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tubes by S, the sink of S tubes may be written -S, sinks always being
reckoned negative, and the general expression for the number of cells in the
system will be S (5p).

(30) The same conclusion may be arrived at by observing that unity of
work is done on each cell. Now in each source S, S units of fluid are
expelled against a pressure p, so that the work done by the fluid in over-
coming resistance is Sj?. At each sink in which S' tubes terminate, S' units
of fluid sink into nothing under pressure p' ; the work done upon the fluid by



174 ON Faraday's lines of force.

the pressure is therefore S' p\ The whole work done by the fluid may there-
fore be expressed by

W = tSp^tS'p,
or more concisely, considering sinks as negative sources,

W = t(Sp).

(31) Let S represent the rate of production of a source in any medium,
and let p be the pressure at any given point due to that source. Then if we
superpose on this another equal source, every pressure will be doubled, and
thus by successive superposition we find that a source nS would produce a
pressure np, or more generally the pressure at any point due to a given
source varies as the rate of production of the source. This may be expressed
by the equation

p = RS,

where R is a, coefficient depending on the nature of the medium and on the
positions of the source and the given point. In a uniform medium whose
resistance is measured by k,

R may be called the coefficient of resistance of the medium between the source
and the given point. By combining any number of sources we have generally

p = %{RS),

(32) In a uniform medium the pressure due to a source S

k S

At another source S' at a distance r we shall have

a, k SS' CI f

if 2^' he the pressure at S due to S\ If therefore there be two systems of
sources X{S) and %{S'), and if the pressures due to the first be p and to the
second p', then

2(S» = 2{S/).

For every term S'p has a term Sp' equal to it.



ON Faraday's lines of force. 175

(33) Suppose that in a uniform medium the motion of the fluid is every-
where parallel to one plane, then the surfaces of equal pressure will be
perpendicular to this plane. If we take two parallel planes at a distance equal
to k from each other, we can divide the space between these planes into unit
tubes by means of cylindric surfaces perpendicular to the planes, and these
together with the surfaces of equal pressure will divide the space into cells of
which the length is equal to the breadth. For if h be the distance between
consecutive surfaces of equal pressure and s the section of the unit tube, we
have by (13) s = kh.

But s is the product of the breadth and depth ; but the depth is k,
therefore the breadth is h and equal to the length.

If two systems of plane curves cut each other at right angles so as to
divide the plane into little areas of which the length and breadth are equal,
then by taking another plane at distance k from the first and erecting
cyhndric surfaces on the plane curves as bases, a system of cells will be
formed which will satisfy the conditions whether we suppose the fluid to run
along the first set of cutting lines or the second*.



Application of the Idea of Lines of Force.

I have now to shew how the idea of lines of fluid motion as described
above may be modified so as to be apphcable to the sciences of statical elec-
tricity, permanent magnetism, magnetism of induction, and uniform galvanic
currents, reserving the laws of electro-magnetism for special consideration.

I shall assume that the phenomena of statical electricity have been ah*eady
explained by the mutual action of two opposite kinds of matter. If we consider
one of these as positive electricity and the other as negative, then any two
particles of electricity repel one another with a force which is measured by the
product of the masses of the particles divided by the square of their distance.

Now we found in (18) that the velocity of our imaginary fluid due to a
source *S at a distance r varies inversely as r". Let us see what will be the
effect of substituting such a source for every particle of positive electricity. The
velocity due to each source would be proportional to the attraction due to the
corresponding particle, and the resultant velocity due to all the sources would

* See Cambridge and Dublin MalJiematical Jownal, Vol. in. p. 286.



176 ON Faraday's lines of force.

be proportional to the resultant attraction of all the particles. Now we may find
the resultant pressure at any point by adding the pressures due to the given
sources, and therefore we may find the resultant velocity in a given direction
from the rate of decrease of pressure in that direction, and this will be
proportional to the resultant attraction of the particles resolved in that direction.
Since the resultant attraction in the electrical problem is proportional to
the decrease of pressure in the imaginary problem, and since we may select
any values for the constants in the imaginary problem, we may assume that the
resultant attraction in any direction is numerically equal to the decrease of
pressure in that direction, or

ax
By this assumption we find that if F be the potential,
dV=Xdx+ Ydy + Zdz= -dp,
or since at an infinite distance F= and p = 0, V= —p.
In the electrical problem we have



7. Q

In the fluid p = S [-



^ r



S= -jr dm.



If k be supposed very great, the amount of fluid produced by each source
in order to keep up the pressures will be very small.

The potential of any system of electricity on itself will be

If t (dm), X (dm') be two systems of electrical particles and p, p' the potentials
due to them respectively, then by (32)

or the potential of the first system on the second is equal to that of the second
system on the first.



ON Faraday's lines of force. 177

So that in the ordinary electrical problems the analogy in fluid motion is
of this kind :

V=-p,

dm = -— S,
Ait

whole potential of a system = -XVdm^— W, where W is the work done by

the fluid in overcoming resistance.

The lines of forces are the unit tubes of fluid motion, and they may be
estimated numerically by those tubes.



Theory of Dielectrics,

The electrical induction exercised on a body at a distance depends not
only on the distribution of electricity in the inductric, and the form and posi-
tion of the inducteous body, but on the nature of the interposed medium, or
dielectric. Faraday* expresses this by the conception of one substance having
a greater inductive capacity, or conducting the lines of inductive action more
freely than another. If we suppose that in our analogy of a fluid in a resisting
medium the resistance is diflerent in difierent media, then by making the
resistance less we obtain the analogue to a dielectric which more easily conducts
Faraday's lines.

It is evident from (23) that in this case there will always be a:n apparent
distribution of electricity on the surface of the dielectric, there being negative
electricity where the lines enter and positive electricity where they emerge. In
the case of the fluid there are no real sources on the surface, but we use
them merely for purposes of calculation. In the dielectric there may be no
real charge of electricity, but only an apparent electric action due to the surface.

If the dielectric had been of less conductivity than the surrounding medium,
we should have had precisely opposite eflects, namely, positive electricity where
lines enter, and negative where they emerge.



* Series xi.
VOL. I.



23



178 ON Faraday's lines of force.

If the conduction of the dielectric is perfect or nearly so for the small
quantities of electricity with which we have to do, then we have the case of
(24). The dielectric is then considered as a conductor, its surface is a surface
of equal potential, and the resultant attraction near the surface itself is per-
pendicular to it.

Theory of Permanent Magnets.

A magnet is conceived to be made up of elementary magnetized particles,
each of which has its own north and south poles, the action of which upon
other north and south poles is governed by laws mathematically identical with
those of electricity. Hence the same application of the idea of lines of force
can be made to this subject, and the same analogy of fluid motion can be
employed to illustrate it.

But it may be useful to examine the way in which the polarity of the
elements of a magnet may be represented by the unit cells in fluid motion.
In each unit cell unity of fluid enters by one face and flows out by the opposite
face, so that the first face becomes a unit sink and the second a unit source
with respect to the rest of the fluid. It may therefore be compared to an
elementary magnet, having an equal quantity of north and south magnetic
matter distributed over two of its faces. If we now consider the cell as forming
part of a system, the fluid flowing out of one cell will flow into the next, and
so on, so that the source will be transferred from the end of the cell to the
end of the unit tube. If all the unit tubes begin and end on the bounding
surface, the sources and sinks will be distributed entirely on that surface, and in
the case of a magnet which has what has been called a solenoidal or tubular
distribution of magnetism, all the imaginary magnetic matter will be on the
surface^".

Theory of Paramagnetic and Diamagnetic Induction.

Faraday t has shewn that the effects of paramagnetic and diamagnetic bodies
in the magnetic field may be explained by supposing paramagnetic bodies to

* See Professor Thomson On the Matliematical Theory of Magnetism, Chapters in. and v. Ph^.
Trans. 1851.

t Experimental Researches (3292).



ON FARADAY S LINES OF FORCE. 179

conduct the lines of force better, and diamagnetic bodies worse, than the
surrounding medium. Bj referring to (23) and (26), and supposing sources to
represent north magnetic matter, and sinks south magnetic matter, then if a
paramagnetic body be in the neighbourhood of a north pole, the lines of force
on entering it will produce south magnetic matter, and on leaving it they will
produce an equal amount of north magnetic matter. Since the quantities of
magnetic matter on the whole are equal, but the southern matter is nearest
to the north pole, the result will be attraction. If on the other hand the body
be diamagnetic, or a worse conductor of lines of force than the surrounding
medium, there will be an imaginary distribution of northern magnetic matter
where the lines pass into the worse conductor, and of southern where they pass
out, so that on the whole there will be repulsion.

"We may obtain a more general law from the consideration that the poten-
tial of the whole system is proportional to the amount of work done by the
fluid in overcoming resistance. The introduction of a second medium increases
or diminishes the work done according as the resistance is greater or less than
that of the first medium. The amount of this increase or diminution will vary
as the square of the velocity of the fluid.

Now, by the theory of potentials, the moving force in any direction is
measured by the rate of decrease of the potential of the system in passing along
that direction, therefore when ¥, the resistance within the second medium, is
greater than k, the resistance in the surrounding medium, there is a force tend-
ing from places where the resultant force v is greater to where it is less, so
that a diamagnetic body moves from greater to less values of the resultant
force *.

In paramagnetic bodies V is less than k, so that the force is now from
points of less to points of greater resultant magnetic force. Since these results
depend only on the relative values of k and k', it is evident that by changing
the surrounding medium, the behaviour of a body may be changed from para-
magnetic to diamagnetic at pleasure.

It is evident that we should obtain the same mathematical results if we
had supposed that the magnetic force had a power of exciting a polarity in
bodies which is in the same direction as the lines in paramagnetic bodies, and



* Experimental Heaearchei (2797), (2798). See Thomson, Canibridge and Dublin Mathe)naticcU
Journal, May, 1847.

23—2



180 ON Faraday's lines of force.

in the reverse direction in diamagnetic bodies*. ' In fact we have not as yet
come to any facts which would lead us to choose any one out of these three
theories, that of lines of force, that of imaginary magnetic matter, and that of
induced polarity. As the theory of lines of force admits of the most precise,
and at the same time least theoretic statement, we shall allow it to stand for
the present.

TJieory of Magnecrystallic Induction.

Ihe theory of Faraday t with respect to the behaviour of crystals in the
magnetic field may be thus stated. In certain crystals and other substances the
lines of magnetic force are conducted with difierent facility in different directions.
The body when suspended in a uniform magnetic field will turn or tend to turn
into such a position that the lines of force shall pass through it with least resist-
ance. It is not difficult by means of the principles in (28) to express the laws
of this kind of action, and even to reduce them in certain cases to numerical
formulae. The principles of induced polarity and of imaginary magnetic matter
are here of Httle use; but the theory of lines of force is capable of the most
perfect adaptation to this class of phenomena.



Theory of the Conduction of Current Electricity.

It is in the calculation of the laws of constant electric currents that the
theory of fluid motion which we have laid down admits of the most direct appU-
cation. In addition to the researches of Ohm on this subject, we have those
of M. Kirchhoff, Ann. de Chim. xli. 496, and of M. Quincke, XLvn. 203, on the
Conduction of Electric Currents in Plates. According to the received opinions
we have here a current of fluid moving uniformly in conducting circuits, which
oppose a resistance to the current which has to be overcome by the application
of an electro-motive force at some part of the circuit. On account of this
resistance to the motion of the fluid the pressure must be diflerent at difierent
points in the circuit. This pressure, which is commonly called electrical tension,

♦ Uxp. Ees. (2429), (3320). See Weber, PoggendorflF, lxxxvil p. H5. Prof. TyndaU, Fhxi.
Trans. 1856, p. 237.

t Fxp. Res. (2836), &c.



ON FAHADAYS LINES OF FORCE. 181

is found to be physically identical with the potential in statical electricity, and
thus we have the means of connecting the two sets of phenomena. If we knew
what amount of electricity, measured statically, passes along that current which
we assume as our unit of current, then the connexion of electricity of tension
with current electricity would be completed*. This has as yet been done only
approximately, but we know enough to be certain that the conducting powers of
diflferent substances differ only in degree, and that the difference between glass
and metal is, that the resistance is a great but finite quantity in glass, and a
small but finite quantity in metal. Thus the analogy between statical electricity
and fluid motion turns out more perfect than we might have supposed, for there
the induction goes on by conduction just as in current electricity, but the quan-
tity conducted is insensible owing to the great resistance of the dielectricst.



On Electro-motive Forces.

When a uniform current exists in a closed circuit it is evident that some
other forces must act on the fluid besides the pressures. For if the current
were due to difference of pressures, then it would flow from the point of
greatest pressure in both directions to the point of least pressure, whereas in
reahty it circulates in one direction constantly. We must therefore admit the
existence of certain forces capable of keeping up a constant current in a closed
circuit. Of these the most remarkable is that which is produced by chemical
action. A cell of a voltaic battery, or rather the surface of separation of the
fluid of the ceU and the zinc, is the seat of an electro-motive force which
can maintain a current in opposition to the resistance of the circuit. If we
adopt the usual convention in speaking of electric currents, the positive current
is from the fluid through the platinum, the conducting circuit, and the zinc,
back to the fluid again. If the electro-motive force act only in the surface of
separation of the fluid and zinc, then the tension of electricity in the fluid
must exceed that in the zinc by a quantity depending on the nature and
length of the circuit and on the strength of the current in the conductor.
In order to keep up this difference of pressure there must be an electro-motive
force whose intensity is measured by that difference of pressure. If F be the
electro-motive force, / the quantity of the current or the number of electrical

♦ See Exp. Ees. (371). t Hxp. Ret. Vol iii. p. 513.



182 ON Faraday's lines of force.

units delivered in unit of time, and K a quEfntity depending on the length
and resistance of the conducting circuit, then

F=IK=p-p\
where p is the electric tension in the fluid and p' in the zinc.

If the circuit be broken at any point, then since there is no current the
tension of the part which remains attached to the platinum will be p, and
that of the other will be p, p-p or F afibrds a measure of the intensity
of the current. This distinction of quantity and intensity is very useful *,
but must be distinctly understood to mean nothing more than this : — The
quantity of a current is the amount of electricity which it transmits in unit
of time, and is measured by / the number of unit currents which it contains.
The intensity of a current is its power of overcoming resistance, and is
measured by F or IK, where K is the resistance of the wliole circuit.

The same idea of quantity and intensity may be applied to the case of

magnetism f. The quantity of magnetization in any section of a magnetic

body is measured by the number of lines of magnetic force which pass through

it. The intensity of magnetization in the section depends on the resisting

power of the section, as well as on the number of lines which pass through

it. If h be the resisting power of the material, and S the area of the section,

and / the number of lines of force which pass through it, then the whole

intensity throughout the section

h
= F=I-

When magnetization is produced by the influence of other magnets only,
we may put p for the magnetic tension at any point, then for the whole
magnetic solenoid



F=l(^dx = IK=p-p,



When a solenoidal magnetized circuit returns into itself, the magnetization
does not depend on difference of tensions only, but on some magnetizing force
of which the intensity is F.

If i be the quantity of the magnetization at any point, or the number of
lines of force passing through unit of area in the section of the solenoid, then

* Hxp. Res. Vol. HI. p. 519. t Exp. Res. (2870), (3293).



ON Faraday's lines of force. 183

the total quantity of magnetization in the circuit is the number of lines which
pass through any section, I=Xidydz, where dydz is the element of the section,
and the summation is performed over the whole section.

The intensity of magnetization at any point, or the force required to
keep up the magnetization, is measured by Jci=f, and the total intensity of
magnetization in the circuit is measured by the sum of the local intensities all
round the circuit,

F=t(fdx),

where dx is the element of length in the circuit, and the summation is extended
round the entire circuit.

In the same circuit we have always F = IK, where K is the total resistance
of the circuit, and depends on its form and the matter of which it is
composed.



On the Action of closed Currents at a Distance.

The mathematical laws of the attractions and repulsions of conductors have
been most ably investigated by Ampere, and his results have stood the test of
subsequent experiments.

From the single assumption, that the action of an element of one current
upon an element of another current is an attractive or repulsive force acting
in the direction of the line joining the two elements, he has determined by
the simplest experiments the mathematical form of the law of attraction, and
has put this law into several most elegant and useful forms. We must
recollect however that no experiments have been made on these elements of
currents except under the form of closed currents either in rigid conductors
or in fluids, and that the laws of closed currents can only be deduced from
such experiments. Hence if Ampere's formulae applied to closed currents give
true results, their truth is not proved for elements of currents unless we
assume that the action between two such elements must be along the line which
joms them. Although this assumption is most warrantable and philosophical in
the present state of science, it wiQ be more conducive to freedom of investi-
gation if we endeavour to do without it, and to assume the laws of closed currents
as the ultimate datum of experiment.



384 ON fahaday's lines of force.

Ampere has shewn that when currents are combined according to the law
of the parallelogram of forces, the force due to the resultant current is the
resultant of the forces due to the component currents, and that equal and
opposite currents generate equal and opposite forces, and when combined
neutralize each other.

He has also shewn that a closed circuit of any form has no tendency to
turn a moveable circular conductor about a fixed axis through the centre of
the circle perpendicular to its plane, and that therefore the forces in the case
of a closed circuit render Xdx + Ydy + Zdz a complete differential.

Finally, he has shewn that if there be two systems of circuits similar
and similarly situated, the quantity of electrical current in corresponding
conductors being the same, the resultant forces are equal, whatever be the
absolute dimensions of the systems, which proves that the forces are, cceteris
paribus, inversely as the square of the distance.

From these results it follows that the mutual action of two closed currents
whose areas are very small is the same as that of two elementary magnetic
bars magnetized perpendicularly to the plane of the currents.

The direction of magnetization of the equivalent magnet may be pre-
dicted by remembering that a current travelling round the earth from east
to west as the sun appears to do, would be equivalent to that magnetization
which the earth actually possesses, and therefore in the reverse direction to
that of a magnetic needle when pointing freely.

If a number of closed unit currents in contact exist on a surface, then at
aU points in which two currents are in contact there will be two equal and
opposite currents which will produce no effect, but all round the boundary of the
surfeice occupied by the currents there will be a residual current not neutralized
by any other; and therefore the result will be the same as that of a single
unit current round the boundary of all the currents.

From this it appears that the external attractions of a shell uniformly
magnetized perpendicular to its surface are the same as those due to a current
round its edge, for each of the elementary currents in the former case has
the same effect as an element of the magnetic shell.

If we examine the Unes of magnetic force produced by a closed current,
we shall find that they form closed curves passing round the current and
embracing it, and that the total intensity of the magnetizing force all along
the closed line of force depends on the quantity of the electric current only.



ON FARADAY 3 LINES OF FORCE. 185

The number of unit lines* of magnetic force due to a closed current depends
on the form as well as the quantity of the current, but the number of unit
cells t in each complete line of force is measured simply by the number of unit
currents which embrace it. The unit cells in this case are portions of space in
which unit of magnetic quantity is produced by unity of magnetizing force.
The length of a cell is therefore inversely as the intensity of the magnetizing
force, and its section inversely as the quantity of magnetic induction at that
point.

The whole number of cells due to a given current is therefore proportional
to the strength of the current multiplied by the number of lines of force
which pass through it. If by any change of the form of the conductors the
number of cells can be increased, there will be a force tending to produce that



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