James Clerk Maxwell.

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Font size 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.

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 551

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

Then, if sliding the moveable conductor in a given direction increases the
value of 3/, 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 direc-
tion 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 con-
ductor 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 ^vill 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 Hnes of magnetic force, and
on this surface 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

552 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

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 com-
pletely 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.

Srdly. 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 the amount of work done by these forces is equal to
the current in the circuit multiplied 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
Avill 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 coefiicient depending on the magnetic nature of the
medium, and called the coefiicient of magnetic induction.

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

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

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 553

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 mag-
netic 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 diflferent
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 magnetic 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 con-
ductors 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
surfiice will have a series of values in arithmetical progression, differing by the
work done in passing completely round one of the currents in the field.

The equipotential surfaces w^ill not be continuous closed surfaces, but some
of them 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 = A-ny, 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 A-ny 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.

554 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

PART III.
GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD.

(53) Let US assume three rectangular directions in space as the axes of
X, ]), and z, and let all quantities having direction be expressed by their com-
ponents in these three directions.

Electrical Cui^ents (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 p, then
j) 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 x, 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 accom-
panied 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 f 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 p', q, r\ so that

P

= P + i

dt
dg

dh
= ' + dt

(A).

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 555

Electromotive Force (P, Q, R).

(56) Let P, Q, R represent the components of tlie electromotive force at
any point. Then P represents the difference of potential per unit of length in
a conductor placed in the direction of x at the given point. We may suppose
an indefinitely short wire placed parallel to a: 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 / be the length of the wire, the difference of potential at its ends
will be PI, and if C be the capacity of each of the small conductors the charge
on each will be ^CPl. Since the capacities of moderately large conductors,
measured on the electromagnetic system, are exceedingly small, ordinary electro-
motive forces arising from electromagnetic actions could hardly be measured in
this way. In practice such measurements are always made with long conductors,
forming closed or nearly closed circuits.

Electromagnetic Momentum (F, G, H).

(57) Let F, G, H 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 = lPdt.

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

- f. ^=-f - - f (-)•

Electromagnetic Momentum of a Circuit.

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

/(4>4^-^^^ (-)

70—2

556 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

round the circuit, we shall get the total electromagnetic momentum of the circuit,
vv the number of lines of magnetic force which pass through it, the variations
of which measure the total electromotive force in the circuit. This electromag-
netic 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 di/ dz, then its electro-
mao-uetic momentum is

\ dy dz

and this is the number of lines of magnetic force which pa^ss through the
area dij dz.

Magnetic Force {a, /3, y).

(59) Let a, /S, y represent the force acting on a unit magnetic pole placed
at the given point resolved in the directions of x, y, and z.

Coefficient of Magnetic Induction (/x).

(60) Let /x 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 [xa (/x is a quantity dependmg 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,

dH

dG]

^-dy-

~ dl

o dP

dH

^^ = Tz-

~d^

dG

dF

i'y^ dx -

"d-y\

(B).

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 557

Equations of Currents.

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

adx + fidi/ + ydz = d(f> (31)

a complete differential of <f), the magnetic potential.

The quantity (f> 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 (f) corre-
sponding to a passage completely round a current of strength c being inc.

Hence if there is no electric current,

dy dz

but if there is a current p,

(C).

o- M 1 da dy ^ , \

Similarly, -i^~ -f = '^'^^ ^

dx dy~
We may call these the Equations of Currents.

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

f=/(^S+«?l-^3'^ (-)■

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

558 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

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

^'^+G^ + Hf)ds = Lu + Mv (33),

da ds ds

It

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

^=-^ilAL + Mi^ (34).

(35),

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

p^_dF_d^'
dt dx

^~ dt dy .

P_ d^_chp\
^-~ dt dz]

where ^ is a function of x, y, z, and t, 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 V' is, that it represents the electnc
liotential at each point of space.

Electromotive Force on a Moving Conductor.

(64) Let a short straight conductor of length a, parallel to the axis of

X, move with a velocity whose components are ^, -^, ^, and let its ex-

ds
tremities sHde along two parallel conductors with a velocity ^. Let us find

the alteration of the electromagnetic momentum of the circuit of which this

arrangement forms a part.

dx dy dz
In unit of time the moving conductor has travelled distances ^ » ;^ > ^

aloncr the directions of the three axes, and at the same time the lengths of

the parallel conductors included in the circuit have each been increased by ^ .

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 559

Hence the quantity

lis as

/(

F''-UG''l + Jlf^)

will be Increased by the following increments,

/dFdx dF dy , dF dz\ , ^ ^- c ^ ^

ai-i r- + -i r- + -7 - rK ciue to motion of conductor,

\dx dt dy dt dz dtj

dsfdFdx dGdiidHdz\ , . i ^i • c ■ -,

-a-ri-,- -J- + -i- -i + —1 r > due to lengthening ot circuit.

dt \dx ds dx ds dx dsj ^ ^

The total increment will therefore be

fdF_d^dy_ (dH_clF\dz
''[dy dx) cit ^\dx dzjdt'

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

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

^^-l-'^^s (-)•

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

Equations of Electromotive Force.

-K4'-

-4;)-

dF

' dt ~

dxjj
'dx

<?=.(4:-

-r%-

dG

~ dt '

dxfj
'dy

^-(^t-

-4)-

dll

' dt '

d\\i
"dz

(D).

The fii-st term on the right-hand side of each equation represents the electro-
motive force arising from the motion of the conductor itself This electromotive

560 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

force is perpendicular 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 electro-
motive force due to the motion of the conductor, 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 shews the effect of the electric potential \jj. 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.

Electric Elasticity.

{QQ) Wlien 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 electrified. The amount of this electrification depends on the electro-
motive force and on the nature of the substance, and, in solids having a structure
defined by axes, on the direction 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 Electmc Elasticity,

Q = kg\ (E).

R = kh

Electric Resistance.

(Q7) When an electromotive force acts on a conductor it produces a current
of electricity through it. This effect is additional to the electric displacement
already considered. In soHds of complex structure, the relation between the
electromotive force and the current depends on their direction through the solid.

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 561

In isotropic substances, which alone we shall here consider, if p is the specific
resistance referred to unit of volume, we may write the

Equations of Electric Resistance,

Q=-pq\ (F)-

R=-pr\

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 electri-
fication of the different parts of the field not neutralizing each other, we may
write the

Equation of Free Electncity,

df da dh ^ ,^.

«+i+i+ar=° (G).

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

Equation of Continuity,

dt^dx^Ty^di-^ W-

(70) In these equations of the electromagnetic field we have assumed twenty
variable quantities, namely.

For Electromagnetic Momentum iF G H

„ Magnetic Intensity a /3 y

„ Electromotive Force P Q R

„ Current due to true Conduction p q r

„ Electric Displacement / (j h

„ Total Current (including variation of displacement) p' q r

„ Quantity of Free Electricity c

„ Electric Potential ■^I'

562 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

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, however, 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 = it{Fp+Gq+Hr')dV (37)

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

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

Integrating by parts, and remembering that a, /8, y vanish at an infinite
distance, the expression becomes

l^r [dH dG\ ^(dF dH\ ^ IdG dF\\,^

where the integration is to be extended over aU space. Referring to the equa-
tions of Magnetic Force (B), p. 556, this becomes

E=~t{a.,Ma + l3.ti^ + y.lJiy}dV (38),

A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 563

where a, ^, y are the components of magnetic intensity or the force on a unit
magnetic pole, and /la, /x^, /xy 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 /x 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

Stt
per unit of volume, where / 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 dis-
placement, f, is got by integrating

\pdf

from P = to the given value of P.

Since P = ^f, equation (E), this quantity becomes

mf=ikr=^pf.

Hence the intrinsic energy of any part of the field, as existing in the
form of electric displacement, is

^t(Pf+Qg + Rh)dV.
The total energy existing in the field is therefore

E = ^{^{a^.a + ^y.^ + yiiy) + \{Pf +Qg + Rh)^^ (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 n 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

♦ "On Physical Lines of Force," Philosophical Magazine, 1861—62. (In this voL p. 451.)

71—2

in

564 A DYNAMICAL THEORY OF THE ELEOTROMAGNETIC FIELD.

using such words as electric momentum and electric elasticity in reference to
the known phenomena of the induction of currents and the polarization of
dielectrics, I wish merely to direct the mind of the reader to mechanical pheno-
mena which will assist him in understanding the electrical 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 under-
stood 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 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 efiects 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 hypothesis, being deduced from experimental facts of three kinds :

1. The induction of electric currents by the increase or diminution of
neighbouring currents according to the changes in the lines of force passing
through the circuit.

2. The distribution of magnetic intensity according to the variations of a
magnetic potential.

3. The induction (or influence) of statical electricity through dielectrics.

We may now proceed to demonstrate fi:om these principles the existence
and laws of the mechanical forces which act upon electric currents, magnets, and
electrified bodies placed in the electromagnetic field.

A DYNAMICAL THEORY OF THE ELECTRO JIAGNETIC FIELD. 565

PART IV.

MECHANICAL ACTIONS IN THE FIELD.

Mechanical Force on a Moveable Conductor.

(76) We have shewn (§§ 34 & 35) that the work done by the electro-

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