James Clerk Maxwell.

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Font size term gives a mechanical explanation of the force acting on a north or south
pole in the magnetic field.

We now proceed to examine the second term,

Here a^' + ^ + y* is the square of the intensity at any part of the field, and
ft, is the magnetic inductive capacity at the same place. Any body therefore

ON PHYSICAL LINES OF FORCE, 4GI

placed in the field will be urged towards places of stronger magnetic intensity
with a force depending partly on its own capacity for magnetic induction, and
partly on the rate at which the square of the intensity increases.

If the body be placed in a fluid medium, then the medium, as well as the
body, will be urged towards places of greater intensity, so that its hydrostatic
pressure will be increased in that direction. The resultant effect on a body
placed in the medium will be the difference of the actions on the body and
on the portion of the medium which it displaces, so that the body will tend
to or from places of greatest magnetic intensity, according as it has a greater
or less capacity for magnetic induction than the surrounding medium.

In fig. 4 the lines of force are represented as converging and becoming
more powerful towards the right, so that the magnetic tension at B is stronger
than at A, and the body AB will be urged to the right. If the capacity for
magnetic induction is greater in the body than in the surrounding medium, it
will move to the right, but if less it will move to the left.

Fig. 4. Fig. 5.

We may suppose in this case that the lines of force are converging to a
magnetic pole, either north or south, on the right hand.

In fig. 5 the Hues of force are represented as vertical, and becoming more
numerous towards the right. It may be shewn that if the force increases
towards the right, the lines of force will be curved towards the right. The
effect of the magnetic tensions wiU then be to draw any body towards the right
with a force depending on the excess of its inductive capacity over that of the
surrounding medium.

We may suppose that in this figure the lines of force are those surrounding
an electric current perpendicular to the plane of the paper and on the right
hand of the figure.

These two iUustrations will shew the mechanical effect on a paramagnetic
or diamagnetic body placed in a field of varying magnetic force, whether the
increase of force takes place along the lines or transverse to them. The form

462 ON PHYSICAL LINES OF FORCE.

of the second term of our equation indicates the general law, which is quite
independent of the direction of the lines of force, and depends solely on the
manner in which the force varies from one part of the field to another.

"We come now to the third term of the value of X,

1 fd/B da.\

^^ 47r \dx dy,

Here y^^ is, as before, the quantity of magnetic induction through unit of area
perpendicular to the axis of y, and -J- — -j- ^^ ^ quantity which would disap-
pear if adx + ^dy + ydz were a complete differential, that is, if the force acting
on a unit north pole were subject to the condition that no work can be done
upon the pole in passing round any closed curve. The quantity represents the
work done on a north pole in travelHng round unit of area in the direction
from +x to +y parallel to the plane of xy. Now if an electric current whose
strength is r is traversing the axis of z, which, we may suppose, points
vertically upwards, then, if the axis of x is east and that of y north, a unit
north pole will be urged round the axis of z in the direction from x to y, so

that in one revolution the work done will be = 47rr. Hence t- ( -t^ — 7- ) repre-

477 \dy

Att \dx dy/

sents the strength of an electric current parallel to z through unit of area ; and
if we write

dz] P' 4,w\dz dx)~^- 4n\dx dyj~^ ^^''

then p, q, r will be the quantity of electric current per unit of area perpen-
dicular to the axes of x, y, and z respectively.

The physical interpretation of the third term of X, —fi^r, is that if /xyS is
the quantity of magnetic induction parallel to y, and r the quantity of electricity
flowing in the direction of z, the element will be urged in the direction of —x,
transversely to the direction of the current and of the lines of force; that is,
an ascending current in a field of force magnetized towards the north would
tend to move west.

To illustrate the action of the molecular vortices, let sn be the direction
of magnetic force in the field, and let C be the section of an ascending mag-
netic current perpendicular to the paper. The lines of force due to this current

ON PHYSICAL LINES OF FORCE.

463

will be circles drawn in the opposite direction from that of the hands of a

watch ; that is, in the direction nwse. At c the lines of force

will be the sum of those of the field and of the current, and

at w they will be the difference of the two sets of lines ; so

that the vortices on the east side of the current will be more

powerful than those on the west side. Both sets of vortices have

their equatorial parts turned towards C, so that they tend to

expand towards C, but those on the east side have the greatest

effect, so that the resultant effect on the current is to urge it towards the west

The fourth term,

^da dy

Fig. 6.

1 da

or ^-iiyq

(10),

may be interpreted in the same way, and indicates that a current q in the
direction of y, that is, to the north, placed in a magnetic field in which the
lines are vertically upwards in the direction of z, will be urged towards the ecLnt.

The fifth term,

dx

(n),

merely implies that the element wiQ be urged in the direction in which the
hydrostatic pressure p^ diminishes.

We may now write down the expressions for the components of the resultant
force on an element of the medium per unit of volume, thus :

^"^"^^^ ^('^)"''^'' + ''>'^"^ (^^)'

fiyp + n-tar —

dp,
dy

(13),

The first term of each expression refers to the force acting on magnetic

poles.

The second term to the action on bodies capable of magnetism by induction.
The third and fourth terms to the force acting on electric currents.
And the fifth to the effect of simple pressure.

464 ON PHYSICAL LINES OF FORCE.

Before going further in the general investigation, we shall consider equations
(12, 13, 14), in particular cases, corresponding to those simplified cases of the
actual phenomena which we seek to obtain in order to determine their laws by
experiment.

We have found that the quantities p, q, and r represent the resolved parts
of an electric current in the three co-ordinate directions. Let us suppose in the
first instance that there is no electric current, or that p, q, and r vanish. We
have then by (9),

^_^ = ^-^ = ^-^ = (15)

dy dz ' dz dx ' dx dy ^ ''

whence we learn that adx + /3dy + ydz = d<l) (16),

is an exact differential of <^, so that

-t ^ = f • r = f (m:

fi is proportional to the density of the vortices, and represents the " capacity
for magnetic induction" in the medium. It is equal to 1 in air, or in whatever
medium the experiments were made which determined the powers of the magnets,
the strengths of the electric currents, &c.

Let us suppose fi constant, then

m

=h{T>'^^4^^^^4M - rA?^^9^'^) (-)

represents the amount of imaginary magnetic matter in unit of volume. That
there may be no resultant force on that unit of volume arising from the action
represented by the first term of equations (12, 13, 14), we must have m = 0, or

'J-g^-S = o (-)•

Now it may be shewn that equation (19), if true within a given space,
implies that the forces acting within that space are such as would result from
a distribution of centres of force beyond that space, attracting or repelling
inversely as the square of the distance.

Hence the lines of force in a part of space where fi is uniform, and where
there are no electric currents, must be such as would result from the theory
of "imaginary matter" acting at a distance. The assumptions of that theory
are unlike those of ours, but the results are identical

dr LL r' ^ ''

ON PHYSICAL LINES OF FORCE. 465

Let us first take the case of a single magnetic pole, that is, one end of
a long magnet, so long that its other end is too far off to have a perceptible
influence on the part of the field we are considering. The conditions then are,
that equation (18) must be fulfilled at the magnetic pole, and (19) everywhere
else. The only solution under these conditions is

't'= - ,l (^«).

where r is the distance from the pole, and m the strength of the pole.
The repulsion at any point on a unit pole of the same kind is

d(f> _'in 1

In the standard medium /i = 1 ; so that the repulsion is simply — in that

medium, as has been shewn by Coulomb.

In a medium having a greater value of fi (such as oxygen, solutions of
salts of iron, &c.) the attraction, on our theory, ought to be less than in air,
and in diamagnetic media (such as water, melted bismuth, &c.) the attraction
between the same magnetic poles ought to be greater than in air.

The experiments necessary to demonstrate the difference of attraction of two
magnets according to the magnetic or diamagnetic character of the medium in
which they are placed, would require great precision, on account of the limited
range of magnetic capacity in the fluid media known to us, and the small
amount of the difference sought for as compared with the whole attraction.

Let us next take the case of an electric current whose quantity is C,
flowing through a cylindrical conductor whose radius is R, and whose length is
infinite as compared with the size of the field of force considered.

Let the axis of the cylinder be that of z, and the direction of the current
positive, then within the conductor the quantity of current per unit of area is

C 1 /d^ da\

) (22):

ir-R* Air \dx dy^
80 that within the conductor

o-=-2^,y, /3 = 2-^a:, y = (23).

VOL. L 59

466 ON PHYSICAL LINES OF FORCE.

Beyond the conductor, in the space round it,

«^ = 2Ctan-' ^ (24),

« = i=-^^^.- ^ = g = ^^^-^.' r = f = (25).

If p — sjdi^-^y^ is the perpendicular distance of any point from the axis of
the conductor, a unit north pole will experience a force = — , tending to move

it round the conductor in the direction of the hands of a watch, if the observer
view it in the direction of the current.

Let us now consider a current running parallel to the axis of z in the
plane of xz at a distance p. Let the quantity of the current be c', and let

the length of the part considered be I, and its section 5, so that - is its

strength per unit of section. Putting this quantity for p in equations (12, 13,
14), we find

^= -M^ "-

per unit of volume; and multiplying by Is, the volume of the conductor con-
sidered, we find

X= -p.^c'1

= -2.f (26),

shewing that the second conductor will be attracted towards the first with a
force inversely as the distance.

We find in this case also that the amount of attraction depends on the

value of /A, but that it varies directly instead of inversely as /i ; so that the

attraction between two conducting wires will be greater in oxygen than in air,
and greater in air than in water.

We shall next consider the nature of electric currents and electromotive
forces in connexion with the theory of molecular vortices.

ON PHYSICAL LINES OF FORCE. 467

PART 11.
The Theory of Molecular Vortices applied to Electric Currents.

We have already shewn that all the forces acting between magnets, sub-
stances capable of magnetic induction, and electric currents, may be mechanically
accounted for on the supposition that the surrounding medium is put into such
a state that at every point the pressures are different in different directions,
the direction of least pressure being that of the observed lines of force, and
the difference of greatest and least pressures being proportional to the square
of the intensity of the force at that point.

Such a state of stress, if assumed to exist in the medium, and to be
arranged according to the known laws regulating lines of force, will act upon
the magnets, currents, &c. in the field with precisely the same resultant forces
as those calculated on the ordinary hypothesis of direct action at a distance.
This is true independently of any particular theory as to the cause of this
state of stress, or the mode in which it can be sustained in the medium. We
have therefore a satisfactory answer to the question, "Is there any mechanical
hypothesis as to the condition of the medium indicated by lines of force, by
which the observed resultant forces may be accounted for?" The answer is,
the hues of force indicate the direction of minimum pressure at every point of
the medium.

The second question must be, "What is the mechanical cause of this
difference of pressure in different directions?" We have supposed, in the first
part of this paper, that this difference of pressures is caused by molecular
vortices, having their axes parallel to the lines of force.

We also assumed, perfectly arbitrarily, that the direction of these vortices
is such that, on looking along a line of force from south to north, we should
see the vortices revolving in the direction of the hands of a watch.

We found that the velocity of the circumference of each vortex must be
proportional to the intensity of the magnetic force, and that the density of
the substance of the vortex must be proportional to the capacity of the medium
for magnetic induction.

We have as yet given no answers to the questions, " How are these vortices
set in rotation?" and "Why are they arranged according to the known laws

59—2

468 ON PHYSICAL LINES OF FORCE.

of lines of force about magnets and currents?" These questions are certainly
of a higher order of difficulty than either of the former ; and I wish to separate
the suggestions I may offer by way of provisional answer to them, from the
mechanical deductions which resolved the first question, and the hypothesis of
vortices which gave a probable answer to the second.

We have, in fact, now come to inquire into the physical connexion of these
vortices with electric currents, while we are still in doubt as to the nature of
electricity, whether it is one substance, two substances, or not a substance at
all, or in what way it differs from matter, and how it is connected with it.

We know that the lines of force are affected by electric currents, and we
know the distribution of those lines about a current ; so that from the force
we can determine the amount of the current. Assuming that our explanation
of the lines of force by molecular vortices is correct, why does a particular
distribution of vortices indicate an electric current? A satisfactory answer to
this question would lead us a long way towards that of a very important one,
"What is an electric current?"

I have found great difficulty in conceiving of the existence of vortices in a
medium, side by side, revolving in the same direction about parallel axes. The
contiguous portions of consecutive vortices must be moving in opposite directions ;
and it is difficult to understand how the motion of one part of the medium
can coexist with, and even produce, an opposite motion of a part in contact
with it.

The only ibnception which has at all aided me in conceiving of this kind of
motion is that of the vortices being separated by a layer of particles, revolving
each on its own axis in the opposite direction to that of the vortices, so that
the contiguous surfaces of the particles and of the vortices have the same
motion.

In mechanism, when two wheels are intended to revolve in the same direc-
tion, a wheel is placed between them so as to be in gear with both, and this
wheel is called an "idle wheel." The hypothesis about the vortices which I
have to suggest is that a layer of particles, acting as idle wheels, is interposed
between each vortex and the next, so that each vortex has a tendency to make
the neighbouring vortices revolve in the same direction with itself

In mechanism, the idle wheel is generally made to rotate about a fixed
axle; but in epicyclic trains and other contrivances, as, for instance, in Siemens's

ON PHYSICAL LINES OF FORCE. 469

governor for steam-engines*, we find idle wheels whose centres are capable of
motion. In all these cases the motion of the centre is the half sum of the
motions of the circumferences of the wheels between which it is placed. Let
us examine the relations which must subsist between the motions of our vortices
and those of the layer of particles interposed as idle wheels between them.

Prop. IV. — To determine the motion of a layer of particles separating two
vortices.

Let the circumferential velocity of a vortex, multiplied by the three direc-
tion-cosines of its axis respectively, be a, ;8, y, as in Prop. II. Let I, m, n be
the direction- cosines of the normal to any part of the surface of this vortex,
the outside of the surface being regarded positive. Then the components of the
velocity of the particles of the vortex at this part of its surface will be

nfi — my parallel to x,

hf — na parallel to y,

ma — l^ parallel to z.

If this portion of the surface be in contact with another vortex whose velocities
are a, ^, y, then a layer of very small particles placed between them will
have a velocity which wiU be the mean of the superficial velocities of the
vortices which they separate, so that if u ia the velocity of the particles in
the direction of x,

u = ^m(y-y)^in{^-fi) (27),

since the normal to the second vortex is in the opposite direction to that of
the first.

Prop. V. — To determine the whole amount of particles transferred across
unit of area in the direction of x in unit of time.

Let Xi, 2/1, Zi be the co-ordinates of the centre of the first vortex, x.,, y„, z.,
those of the second, and so on. Let F,, Fj, &c. be the volumes of the first,
second, &c. vortices, and F the sum of their volumes. Let dS be an element
of the surface separating the first and second vortices, and x, y, z its co-ordinates.
Let p be the quantity of particles on every unit of surface. Then if p be the
whole quantity of particles transferred across irnit of area in unit of time in

♦ See Goodeve's ElemenU of Mechanism, p. 118.

470 ON PHYSICAL LINES OF FORCE.

the direction of rr, the whole momentum parallel to x of the particles within
the space whose volume is V will be Fp, and we shall have

Vp==tupdS (28),

the summation being extended to every surface separating any two vortices
within the volume V.

Let us consider the surface separating the first and second vortices. Let an
element of this surface be dS, and let its direction-cosines be Zj, m^, n^^ with
respect to the first vortex, and l^, m^, n, with respect to the second; then we
know that

^1 + 4 = 0, mi + ma = 0, ni + n, = (29).

The values of a, ^, y vary with the position of the centre of the vortex ;
so that we may write

with similar equations for )8 and y.

The value of u may be written >—

w = i ^ H {x-x,) + m^ (x-x,)]

+i^H(2/-2/i)+w2(2/-y.)}+i^H (2-^0+^.(2-2;.)}

-l-J^{^i{^-^^) + '^h{x-x,)]-:^-£j{n,{y-y,) + n,{y-y,)]

-if K(2-2.) + n, (.-.,)} (31).

In effecting the summation of %updS, we must remember that round any
closed surface XldS and all similar terms vanish ; also that terms of the form
XlydS, where I and y are measured in different directions, also vanish; but that
terms of the form tlxdS, where I and x refer to the same axis of co-ordinates,
do not vanish, but are equal to the volume enclosed by the surface. The
result is

^^=4''(|-S<'''+''"+*") ^''^'

ON PHYSICAL LINES OF FORCE. 471

or dividing by F= F,+ F,4-&c.,

i^l-f) '^^)-

If we make P = 7r (3^).

then equation (33) will be identical with the first of equations (9), which give
the relation between the quantity of an electric current and the intensity of
the lines of force surrounding it.

It appears therefore that, according to our hypothesis, an electric current
is represented by the transference of the moveable particles interposed between
the neighbouring vortices. We may conceive that these particles are very small
compared with the size of a vortex, and that the mass of all the particles
together is inappreciable compared with that of the vortices, and that a great
many vortices, with their surrounding particles, are contained in a single complete
molecule of the medium. The particles must be conceived to roll without sliding
between the vortices which they separate, and not to touch each other, so that,
as long as they remain within the same complete molecule, there is no loss of
energy by resistance. When, however, there is a general transference of par-
ticles in one direction, they must pass from one molecule to another, and in
doing so, may experience resistance, so as to waste electrical energy and generate
heat.

Now let us suppose the vortices arranged in a medium in any arbitraiy

manner. The quantities j^ — ~r > &c. will then in general have values, so that

there will at first be electrical currents in the medium. These will be opposed
by the electrical resistance of the medium ; so that, unless they are kept up
by a continuous supply of force, they will quickly disappear, and we shall then

have j^ "~ ;j~ = ^> ^^•'> ^^^^ is, adx + fidy + ydz will be a complete difierential

(see equations (15) and (16)); so that our hypothesis accounts for the distri-
bution of the lines of force.

In Plate VIII. p. 488, fig. 1, let the vertical circle EE represent an
electric current flowing from copper C to zinc Z through the conductor EE',
as shewn by the arrows.

472 ON PHYSICAL LINES OF FORCE.

Let the homontal circle MM' represent a line of magnetic force embracing
the electric circuit, the north and south directions being indicated by the lines
SN and NS.

Let the vertical circles V and V represent the molecular vortices of which
the line of magnetic force is the axis. V revolves as the hands of a watch,
and F' the opposite way.

It will appear from this diagram, that if V and V were contiguous vortices,
particles placed between them would move downwards ; and that if the particles
were forced downwards by any cause, they would make the vortices revolve as
in the figure. We have thus obtained a point of view from which we may
regard the relation of an electric current to its lines of force as analogous to
the relation of a toothed wheel or rack to wheels which it drives.

In the first part of the paper we investigated the relations of the statical
forces of the system. We have now considered the connexion of the motions
of the parts considered as a system of mechanism. It remains that we should
investigate the dynamics of the system, and determine the forces necessary to
produce given changes in the motions of the different parts.

Prop. VI. — To determine the actual energy of a portion of a medium due
to the motion of the vortices within it.

Let a, /8, y be the components of the circumferential velocity, as in Prop. II.,
then the actual energy of the vortices in unit of volume will be proportional
to the density and to the square of the velocity. As we do not know the
distribution of density and velocity in each vortex, we cannot determine the
numerical value of the energy directly; but since /x also bears a constant
though unknown ratio to the mean density, let us assume that the energy
in unit of volume is

where (7 is a constant to be determined.
Let us take the case in which

«=g. ^=f- y=t (^^)-

Let <l> = <f>i + <f> (36),

ON PHYSICAL LINES OF FORCE. 473

then <^i is the potential at any point due to the magnetic system m„ and <^„
that due to the distribution of magnetism represented by m^. The actual
energy of all the vortices is

/; = 2C/x(a' + /8' + y)dF (38),

the integration being performed over all space.

This may be shewn by integration by parts (see Green's * Essay on Elec-
tricity,' p. 10) to be equal to

E= -4:iTCt{cf>,m,-h(f>,'m, + <f>,m, + (j>,m,)dV (39).

Or since it has been proved (Green's 'Essay/ p. 10) that

t<l>,m,dV=t<f>^m,dV,

E=^-4:7rC{(fy{m, + <j),vi, + 2<f),m,)dV (40).

Now let the magnetic system m^ remain at rest, and let w, be moved
parallel to itself in the direction of x through a space Sx; then, since ^i
depends on m^ only, it will remain as before, so that ^iTti^ will be constant ;
and since <f>j depends on m, only, the distribution of (j), about m^ will remain
the same, so that ^^rrij will be the same as before the change. The only part

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