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the true and only one. The potential inside the sphere must really be constant
and equal to that at the surface.

We must therefore find the images of the external electrified points, that
is, for every point at distance b from the centre we must find a point on the

same radius at a distance j- , and at that point we must place a quantity
= — e , of imaginary electricity.

At the centre we must put a quantity E' such that
K = E + e,^ + e,^- + kc.;

then if i^ be the distance from the centre, r„ r^, &c. the distances from the
electrified points, and r\, r\, &c. the distances from their images at any point
outside the sphere, the potential at that point will be

E e, ( a \ ci\ e, /a b, a\ , .



212 ON Faraday's lines of force.

This is the value of the potential outside the sphere. At the surface we



have



K = a and — = -7- , — = -7- , ac.



so that at the surface






and this must also be the value oi p for any point within the sphere.

For the application of the principle of electrical images the reader is referred
to Prof Thomson's papers in the Cambridge and Dublin Mathematical Journal.

The only case which we shall consider is that in which A = /, and b^ is infi-
nitely distant along the axis of x, and j&=0.

The value p outside the sphere becomes then

and inside ^ = 0.



II. On the effect of a paramagnetic or diam/xgnetic sphere in a uniform field oj

magnetic force'^.

The expression for the potential of a small magnet placed at the origin of
co-ordinates in the direction of the axis of x is



dx \rj~



i:^i'-]=-lm^



The eflPect of the sphere in disturbing the lines of force may be supposed
as a first hypothesis to be similar to that of a small magnet at the origin,
whose strength is to be determined. (We shall find this to be accurately true.)

* See Prof. Thomson, on the Theory of Magnetic Induction, PhiL Mag. March, 1851. The induc-
tive capadiy of the sphere, according to that paper, is the ratio of the qv/iTiiUy of magnetic induction

(not the intensity) within the sphere to that without It is therefore equal to j^T = 2k k ' ^^^^^'
ing to our notation.



ON Faraday's lines of force. 213

Let the value of the potential undisturbed by the presence of the sphere be

'p = Ix.
Let the sphere produce an additional potential, which for external points is

, . a'

and let the potential within the sphere be

Pi = Bx.

Let k' be the coefficient of resistance outside, and k inside the bphere, then
the conditions to be fulfilled are, that the interior and exterior potentials should
coincide at the surface, and that the induction through the surface should be the
same whether deduced from the external or the internal potential. Putting
a; = rcos^, we have for the external potential

P = //r + ^^')cos^,

and for the internal

p^ = Brco%dy

and these must be identical when r = a, or

I+A = B.

The induction through the surface in the external medium is

and that through the interior surface is

and .•. i(7-2^) = i£.



These equations give



A = ^^f^J, B= ^^



2k + k' ' ik + k'



The effect outside the sphere is equal to that of a little magnet whose
length is I and moment ml, provided



214



ON Faraday's lines of force.



Suppose this uniform field to be that due to terrestrial magnetism, then,
if k is less than k' as in paramagnetic bodies, the marked end of the equi-
valent magnet will be turned to the north. If A; is greater than F as in
diamagnetic bodies, the unmarked end of the equivalent magnet would be turned
to the north.



III. Magnetic Jield of variable Intensity.



Now suppose the intensity in the undisturbed magnetic field to vary in
magnitude and direction from one point to another, and that its components
in X, y, z are represented by a, /8, y, then, if as a first approximation we re-
gard the intensity within the sphere as sensibly equal to that at the centre,
the change of potential outside the sphere arising from the presence of the
sphere, disturbing the lines of force, will be the same as that due to three
small magnets at the centre, with their axes parallel to x, y, and z, and their
moments equal to

k-k' 3 k-k' 5^ k-k'

2kTk'^^' 2FFF^^' 2FfF"^-

The actual distribution of potential within and without the sphere may be
conceived as the result of a distribution of imaginary magnetic matter on the
surface of the sphere ; but since the external effect of this superficial magnetism
is exactly the same as that of the three small magnets at the centre, the
mechanical effect of external attractions will be the same as if the three ma^ets
really existed.

Now let three small magnets whose lengths are l^, k, k, and strengths
m„ m^, m„ exist at the point x, y, z with their axes parallel to the axes of
then resolving the forces on the three magnets in the direction of X, we



X, y, z
have



X = 'm^



da Zi



•a +



da l^
dx 2



Y +'in.-{



a +



a +



da I,
dy 2

da It

dy2\



■+«i.



da /g"



a +



da Zj
dz 2.



J da T da , da



ON Faraday's lines of force. 215



Substituting the values of the moments of the imaginary magnets
J da ^(IB dy\ k-k' a' d , , r>^ , 2\



2k + k'

The force impelling the sphere in the direction of x is therefore dependent
on the variation of the square of the intensity or (a' + ^ + y), as we move along
the direction of x, and the same is true for y and z, so that the law is, that
the force acting on diamagnetic spheres is from places of greater to places of
less intensity of magnetic force, and that in similar distributions of magnetic
force it varies as the mass of the sphere and the square of the intensity.

It is easy by means of Laplace's CoeflBcients to extend the approximation
to the value of the potential as far as we please, and to calculate the attrac-
tion. For instance, if a north or south magnetic pole whose strength is M, be
placed at a distance b from a diamagnetic sphere, radius a, the repulsion will be

When r is small, the first term gives a sufficient approximation. The repul-



sion is then as the square of the strength of the pole, and the mass of the
sphere directly and the fifth power of the distance inversely, considering the
pole as a point.



IV. Tivo Spheres in uniform jield.

Let two spheres of radius a be connected together so that their centres are
kept at a distance h, and let them be suspended in a uniform magnetic field,
then, although each sphere by itself would have been in equilibrium at any part
of the field, the disturbance of the field will produce forces tending to make the
balls set in a particular direction.

Let the centre of one of the spheres be taken as origin, then the undis-
turbed potential is

p = Ir cos dy ■

and the potential due to the sphere is

^ k — k' a? a



216



ON Faraday's lines of force.



The whole potential is therefore equal to

l(r +



'2jc + k'



dp
dr



,^003 0= p..



dp
dr



\ldp



Idp
rdS

1



dp\






|=«-



T^^m'Bdi



^'{i+^'^*(i-3-«''')+i5r^'(i+3-''')}



This is the value of the square of the intensity at any point. The moment
of the couple tending to turn the combination of balls in the direction of the
original force

L = l^a^i7;fn?<n when r = h,



dd \2k + k'



L^^P



k-k'



2k-\-k'



k — k' a\ . ^^



This expression, which must be positive, since h is greater than a, gives the
moment of a force tending to turn the line joining the centres of the spheres
towards the original lines of force.

Whether the spheres are magnetic or diamagnetic they tend to set in the
axial direction, and that without distinction of north and south. If, however,
one sphere be magnetic and the other diamagnetic, the line of centres will set
equatoreally. The magnitude of the force depends on the square of (k — k'), and
is therefore quite insensible except in iron*.



V. Two Spheres between the poles of a Magnet.

Let us next take the case of the same balls placed not in a uniform field
but between a north and a south pole, ±M, distant 2c from each other in the
direction of x.



* See Prof. Thomson in Phil. Mag. March, 1851.



ON Faraday's lines of force. 217

The expression for the potential, the middle of the line joining the poles
being the origin, is

p=m(, ' —, ' )■

Wc* + i^-2crcos0 Vc' + ?-' + 2crcos^/
From this we find as the value of P,

P = i^7l_3!:+9^,cos-<^):

c* \ C^ & ]

.'. I~= - 18 ^^V sin 2^.

and the moment to turn a pair of spheres (radius a, distance 2h) in the
direction in which is increased is

-^'wvk'-^''''^^'

This force, which tends to turn the line of centres equatoreally for diamagnetic
and axially for magnetic spheres, varies directly as the square of the strength of
the magnet, the cube of the radius of the spheres and the square of the dis-
tance of their centres, and inversely as the sixth power of the distance of the
poles of the magnet, considered as points. As long as these poles are near each
other this action of the poles will be much stronger than the mutual action of
the spheres, so that as a general rule we may say that elongated bodies set
axially or equatoreally between the poles of a magnet according as they are mag-
netic or diamagnetic. If, instead of being placed between two poles very near
to each other, they had been placed in a uniform field such as that of terrestrial
magnetism or that produced by a spherical electro-magnet (see Ex. viii.), an
elongated body would set axially whether magnetic or diamagnetic.

In all these cases the phenomena depend on k — k', so that the sphere con-
ducts itself magnetically or diamagnetically according as it is more or less
magnetic, or less or more diamagnetic than the medium in which it is placed.



VI. On the Magnetic Phenomena of a Sphere cut from a substance whose
coefficient of resistance is diffierent in different directions.

Let the axes of magnetic resistance be parallel throughout the sphere, and

let them be taken for the axes of x, y, z. Let K, k„ k„ be the coefficients of

resistance in these three directions, and let k' be that of the external medium,

VOL. I. 28



218 ON FARADAY S LINES OF FORCE.

and a the radius of the sphere. Let / be the undisturbed magnetic intensity
of the field into which the sphere is introduced, and let its direction- cosines
be I, m, n.

Let us now take the case of a homogeneous sphere whose coefficient is ^,
placed in a uniform magnetic field whose intensity is II in the direction of x.
The resultant potential outside the sphere would be

and for internal points

So that in the interior of the sphere the magnetization is entirely in the direc-
tion of X. It is therefore quite independent of the coefficients of resistance in
the directions of x and y, which may be changed from X\ into k^ and ^3 with-
out disturbing this distribution of magnetism. We may therefore treat the sphere
as homogeneous for each of the three components of /, but we must use a
different coefficient for each. We find for external points

and for internal points

The external effect is the same as that which would have been produced
if the small magnet whose moments are

te§'^"'' ^™^"'' te^'"-^"*'

had been placed at the origin with their directions coinciding with the axes of
Xy y, z. The effect of the original force / in turning the sphere about the axis
of x may be found by taking the moments of the components of that force
on these equivalent magnets. The moment of the force in the direction of y
acting on the third magnet is

and that of the force in z on the second magnet is

2k^-\-k



ON FARADAY S LINES OF FORCE. 219

The whole couple about the axis of a; is therefore

tending to turn the sphere round from the axis of y towards that of z. Sup-
pose the sphere to be suspended so that the axis of x is vertical, and let /
be horizontal, then if 6 be the angle which the axis of y makes with the
direction of /, m = cos 6, n= — sin 0, and the expression for the moment becomes

f TT^T^ hT}? i' \ ^'«' sin 2d,

tending to increase 0. The axis of least resistance therefore sets axially, but
with either end indifferently towards the north.

Since in all bodies, except iron, the values of k are nearly the same as in
a vacuum, the coefficient of this quantity can be but little altered by changing
the value of k' to k, the value in space. The expression then becomes

i^^^/Vsin2(9,

independent of the external medium'".



VII. Permanent magnetism in a spherical shell.

The case of a homogeneous shell of a diamagnetic or paramagnetic substance
presents no difficulty. The intensity within the shell is less than what it would
have been if the shell were away, whether the substance of the shell be dia-
magnetic or paramagnetic. When the resistance of the shell is infinite, and when
it vanishes, the intensity within the sheU is zero.

In the case of no resistance the entire effect of the shell on any point,
internal or external, may be represented by supposing a superficial stratum of

♦ Taking the more general case of magnetic induction referred to in Art. (28), we find, in the
expression for the moment of the magnetic forces, a constant term depending on T, besides those
terms which dejjend on sines and cosines of 6. The result is, that in every complete revolution in
the negative direction round the axis of T, a certain jMJsitive amount of work is gained ; but, since
no inexhaustible source of work can exist in nature, we must admit that T-0 in all substances,
with resf>ect to magnetic induction. This argument does not hold in the case of electric conduction,
or in the case of a body through which heat or electricity is passing, for such states are main-
tained by the continual expenditure of work. See Prof Thomson, Phil. Mag. March, 1851, p. 186.

28—2



220 ON Faraday's lines of force.

magnetic matter spread over the outer surface, the density being given by the
equation

p = 3/ cos d.
Suppose the shell now to be converted into a permanent magnet, so that the
distribution of imaginary magnetic matter is invariable, then the external poten-
tial due to the shell will be

p = —I—CO3 0,

and the internal potential Pi— ~ ^*' ^^^ 0.

Now let us investigate the eflfect of filling up the shell with some substance
of which the resistance is k, the resistance in the external medium being k".
The thickness of the magnetized shell may be neglected. Let the magnetic
moment of the permanent magnetism be la^, and that of the imaginary super-
ficial distribution due to the medium k = Aa\ Then the potentials are

external p' = {I-\-A)~ cos 6, internal ^, = (/+ ^ ) r cos 0.

The distribution of real magnetism is the same before and after the introduc-
tion of the medium k, so that

l/+|/=i(/+4)+|(/+^),

The external efiect of the magnetized shell is increased or diminished according
as A; is greater or less than k'. It is therefore increased by filling up the shell
with diamagnetic matter, and diminished by filling it with paramagnetic matter,
such as iron.



VIII. Electro-magnetic spherical shell.

Let us take as an example of the magnetic effects of electric currents,
an electro-magnet in the form of a thin spherical sheU. Let its radius be a,
and its thickness t, and let its external effect be that of a magnet whose
moment is /a*. Both within and without the shell the magnetic effect may be
represented by a potential, but within the substance of the shell, where there



ON FARADAY S LINES OF FORCE. 221

are electric currents, the magnetic effects cannot be represented by a potential.
Let p', pi be the external and internal potentials,

p' = 1 -^cosd, p^ = Ar cos 0,

and since there is no permanent magnetism, -^ = -^- , when r = a,

A=-2L

If we draw any closed curve cutting the shell at the equator, and at some
other point for which is known, then the total magnetic intensity round this
curve will be Sla cos 0, and as this is a measure of the total electric current which
flows through it, the quantity of the current at any point may be found by
differentiation. The quantity which flows through the element tcW is — 3/a sin 0d0,
so that the quantity of the current referred to unit of area of section is

-3l^sm0.
t

If the shell be composed of a wire coiled round the sphere so that the number
of coils to the inch varies as the sine of 0, then the external effect will be
nearly the same as if the shell had been made of a uniform conducting sub-
stance, and the currents had been distributed according to the law we have just
given.

If a wire conducting a current of strength /, be wound round a sphere
of radius a so that the distance between successive coUs measured along the

2a
axis of cc is — , then there wiU be n coils altogether, and the value of /, for

the resulting electro-magnet will be

The potentials, external and internal, will be

P=I,Q^ 003 0, p,= ■

The interior of the shell is therefore a uniform magnetic field.






P =I,Q ^ cos^, p,= -21,- -cos^.



ON FARADAY S LINES OF FORCE.



IX. Effect of the core of the electro-magnet.

Now let us suppose a sphere of diamagnetic or paramagnetic matter intro-
duced into the electro-magnetic coil. The result may be obtained as in the
last case, and the potentials become

., J n Zk' a? ^ .J. n Sk r

The external effect is greater or less than before, according as yfc' is greater
or less than k, that is, according as the interior of the sphere is magnetic or
diamagnetic with respect to the external medium, and the internal effect is
altered in the opposite direction, being greatest for a diamagnetic medium.

This investigation explains the effect of introducing an iron core into an
electro-magnet. If the value of k for the core were to vanish altogether, the
effect of the electro-magnet would be three times that which it has without
the core. As k has always a finite value, the effect of the core is less than this.

In the interior of the electro-magnet we have a uniform field of magnetic
force, the intensity of which may be increased by surrounding the coil with a
shell of iron. If k' = 0, and the shell infinitely thick, the effect on internal points
would be tripled.

The effect of the core is greater in the case of a cylindric magnet, and
greatest of aU when the core is a ring of soft iron.



X. Electro-tonic functions in spherical dectro-magnet.

Let us now find the electro-tonic functions due to this electro-magnet.
They will be of the form

ao = 0, ^^ — oiZ, y^= —<»y,

where tu is some function of r. Where there are no electric currents, we must
have ttj, 6j, Cj each = 0, and this implies

d /_ . doi\ ^

the solution of which is



ON Faraday's lines of force. 223

Within the shell co cannot become infinite ; therefore oi = C^ is the solution,
and outside a must vanish at an infinite distance, so that

is the solution outside. The magnetic quantity within the shell is found by last
article to be

therefore within the sphere

Ln 1



* 2a 3^ + ^"

Outside the sphere we must determine w so as to coincide at the surface
with the internal value. The external value is therefore

= _:?> 1 a'
^ 2a 3k + k' r' '

where the shell containing the currents is made up of n coils of wire, con-
ducting a current of total quantity /j.

Let another wire be coiled round the shell according to the same law, and
let the total number of coils be n ; then the total electro-tonic intensity EI^
round the second coil is found by integrating

EI^ = I (oa sin 6ds,



-i:



along the whole length of the wire. The equation of the wire is

/, <^

cos = -y- .

nv

where n' is a large number; and therefore

ds = a sin 6d<^,

= — ariTT sin- Odd,

T?T ^'"' 2 / 27r ,j 1

.*. EI^= -— (oan = — — ann 1



3 """ "" 3 '"""^ 3k + k"
E may be called the electro-tonic coeflBcient for the particular wire.



224 ON Faraday's lines of force.



XI. Spherical electro-magnetic CoU-Machine.

We have now obtained the electro-tonic function which defines the action
of the one coil on the other. The action of each coil on itself is found by-
putting n* or n* for nn\ Let the first coil be connected with an apparatus
producing a variable electro-motive force F. Let us find the efiects on both
wires, supposing their total resistances to be i2 and R, and the quantity of
the currents / and /'.

Let N stand for -^ (sk+k") ' *^^^ *^® electro-motive force of the first

wire on the second is

dl



That of the second on itself is



Nnn , .
at



-^<-



The equation of the current in the second wire is therefore

-iyr„n'f-iyr«-f=ij'i' (i).

The equation of the current in the first wire is

-Nn'^^^-Nnn'§ + F=RI. (2).

EHminating the differential coefficients, we get

n n' ~ n*

^^ ^[r^r] di + ^-E^^RW (^)'

from which to find / and F. For this purpose we require to know the value
of i^ in terms of t.

Let us first take the case in which F is constant and / and T initially = 0.
This is the case of an electro-magnetic coil-machine at the moment when the
connexion is made with the galvanic trough.



ON Faraday's lines of force. 225

Putting ^T for ^ [ji + j^J "^^ ^^

The primary current increases very rapidly from to >, , and the secondary
commences at - jy — and speedily vanishes, owing to the value of t being
generally very small

The whole work done by either current in heating the wire or in any other
kind of action is found from the expression



PRdt.



The total quantity of current is

^ Idt.



f.



For the secondary current we find



/;



'-"-S;. f."-m'r



The work done and the quantity of the current are therefore the same as
if a current of quantity F = —jrr- had passed through the wire for a time t, where



- (^a-



This method of considering a variable current of short duration is due to
Weber, whose experimental methods render the determination of the equivalent
current a matter of great precision.

Now let the electro-motive force F suddenly cease while the current in the
primary wire is /<, and in the secondary = 0. Then we shall have for the subse-
quent time

, . -^ „ /„ Rn -f



226 ON fahaday's lines of force.

R n
The equivalent currents are ^I^ and ^I^ -^ — , and their duration is t.

When the communication with the source of the current is cut off, there
will be a change of E. This will produce a change in the value of t, so that
if i2 be suddenly increased, the strength of the secondary current will be increased,
and its duration diminished. This is the case in the ordiaaiy coU-machines. The
quantity N depends on the form of the machine, and may be determined by
experiment for a machine of any shape.



XII. Spherical shell revolving in magnetic field.

Let us next take the case of a revolving shell of conducting matter under
the influence of a uniform field of magnetic force. The phenomena are explained
by Faraday in his Experimental Researches, Series ii., and references are there
given to previous experiments.

Let the axis of z be the axis of revolution, and let the angular velocity
be 6). Let the magnetism of the field be represented in quantity by /, inclined
at an angle 6 to the direction of z, in the plane of zx.

Let R be the radius of the spherical sheU, and T the thickness. Let the
quantities Oj, ^o* yoj.he the electro-tonic functions at any point of space; a^, \, c„
«i» Aj 7i symbols of magnetic quantity and intensity; a^, h^, c„ a,, 13,, y, of
electric quantity and intensity. Let p, be the electric tension at any point,



^'+*a.l



(1).






ON Faraday's lines op roRCE. 227

The expressions for a,, ^„ y, due to the magnetifim of the field are

^, = 5, + 2 (2 Bin ^ - a; cos ^),

A^, B,, Co being constants; and the velocities of the particles of the revolving
sphere are

dx dy dz ^



We have therefore for the electro-motive forces

An dt 4iT 2



a>=-7Z-^= - 7^008^0)0;,



_ 1 d^o I I n

$,= P = — -:— 7T cos uayy,

^* 47r dt An 2 ^'



1 / .



' 4n dt An 2

Returning to equations (1), we get

^db, dct\ dfii <^y»



j^ (db^ _dc,\d§, _dy,^^
\dz dy) dz dy '

\dx dz I dx dz An 2

^ /da, _ dbA ^ ^ _ ^^ ^ q
dy dx) ' '



^dy dx) dy dx

From which with equation (2) we find

11/..
ttj = - 7- -7- -7 sin C/a>;
k An A

h, = 0,

I 1 I . a

C, = T T- T Sin U(OX,

k An A



p, = - — - loi {(x* + 2/*) cos ^ - a:s sin $].



228 ON Faraday's lines of force.

These expressions would determine completely the motion of electricity in
a revolving sphere if we neglect the action of these currents on themselves.
They express a system of circular currents about the axis of y, the quantity
of current at any point being proportional to the distance from that axis.



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