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or discontinuous function of x, y, z, then the equation in p

has one, and only one solution, in which p is always finite and vanishes at
an infinite distance.

The proof of this theorem, by Prof W. Thomson, may be found in the
Cambridge and Dublin Mathematical Journal, Jan. 1848.

ON FARADAY S LINES OF FORCE, 197

If a, /3, y be the electro-motive forces, p the electric tension, and Ic the
coefficient of resistance, tlien the above equation is identical with the equation
of continuity

da^ ,dh,dc,

ax dy dz r '

and the theorem shews that when the electro-motive forces and the rate of
production of electricity at every part of space are given, the value of the
electric tension is determinate.

Since the mathematical laws of magnetism are identical with those of elec-
tricity, as far as we now consider them, we may regard a, /8, y as magnetizing
forces, p as magnetic tension, and p as real magnetic density, k being the
coefficient of resistance to magnetic induction.

The proof of this theorem rests on the determination of the minimum value

where V is got from the equation

d'V d'V d'V ,

and p has to be determined.

The meaning of this integral in electrical language may be thus brought
out. If the presence of the media in which k has various values did not
affect the distribution of forces, then the '^quantity" resolved in x would be

simply -7— and the intensity k -^ . But the actual quantity and intensity are

J- (a — j-j and a— ^, and the parts due to the distribution of media alone
are therefore

1 / dp\ dV , dp , dV
T {°'-~ji — 7- and a — ~ — k -i- .
fc \ ax) dx dx dx

Now the product of these represents the work done on account of this
distribution of media, the distribution of sources being determined, and taking
in the terms in y and z we get the expression Q for the total work done

198 ON Faraday's lines or force.

by that part of the whole effect at any point which is due to the distribution
of conducting media, and not directly to the presence of the sources.

This quantity Q is rendered a minimum by one and only one value of p,
namely, that which satisfies the original equation.

Theorem V.
If a, h, c be three functions of x, y, % satisfying the equation

da db ^ _r.
dx dy dz~ '

it is always possible to find three functions a, /3, y which shall satisfy the equa-
tions

dz dy '

i-

da

-h,

da

dfi

-Tx'-

= c.

Let A = Icdy, where the integration is to be performed upon c considered
as a function of y, treating x and z as constants. Let B='\adz, C^\hdx,
A' = \hdz, R = \cdx, C' = \ady, integrated in the same way.

Then

will satisfy the given equations ; for

d§^_dy^fda^^^fdc^^__fdb^^_^fda ,
dz dy J dy J dz Jdy J dy ^'

and 0=\ - j-dx+\-f- dx+ l-j- dx;

d3 dy (da , (da , (da ,

= a.

ON Faraday's lines of force. 199

In the same way it may be shewn that the values of a, ^, y satisfy
the other given equations. The function i/; may be considered at present as
perfectly indeterminate.

The method here given is taken from Prof. W. Thomson's memoir on
Magnetism {Phil Trans. 1851, p. 283).

As we cannot perform the required integrations when a, h, c are discon-
tinuous functions of x, y, z, the following method, which is perfectly general
though more compUcated, may indicate more clearly the truth of the proposition.

Let A, B, C be determined from the equations
d'A d'A d'A

^ + ^^ + £^ + 6 =
dor dy^ dz' '

d'Cd'Ccr-c^ ^

by the methods of Theorems I. and II., so that A, B, C are never infinite,
and vanish when x, y, or z is infinite.
Also let

then

a =

dB

-dz'

dC
-dy-

d^
^dx'

0-

dC
~ dx'

dA

"dz'

dy

7-

dA
-dy-

dB
~dx'

drP

^Tz'

^

dB

-dy'

dC\ fd'A d'A
- dz) W "^ clf "^

d"-A
dz\

d^/dA dB dC\,^
dx\dx dy dzj

dx\dx dy

If we find similar equations in y and z, and differentiate the first by x,
the second by y, and the third by z, remembering the equation between
a, b, c, we shall have

/c?^ d^ dr\fdA dB cZC\

\dxr dif' dz^]\dx dy dz

200 ON Faraday's lines of force.

and since A, B, C are always finite and vanish at an ir finite distance, the
only solution of this equation is

dA dB dC^^
dx dy dz *

and we have finally

d§ _dY_
dz dy~ '

with two similar equations, shewing that a, /9, y have been rightly determined.

The function i/» is to be determined from the condition

dx^ dy^ dz~ [dx" '^dy'^ dz') ^ '
if the left-hand side of this equation be always zero, xp must be zero also.

Theorem YI,

Let a, h, c he any three functions of x, y, z, it is possible to find three
functions a, /8, y and a fourth V, so that

dx dy dz '

and =^_^ ^

dz dy dx '

dx dz dy '

dy dx dz

Let

da dh dc

di + Ty + dz^-^^'P'

and let V be found from the equation

d^V d'V d'V

ON fabaday's lines of force. 201

then

a'^a-

dV
dx'

h' = h-

dV

c=c-

dV

da'
-dbc

dh'

dc ^
dz

satisfy the condition

and therefore we can find three tunctions A, B, C, and from these a, ^, y, so as
to satisfy the given equations.

Theorem VIL

The integral throughout infinity

Q = jjj (a,a, + hfi, + c^y,) dxdydz,
where a}>fi^, a^{y^ are any functions whatsoever, is capable of transformation into

Q=+ lll{^n>P^ - (^o«2 + A^2 + roC,)} dxdydz,
in which the quantities are found from the equations

dcL dh, d€<
^■'dy^fz^'^P^-''^

ojSoyo^ axe determined from ap^c^ by the last theorem, so that

^ dz dy dx '
a}>/:^ are found from cgSiyi by the equations

and p is found from the equation

d'p d'pd'p^^ , .

vol. l 26

202 ON Faraday's lines of force.

For, if we put a, in the form

dz dy dx '

and treat h^ and c, similarly, then we have by integration by parts through
infinity, remembering that all the functions vanish at the limits,

or <? = + ///{(47r V) - (aA + A&. + y.c,)] dxdydz,
and by Theorem III.

Ill Vp dxdydz = lUppdxdydz,
so that finally

Q = lll{^7rpp - (a„a, + A^2 + y«cj} dxdydz.

If afi^c^ represent the components of magnetic quantity, and a^iyi those
of magnetic intensity, then p will represent the real magnetic density, and p
the magnetic potential or tension. aJ)iCi will be the components of quantity
of electric currents, and a^^.y^ will be three functions deduced from afi^c^,
which will be found to be the mathematical expression for Faraday's Electro-
tonic state.

Let us now consider the bearing of these analytical theorems on the
theory of magnetism. Whenever we deal with quantities relating to magnetism,
we shall distinguish them by the suffix d). Thus aj^iC, are the components
resolved in the directions of x, y, z of the quantity of magnetic induction acting
through a given point, and aJS^yi are the resolved intensities of magnetization
at the same point, or, what is the same thing, the components of the force
which would be exerted on a unit south pole of a magnet placed at that
point without disturbing the distribution of magnetism.

The electric currents are found from the magnetic intensities by the equations

djB, dy, ,
dz dy

When there are no electric currents, then

a^dx + P^dy -f y^dz = dp, ,

ON Faraday's lines of force. 203

a perfect differential of a function of x, y, z. On the principle of analogy we
may call jo, the magnetic tension.

The forces which act on a mass m of south magnetism at any point are

in the direction of the axes, and therefore the whob work done during any
displacement of a magnetic system is equal to the decrement of the integral

Q = ll\p,p4xdydz
throughout the system.

Let us now call Q the total potential of the system on itself. The increase
or decrease of Q will measure the work lost or gained by any displacement
of any part of the system, and will therefore enable us to determine the
forces acting on that part of the system.

By Theorem III. Q may be put under the form

Q = + ^ j I (ctio, + hSi + c,y,) dxdydz

in which a^iji are the differential coefficients of p^ with respect to x, y, z
respectively.

If we now assume that this expression for Q is true whatever be the
values of Oj, )8„ yi, we pass from the consideration of the magnetism of permanent
magnets to that of the magnetic effects of electric currents, and we have then
by Theorem VII.

So that in the case of electric currents, the components of the currents have
to be multiplied by the functions a„, ySj, yo respectively, and the summations of
all such products throughout the system gives us the part of Q due to those
currents.

We have now obtained in the functions a,,, Aj yo the means of avoiding
the consideration of the quantity of magnetic induction which passes through
the circuit. Instead of this artificial method we have the natural one of con-
sidering the current with reference to quantities existing in the same space
with the current itself. To these I give the name of Electro-tonic functions, or
components of the Electro-tonic intensity.

2G— 2

204 ON Faraday's lines of force.

Let us now consider the conditions of the conduction of the electric
currents within the medium during changes in the electro-tonic state. The
method which we shall adopt is an appHcation of that given by Helmholtz in
his memoir on the Conservation of Force*.

Let there be some external source of electric currents which would generate
in the conducting mass currents whose quantity is measured by a^, h^, c, and
their intensity by cu,, /Sa, y^.

Then the amount of work due to this cause in the time dt is
dt lll{a^(h + hS^ + c^y^ dxdydz
in the form of resistance overcome, and

^ ^ J j J (^2^0 4- 6 A + c,yo) dxdydz

in the form of work done mechanically by the electro-magnetic action of these
currents. If there be no external cause producing currents, then the quantity
representing the whole work done by the external cause must vanish, and we
have

dt \\ \(a,a^ + hS, + c.y,) dxdydz + 4^ ^ I I I («**o + ^So + c^Jo) dxdydz,

where the integrals are taken through any arbitrary space. We must therefore
have

for every point of space ; and it must be remembered that the variation of
Q is supposed due to variations of a^, ySo, y^, and not of a^, \, c^. We must
therefore treat a^, 63, c^ as constants, and the equation becomes

In order that this equation may be independent of the values of a^, b^, Cj,
each of these coefficients must = ; and therefore we have the following
expressions for the electro-motive forces due to the action of magnets and
currents at a distance in terms of the electro-tonic functions,

°^~ ATrdt' ^^~ Andt' '^'~ An dt '

* Translated in Taylor's N'ew Scientific Memoirs, Part 11.

ON Faraday's lines of force. 205

It appears from experiment that the expression -jj refers to the change

of electro-tonic state of a given particle of the conductor, whether due to
change in the electro-tonic functions themselves or to the motion of the particle.

If Oo be expressed as a function of x, y, z and t, and \£ x, y, z be the
co-ordinates of a moving particle, then the electro-motive force measured in the
direction of a; is

_ _ Jl (^' dx da^dy da,dz doA
°^~ 477 \dx dt dy dt dz dt dtj

The expressions for the electro-motive forces in y and z are similar. The
distribution of currents due to these forces depends on the form and arrange-
ment of the conducting media and on the resultant electric tension at any
point.

The discussion of these functions would involve us in mathematical formulae,
of which this paper is already too full. It is only on account of their physical
importance as the mathematical expression of one of Faraday's conjectures that I
have been induced to exhibit them at all in their present form. By a more
patient consideration of their relations, and with the help of those who are
engaged in physical inquiries both in this subject and in others not obviously
connected with it, I hope to exhibit the theory of the electro-tonic state in a
form in which all its relations may be distinctly conceived without reference to
analytical calculations.

Summary of the Theory of the Electro-tonic State.

We may conceive of the electro-tonic state at any point of space as a
quantity determinate in magnitude and direction, and we may represent the
electro-tonic condition of a portion of space by any mechanical system which
has at every point some quantity, which may be a velocity, a displacement, or
a force, whose direction and magnitude correspond to those of the supposed
electro-tonic state. This representation involves no physical theory, it is only
a kind of artificial notation. In analytical investigations we make use of the
three components of the electro-tonic state, and call them electro-tonic functions.
We take the resolved part of the electro-tonic intensity at every point of a

206 ON Faraday's lines of force.

closed curve, and find by integration what we may caU the entire electro-tonic
intensity round the curve.

Prop. I. If on any surface a closed curve be drawn, and if the surface
within it he divided into small areas, then the entire intensity round the closed
curve is equal to the sum of the intensities round each of the small areas, all
estimated in the same direction.

For, in going round the small areas, every boundary line between two of
them is passed along twice in opposite directions, and the intensity gained in
the one case is lost in the other. Every eflfect of passing along the interior
divisions is therefore neutraUzed, and the whole efiect is that due to the
exterior closed curve.

Law I. The entire dectro-tonic intensity round the boundary of an element of
surface measures the quantity of magnetic induction which passes through that
surface, or, in other words, the number of lines of magnetic force which pass
through that surface.

By Prop. I. it appears that what is true of elementary surfaces is true also
of surfaces of finite magnitude, and therefore any two surfaces which are
bounded by the same closed curve will have the same quantity of magnetic
induction through them.

Law II. The magnetic intensity at any point is connected with the quantity
of magnetic induction by a set of linear equations, called the equations of con-
duction*.

Law III. The entire magnetic intensity round the boundary of any surface
measures the quantity of electric current which passes through that surface.

Law IV. The quantity and intensity of electric currents are connected by a
system of equations of conduction.

By these four laws the magnetic and electric quantity and intensity may be
deduced from the values of the electro-tonic functions. I have not discussed
the values of the units, as that will be better done with reference to actual
experiments. We come next to the attraction of conductors of currents, and to
the induction of currents within conductors.

* See Art. (28).

ON FARADAY S LINES OF FORCE. 207

Law v. The total electro-magnetic potential of a closed current is measxired
by the product of the quantity of the current multiplied by the entire electro-tonic
intensity estimated in t/ie same direction round the circuit.

Any displacement of the conductors which would cause an increase in the
potential will be assisted by a force measured by the rate of increase of the
potential, so that the mechanical work done during the displacement will be
measured by the increase of potential.

Although in certain cases a displacement in direction or alteration of inten-
sity of the current might increase the potential, such an alteration would not
itself produce work, and there will be no tendency towards this displacement,
for alterations in the current are due to electro-motive force, not to electro-
magnetic attractions, which can only act on the conductor.

Law VI. The electro-motive force on any element of a conductor is measured
by the instantaneous rate of change of the electro-tonic intensity on that element,
whether in magnitude or direction.

The electro-motive force in a closed conductor is measured by the rate of
change of the entire electro-tonic intensity round the circuit referred to unit
of time. It is independent of the nature of the conductor, though the current
produced varies inversely as the resistance ; and it is the same in whatever
way the change of electro-tonic intensity has been produced, whether by motion
of the conductor or by alterations in the external circumstances.

In these six laws I have endeavoured to express the idea which I believe to
be the mathematical foundation of the modes of thought indicated in the Ex-
perimental Researches. I do not think that it contains even the shadow of a
true physical theory; in fact, its chief merit as a temporary instrument of
research is that it does not, even in appearance, account for anything.

There exists however a professedly physical theory of electro-dynamics, which
is so elegant, so mathematical, and so entirely different from anything in this
paper, that I must state its axioms, at the risk of repeating what ought to
be well known. It is contained in M. W. Weber's Electro-dynamic Measure-
ments, and may be found in the Transactions of the Leibnitz Society, and of the
Royal Society of Sciences of Saxony*. The assumptions are,

* When this was written, I was not aware that part of M. Weber's Memoir is translated in
Taylor's Scientific Memoirs, VoL v. Art. xiv. The value of his researches, both experimental and
theoretical, renders the study of his theory necessary to every electrician.

208 ON Faraday's lines of force.

(1) That two particles of electricity when in motion do not repel each other
with the same force as when at rest, but that the force is altered by a quantity
depending on the relative motion of the two particles, so that the expression for
the repulsion at distance r is

eeV, dr

(2) That when electricity is moving in a conductor, the velocity of the
positive fluid relatively to the matter of the conductor is equal and opposite to
that of the negative fluid.

(3) The total action of one conducting element on another is the resultant
of the mutual actions of the masses of electricity of both kinds which are
in each.

(4) The electro-motive force at any point is the difference of the forces
acting on the positive and negative fluids.

From these axioms are deducible Ampere's laws of the attraction of
conductors, and those of Neumann and others, for the induction of currents.
Here then is a really physical theory, satisfying the required conditions better
perhaps than any yet invented, and put forth by a philosopher whose experi-
mental researches form an ample foundation for his mathematical investigations.
What is the use then of imagining an electro-tonic state of which we have
no distinctly physical conception, instead of a formula of attraction which we
can readily understand ? I would answer, that it is a good thing to have
two ways of looking at a subject, and to admit that there are two ways of
looking at it. Besides, I do not think that we have any right at present to
understand the action of electricity, and I hold that the chief merit of a
temporary theory is, that it shall guide experiment, without impeding the
progress of the true theory when it appears. There are also objections to
making any ultimate forces in nature depend on the velocity of the bodies
between which they act. If the forces in nature are to be reduced to forces
acting between particles, the principle of the Conservation of Force requires
that these forces should be in the line joining the particles and functions of
the distance only. The experiments of M. Weber on the reverse polarity of
diaraagnetics, which have been recently repeated by Professor Tyndall, establish
a fact which is equally a consequence of M. Weber's theory of electricity and
of the theory of lines of fcH-ce.

ON FARADAY S LINES OF FORCE. 209

With respect to the history of the present theory, I may state that the
recognition of certain mathematical functions as expressing the "electro-tonic
state " of Faraday, and the use of them in determining electro-dynamic
potentials and electro-motive forces is, as far as I am aware, original ; but the
distinct conception of the possibility of the mathematical expressions arose in
my mind from the perusal of Prof W. Thomson's papers "On a Mechanical
Representation of Electric, Magnetic and Galvanic Forces," Cambridge and
Dublin Mathematical Journal, January, 1847, and his "Mathematical Theory of
Magnetism," Philosophical Transactions, Part I. 1851, Art. 78, &c. As an
instance of the help which may be derived from other physical investigations,
I may state that after I had investigated the Theorems of this paper
Professor Stokes pointed out to me the use which he had made of similar
expressions in his "Dynamical Theory of Diffraction," Section 1, Camhndge
Transactions, Vol. ix. Part 1. Whether the theory of these functions, consi-
dered with reference to electricity, may lead to new mathematical ideas to be
employed in physical research, remains to be seen. I propose in the rest of
this paper to discuss a few electrical and magnetic problems with reference to
spheres. These are intended merely as concrete examples of the methods of
which the theory has been given ; I reserve the detailed investigation of cases
chosen with special reference to experiment till I have the means of testing
their results.

Examples.
I. Theory of Electrical Images.

The method of Electrical Images, due to Prof W. Thomson"", by whicli
the theory of spherical conductors has been reduced to great geometrical sim-
plicity, becomes even more simple when we see its connexion with the methods
of this paper. We have seen that the pressure at any point in a uniform
medium, due to a spherical shell (radius = a) giving out fluid at the rate of

a"
AnPa^ units in unit of time, is ^P— outside the shell, and kPa inside it,

r

where r is the distance of the point from the centre of the shell.

* See a series of papers "On the Mathematical Theory of Electricity," in the Cambridge and
Dublin Math. Jour., beginning March, 1848.

VOL L 27

210 ON Faraday's lines of force.

If there be two shells, one giving out fluid at a rate inPa\ and the
other absorbing at the rate of iirFa\ then the expression for the pressure will
be, outside the shells,

J^ r r

where r and / are the distances from the centres of the two shells. Equating
this expression to zero we have, as the surface of no pressure, that for which

/ _ Fa''
r ~ Pa'

Now the surface, for which the distances to two fixed points hav^e a given

ratio, is a sphere of which the centre is in the line joining the centres of
the shells CC produced, so that

Pa'lt-F^'

If at the centre of this sphere we place another source of the fluid, then
the pressure due to this source must be added to that due to the other two;
and since this additional pressure depends only on the distance from the centre,
it will be constant at the surface of the sphere, where the pressure due to
the two other sources is zero.

We have now the means of arranging a system of sources within a given
sphere, so that when combined with a given system of sources outside the
sphere, they shall produce a given constant pressure at the surface of the sphere.

Let a be the radius of the sphere, and p the given pressure, and let the
given sources be at distances 6„ h„ &c. from the centre, and let their rates of
production be 4.TrP„. 47rP„ &c.

Then if at distances ^ , ?- , &c. (measured in the same direction as h„ \, &c.
from the centre) we place negative sources whose rates are

-47rP,?, -477P,^, &c.,

0, Oj

ON Faraday's lines of force. 211

the pressure at the surface r = a will be reduced to zero. Now placing a source
477-^ at the centre, the pressure at the surface will be uniform and equal to />.

The whole amount of fluid emitted by the surface r = a may be found by
adding the rates of production of the sources within it. The result is

To apply this result to the case of a conducting sphere, let us suppose
the external sources inP^, AnP^ to be small electrified bodies, containing e„ e,
of positive electricity. Let us also suppose that the whole charge of the con-
ducting sphere is =E previous to the action of the external points. Then all
that is required for the complete solution of the problem is, that the surface
of the sphere shall be a surface of equal potential, and that the total charge
of the surface shall be E.

If by any distribution of imaginary sources within the spherical surface we
can effect this, the value of the corresponding potential outside the sphere is

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