2 2 dt 3 
The total counter current of self-induction at any point is 

J {" -PJ *t- - 1 + j -#1 + y~ f 2 2 2 W 2 / + &c * 
from £=0 to £=oo . 

When £=0, ^=0, .\ l-^-j =P, (^l ^O* &c. 

When t=co 9J? =j> ... (f )^ = 9 (^=0. &c * 

n r /P \ 1 1 jU/TT 2 ^T |A 3 1 rf 2 T 

2*(j-p)«Z«Ztf= -T*r 2 + 2 ^ ^ + -pr 1^73 T^°+ &c ' 

from t=0 to =oc* 

When £=0, J9=0 throughout the section, .*. (-^A =P? ( *^i" ) ^O? & c * 

/dT\ /d q T\ 

When t=oo , p=0 throughout . . * . .% [-$) =0, (-^2) =0, &c. 



\ / 00 \ / 00 



Also if I be the length of the wire, and R its resistance, 



R=-% 



P/ 

and if C be the current when established in the wire, C= -g« 
The total counter current may be written 

i(T„-T.)-!4c=-^by§(86). 



PEOFESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 511 

Now if the current instead of being variable from the centre to the circumference of 
the section of the wire had been the same throughout, the value of F would have been 



F=T+^y(l~- 2 ), 






where y is the current in the wire at any instant, and the total countercurrent would 

have been 

n'ldF , Z /rp ^ 3 l n L'C 
^^■2«-rfr== s (T w -T )-- if 6 g C==- s -,say. 

Hence 

1j = 1j — •4^6, 

or the value of L which must be used in calculating the self-induction of a wire for 
variable currents is less than that which is deduced from the supposition of the current 
being constant throughout the section of the wire by ^pl, where I is the length of the 
wire, and p is the coefficient of magnetic induction for the substance of the wire. 

(116) The dimensions of the coil used by the Committee of the British Association 
in their experiments at King's College in 1864 were as follows : — 

metre. 

Mean radius ....... =#='158194 

Depth of each coil =# = '01608 

Breadth of each coil .... =c = *01841 

Distance between the coils . . . =-02010 

Number of windings . . . . %z=313 

Diameter of wire ..... =-00126 

The value of L derived from the first term of the expression is 437440 metres. 
The correction depending on the radius not being infinitely great compared with the 
section of the coil as found from the second term is —7345 metres. 

The correction depending on the diameter of the wire is "] .44007 

per unit of length J 

Correction of eight neighbouring wires +*0236 

For sixteen wires next to these +-0008 

Correction for variation of current in different parts of section — -2500 

Total correction per unit of length -22437 

Length 311*236 metres. 

Sum of corrections of this land . 70 

Final value of L by calculation . 430165 

This value of L was employed in reducing the observations, according to the method 
explained in the Eeport of the Committee*. The correction depending on L varies 
as the square of the velocity. The results of sixteen experiments to which this correc- 
tion had been applied, and in which the velocity varied from 100 revolutions in 
seventeen seconds to 100 in seventy-seven seconds, were compared by the method of 

* British Association Beports, 1863, p. 169. 






512 PEOEESSOE CLEEK MAXWELL ON THE ELECTEOMAGNETIC FIELD. 

least squares to determine what further correction depending on the square of the 
velocity should be applied to make the outstanding errors a minimum. 

The result of this examination showed that the calculated value of L should be 
multiplied by 1*0618 to obtain the value of L, which would give the most consistent 
results. 

We have therefore L by calculation .......... 430165 metres. 

Probable value of L by method of least squares ...... 456748 „ 

Eesult of rough experiment with the Electric Balance (see § 46) 410000 „ 

The value of L calculated from the dimensions of the coil is probably much more 
accurate than either of the other determinations.