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change, so that there is always a force urging a conductor transverse to the
lines of magnetic force, so as to cause more lines of force to pass throuo-h the
closed circuit of which the conductor forms a part.

The number of cells due to two given currents is got by multiplying
the number of lines of inductive magnetic action which pass through each by
the quantity of the currents respectively. Now by (9) the number of lines
which pass through the first current is the sum of its own lines and those
of the second current which would pass through the first if the second current
alone were in action. Hence the whole number of cells will be increased by
any motion which causes more lines of force to pass through either circuit,
and therefore the resultant force will tend to produce such a motion, and the
work done by this force during the motion will be measured by the number
of new cells produced. All the actions of closed conductors on each other may
be deduced from this principle.

On Electric Currents prodiiced by Induction.

Faraday has shewn| that when a conductor moves transversely to the lines
of magnetic force, an electro-motive force arises in the conductor, tending to
produce a current in it. If the conductor is closed, there is a continuous
current, if open, tension is the result. If a closed conductor move transversely
to the lines of magnetic induction, then, if the number of lines which pass

♦ Hxp. Rea. (3122). See Art. (6) of this paper. t Art. (13).

X Exp. lies. (3077), &c.

VOL. I. 24



186 ON Faraday's lines of force.

through it does not change during the motion, the electro-motive forces in the
circuit will be in equilibrium, and there will be no current. Hence the electro-
motive forces depend on the number of lines which are cut by the conductor
during the motion. If the motion be such that a greater number of lines pass
through the circuit formed by the conductor after than before the motion,
then the electro-motive force will be measured by the increase of the number
of lines, and will generate a current the reverse of that which would have
produced the additional Hnes. When the number of lines of inductive magnetic
action through the circuit is increased, the induced current will tend to diminish
the number of lines, and when the number is diminished the induced current
will tend to increase them.

That this is the true expression for the law of induced currents is shewn
from the fact that, in whatever way the number of lines of magnetic induction
passing through the circuit be increased, the electro-motive effect is the same,
whether the increase take place by the motion of the conductor itself, or of other
conductors, or of magnets, or by the change of intensity of other currents, or
by the magnetization or demagnetization of neighbouring magnetic bodies, or
lastly by the change of intensity of the current itself.

In all these cases the electro-motive force depends on the change in the
number of lines of inductive magnetic action which pass through the circuit*.

* The electro-magnetic forces, which tend to produce motion of the material conductor, must be
carefully distinguished from the electro-motive forces, which tend to produce electric currents.

Let an electric current be passed through a mass of metal of any form. The distribution of
the currents within the metal will be determined by the laws of conduction. Now let a constant
electric cuiTent be passed through another conductor near the first. If the two currents are in the
same direction the two conductors will be attracted towards each other, and would come nearer if
not held in their positions. But though the material conductors are attracted, the currents (which
are free to choose any course within the metal) will not alter their original distribution, or incline
towards each other. For, since no change takes place in the system, there will be no electro-motive
forces to modify the original distribution of currents.

In this case we have electro-magnetic forces acting on the material conductor, without any
electi"o-motive forces tending to modify the current which it can-ies.

Let us take as another example the case of a linear conductor, not forming a closed circuit,
and let it be made to traverse the lines of magnetic force, either by its own motion, or by changes
in the magnetic field. An electro-motive force wiU act in the direction of the conductor, and, as it
cannot produce a current, because there is no circuit, it will produce electric tension at the extremi-
ties. There will be no electro-magnetic attraction on the material conductor, for this attraction
depends on the existence of the cun-ent within it, and this is prevented by the circuit not being closed.

Here then we have the opposite case of an electro-motive force acting on the electricity in the
conductor, but no attraction on its material particles.



ON FARADAY 8 LINES OF FORCE. 187

It is natural to suppose that a force of this kind, which depends on a
change in the number of lines, is due to a change of state which is measured
by the number of these lines. A closed conductor in a magnetic field may-
be supposed to be in a certain state arising from the magnetic action.
As long as this state remains unchanged no effect takes place, but, when the
state changes, electro-motive forces arise, depending as to their intensity and
direction on this change of state. I cannot do better here than quote a
passage from the first series of Faraday's Experimental Researches, Art. (60).

"While the wire is subject to either volta-electric or magno-electric
induction it appears to be in a peculiar state, for it resists the formation of
an electrical current in it ; whereas, if in its common condition, such a current
would be produced; and when left uninfluenced it has the power of originating a
current, a power which the wire does not possess under ordinary circumstances.
This electrical condition of matter has not hitherto been recognised, but it
probably exerts a very important influence in many if not most of the phe-
nomena produced by currents of electricity. For reasons which will immediately
appear (7) I have, after advising with several learned friends, ventured to
designate it as the electro-tonic state." Finding that all the phenomena could
be otherwise explained without reference to the electro-tonic state, Faraday in
his second series rejected it as not necessary ; but in his recent researches ■'"'
he seems still to think that there may be some physical truth in his
conjecture about this new state of bodies.

The conjecture of a philosopher so familiar with nature may sometimes be
more pregnant with truth than the best established experimental law disco-
vered by empirical inquirers, and though not bound to admit it as a physical
truth, we may accept it as a new idea by which our mathematical conceptions
may be rendered clearer.

In this outline of Faraday's electrical theories, as they appear from a
mathematical point of view, I can do no more than simply state the mathe-
matical methods by which I believe that electrical phenomena can be best
comprehended and reduced to calculation, and my aim has been to present the
mathematical ideas to the mind in an embodied form, as systems of lines or
surfaces, and not as mere symbols, which neither convey the same ideas, nor
readily adapt themselves to the phenomena to be explained. The idea of the
electro-tonic state, however, has not yet presented itself to my mind in such a

* (3172) (3269).

24—2



188 ON Faraday's lines of force.

form that its nature and properties may be clearly explained witliout reference
to mere symbols, and therefore I propose in the following investigation to use
symbols freely, and to take for granted the ordinary mathematical operations.
By a careful study of the laws of elastic solids and of the motions of viscous
fluids, I hope to discover a method of forming a mechanical conception of this
electro- tonic state adapted to general reasoning*.



Part II.

On Faraday's " Electro^tonic State"

When a conductor moves in the neighbourhood of a current of electricity,
or of a magnet, or when a current or magnet near the conductor is moved, or
altered in intensity, then a force acts on the conductor and produces electric
tension, or a continuous current, according as the circuit is open or closed. This
current is produced only by changes of the electric or magnetic phenomena sur-
rounding the conductor, and as long as these are constant there is no observed
effect on the conductor. Still the conductor is in different states when near- a
current or magnet, and when away from its influence, since the removal or
destruction of the current or magnet occasions a current, which would not have
existed if the magnet or current had not been previously in action.

Considerations of this kind led Professor Faraday to connect with his

discovery of the induction of electric currents the conception of a state into
which all bodies are thrown by the presence of magnets and currents. This
state does not manifest itself by any known phenomena as long as it is undis-
turbed, but any change in this state is indicated by a current or tendency
towards a current. To this state he gave the name of the " Electro-tonic
State," and although he afterwards succeeded in explaining the phenomena
which suggested it by means of less hypothetical conceptions, he has on several
occasions hinted at the probability that some phenomena might be discovered
which would render the electro-tonic state an object of legitimate induction.
These speculations, into which Faraday had been led by the study of laws
which he has well established, and which he abandoned only for want of experi-

* See Pro£ W. Thomson On a Mechanical Representation of Electric, Magnetic and Galvanic
Forces. Camvb. and Dub. Math. Jour. Jan. 1847.



ON f^vraday's lines of forcr 189

mental data for the direct proof of the unknown state, have not, I think, been
made the subject of mathematical investigation. Perhaps it may be thought
that the quantitative determinations of the various phenomena are not suffi-
ciently rigorous to be made the basis of a mathematical theory ; Faraday,
however, has not contented himself with simply stating the numerical results of
his experiments and leaving the law to be discovered by calculation. Where
he has perceived a law he has at once stated it, in terms as unambiguous as
those of pure mathematics ; and if the mathematician, receiving this as a physical
truth, deduces from it other laws capable of being tested by experiment, he
has merely assisted the physicist in arranging his own ideas, which is con-
fessedly a necessary step in scientific induction.

In the following investigation, therefore, the laws established by Faraday
will be assumed aa true, and it will be shewn that by following out his
speculations other and more general laws can be deduced from them. If it
should then appear that these laws, originally devised to include one set of
phenomena, may be generalized so as to extend to phenomena of a different
class, these mathematical connexions may suggest to physicists the means of
establishing physical connexions; and thus mere speculation may be turned to
account in experimental science.

On Quantity and Intensity as Properties of Electric Currents.

It is found that certain effects of an electric current are equal at what-
ever part of the circuit they are estimated. The quantities of water or of
any other electrolyte decomposed at two different sections of the same circuit,
are always found to be equal or equivalent, however different the material and
form of the circuit may be at the two sections. The magnetic effect of a
conducting wire is also found to be independent of the form or material of
the wire in the same circuit. There is therefore an electrical effect which is
equal at every section of the circuit. If we conceive of the conductor as the
channel along which a fluid is constrained to move, then the quantity of fluid
transmitted by each section will be the same, and we may define the quantity
of an electric current to be the quantity of electricity which passes across a
complete section of the current in unit of time. We may for the present
measure quantity of electricity by the quantity of water which it would decom-
pose in unit of time.



190 ON FABADAYS LINES OF FORCE.

In order to express mathematically the electrical currents in any conductor,
we must have a definition, not only of the entire flow across a complete section,
but also of the flow at a given point in a given direction.

Def. The quantity of a current at a given point and in a given direction
is measured, when uniform, by the quantity of electricity which flows across
unit of area taken at that point perpendicular to the given direction, and when
variable by the quantity which would flow across this area, supposing the flow
uniformly the same as at the given point.

In the following investigation, the quantity of electric current at the point
(xyz) estimated in the directions of the axes x, y, z respectively will be denoted
by Oj, 5j, C3.

The quantity of electricity which flows in unit of time through the ele-
mentary area dS

= dS (la^ + ?nZ)2 + nc^),
where I, m, n are the direction-cosines of the normal to dS.

This flow of electricity at any point of a conductor is due to the electro-
motive forces which act at that point. These may be either external or internal.

External electro- motive forces arise either from the relative motion of currents
and magnets, or from changes in their intensity, or from other causes acting
at a distance.

Internal electro-motive forces arise principally from diSerence of electric
tension at points of the conductor in the immediate neighbourhood of the point
in question. The other causes are variations of chemical composition or of tem-
perature in contiguous parts of the conductor.

Let Pi represent the electric tension at any point, and X^, F,, Z, the sums
of the parts of all the electro-motive forces arising from other causes resolved
parallel to the co-ordinate axes, then if Og, ySj, y^ be the efiective electro-motive
forces

"^-^^'dx



dp,
^'-^'"dy

dp,

y^^^'^-d^



(A).



ON Faraday's lines of force. 191

Now the quantity of the current depends on the electro-motive force and
on the resistance of the medium. If the resistance of the medium be uniform
in all directions and equal to k^,

a^ = Jc,a„ ^, = kK y2 = Kc2 (B),

but if the resistance be different in different directions, the law will be more
complicated.

These quantities Oj, /3j, y., may be considered as representing the intensity
of the electric action in the directions of x, y, z.

The intensity measured along an element da of a curve is given by

€ = Za + mji + ny,
where Z, m, n are the direction-cosines of the tangent.

The integral JecZcr taken with respect to a given portion of a curve line,
represents the total intensity along that line. If the curve is a closed one, it
represents the total intensity of the electro-motive force in the closed curve.

Substituting the values of a, /8, y from equations (A)

l^da- = l{Xdx + Ydy + Zdz) -p + a

If therefore {Xdx+ Ydy + Zdz) is a complete differential, the value of Jedo- for
a closed curve will vanish, and in all closed curves

leda- = l{Xdx+Ydy + Zdz),

the integration being effected along the curve, so that in a closed curve the
total intensity of the effective electro- motive force is equal to the total intensity
of the impressed electro-motive force.

The total quantity of conduction through any surface is expressed by

\edS,
where

e = la + mh + nc,

I, m, n being the direction- cosines of the normal,

. •. \edS = l\adydz + ^bdzdx + \\cdxdy,

the integrations being effected over the given surface. AVhen the surface is a
closed one, then we may find by integration by parts



w.=///(:-



7a dh dc\ , , ,



192 ON FARADAY S LINES OF FORCE.

If we make

da dh ^ d.c /^v

Tx + dy+di^^^P (^)'

\edS= iirlWpdxdydz,

where the integration on the right side of the equation is effected over every
part of space within the surface. In a large class of phenomena, including all
cases of uniform currents, the quantity p disappears.



Magnetic Quantity and Intensity.

From his study of the lines of magnetic force, Faraday has been led to
the conclusion that in the tubular surface ■''' formed by a system of such lines,
the quantity of magnetic induction across any section of the tube is constant,
and that the alteration of the character of these lines in passing from one
substance to another, is to be explained by a difference of inductive capacity
in the two substances, which is analogous to conductive power in the theory
of electric currents.

In the following investigation we shall have occasion to treat of magnetic
quantity and intensity in connection with electric. In such cases the magnetic
symbols wiU be distinguished by the sufiix 1, and the electric by the suffix 2.
The equations connecting a, h, c, h, a, /8, y, p, and p, are the same in form as
those which we have just given, a, 6, c are the symbols of magnetic induction
with respect to quantity ; k denotes the resistance to magnetic induction, and
may be different in different directions ; a, /8, y, are the effective magnetiang
forces, connected with a, h, c, by equations (B) ; p is the magnetic tension or
potential which will be afterwards explained ; p denotes the density of real
magnetic matter and is connected with a, h, c by equations (C). As all the
details of magnetic calculations will be more intelligible after the exposition of the
connexion of magnetism with electricity, it will be sufficient here to say that
all the definitions of total quantity, with respect to a surface, the total intensity
to a curve, apply to the case of magnetism as well as to that of electricity.

* Exp. Res. 3271, definition of " Sphondyloid."



ON Faraday's lines of force. 193



Electro-magnetism.



Ampere has proved the following laws of the attractions and repulsions of
electric currents :

I. Equal and opposite currents generate equal and opposite forces.

II. A crooked current is equivalent to a straight one, provided the two
currents nearly coincide throughout their whole length.

IIL Equal currents traversing similar and similarly situated closed curves
act with equal forces, whatever be the linear dimensions of the circuits.

IV. A closed current exerts no force tending to turn a circular conductor
about its centre.

It is to be observed, that the currents with which Ampere worked were constant
and therefore re-entering. All his results are therefore deduced from experiments
on closed currents, and his expressions for the mutual action of the elements
of a current involve the assumption that this action is exerted in the direction
of the line joining those elements. This assumption is no doubt warranted by the
universal consent of men of science in treating of attractive forces considered
as due to the mutual action of particles ; but at present we are proceeding
on a different principle, and searching for the explanation of the phenomena,
not in the currents alone, but also in the surrounding medium.

The first and second laws shew that currents are to be combined like
velocities or forces.

The third law is the expression of a property of all attractions which may
be conceived of as depending on the inverse square of the distance from a fixed
system of points ; and the fourth shews that the electro-magnetic forces may
always be reduced to the attractions and repulsions of imaginary matter properly
distributed.

In fact, the action of a very small electric circuit on a point in its neigh-
bourhood is identical with that of a small magnetic element on a point outside
it. If we divide any given portion of a surface into elementary areas, and
cause equal currents to flow in the same direction round all these Httle areas,
the effect on a point not in the surface will be the same as that of a shell
coinciding with the surface, and uniformly magnetized normal to its surface.
But by the first law all the currents forming the little circuits will destroy

VOL. L 25



194 ON FARADAY S LINES OF FORCE.

one another, and leave a single current running round the bounding line. So
that the magnetic effect of a uniformly magnetized shell is equivalent to that
of an electric current round the edge of the shell. If the direction of the current
coincide with that of the apparent motion of the sun, then the direction of
magnetization of the imaginary shell will be the same as that of the real mag-
netization of the earth*.

The total intensity of magnetizing force in a closed curve passing through
and embracing the closed current is constant, and may therefore be made a
measure of the quantity of the current. As this intensity is independent of the
form of the closed curve and depends only on the quantity of the current which
passes through it, we may consider the elementary case of the current which
Hows through the elementary area dydz.

Let the axis of x point towards the west, z towards the south, and y
upwards. Let x, y, z be the coordinates of a point in the middle of the area
dydz, then the total intensity measured round the four sides of tlie element is



(A*Si)*


('■*


t' 1') *.
dy 2j


{*-


■ff)^^.


('-


■tf)<'^-


[dz


-©''^*-



Total intensity =

The quantity of electricity conducted through the elementary area dydz is
adydz, and therefore if we define the measure of an electric current to be the
total intensity of magnetizing force in a closed curve embracing it, we shall have

^^^dl,_dy,
' dz dy '



h,.



dy^ dai
dx dz



_da,_d£,
' dy dx



-■ See Experimental Researches (3265) for the relations between the electrical and magnetic circuit,
considered as mutiudly embracing curves.



ON Faraday's lines of force. 195

These equations enable us to deduce the distribution of the currents of
electricity whenever we know the values of a, y3, y, the magnetic intensities.
If a, /3, y be exact differentials of a function of x, y, z with respect to x, y
and 2 respectively, then the values of a,, h^, c, disappear; and we know that the
magnetism is not produced by electric currents in that part of the field which
we are investigating. It is due either to the presence of permanent magnetism
within the field, or to magnetising forces due to external causes.

We may observe that the above equations give by differentiation

^ + ^'4.^^ =

dx dy dz *

which is the equation of continuity for closed currents. Our investigations are
therefore for the present limited to closed currents ; and we know little of the
magnetic effects of any currents which are not closed.

Before entering on the calculation of these electric and magnetic states it
may be advantageous to state certain general theorems, the truth of which may
be established analytically.

Theorem I.

The equation

d'V d^V d'V ^ ^

d^-^W'^^'^ ^^^ '

(where V and p are functions of x, y, z never infinite, and vanishing for all points
at an infinite distance), can be satisfied by one, and only one, value of V. See
Art. (17) above.



Theorem II.

The value of V which will satisfy the above conditions is found by inte-
grating the expression

pdxdydz



///,



where the limits of x, 3/, 2 are such as to include every point of space where />
is finite.

25—2



196 ON Faraday's lines of force.

The proofs of these theorems may be found in any work on attractions or

electricity, and in particular in Green's Essay on the Application of Mathematics

to Electricity. See Arts. 18, 19 of this paper. See also Gauss, on Attractions^
translated in Taylor's Scientijtc Memoirs.



Theorem III.
Let U and V be two functions of x, y, z, then

d'U d'U d'-U\ J., , ,

where the integrations are supposed to extend over all the space in which U
and V have values differing from 0. — (Green, p. 10.)

This theorem shews that if there be two attracting systems the actions
between them are equal and opposite. And by making U= V we find that
the potential of a system on itself is proportional to the integral of the square
of the resultant attraction through all space ; a result deducible from Art. (30),
since the volume of each cell is inversely as the square of the velocity (Arts.
12, 13), and therefore the number of cells in a given space is directly as the
square of the velocity.

Theorem IV.

Let a, /8, y, p be quantities finite through a certain space and vanishing
in the space beyond, and let k be given for all parts of space as a continuous



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