Download On Faradays Lines of Force

On Faraday s
III.
By
Lines of Force.
Clerk Maxwkll, B.A. Fellow of
J.
Trinity College, Cambridge.
[Read Dec.
The
11, 1856.]
of electricity on the surface of conductors have been
duction of galvanism and
No
science.
between
electrical
theory
electricity at
into
fallen
with
relation
can now be put forth, unless
rest
current electricity, but
and
inductive effects of electricity in both states.
laws, the mathematical form of which
is
it
have been reduced
the other
parts of
the
shews the connexion not
between the attractions and
Such a theory must accurately
those
satisfy
known, and must afford the means of calculating the
where the known formulae are inapplicable.
effects in the limiting cases
esta-
the theory of the con-
mutual attraction of conductors
that of the
but have not
formulDe,
data are wanting;
the experimental
while in other parts
to matliematical
The
analytically
some parts of the mathematical theory of magnetism are
deduced from experiment;
only
and Feh.
present state of electrical science seems peculiarly unfavourable to speculation.
laws of the distribution
blished,
10, 1855,
appreciate the requirements of the science, the student must
In order therefore to
make himself
materially interferes with further progress.
The
first
with a
familiar
considerable body of most intricate mathematics, the mere retention of which in the
memory
process therefore in the effectual study of
the science, must be one of simplification and reduction of the results of previous investigation to a
form
in
which the mind can grasp them.
The
results of this simplification
the form of a purely mathematical formula or of a physical hypothesis.
entirely lose sight of the
phenomena
to
be explained; and though we
In the
may
may
first
trace
take
case
out
we
the
consequences of given laws, we can never obtain more extended views of the connexions of
the subject.
If,
on the other hand, we adopt a physical hypothesis, we see the phenomena only
through a medium, and are
liable to that blindness to facts
a partial explanation encourages.
which allows the mind
committed
to
We
it is
in
assumption which
must therefore discover some method of investigation
at every step to lay hold of a clear physical conception, without
any theory founded
borrowed, so that
and rashness
neither
drawn
being
on the physical science from which that conception
is
aside from the subject in pursuit of analytical subtleties,
nor carried beyond the truth by a favourite hypothesis.
4—2
Mr maxwell, ON FARADAY'S LINES OF FORCE.
28
In order to obtain physical ideas without adopting a physical theory we must make oura physical analogy I
Thus
illustrate the other.
the mathematical sciences are founded on relations between
all
physical laws and laws of numbers, so that the aim of exact science
is
to reduce the
of nature to the determination of quantities by operations with numbers.
most universal of
all analogies to a very
partial
one,
we
mathematical form between two different phenomena giving
The
changes of direction which light undergoes
which intense forces
problems
Passing from the
same resemblance
rise to a physical
theory of light.
from one medium
to another,
to the direction,
and not
to the
and
velocity of motion, was long believed to be the true explanation of the refraction of light;
we
still find it useful in the solution of certain
in
moving through a narrow space
This analogy, which extends only
act.
the
find
in passing
are identical witii the deviations of the path of a particle in
in
mean that
between the laws of one science and those of another which makes each of
partial similarity
them
By
the existence of physical analogies.
selves familiar with
problems, in which we employ
it
without danger,
The other analogy, between light and the vibrations of an elastic
but, though its importance and fruitfulness cannot be overfarther,
much
medium, extends
estimated, we must recollect that it is founded only on a resemblance inform between the laws
as an artificial method.
of
lio-ht
By
and those of vibrations.
stripping
observation, but probably deficient both
method.
I
of
its
physical dress and reducing
we might obtain a system of truth
a theory of " transverse alternations,"
its
it
in the vividness of its
strictly
it
to
founded on
conceptions and the fertility of
have said thus much on the disputed questions of Optics, as a preparation
for the discussion of the almost universally admitted theory of attraction at a distance.
We
have
all
acquired the mathematical conception of these attractions.
about them and determine
distinct mathematical significance,
phenomena.
.the
There
is
appropriate forms
their
and
no formula
their results are
in applied
found
uniform media appear at
first
those relating to attractions.
heat, conductivity.
matical laws of
tlie
The
sight
The
a
at
distance.
among
viorA force
is
be in accordance with natural
in the
the most different in
foreign to the subject.
in
minds of men than that of
The laws of the conduction of heat
quantities which enter into
uniform motion of heat
can reason
mathematics more consistent with nature than
formula of attractions, and no theory better established
the action of bodies on one another
to
We
These formula have a
or formula;.
tiieir
in
physical relations from
them are temperature, flow of
Yet we find that the mathe-
Iiomogeneous media are identical in form with
those of attractions varying inversely as the square of the distance.
We
have only to substitute
source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any
point,
and temperature
for jwtential,
and the solution of a problem
in attractions is transformed
into that of a problem in heat.
This analogy between the formulae of heat and attraction was,
I believe, first pointed
out
by Professor William Thomson in the Cambridge Math. Journal, Vol. III.
Now the conduction of heat is supposed to proceed by an action between contiguous
parts of a medium, while the force of attraction is a relation between distant bodies, and
yet, if
we knew nothing more than
is
expressed in the mathematical formulfe,
be nothing to distinguish between the one set of phenomena and the other.
tiiere
would
Mr.
It
that if
true,
is
maxwell, ON FARADAY'S LINES OF FORCE.
we introduce
otlier considerations
subjects will assume very different
their laws
may
remain, and
will
aspects,
but
and observe additional
the two
facts,
mathematical resemblance of some of
tiie
be made useful
still
29
exciting appropriate mathematical
in
ideas.
It
is
by the use of analogies of
mind,
in
a convenient and manageable form, those mathematical ideas which
to the
study of the phenomena of
kind that I
this
The methods
electricity.
by the processes of reasoning which are found
though they have
have attempted
in
to
the researches of Faraday *, and which,
been interpreted mathematically by Prof.
those employed by the professed mathematicians.
in
which
it
I
evident that I
am
Thomson and
By
strict
I
adopt,
I
iiope
not attempting to establish any pliysical theory of a science
my
design
is
to
application of the ideas and methods of Faraday, the connexion of
the very different orders of
the niatlietnatical mind.
others, are very
when compared with
method which
the
have hardly made a single experiment, and that the limit of
shew how, by a
are necessary
are generally those suggested
generally supposed to be of an indefinite and unmathematical character,
to render
bring before the
phenomena which he has discovered may be
I shall therefore avoid as
much
clearly placed before
as I can the introduction of anything
which does not serve as a direct illustration of Faraday's methods, or of the mathematical
deductions which
I shall use
may be made from
them.
In
treating the
Faraday's mathematical methods as well as his ideas.
subject requires
it,
I shall use analytical notation,
simpler parts of the subject
When
the complexity of the
confining myself to the development
still
of ideas originated by the same philosopher.
I
have
When
and placed
in
the first place to explain and illustrate
a body
in
is
electrified in
any given position,
If the small body be
now
the idea of "lines
of force."
any manner, a small body charged with positive
urging
experience a force
will
negatively electrified,
it
will
in
it
electricity,
a certain direction.
be urged by an equal force
in
a
the north or south poles of
a
direction exactly opposite.
The same
small magnet.
relations hold between
If the
north pole
a magnetic
urged
is
in
body and
one direction, the south pole
is
urged
in
the opposite direction.
In this way we might find a line passing through any point of space, such that
it
represents
the direction of the force acting on a positively electrified particle, or on an elementary north
pole,
and the reverse direction of the force on a negatively
south pole.
at
electrified
Since at every point of space such a direction
any point and draw a
line
so that,
we go along
as
it,
its
account a line of force.
filled
all
We
might
by
passes,
this
their direction
curve
if
any point
XXXVIII.
of
tlie
shall
will indicate the
that of
lines
tlie
of force,
force at
point.
• See especially Series
we commence
and might be called on that
same way draw other
in the
space with curves indicating
it
found,
direction at
always coincide with that of the resultant force at that point,
direction of that force for every point through which
particle or an elementary
may be
Experimental Researches, and Phil. Mag. 1852.
till
we had
any assigned
so
maxwell, ON FARADAY'S LINES OF FORCE.
Mb.
We
should thus obtain a geometrical model of the physical plienomcna,
us the direction of
tell
but as
any point.
according to any given law, by regulating
tubes.
This method
a tube
is
and
velocity vary
way we might
in this
by the motion of the
of representing the intensity of a force
by the
fluid in these
velocity of an imaginary
applicable to any conceivable system of forces, but
the case in
simplification in
direction
its
the ve-
since
tiien,
we may make the
section of the tube,
tlie
represent the intensity of the force as well as
fluid in
these curves not as mere lines,
an incompressible fluid,
inversely as the section of the tube,
is
some method of indicating
require
still
we consider
If
tubes of variable section carrying
fine
of the fluid
locity
but we should
force,
tlie
the intensify of the force at
which would
it
is
capable of great
which the forces are such as can be explained by the hypothesis
of attractions varying inversely as the square of the distance, such as those observed in electrical
and magnetic phenomena.
will generally
In the case of a perfectly arbitrary system of forces, there
be interstices between the tubes; but
in the case of electric
The
possible to arrange the tubes so as to leave no interstices.
it is
motion of a
surfaces, directing the
commence
fluid
filling
up the whole
the investigation of the laws of these forces
by
and magnetic forces
tubes will then be mere
It has been usual to
space.
assuming that the pJienomena
at once
between certain points.
are due to attractive or repulsive forces acting
We
may however
obtain a different view of the subject, and one more suited to our more difficult inquiries,
by adopting
for the definition of the forces of
which we
treat, that
they
magnitude and direction by the uniform motion of an incompi-essible
propose,
I
then,
to
first
can be clearly conceived
describe
a
may be
represented in
fluid.
method by which the motion of such
a
fluid
secondly to trace the consequences of assuming certain conditions
;
of motion, and to point out the application of the method to some of the less complicated
phenomena of
electricity,
magnetism, and galvanism
of these methods, and the introduction
attractions
By
fluid,
shew how by an extension
lastly to
to
Faraday,
the laws
of the
as
to
physical nature of
the
electricity,
or adding
anything to
has been already proved by experiment.
referring everything to the purely geometrical idea of the motion of an imaginary
I
hope
premature
mere
and
and inductive actions of magnets and currents may be clearly conceived, without
making any assumptions
that which
;
of another idea due
to
attain generality
and
precision,
theory professing to explain
speculation
which
I
have collected
and
the cause of
are
found
to
avoid the dangers arising from a
the
to
phenomena.
be
of
the
If
any use
to
results
of
experimental
philosophers, in arranging and interpreting their results, they will have served their purpose,
and a mature theory,
in
which physical facts
will
be physically explained, will be formed
by those who by interrogating Nature herself can obtain the only true
questions which the mathematical theory
I.
(1)
The
solution
of the
suggests.
Theory of the Motion of an incompressible Fhiid,
substance here treated of must not be assumed to possess any of the properties
of ordinary fluids except those of freedom of motion and resistance to compression.
It is not
Mr maxwell, ON FARADAY'S LINES OF FORCE.
even a liypothctical fluid which
is
31
introduced to explain actual phenomena.
It
is
merely a
may he employed for cstahlisliing certain theorems in
a way more intclligihle to many minds and more applicahle to physical
which algebraic symbols alone are used.
The use of the word " Fluid"
collection of imaginary properties wliich
pure mathematics
in
problems than that
in
not lead us into error,
will
if
we remember
that
it
denotes a purely imaginary substance with
the following property
The portion ofjluid
ivhich at
any instant occupied a given volume,
will at
any succeed-
ing instant occupy an equal volume.
This law expresses the incompressibility of the
measure of
quantity, namely
its
volume.
direction of motion of
tlie fluid
its
The
and furnishes us with a convenient
fluid,
unit of quantity of the fluid will therefore
be the unit of volume.
The
(2)
the space which
may
occupies, but since the direction
be different at diS'erent points of
we
determinate for every such point,
is
conceive a line to begin at any point and to be continued so that every element of the line
indicates
a
it
will in general
manner
by
its
direction the direction of motion at that point of space.
Lines drawn in such
that their direction always indicates the direction of fluid motion are called lines of
fluid motion.
If the motion of the fluid be what
velocity of the motion at any fixed
is
called steady motion, that
is,
if
the direction and
point be independent of the time, these curves will repre-
sent the paths of individual particles of the fluid, but if the motion be variable this will not
generally be the case.
The
come under our
cases of motion which will
notice will be those of
steady motion.
(3)
and
if
If upon any surface which cuts the lines of fluid motion
from every point of
this
curve we draw a
generate a tubular surface which we
may
call
line
we draw a closed curve,
of motion, these lines of motion will
a tube of fluid motion.
Since this surface
generated by lines in the direction of fluid motion no part of the fluid can flow across
that this imaginary surface
(4)
The
is
as
impermeable
quantity of fluid which in unit of time crosses any fixed section of the tube
and no part runs tlirough the
the second section
is
sides of the tube, therefore
equal to that which enters through the
For
tlie fluid is
the quantity
is
incompressible,
which escapes from
first.
If the tube be such that unit of volume passes through any section in unit of time
called a unit tube
so
to the fluid as a real tube.
the same at whatever part of the tube the section be taken.
(5)
it,
is
it
is
offluid motion.
In what follows, various units
will
be referred
to,
and a
finite
number of
surfaces will be drawn, representing in terms of those units the motion of the fluid.
in
order to define the motion in every part of the
to
be drawn at indefinitely small intervals
would involve continual reference
;
fluid,
but since
lines or
Now
an infinite number of lines would liave
tlie
to the theory of limits,
description of such a system of lines
it
has been thought better to suppose
Mk maxwell, on FARADAY'S LINES OF FORCE.
32
the lines
drawn
as small as
we
To
(fi)
at intervals
depending on the assumed
surface which cuts
tlie
all
curves intersecting the
The
From
These
motion be drawn.
Similar impermeable surfaces
curves of
which, by their intersection with the
unity of time,
this
any
it
draw a second system of
every point in a curve of the
lines
may
first
system
form a surface through which no
will
be drawn for
all
the curves of the
first
let
fluid
system.
system, will form quadrilateral tubes, which will be
first
Since each quadrilateral of the cutting surface transmits unity of fluid
transmit unity of fluid through any of its*
every tube in the svstem will
The motion
sections in unit of time.
determined by
the same surface
second system will give rise to a second system of impermeable surfaces,
tlie
tubes of fluid motion.
in
a system of unit tubes.
quadrilaterals formed by the intersection of the two systems
tlie
of curves shall be unity in unit of time.
a line of fluid
unit.
system, and so arranged that the quantity of fluid which crosses
first
the surface within each of
assume the unit
to
of fluid motion, and draw upon
lines
On
system of curves not intersecting one another.
passes.
by means of
define the motion of the whole fluid
fixed
and afterwards
by taking a small submultiple of the standard
please
Take any
unit,
at every part of the
of the fluid
system of unit tubes
for
;
the direction of motion
through the point in question, and the velocity
is
space
is
it
occupies
that of the
is
tube
reciprocal of the area of the section
the
of the unit tube at that point.
We
(7)
have now obtained a geometrical construction wliich completely defines the
motion of the
next to shew
motion of
A
these
fluid
by dividing the space
how by means
it
occupies into a system of unit tubes.
of these tubes we
have
may
ascertain various points relating to the
may
begin and end at different points, and
tlie fluid.
unit tube
may be
may
either return into itself, or
either in the
In the
that space.
first
boundary of the space
case there
is
in
which we investigate the motion, or within
continual circulation of fluid in the tube, in
a
second the fluid enters at one end and flows out at the other.
may be supposed
are in the bounding surface, the fluid
from an unknown source, and to flow out
origin of
We
tiie
tube or
its
to
tiie
If the extremities of the tube
be continually supplied from without
at the other into an
termination be within the space
unknown
reservoir;
but
if
the
under consideration, then we must
conceive the fluid to be supplied by a source within that space, capable of creating and emitting
unity of fluid in unity of time, and to be afterwards swallowed
receiving and destroying the same
There
created,
it
and reduced
up by a sink capable of
continually.
nothing self-contradictory in the conception of these sources where the fluid
is
and sinks where
have made
amount
it is
annihilated.
incompressible, and
to nothing at others.
The
properties of the fluid are at our disposal,
now we suppose
The
it
produced from
notiiiiig at certain points
places of production will be called sources, and their
numerical value will be the number of units of fluid which they produce in unit of time.
places of reduction will, for want of a better name, be called sinks, and will be estimated
number of
units of fluid
absorbed
is
we
in unit
of time.
sources, a source being understood to be a sink
when
Both places
its
sign
is
will
negative.
The
by the
sometimes be called
Mn MAXWELL, ON FARADAY'S LINES OF FORCE.
It is evident that tlie
(8)
the
by
number of
that of
motion in the tubes.
A
carries in.
the other,
in
as
and the direction in which the
it,
measured by
determined
and
must carry
therefore
much
as
fluid out of the surface as it
tube which begins within the surface and ends without
it
will carry
it
out unity
will carry in the
same
In order therefore to estimate the amount of fluid which flows out of the closed
quantity.
we must subtract the number of tubes which end within the
surface,
is
fluid passes is
If the surface be a closed one, tiien any tube whose ter-
and one which enters the surface and terminates within
of fluid;
of tubes which begin there.
we
If
which passes any fixed surface
fluid
on the same side of the surface must cross the surface as many times in the one
lie
direction
unit tubes which cut
its
minations
amount of
33
If the result
is
the quantity of fluid produced within the surface
minus the number of unit
sinks,
and
number
negative the fluid will on the whole flow inwards.
beginning of a unit tube a unit source, and
call the
surface from the
its
termination a unit sink, then
estimated by the
is
number of
unit sources
must flow out of the surface on account of the
this
inconipressibility of the fluid.
In speaking of these unit tubes, sources and sinks, we must remember what was stated
(5)
to the
as
magnitude of the
number we may
If
(9)
and
distribute
we know
we conceive
if
them according
sum of
to
a third case in which
and velocity of the
the direction
the two former cases
at
fluid at
any surface
crosses
fluid
any point
corresponding points,
in the third
then
is
the
case will be the algebraic
the quantities which pass the same surface in the two former cases.
which the
their
any law however complicated.
which passes a given fixed surface
fluid
and increasing
size
in
the direction and velocity of the fluid at any point in two different cases,
the resultant of the velocities in
amount of
and how by diminishing their
unit,
For the
rate at
the resolved part of the velocity normal to the surface,
is
and the resolved part of the resultant
is
equal to the
sum of
the resolved parts of the com-
ponents.
Hence
the
number of
unit tubes which cross the surface outwards in the third case
be the algebraical sum of the numbers which cross
it
sum
as we
of sources within any closed surface will be the
may be taken
Since the closed surface
of sources and sinks
in the
II.
two former
Theory of
as small
in the third case arises
The
which the
the
fluid
fluid is
will in general
uniform motion
we may
it is
and the number
evident that the distribution
of an
I.
medium.
its
motion
is
opposed by the
medium through
This resistance depends on the nature of the medium,
depend on the direction
Part
imponderable incompressible Jluid through a
conceive to be due to the resistance of a
supposed to flow.
in which the fluid moves, as well as
restrict ourselves to the case of a
all directions.
Vol. X.
please,
here supposed to have no inertia, and
is
For the present we may
the same in
cases,
cases.
action of a force which
and
two former
of the numbers in the two former cases.
from the simple superposition of the distributions
resisting
(10)
in the
must
The law which we assume
is
on
its velocity.
uniform medium, whose resistance
as follows.
5
is
Mr maxwell, ON FARADAY'S LINES OF FORCE.
34
Any
retarding force proportional to
volume of the
unit of
velocity
fluid in
by
may neutralise the effect of
may make as small as we please,
two of its faces.
Then the resistance
difference of pressures
is
the resistance.
Conceive a cubical unit of fluid (which we
(5)),
and
let
move
it
so that
it,
tlie
in a direction perpendicular to
be kv, and therefore the difference of pressures on the
will
the pressure diminishes in
the direction
measured along the
motion
line of
ference of pressure at
its
is
supposed
the surface (p) of equal pressure.
call
corresponding to the pressures
pressure belonging to
unit of pressure
is
we measure a length equal
vary continuously in the
to
equal to a given pressure p will
is
is
kv, so that
of motion at the rate of kv for every unit of length
so that if
;
and second faces
first
to
k
units, the
dif-
extremities will be kvh.
Since the pressure
(11)
which the pressure
In order, therefore, that
must be a greater pressure behind any portion of the
there
fluid than there
bv
be a force equal to kv acting on
v, then the resistance will
a direction contrary to that of motion.
may be kept up,
in front of
directhj opposed by a
is
its velocity.
If the velocity be represented
the
medium
portion of the fluid moving thrmirih the resisting
it,
lie
at
may
If a series of these surfaces be constructed in the fluid
then the
0, 1, 2, 3 &c.,
number
and the surface niav be referred
that pressure which
fluid, all the points
on a certain surface which we
is
of the surface will indicate the
The
to as the surface 0, 1, 2 or 3.
produced by unit of force acting on unit of surface.
In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force
same proportion.
in the
(12)
'
It is easy to see that
of fluid motion
lines
;
for
if
tliese
surfaces of equal pressure
the fluid were
to
move
resistance to its motion which could not be balanced
remember
to be
considered.)
it,
by any
difference of pressures.
so that the resistances
There are therefore two
to the
any other direction, there would be a
in
that the fluid here considered has no inertia or mass,
only which are formally assigned to
must be perpendicular
sets of surfaces
and
tliat its
(We must
properties are those
and pressures are the only things
which by their intersection form
the system of unit tubes, and the system of surfaces of equal pressure cuts both the others at
Let h be the distance between two consecutive surfaces of equal pressure mea-
right angles.
sured along a line of motion, then since the difference of pressures
kvh =
which determines the relation of v
Let
s
to
/;,
1,
so that one can be found
when the other
is
known.
definition of a unit tube
vs
find
by the
last
(13)
The
and section
s.
=
\,
equation
s
= kh.
surfaces of equal pressure cut the unit tubes into portions whose length
These elementary portions of unit tubes
each of them unity of volume of fluid passes from
unit
1,
be the sectional area of a unit tube measured on a surface of equal pressure, then since
bv the
we
=
of time, and
will
a pressure
be
/)
called
to
therefore overcomes unity of resistance in that time.
overcoming resistance
is
therefore
unity in every
cell
in
unit
a pressure
The work
every unit of time.
is
cells.
(p-1)
h
In
in
spent in
35
Mr maxwell, ON FARADAY'S LINES OF FORCE.
If
(14)
may be
tubes
known, the direction and magnitude of
pressure are
may be
of the fluid at any point
the velocity
unit
of equal
surfaces
tlie
found, after
and the beginnings and endings of these tubes ascertained
constructed,
and marked out as the sources whence the
fluid is derived,
In order to prove the converse of
that
may be
pressure at every point
if
we
down
lay
two
at every point in the fluid in
the pressure at any point
the two former
it
disappears.
certain preliminary propositions.
sum
the
is
different cases,
and
of the pressures at
then the velocity at any point in the third
cases,
the distribution of sources
of the velocities in the other two, and
the resultant
is
and the sinks where
the distribution of sources be given, the
if
we must
found,
take a third case in which
corresponding points in
case
this,
we know the pressures
If
(15)
the complete system of
wliicli
is
that due to the simple superposition of the sources in the two former cases.
For the
in
that
any direction
velocity in
direction
;
that
so
if
proportional to
is
new system resolved
the
in the
the same
in
The
two original systems.
It
in
by
follows from this,
new
the
system, the
(<)),
sum
It
all
is
evident
in
tliat
will therefore
are simply combined
which
in
the pressure
two given systems the distribution of sources
will
is
old ones, and that
difference
the
of pressure
be got by changing the sign of
first.
If the pressure at every point of a closed surface be the same and equal to p^
(16)
if
tlie
form the third.
to
the sources in the second system and adding them to those in the
and
resolved parts
be the resultant
former systems.
of the corresponding quantities in
system
the
the two
rates, the velocity
sum of the
the
new system
tlie
in
combined
the
that the quantity of fluid which crosses any fixed surface
the sources of the two original systems
in the
sum of
direction will be
velocity in
of the velocities at corresponding points
is,
rate of decrease of the pressure
two systems of pressures be added together, since the rate
of decrease of pressure along any line will be the
in
tlie
there be no sources or sinks within the surface, then there will be no motion
and the pressure within
fluid within the surface,
For
if
there be motion of
fluid
tlie
it
will
of the
be uniform and equal to p.
within the
there will
surface
be tubes of fluid
motion, and these tubes must either return into themselves or be terminated either within
the
surface or at
its
Now
boundary.
pressure to places of less
pressure,
it
since
the
fluid
always flows from places of greater
cannot flow in a I'e-entering curve;
there are
since
no sources or sinks within the surface, the tubes cannot begin or end except on the surface
and since the pressure
in
tubes havinjr both
surface,
at
points
all
extremities on
If the
distribution
pressure at
surface.
if
of sources within
two
is
the same, there can be no
Hence
there
is
p, the pressure at any point within
every point of a
pressures can exist within the
For
tiie
is
;
motion
no motion within the
and therefore no difference of pressure which would cause motion, and since the
pressure at the bounding surface
(17)
of the surface
the surface
given
closed
be also known,
it
is
surface be
also p.
known,
and the
then only one distribution
of
surface.
different distributions of pressures satisfying these conditions could be found,
a third distribution
could be formed in
which
the pressure at any
point
should
be the
5—2
Mr MAXT\'ELL, ON FARADAY'S LINES OF FORCE.
36
difference of the pressures in tlie
at
the
in
surface
at tlie
Then bv
but this
and there
cases are the same,
centre of which
in the
pressures
is
in
a unit source
the source being supposed
The
the sources within
all
the
must be zero;
distribution
third
the two former cases,
and therefore these
at
any point of an
placed, the pressure
is
body of
infinite
an
at
infinite
fluid
distance from
be zero.
to
flow out
will
fluid
and
only one distribution of pressure possible.
Let us next determine the pressure
(18)
zero,
every point in the
at
the difference of the
is
would be
the pressure
distributions,
(15).
pressure
the
(l6)
this case, since the pressures
are the same in both
it
distribution
third
by
surface would vanish,
In
two former distributions.
the surface and the sources within
from
and since unity of volume
symmetrically,
centre
the
flows out of every spherical surface surrounding the point in unit of time, the velocity at a
distance r from the source will be
1
=
V
.
47rr^
k
The
when r
of decrease of pressure
rate
the actual pressure at
infinite,
is
therefore kv or
is
any point
,
he
will
and since the pressure =
p=
.
47rr
The
pressure
It
evident that
is
therefore inversely
is
proportional to
the pressure due to a
the
distance from
the source.
unit sink will be negative and equal to
k
47rr
source formed by the coalition of
If we have a
pressure will be
])
=
,
so
5
unit
sources,
then
the resulting
that the pressure at a given distance varies as the resistance
iTTl'
and number of sources conjointly.
If a
(19)
number of sources and
sinks coexist in the fluid, then in order to determine
add the pressures which each source or
the resultant pressure
we have only
For by
be a solution of the problem, and by (17)
By
this
this will
(lo)
method we
method of
(I-i)
of pressures
is
We
(20)
space and
in
we can determine the
if
we
will
be the only one.
distribution of sources to which a given distribution
due.
have next to
shew that
if
we conceive any imaginary surface
the lines of motion
of the
this surface a distribution of sources
any way the motion of the
For
it
sink produces.
can determine the pressures due to any distribution of sources, as by the
intersecting
on one side of
to
describe the
fluid
fluid,
we may
as
substitute for
upon the surface
itself
fixed in
the fluid
without altering
on the other side of the surface.
system
of unit
tubes which
defines
the
motion of the
fluid,
and wherever a tube enters through the surface place a unit source, and wherever a tube
goes out through the surface place a unit sink, and at the same time render the surface
impermeable to the
fluid,
the motion of the fluid in
the tubes
will
go on as before.
Mr maxwell, ON FARADAY'S LINES OF FORCE.
37
If the system of pressures and the distribution of sources which produce them
(21)
be known
in
a
medium whose
same system of pressures
resistance
then in order to produce the
k,
medium whose resistance is unity, the rate of production
For the pressure at any point due to a given
k.
a
in
measured by
is
at each source must be multiplied by
source varies as the rate of production and the resistance conjointly; therefore
must vary inversely
be constant, the rate of production
On
(22)
the conditions to be fulfilled at a surface
coe^cients of resistance are k
and
medium
both media,
The
resistance.
which separates two media whose
the
that
quantity
medium
The
to the other.
velocity normal to the surface
and therefore the rate of diminution of pressure
of the tubes
direction
of motion
the same
and the surfaces of equal pressure
will
is
this refraction will be, that
is
it
the normal to the
the tangent of the angle of
to
as k' is to k.
Let the space within a given closed surface be
and
with a
filled
the pressures at any point of this
medium
that
given distribution of sources within and without the surface be given
exterior
to
it,
let
required
it is
;
the same system of pressures
to determine a distribution of sources which would produce
medium whose
difi'erent
compound system due
from
to a
a
out
the
and that the tangent of the angle of incidence
(23)
in
is
proportional
the direction of incidence and
takes place in the plane passing through
refraction
flows
to
be altered after passing through the surface, and the law of
surface,
which
of fluid
at any point flows into the other, and that the pressure varies con-
tinuously from one
in
the pressure
k'.
These are found from the consideration,
of the one
if
as the resistance.
coefficient of resistance is unity.
Construct the tubes of fluid motion, and wherever a unit tube enters either medium
place
a unit source,
and wherever
it
leaves
it
Then
unit sink.
place a
if
we make
the
surface impermeable all will go on as before.
Let the resistance of the exterior medium be measured by
by
k'.
Then
(including
if
those
we multiply
in
and that of the
interior
the rate of production of all the sources in the exterior
medium
the surface),
by
k,
and make the
pressures will remain as before, and the same
multiply
all
the sources in
it
by
k',
will
k,
coefficient of resistance unity,
be true
of the
interior
including those in the surface, and
medium
make
if
the
we
resistance
its
unity.
Since the pressures on
permeable
We
if
we
both sides of the surface are now equal, we
suppose
it
please.
have now the original system of pressures produced
combination of three systems of sources.
multiplied by k, the second
is
The
medium had an equal
but now the source
is
first
of these
is
multiplied
medium on
by k and the sink by
external unit source on the surface, a source
sources the original system of pressures
=
(k
the
by
medium by
a
given external system
k',
and the third
is
the
In the original case every source in the
itself.
sink in the internal
a uniform
in
the given internal system multiplied
system of sources and sinks on the surface
external
may
—
k').
may be produced
k',
the other side of the surface,
so that the result
By means
in a
is
for every
of these three systems of
medium
for
which
k=
1.
Mr maxwell, ON FARADAY'S LINES OF FORCE.
38
Let there be no resistance
(24)
let
=
k'
0,
in addition to the given
and sinks within the surface
supposing the
medium
p
uniform pressure
which
external and internal sources, and
by
we can render the pressure
pressures so found for the external medium, together with the
tlie
medium,
in the internal
will
be the true and only distribution of pressures
possible.
is
For
is,
uniform and equal to p, and the
is
the same within and without the surface,
the surface uniform,
that
surface,
closed
If by assuming any distribution of pairs of sources
also p.
is
the
witliin
then the pressure within the closed surface
pressure at the surface itself
at
medium
the
in
if
two such distributions could be found by taking
different
imaginary distributions
of pairs of sources and sinks within the medium, then by taking the difference of the two
we should have the pressure of the bounding
for a third distribution,
the
new system and
many
as
sources as sinks within
and therefore whatever
it,
medium without
that
The
sources surrounding the internal medium.
the surface
at
the fluid must be outwards from every point of the surface
But
inwards towards the surface.
because
if
any
fluid
is
by
pressure
if it
neitlier
positive,
zero,
the motion of
be negative,
must flow
it
of these cases
is possible,
at
zero in the third system.
is
all
points in the
boundary of
therefore zero, and there are no sources,
tlie
medium
internal
and therefore the pressure
in
is
the
third case
everywhere zero,
(16).
The
is
has been shewn that
it
;
is
infinite
is
enters the surface an equal quantity must escape, and therefore the
pressure at the surface
The
pressure at infinity
If the pressure at the surface
constant.
is
flows
point.
and we have an
sources destroy one another,
the
all
fluid
some other
in at any point of the surface, an equal quantity must flow out at
In the external medium
surface constant in
no
pressure
resistance,
the bounding surface of the internal
in
therefore
it
is
difference of pressures in the
zero tliroughout
two given
one distribution of pressure which
is
;
medium
is
and there
also zero,
but the pressure in the third case
and there
cases, therefore these are equal,
is
is
the
only
possible, namely, that due to the imaginary distribution
of sources and sinks.
When
(25)
through
of fluid
impermeable
If
the resistance
or into
it
to the fluid,
infinite in
is
The bounding
it.
and the tubes of
fluid
surface
medium, there can be no passage
may
be considered as
therefore
motion will run along
it
without cutting
by assuming any arbitrary distribution of sources within the surface
the given sources
velocities
as in
in
the
the outer
case
true one.
For
since
no
medium, we can
fluid passes
from
what
it
also
small,
would be
if
the condition of there being
fulfil
through the surface, the tubes
may be
If the extent of the internal
in the two media be
addition to
system of pressures in the outer medium will be the
the
independent of those outside, and
(26)
in
it.
medium, and by calculating the resulting pressures and
of a uniform
no velocity across the surface,
altered
the internal
then
in
the
interior
are
taken away without altering the external motion.
medium be
small,
the position
the external
and
of the
medium
if
the difference of resistance
unit tubes
filled
will
not
the whole space.
be much
Mr maxwell, ON FARADAY'S LINES OF FORCE.
On
this
supposition
of the internal
medium
we can
easily calculate the
39
kind of alteration which the introduction
produce; for wherever a unit tube enters the surface we must
will
conceive a source producing fluid at a rate
——
-
-k
—--—
,
and wherever a tube leaves
we must
it
k'
place a sink annihilating fluid at the rate
that the resistance in botii media
,
then calculating pressures on the supposition
k the same as
is
in
medium, we shall obtain
we may get a better result by
the external
the true distribution of pressures very approximately, and
repeating the process on the system of pressures thus obtained.
abrupt change from one
If instead of an
(27)
coefficient of resistance
to another
take a case in which the resistance varies continuously from point to point,
medium
the
as if
it
were composed of thin
shells each of
we may
may
treat
the case as if the resistance were equal to unity throughout, as in
sources will then be distributed
whenever the motion
positive
when
is
less to places
the same in
The
(23).
and
be
will
of greater resistance, and negative
whatever direction the motion of the fluid takes place
a case in which the resistance
will not
in
resistance at
is
diff'erent in
different directions.
velocity
at
any point,
and
a,
ft,
the same point, these quantities will be connected
linear equations,
;
but we
medium to be
may conceive
In such cases the lines of
general be perpendicular to the surfaces of equal pressure.
be the components of the
y
the
If a,
components
of
h,
the
and
will
be referred to
conductivity.
In order to
that name.
b
c
In these equations there are
simplify the equations,
let
nine
= P^a + Q3/3 + ^,7,
= P,ft + Q,7 + R,a,
= P37 + Q.,a + R,ft.
independent
coefficients
+
i?j
=
2^,,
Q^-R^ =
&c
4:T'={Q,- BJ' +
where
I,
m,
The
n
of
us put
Q,
{Q, -
are direction cosines of a certain
2lT,
&c.
R.f +
(Q,
-
R^,
fixed line in
space.
equations then become
= P^a-^ S,ft + S,y + (n/3 - m 7) T,
b = P.,ft+S,y + S,a + (ly -na)T,
c^ P^y + S,a + S,ft - {ma - lft)T.
transformation of coordinates we may get rid of
a
By
The
c
by the following system of
which may be called " equations of conduction"
a
and
whole medium,
Hitherto we have supposed the resistance at a given point of the
(28)
by
tlie
we
shells,
the contrary direction.
in
motion
continuously throughout
from places of
By
which has uniform resistance.
properly assuming a distribution of sources over the surfaces of separation of the
we
treat
the ordinary
equations then become
the coefficients
marked
.S*.
Mr maxwell, ON FARADAYS LINES OF FORCE.
40
P^a+
a =
= P,'/3 + (I'y - na)T,
= P^y +{m'a - l'(i)T,
b
C
where
If
t,
m, n
(«'/3- 7n'y)T,
new
are the direction cosines of the fixed line with reference to the
axes.
we make
—
dx'
u
^
- ,
—
=
ana z = -—
dz
,
dy'
the equation of continuity
da
db
—
+ —
dx
dy
dc
^
= 0,
dz
becomes
p;^+p;"j^
"
'
and
x = ^/Pi^,
we make
if
y
the ordinary equation
of the coefficient T.
o,
^ =
v
rib-
= V-P^V^
drp
d'p
dp
rfp
dtr
d(^-
^3^1
0,
of conduction.
It appears therefore that the distribution
this
'
^ +-—+-_ =
then
=
+ p;"^
dy'
-
'
dx'-
of pressures
Thomson has shewn how
Professor
is
to
not
altered
conceive a substance in which
determines a property having reference to an axis,
coefficient
of Pj, P,, P,
by the existence
which
unlike
the
dipolar.
is
For further information on the equations of conduction,
see
Professor
Thomson on
On
Stokes
Conduction of Heat in Crystals (Cambridge and Dublin Math. Jo?crn.), and
the
Professor
Dynamical Theory of Heat, Part V. (Transactions of Royal Society of
the
Edinburgh, Vol.
axes
XXI.
Part
It is evident that all
I.)
that has been proved in (u), (15), (16), (17), with respect to the
superposition of different distributions of pressure, and there being only one distribution of
pressures corresponding to a given distribution of sources, will be true also in the case in
which the resistance varies from point to point, and the resistance at the same point
different in
to
different
For
directions.
if
we examine the proof we
shall find
is
applicable
it
such cases as well as to that of a uniform medium.
(29)
We
now
are prepared to prove certain general propositions which are true in
most general case of a medium whose resistance
is
different in different directions
the
and varies
from point to point.
We
may by
tlie
method of
(28),
when
the distribution of pressures
surfaces of equal pressure, the tubes of fluid motion,
that since in
each
cell
into
which a unit tube
unity of fluid passes from pressure
done by the
fluid in each cell in
The number
p
to pressure
overcoming
of cells in eacli unit tube
pressure through which
the endp', then the
it
passes.
number of
is
is
and the sources and
known, construct the
sinks.
It
evident
—
(}>
1)
in
unit of time, unity of
work
is
resistance.
is
determined by the number of surfaces of equal
If the pressure at the beginning of the tube be
cells in it will
is
divided by the surfaces of equal pressure
be p
- jy
.
Now
if
jy
and
at
the tube had extended from the
Mr maxwell, ON FARADAY'S LINES OF FORCE.
source to a place where the pressure
the tube had
come from the
is
the difference of these.
is
the
Therefore
we
if
number of
numhcr of
zero, the
due
to
we must cut
we denote the source of
S
S
may
be written
the
-
S', sinks
if
always
system will
cells in the
(Sp).
The same
(30)
cell.
Now
work
done
each
Now
from the number previously obtained.
off S'p' cells
being reckoned negative, and the general expression for the number of
be
if
number
the true
from which S tubes proceed to be p, Sp
the source S; but if S' of the tubes terminate in a sink at
tubes by S, the sink of S' tubes
.S*
would have been p, and
cells
number would have been p, and
sink to zero, the
find the pressure at a source
cells
a pressure j), then
is
41
in
by
conclusion
may be
each source S,
the
fluid
S
in
by observing
arrived at
that unity of
work
done on
is
units of fluid are expelled against a pressure p, so that
overcoming resistance
is
At
.S'p.
each sink
which
in
S' tubes terminate, S' units of fluid sink into nothing under pressure p'; the work done upon
the fluid
by the pressure
is
The whole work done by
therefore S'p.
the fluid
may
therefore
be expressed by
W=I.Sp-1S'p',
or more, concisely, considering sinks as negative sources,
W=1{Sp).
Let
(31)
S
represent the rate of production of a source in any medium, and
the pressure at any given point due to that source.
Then
we superpose on
if
equal source, every pressure will be doubled, and thus by successive superposition
a source
nS would produce
more generally the pressure
a pressure np, or
a given source varies as the rate of production of the source.
at
let
p
be
this
another
we
find that
any point due
to
This may be expressed by the
equation
p==RS,
where
R
is
a coefficient depending on the nature of the
source and the given point.
hS
R
medium and on
In a uniform medium whose resistance
is
the positions of the
measured by
k,
k
„
may be called the coefficient of resistance of the medium between
By combining any number of sources we have generally
p = ^{RS).
the source and the given
point.
(32)
In a uniform medium the pressure due to a source
p=
At another source S'
at a distance r
we
if
p' be the pressure at
S
due
to S".
2(5"), and if the pressures due to the
S
—
47r
r
.
have
shall
o.
—k
S
k
S^
47r
r
If therefore there be two systems of sources
first
be
^{S'p)
For every term S'p has a term Sp' equal to
Vol. X. Paet I.
p and to the
= -EiSp).
second
p,
1{S) and
then
it.
6
Mr maxwell, ON FARADAY'S LINES OF FORCE.
42
Suppose tbat
(33)
to
in a
uniform medium
tlie
one plane, then the surfaces of equal pressure
motion of the fluid
will
is
everywhere parallel
be perpendicular to
tliis
If
plane."
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 leno-th
is
equal to the breadth.
For
h be the distance between consecutive surfaces of
if
equal pressure and s the section of the unit tube, we have by (l3) s
But
is
the product of the breadth and depth
s is
but the depth
;
=
is
kh.
k, therefore the breadth
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
distance k from the
of
which the length and breadth are equal, then by taking another plane at
first
and erecting cylindric surfaces on the plane curves
be formed which
cells will
as bases, a system
whether we suppose the
will satisfy the conditions
fluid to
run
alonsr 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
motion as described above
lines of fluid
may be
modified so as to be applicable to the sciences of statical electricity, 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
the mutual action of two opposite kinds of matter.
electricity
have been already explained by
statical electricity
we consider one of
If
and the other as negative, then any two particles of
a force which
is
electricity repel
these as positive
one another with
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
r varies inversely as
r".
Let us see what
every particle of positive electricity.
The
our imaginary fluid due to a source
will
be the
velocity
effect
therefore
tlie
resultant velocity
be proportional to the resultant attraction of
find the resultant pressure at
we may
aX
a.
distance
due to each source would be proportional to
the attraction due to the corresponding particle, and
sources would
S
of substituting such a source for
all
the particles.
any point by adding the pressures due
due
to all
Now we may
to tlie given sources,
find the resultant velocity in a given direction from
the
and
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
pressure in the imaginary problem, and since
the imaginary problem,
we may assume
we may
is
select
proportional to the decrease of
any values
rically equal to the decrease of pressure in that direction, or
A.
— -——
dai
•
for the constants in
that the resultant attraction in any direction
See Cambridge and Dublin Malhemalical Journal, Vol. III.
p. 28li.
is
nume-
Mn MAXWELL, ON FARADAY'S LINES OF FORCE.
By
this
assumption we find that
if
Vhe
dV= Xdx
V=
or since at an infinite distance
43
the potential,
+ Ydy + Zdz = - dp,
p = 0, V = —p.
and
In the electrical problem we have
In the fluid p =
2
(
—
-)
;
'.
—k dm.
S=
4nr
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
2(pd«)= ^,npS) = ^W.
iir
If
^(dm), 2 (dm') be two systems of
respectively, then
by
electrical particles
—
2ipS') =
,
47r
the potential of the
on the
and pp' the potentials due
to
them
(32)
S(pcZm')=
or
47r
first
~,
S (p'^ = 2 (p'rfm),
47r
system on the second
equal to that of the second system
is
first.
So that
problems the analogy
in the ordinary electrical
in fluid
motion
is
of this kind
:
F=-p,
X=-'/
dx
dm —
=
ku,
—k S,
47r
whole potential of a system =
coming
— "LVdm =
— W, where W
is
the
work done by the
fluid in over-
resistance.
The
by those
lines of force are the unit tubes of fluid motion,
and they may be estimated numerically
tubes.
Theory of Dielectncs.
The
electrical induction exercised
on a body at a distance depends not only on the
distri-
bution of electricity in the inductric, and the form and position of the inducteous body, but on
the nature of the interposed
medium, or
dielectric.
" Series
Faraday * expresses
this
by the conception
XI.
6—2
Mr maxwell, ON FARADAY'S LINES OF FORCE.
44
of one substance having a greater inductive capacity, or conducting the lines of inductive
more
action
medium
than another.
freely
the resistance
is
we suppose that
If
in
our analogy of a fluid in a resisting
different in different media, then
by making the
resistance less
we
obtain the analogue to a dielectric which more easily conducts Faraday's lines.
It
is
electricity
evident from (23) that in this case there will always be an apparent distribution of
dielectric, there
on the surface of the
and positive
electricity
being negative
electricity
where the
lines enter
In the case of the fluid there are no real sources
where they emerge.
on the surface, but we use them merely for purposes of calculation.
In the dielectric there
be no real charge of electricity, but only an apparent electric action due to the surface.
may
medium, we should
If the dielectric had been of less conductivity than the surrounding
have had precisely opposite
namely, positive electricity where lines enter, and negative
effects,
where they emerge.
If the conduction of the dielectric
tricity
perfect or nearly so for the small quantities of elec-
is
with which we have to do, then we have the case of
considered as a conductor,
its
surface
is
The
(fii).
dielectric is then
a surface of equal potential, and the resultant attrac-
tion near the surface itself is perpendicular to
it.
Theory of Permanent Magneti.
A
its
maofnet
is
conceived to be
own north and south
o-overned
made up of elementary magnetized
the action
poles,
by laws mathematically
motion can be employed
to illustrate
it may be useful to examine
be represented by the unit
may
magnet
by one
made
to this subject,
the
way
cells
in
in
to
Hence the same
is
applica-
and the same analogy of
In each unit
cell
face, so that the first face
sink and the second a unit source with respect to the rest of the
compared
north and south poles
fluid
which the polarity of the elements of a
fluid motion.
and flows out by the opposite
face
which has
it.
But
enters
of which upon other
identical with those of electricity.
of the idea of lines of force can be
tion
particles, each of
fluid.
It
unity of fluid
becomes a unit
may
therefore be
an elementary magnet, having an equal quantity of north and south magnetic
matter distributed over two of
its faces.
system, the fluid flowing out of one
If we
now
consider the cell as forming part of a
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
the bounding surface, the sources and sinks will be distributed entirely
begin and end on
on that surface, and
in
the case of a
tubular distribution of magnetism,
all
magnet which has what has been
called a solenoidal or
the imaginary magnetic matter will be on the surface*.
Theory of Paramagnetic and Diatnagnetic Induction.
Faraday "f has shewn that the
field
may be
•
effects
of paramagnetic and diamagnetic bodies in the magnetic
explained by supposing paramagnetic bodies to conduct the lines of force better.
See Professor
Thomson On
+ Experimental
the
Mathematical Theory of Magnetism, Chapters III.
Researches (3292).
&.
V. Phil. Trans. 1851.
45
Mr maxwell, ON FARADAY'S LINES OF FORCE.
By
and diamagnetic bodies worse, than the surrounding medium.
referring to (23) and (26),
and supposing sources to represent north magnetic matter, and sinks south magnetic matter,
then
a paramagnetic body be in the neighbourhood of a north pole, the lines of force on
if
entering
it
will
produce south magnetic matter, and on leaving
Since the quantities of magnetic matter on
amount of north magnetic matter.
equal, but the southern matter
is
they will produce an equal
it
on the other hand the body be diamagnetic, or a worse conductor of
surrounding medium, there
where the
will
the whole are
If
nearest to the north pole, the result will be attraction.
of force than the
lines
be an imaginary distribution of northern magnetic matter
lines pass into the worse conductor,
and of southern where they pass out, so that on
the whole there will be repulsion.
We
system
may
is
obtain a more general law from the consideration that the potential of the whole
proportional to the amount of work done by the fluid in overcoming resistance.
medium
introduction of a second
ance
is
increases or diminishes the
greater or less than that of the
tion will
first
medium.
vary as the square of the velocity of the
Now, by
the
work done according
The amount
as the resist-
of this increase or diminu-
fluid.
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
the resistance within the second
k',
rounding medium, there
where
it is less,
is
medium,
is
greater
than
Ic,
the resistance in
a force tending from places where the resultant force v
is
when
the sur-
greater to
to less values of the resultant
body moves from greater
so that a diamagnetic
The
force *.
In paramagnetic bodies
k' is less
than k, so that the force
points of greater resultant magnetic force.
of k and k' ,
may
it
is
now from
points of less to
Since these results depend only on the relative values
evident that by changing the surrounding medium, the behaviour of a
body
be changed from paramagnetic to diamagnetic at pleasure.
It is evident that
we should obtain the same mathematical
the magnetic force had a
power of exciting a polarity
as the lines in paramagnetic bodies,
fact
is
we have not
as yet
these three theories,
induced polarity.
come
to
and
in bodies
we had supposed that
results if
which
is
in
the
same
direction
in the reverse direction in diamagnetic bodies
-f.
In
any facts which would lead us to choose any one out of
that of lines of force, that of imaginary magnetic matter, and that of
As the theory of
lines of force
time least theoretic statement, we shall allow
it
admits of the most precise, and at the same
to stand for the present.
Theory of Magnecrystallic Induction.
The
may be
*
son,
1847.
theory of Faraday
thus stated.
\.
with respect to the behaviour of crystals in the magnetic field
In certain crystals and other substances the lines of magnetic force are
Experimental Researches (2797), (2798). See ThomCambridge and Dublin Mathematical Journal, May,
t Exp.
Res.
Ixxxvii. p. 145.
+
(2429), (3320).
See Weber, Poggendorft',
Prof. Tyndall, Phil. Trans. 185G, p. 237.
E.vp. Res. (283G), &c.
Mr maxwell, ON FARADAY'S LINES OF FORCE.
46
conducted with different
The body when suspended
directions.
different
in
facility
in
a
uniform magnetic field will turn or tend to turn into such a position that the lines of force shall
pass through
with least resistance.
it
It
to express the laws of this kind of action,
The
formula2.
of
reduce them in certain cases to numerical
to
and of imaginary magnetic matter are here
principles of induced polarity
use; but the theory of lines of force
little
class of
not difficult by means of the principles in (28)
is
and even
is
capable of the most perfect adaptation to this
phenomena.
Theory of the Conduction of Current
It
the calculation of the laws of constant electric currents that the theory of fluid
in
is
Electricity.
motion which we have laid down admits of the most direct application.
researches of
of
M.
Ohm
on this subject, we have those of
M.
Kirchhoff,
In addition to the
Aim. de
Quincke, xlvii. 203, on the Conduction of Electric Currents
in
CJiim. xli. 496,
and
According to
Plates.
the received opinions we have here a current of fluid moving uniformly in conducting circuits,
which oppose a resistance to the current which has
an electro-motive force at some part of the circuit.
be overcome by the application of
to
On
account of this resistance to the motion
This pressure,
of the fluid the pressure must be different at different points in the circuit.
which
is
commonly
called electrical tension,
in statical electricity,
If we knew what amount of
we assume
electricity
as
electricity,
measured
finite
between
to be physically identical with the potential
statically, passes
phenomena.
sets of
along that current which
This has as yet been done only approximately, but we
*.
(b be certain that the
conducting powers of different substances
degree, and that the difference between
but
found
our unit of current, then the connexion of electricity of tension with current
would be completed
know enough
is
and thus we have the means of connecting the two
glass
quantity in glass, and a small but
electricity
statical
and
fluid
and metal
finite
motion turns
is,
that the
differ
resistance
Thus
quantity in metal.
is
only in
a great
the analogy
more perfect than we might have
out
supposed, for there the induction goes on by conduction just as in current electricity, but
the quantity conducted
is
insensible
On
When
must
act
owing
to
the great resistance of the dielectrics
Electro-motive Forces.
a uniform current exists in a closed circuit
on the
pressures, then
it
fluid
besides the pressures.
For
if
it
is
evident that some other forces
the current were
would flow from the point of greatest pressure
point of least pressure, whereas in
• See
Exp. Res.
(371).
-j-.
reality
it
circulates in
f Exp.
Res.
in
due
to
difference of
both directions to the
one direction constantly.
Vol
III. p. 513.
We
Mr maxwell, ON FARADAY'S LINES OF FORCE.
must therefore admit the existence of
Of
a closed circuit.
in
A
action.
certain forces capable of keeping
most remarkable
tlie
that which
is
is
up a constant current
produced by chemical
of a voltaic battery, or rather the surface of separation of the fluid of the
cell
and the
cell
these
4?
zinc,
the
is
an
seat of
electro-motive force
opposition to the resistance of the circuit.
of electric currents, the positive current
we adopt
If
from the
is
circuit, and the zinc, back to the fluid again.
can maintain a current
whicii
through the platinum, the conducting
fluid
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
exceed that
by
the zinc
in
a quantity
and on the strength of the current
in
If
F
In order to keep up this difference
the conductor.
be the electro-motive force,
of electrical units delivered in
must
depending on the nature and length of the circuit
of pressure there must be an electro-motive force whose intensity
of pressure.
in
the usual convention in speaking
/
measured by that difference
is
the quantity of the current or the
unit of time, and
Ka
number
quantity depending on the length and
resistance of the conducting circuit, then
F = IK = p
where p
is
the electric
the part which
remains attached
F
p — p\ or
p'.
affords a
quantity and intensity
more than
in
this
:
— The
and
unit of time,
The
intensity
or IK,
where
The same
The
quantity
number of
K
is
and p
the
in
zinc.
to
the platinum will be p,
very useful
and
of the
that
the
is
amount of
which
electricity
measured by / the number of unit currents which
its
is
power of overcoming
in
resistance,
and
it
transmits
it
contains.
measured by
is
may be
applied
to
the case of magnetism-j".
any section of a magnetic body
magnetic force which pass through
it.
The
is
measured by the
intensity of magnetization
depends on the resisting power of the section, as well as on the number of
the section, and
/
it.
the
If
k be the
number
F
of the whole circuit.
quantity and intensity
of magnetization
other will
This distinction of
but must be distinctly understood to mean nothing
'",
quantity of a current
is
no current the tension of
is
measure of the intensity of the current.
the resistance
idea of
which pass through
intensity
is
of a current
lines of
in the section
the fluid
in
be broken at any point, then since there
If the circuit
be
tension
- p,
resisting
power of the material, and
of force
of lines
which pass
through
it,
.S*
then
lines
the area of
the
whole
throughout the section
—IWhen
p
for
magnetization
the magnetic
is
tension
produced by
at
Exp. Hes. Vol.
influence of other magnets only,
any point, then
F=
•
the
for
the
whole magnetic solenoid
I f^dj!= IK = p-p'.
III. p. 519.
t
we may put
^^P- ^es. (2870), (3293).
Mr maxwell, ON FARADAY'S LINES OF FORCE.
48
When
depend
If
magnetized circuit returns into
a solenoidal
be the quantity of the magnetization
i
magnetization does
the
itself,
some magnetizing force of which the intensity
on difference of tensions only, but on
any point, or the number of
at
force passing through unit of area in the section of the solenoid, then the
of niao-nctization in the circuit
/ =
"^idyds, where dydx
is
number of
the
is
which pass
lines
total
through
the element of the section, and the summation
is
not
is
F.
lines
of
quantity
any section
performed over
the whole section.
The
mao-netization,
is
of magnetization
intensity
measured by
is
measured by the sum of the
ki
any point,
at
=f, and the
or
force required
the
round the
keep up the
of magnetization in the circuit
total intensity
local intensities all
to
circuit,
F=^ ifdx),
where dx
the
is
the
clement of length in the circuit, and
the summation
is
extended round
entire circuit.
In the same circuit we have always
and depends on
circuit,
On
The mathematical
investigated
From
its
the total
is
it
resistance
of the
composed.
is
Curretits at a Distance.
laws of the attractions and repulsions of conductors have been most ably
have stood the
his results
assumption,
element of another current
the line joining the
K
IK, where
form and the matter of which
the Action of closed
by Ampere, and
the single
F=
is
test
of subsequent experiments.
the action of an element of one
that
an attractive or repulsive force acting
two elements,
he
determined
has
by
current
in
simplest
the
upon an
the direction
of
experiments the
mathematical form of the law of attraction, and has put this law into several most elegant
and useful forms.
We
must
however that no experiments have been made on
recollect
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 only can be deduced from such experiments.
Hence
if
Ampere's formulae applied
currents give true results, their truth
to closed
that the action between
proved for elements of currents unless we assume
must be along the
line
Although
which joins them.
philosophical in the present state of science,
gation
if
ultimate
we endeavour
to
do without
it,
it will
and
to
this
assumption
most warrantable and
be more conducive to freedom of investi-
assume the laws of closed currents as the
that
when currents
are
combined
according to
parallelogram of forces, the force due to the resultant current
to the
has
is
the
law of the
the resultant of the forces
component currents, and that equal and opposite currents generate equal and
opposite forces,
He
not
datum of experiment.
Ampere has shewn
due
is
is
two such elements
and when combined neutralize each other.
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
a complete differential.
in the case
of a closed circuit render
Xdx+ Ydy+Zdz
Mr maxwell, OK FARADAV'S LINES OF FORCE.
has shewn
Finally, he
systems of circuits similar and similarly
corresponding
conductors beino-
same,
the
whatever be the absolute dimensions of the systems, which
resultant forces are equal,
the
there be two
if
quantity of electrical current in
the
situated,
that
49
proves that the forces are, cceferis paribus, inversely as the square of the distance.
From
these results
very small
is
it
follows that the mutual action of two closed currents whose areas are
the same as that of two elementary magnetic bars magnetized perpendicularly
to the plane of the currents.
The
direction of magnetization of the equivalent
that a current travelling round
magnet may be predicted by remembering
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
when pointing
reverse direction to that of a magnetic needle
If a
in
number of
closed unit currents in contact exist on a surface, then at
which two currents are
will
produce no
freely.
but
effect,
points
all
contact there will be two equal and opposite currents which
in
round the boundary of
all
tlie
surface 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
From
this
appears that
it
perpendicular to
the
attractions of
external
those due
surface are the same as
its
each of the elementary currents in the
a
round
to a current
the same
former case has
the currents.
all
uniformly magnetized
shell
edge,
its
as an
effect
for
element of
the magnetic shell.
If
we examine the
lines
of magnetic force produced by a closed current, we shall find
that they form closed curves passing round
the current and embracing
intensity of the magnetizing force all along the closed line of force
tity of the
The number
current only.
electric
closed current depends on
embrace
and that the
The
it.
unit of magnetic quantity
is
unit cells
is
in
number
measured simply by the number of unit
case
this
are
portions
produced by unity of magnetizing
cell is therefore inversely as the intensity of the
total
of unit lines * of magnetic force due to a
the form as well as the quantity of the current, but the
of unit cells f in each complete line of force
currents which
it,
depends on the quan-
force.
magnetizing force, and
its
of space in which
The
length
section
is
of a
inversely
as the quantity of magnetic induction at that point.
The whole number
of cells due to a given current
therefore proportional to the strength
is
of the current multiplied by the number of lines of force which pass through
any change of the form of the conductors the number of
be a force tending to produce that change, so that there
is
transverse to the lines of magnetic force, so as to cause
cells
If by
it.
can be increased, there will
always a force urging a conductor
more
lines of force to pass
through
the closed circuit of which the conductor forms a part.
The number
of cells due to two given currents
respectively.
sum
of
its
own
Now
lines
•
Vol. X.
is
got
by multiplying
the
number of
magnetic action which pass through each by the quantity of the currents
lines of inductive
by
(9) the
number of
lines
which pass through the
first
current
and those of the second current which would pass through the
Erp. Res. (3122).
Pakt
I.
See Art.
(fi)
of this paper.
+
Art. (13).
is
the
first if
the
Mr maxwell, ON FARADAY'S LINES OF FORCE.
50
Hence the whole number of
second current alone were in action.
by any motion which causes more
durino- the motion will be measured
by the number of new
On
All the actions
produced.
cells
may be deduced from
each other
conductors on
of closed
pass through either circuit, and therefore
tend to produce such a motion, and the work done by this force
force will
the resultant
lines of force to
be increased
cells will
this principle.
Electric Currents produced by Induction.
Faraday has shewn * that when a conductor moves transversely to the lines of magnetic
If the
force, an electro-motive force arises in the conductor, tending to produce a current in it.
conductor
is
closed, there
is
a continuous current, if open, tension
conductor move transversely to the lines of magnetic induction, then,
which pass through
circuit will
be
in
If a closed
the result.
is
if tlie
number
of lines
does not change during the motion, the electro-motive forces
it
equilibrium, and there will be no current.
depend on the number of
lines
Hence
in
the
the electro-motive forces
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
the
number of
and
lines,
When
the additional lines.
circuit
the
is
That
is
tlie
increase of
generate a current the reverse of that which would have produced
number of
the
lines of inductive
increased, the induced current will tend to diminish the
number
that, in
will
measured by
magnetic action through the
number
of the lines, and
when
diminished the induced current will tend to increase them.
this is the true expression for the
whatever way the number of
be increased, the electro-motive effect
of the conductor
itself,
lines
is
law of induced currents
is
shewn from the
fact
of magnetic induction passing through the circuit
the same, whether the increase take place
by the motion
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
by the change of intensity of the current itself.
In all these cases the electro-motive force depends on the change
lastly
in the
number of
lines of
inductive magnetic action which pass through the circuit f
•
Erp. Res. (3077), &c.
+ The electro-magnetic
forces,
of the material conductor,
which tend
must be
to
produce motion
carefully distinguished
from the electro-motive forces, which tend
to
produce
electric
currents.
Let an electric current be passed through a mass of metal
The distribution of the currents within the metal
In this case we have electro-magnetic forces acting on the
material conductor, without any electro-motive forces tending
to
modify the current which
it
carries.
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,
An
electro-motive force
of any form.
or by changes in the magnetic field.
Now let a
by the laws of conduction.
constant electric current 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
will act in the direction of the conductor, and, as
will
be determined
electric tension at
it
cannot pro-
no circuit, it will produce
the extremities. There will be no electro-
duce a current, because there
is
the material conductors are attracted, the currents (which are
any course within the metal) will not alter their
magnetic attractionon the material conductor, for this attraction
depends on the existence of the current within it, and this is
prevented by the circuit not being closed.
Here then we have the opposite case of an eleclro-motive
original distributibn, or incline towards each other.
force acting on the electricity in the conductor, but
free to choose
For, since
no change lakes place in the system, there will be no electromotive forces to modify the original distribution of currents.
on
its
material particles.
no attraction
Mr MAXVVELI., on FARADAY'S
It
51
natural to suppose that a force of this kind, which depends on a change in
is
number of
A
OF FORCE.
I,INES
lines, is
due
change of state which
to a
may
closed conductor in a magnetic field
As long
the magnetic action.
measured by the number of these
is
be supposed to be
state arising
from
effect takes place, but,
when
in a certain
unchanged no
as this state remains
tlie
lines.
the state changes, electro-motive forces arise, depending as to their intensity and direction on
change of
this
state.
cannot do better here than quote a passage from the
I
first
series of
Faraday's Experimental Researches, Art. (60).
"While
to
the wire
be in a peculiar
common
in its
is
subject to either volta-electric or magneto-electric induction
state, for
formation of an electrical current in
resists the
it
condition, such a current
would be produced
and when
;
it
;
appears
it
whereas,
uninfluenced
left
it
if
has
the power of originating a current, a power which the wire does not possess under ordinary
This electrical condition of matter has not hitherto been recognised, but
circumstances.
many
probably exerts a very important influence in
by currents of
advising with
Finding that
tonic
state,
all
the
this
learned friends, ventured
new
immediately appear (71)
will
to
his second
still
series
rejected
designate
as
it
as the
I
have, after
electro-tonic
state."
reference to the electro-
not necessary
may be some
think that there
to
it
it
not most of the phenomena produced
phenomena could be otherwise explained without
Faraday in
researches* he seems
about
For reasons which
electricity.
several
if
;
but
in
his
recent
physical truth in his conjecture
state of bodies.
Tlie conjecture of a philosopher so familiar with nature
may sometimes be more pregnant
with truth than the best establisiied experimental law discovered by empirical inquirers, and
though not bound
to
admit
it
as a physical truth,
our mathematical conceptions may be rendered
In
this outline of
view, I can do no
electrical
phenomena can be
and not
as
best
comprehended and reduced
my mind
clearly explained without reference to
operations.
By
fluids, I
state
hope
adapted
use
symbols
new idea by which
to calculation,
The
freely,
in
my
ideas,
aim has
lines or
nor readily adapt
idea of the electro-tonic state, however, has
such a form that
take for
to
and
embodied form, as systems of
its
nature and properties
mere symbols, and therefore
and
I believe that
granted
I
the
propose
may
be
in
the following
ordinary
mathematical
a careful study of the laws of elastic solids and of the motions of viscous
to discover a
to general
method of forming a mechanical conception of
reasoning
Part
or
in an
mere symbols, which neither convey the same
not yet presented itself to
to
as a
mathematical methods by which
state the
themselves to the phenomena to be explained.
investigation
it
Faraday's electrical theories, as they appear from a mathematical point of
more than simply
been to present the mathematical ideas to the mind
surfaces,
we may accept
clearer.
II.
this electro-tonic
•}-.
On Faraday's "Electro-tonic State."
When a conductor moves in tlie neighbourhood of a current of electricity, or of a magnet,
when a current or magnet near the conductor is moved, or altered in intensity, then a force
" (3172) (3269).
tion
of Electric, Magnetic and Galvanic Forces.
Camb. and
I
t See
Prof.
W. Thomson On
a Mechanical Representa-
\
Dub. Math. Jour.
Jan. 1847.
7—2
Mr maxwell, ON FARADAY'S LINES OF FORCE.
52
the conductor and produces electric tension, or a continuous current, according as the
acts on
circuit
This current
open or closed.
is
is
produced only by changes of the
electric or
phenomena surrounding the conductor, and as long as these are constant there
effect
on
conductor.
the
conductor
the
Still
magnet, and when away from
its
is
in
influence, since the
different
states
is
when near
magnetic
no observed
a current or
removal or destruction of the current or
magnet occasions a current, which would not have existed
if
magnet or current had not
the
been previously in action.
Professor Faraday to connect with his discovery of the
Considerations of this kind led
all
bodies are thrown by the
itself
by any known phenomena
in this state is indicated
by a current or tendency
induction of electric currents, the conception of a state into which
presence of magnets and currents.
as long as
it is
This state does not manifest
undisturbed, but any change
towards a current.
To
he gave the name of the " Electro-tonic State," and although
this state
he afterwards succeeded
explaining the phenomena which
in
suggested
by means of
it
hypothetical conceptions, he has on several occasions hinted at the probability
nomena might be discovered which would
Faraday had been
he has well established, and which he abandoned only for
direct proof of the
Perhaps
investigation.
phenomena
unknown
are not
state,
by the study of laws which
led
want of experimental data
for the
have not, I think, been made the subject of mathematical
may be thought
it
that the quantitative determinations of the various
rigorous to be
sufficiently
some phe-
render the electro-tonic state an object of legitimate
Tliese speculations, into which
induction.
tliat
less
made
the
of a mathematical theory
basis
Faraday, however, has not contented himself with simply stating the numerical results of his
experiments and leaving the law to be discovered by calculation.
he has at once stated
it,
in
is
by experiment, he has merely
assisted
;
and
if
the
other laws capable of being
it
own
the physicist in arranging his
ideas,
which
confessedly a necessary step in scientific induction.
In the following investigation, therefore, the laws established by
as
he has perceived a law
terms as unambiguous as those of pure mathematics
mathematician, receiving this as a physical truth, deduces from
tested
Where
true,
and
it
be shewn
will
that
laws can be deduced from them.
If
to include one set of phenomena,
may
class,
these
by following out
it
;
will
be assumed
and more general
should then appear that these laws, originally devised
be generalized so as to extend to phenomena of a different
mathematical connexions
physical connexions
Faraday
his speculations other
may
suggest to physicists the means
and thus mere speculation may be turned
to
of
establishing
account in experimental
science.
On Quantity and
It
is
found that certain
different
sections of the
different
the material
eifect
effects of
The
circuit they are estimated.
Intensity as Properties of Electric
same
an
electric
current are equal at whatever part of the
quantities of water or of any other electrolyte
circuit,
are always found to be
and form of the circuit may be
of a conducting wire
is
Currents.
at
decomposed
at
two
equal or equivalent, however
the two sections.
The magnetic
also found to be independent of the form or material of the wire
Mr IMAXWELL, ON FARADAY'S LINES OF FORCE.
same
in the
circuit.
There
therefore an electrical effect which
is
equal at every section of the
is
If we conceive of the conductor as the channel aloncj which a fluid
circuit.
move, then the quantity of
53
is
constrained to
transmitted by each section will be the same, and we
fluid
may
define the quantity of an electric current to be the quantity of electricity which passes across
a complete section of the current in
We
unit of time.
of electricity by the quantity of water which
In order to express mathematically the
may
for the present
would decompose
it
electrical currents in
in unit
measure quantity
of time.
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
when uniform, by
is
measured,
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
across this area, supposing the flow uniformly the
same
would flow
as at the given point.
In the following investigation, the quantity of electric current at the point {nyz) estimated
in the directions of the axes
The quantity of
.r,
electricity
y,
x respectively
which flows
= dS
will be
in imit of
{la.,
+
mb.2
where Imn are the direction-cosines of the normal
to
+
denoted by
External
These may be
electro-motive
magnets, or from changes
forces
5^
c-i-
time through the elementary area
dS
ncS),
dS.
This flow of electricity at any point of a conductor
which act at that point.
a.^
due
is
to the electro-motive forces
either external or internal.
either
arise
in their intensity, or
from the relative motion
from other causes acting
of currents
and
at a distance.
Internal electro-motive forces arise principally from difference of electric tension at points of
the conductor in the immediate neighbourhood of the point in question.
The
other causes are
variations of cnemical composition or of temperature in contiguous parts of the conductor.
all
Let P2 represent the electric tension at any point, and J^2 J'a -^2 the sums of the parts of
the electro-motive forces arising from other causes resolved parallel to the co-ordinate axes,
then
if
a.j
fi, y-i
be the effective electro-motive forces
02
= ^2 -
dp„
-r-"
do!
(A)
7, = Z2
Now
If the resistance of the
Qj
if
dz
the quantity of the current depends on the electro-motive force and on the resistance
of the medium.
but
-
=
AJjOo,
/3.,
medium be uniform
=
h.hi,
72 =
in all directions
hfii,
These quantities
tlie
a> /Bo 72
directions of wyz.
may be
to
k.^,
(B)
the resistance be difl^erent in different directions, the law will be
action in
and equal
more complicated.
considered as representing the intensity of the electric
Mr maxwell, OX FARADAY'S LINES OF FORCE.
54
The
measured along an element
intensity
e
where Imn
=
+
la
a curve
dcr of
m[i
+ ny,
are the direction-cosines of the tanjjent.
The integral
feda taken with respect to a given portion of a curve
intensity along that line.
If the curve
a closed one,
is
line,
represents the total
represents the total intensity of the
it
electro-motive force in the closed curve.
Substituting the values of 0/87 from equations (A)
fedff
If,
therefore (J^dx
will vanish,
and
+ Ydy + Zdz)
in all closed
= f(Xdx + Ydy +
-p +
C.
a complete differential, the value of
is
jetZcr
for a closed curve
curves
/erfo-
the integration
Zdz)
= f{Xdx + Tdy + Zdz),
being effected along the curve, so that
in a closed
of the effective electro-motive force is equal to the total intensity
curve the
total
intensity
of the impressed electro-
motive force.
The
total quayitity of
conduction through any surface
is
expressed by
fedS,
where
e
=
+ mh +
la
nc,
Imn being the direction-cosines of the normal,
fedS = ffadydz + ffbdzdx +
.*.
the integrations being effected over the given surface.
we may
If
find
ffcd.vdy,
When
the surface
is
a closed one, then
by integration by parts
we make
da
db
dc
dx
dy
dz
fedS
=
4
IT
fjfpdxdydz,
where the integration on the right side of the equation
within the surface.
is
effected over every part
In a large class of phenomena, including
all
of space
cases of uniform currents,
the quantity p disappears.
Magnetic Quantity and Intensity.
From
in the
his study of the lines of magnetic force,
tubular surface* formed by a system of such
across any section of the tube
in
passing from one
is
constant,
and that the
substance to another,
capacity in the two substances, which
electric
Faraday has been led
is
is
to
lines,
to the conclusion that
the quantity of magnetic induction
alteration of the character of these lines
be explained by a difference of inductive
analogous to conductive
currents.
Exp. Res. 3271, de6nition of " Sphondyloid.'
power
in
the theory of
Mr maxwell, ON FARADAY'S LINES OF FORCE.
In
following investigation we shall have occasion to treat of magnetic quantity and
tlie
intensity in connexion with electric.
by the
and
suffix 1,
In such cases the magnetic symbols will be distinguished
by the
the electric
suffix 2.
The
equations connecting a,
p, and p, are the same in form as those which we have just given,
a, b, c
k, denotes the resistance to
magnetic induction with respect to quantity
;
and may be
a, j8, "y, are the effective
nected
different in different directions;
a,
witli
h,
by equations (B)
c,
afterwards explained
by equations
a, b, c
all
;
p,
is
(C).
As
all
b, c,
k, a,
y3,
y,
are the symbols of
magnetic induction,
magnetizing forces, con-
the magnetic tension or potential which will be
p denotes the density of real magnetic matter and
;
the details of magnetic calculations
after the exposition of the connexion of
say that
55
magnetism with
will
connected with
be more intelligible
be
electricity, it will
is
sufficient here to
the definitions of total quantity, with respect to a surface, and total intensity with
respect to a curve, apply to the case of magnetism as well as to that of electricity.
Electro-magnetism.
Ampere has proved
following laws
the
of the
and
attractions
repulsions
of
electric
currents
Equal and opposite currents generate equal and opposite
I.
A
II.
crooked current
is
forces.
equivalent to a straight one, provided the two currents nearly
coincide throughout their whole length.
Equal currents traversing
III.
and similarly situated closed curves act with
similar
equal forces, whatever be the linear dimensions of the circuits.
IV.
A
closed
to
be observed, that the currents with which Ampere worked were constant and
exerts
current
turn a circular conductor about
no force tending to
its centre.
It
is
therefore
currents,
All
re-entering.
and
assumption
results
are
therefore deduced
from
experiments on
this action is
exerted in the direction of the line joining those elements.
no doubt warranted by the universal consent of men of science
is
attractive forces
considered
as
due
to the
mutual
action
of particles;
in treating of
phenomena,
the currents alone, but also in the surrounding medium.
The
The
as
first
and second laws shew that currents are to be combined like velocities or
tliird
law
is
the expression of a property of
all
attractions which
may be
shews that the electro-magnetic forces
may always be reduced
forces.
conceived of
depending on the inverse square of the distance from a fixed system of points
fourth
This
but at present we
are proceeding on a different principle, and searching for the explanation of the
not in
closed
expressions for the mutual action of the elements of a current involve the
his
assumption that
his
;
and the
to the attractions
and
repulsions of imaginary matter properly distributed.
In
fact,
identical
given
with
the action
of a very small electric circuit
that of a small
on a point
magnetic element on a point outside
in its
it.
neighbourhood
If
is
we divide any
portion of a surface into elementary areas, and cause equal currents to flow in the
same direction round
all
these little areas, the effect on a point not in
the surface will be the
Mr maxwell, ON FARADAY'S LINES OF FORCE.
56
same as that of a
surface.
shell
But by the
coinciding with the surface, and uniformly magnetized normal to
first
law
the currents forming the
all
little circuits
another, and leave a single current running round the bounding
effect
of a uniformly magnetized shell
edge of the
the sun,
shell.
line.
its
destroy one
will
So that the magnetic
equivalent to that of an electric current round the
is
If the direction of the current coincide witli that of the apparent motion of
the imaginary shell will be
magnetization of
then the direction of
the
same
as
that of the real magnetization of the earth*.
The
total intensity of
the closed current
current.
As
is
magnetizing force
may
constant, and
this intensity is
in a closed
therefore be
made a measure of
the quantity of the
independent of the form of the closed curve and depends only on
the quantity of the current which passes through
tlie
curve passing through and embracing
it,
we may consider the elementary case of
current which flows through the elementary area dydx.
Let the axis of x point towards the west,
be the position of a point
round the four
quantity
we
Let wys
sides of the element is
Total intensity
The
towards the south, and y upwards.
the middle of the area dydz, then the total intensity measured
in
+
therefore if
sr
=
7>.
dy
(j^- -
of electricity conducted
2 J
-jAdy ds.
through the elementary area dydx
is
adydz, and
define the measure of an electric current to be the total intensity of magnetizing
force in a closed curve
embracing
These equations enable us
we know the values of
a, /3,
it,
we
shall
have
dz
dy
dy.
dai
d.v
dz
da,
rfj3.
dy
d.v
deduce the distribution of the currents of
to
y, the magnetic intensities.
a function of xyz with respect to
• See Experimental Researches (3265)
embracing curves.
.r,
If a,
j8,
y
electricity
be exact differentials of
y and z respectively, then the values of as 62
for the relations
between the
electrical
whenever
and magnetic
circuit,
c>
disappear
considered as mutually
Mr maxwell, OX FARADAY'S
and we know that the magnetism
which we are investigating.
OF FORCE.
57
not produced by electric currents in that part of the
is
due either
It is
LLNTES
to the presence of
field
permanent magnetism within
the field, or to magnetizing forces due to external causes.
We
which
may
is
observe that the above equations give by differentiation
da^
db,
dc.>
o.r
ai/
d~
Our investigations are therefore for
we know little of the magnetic effects of any
the equation of continuity for closed currents.
the present limited to closed currents; and in fact
currents which are not closed.
Before entering on the
advantageous
to
state
calculation
certain
of
general
these
electric
theorems,
and
truth
tlie
magnetic
states
may be
of which
may be
it
established
analytically.
Theorem
The
equation
d'V
V and p
(where
are functions of
distance,) can be satisfied
by
The value
of
d-V
<PV
xyx never
infinite,
Theorem
F which
will satisfy the
—
Qu
x'
\'
+
y —
is
found by integrating the expression
y'
\'
+
s
—
z' r)'
to include every point of space where p
may be found
proofs of these theorems
in
electricity,
and
See Arts.
to Electricity,
See also Gauss, on Attractions, translated in Taylor's Scientific Memoirs.
Theorem
U and V
is finite.
any work on attractions or
Green's Essay on the Application of Mathematics
19 of this Paper.
Let
See Art. (17) above.
pdrdydss
where the limits of xyz are such as
18,
for all points at an infinite
II.
above conditions
rrr
''''•'
in particular in
and vanishing
one, and only one, value of V.
^
The
I.
be two functions of
.ryz,
III.
then
dUdV dUdV\,
JJj^ [d^'-'df^ d^r^^''' =-JjJ[-d^-d:v^^d^^ Tzi^r^y'^''
rr Trr
I d~V
cPV
rf
r\
,
,
,
rrr/dUdV
where the integrations are supposed to extend over
differing
from
(Green, p.
This theorem shews that
equal and opposite.
all
the space in which
Part
if there
And by making
I.
,
U and V have values
10.)
be two attracting systems the actions between them are
^7"=
F we
find that the potential of a
proportional to the integral of the square of the resultant attraction
Vol. X.
,
system on
through
all
itself is
space
8
;
a
Mr maxwell, OX FARADAY'S LINES OF FORCE.
58
result deducible
from Art.
(Arts. 12,
the velocity
!.•?),
(.'50),
volume of each
since the
number of
and therefore the
cell
is
in
cells
inversely as the square of
a given
space
is
directly
as the square of the velocity.
Theorem
Let
and
a, /3,
7, p be quantities
k be wiven
let
then the equation in
for
through a certain space and vanishing
in the
space beyond,
parts of space as a continuous or discontinuous function of .vyz,
p
d
and only
has one,
all
finite
IV.
d l/_
dp\
1/
one solution, in
dp\
which
is
p
d
dp\
If
always
finite
and vanishes
at
an
infinite
distance.
The
Dublin Math.
If
W. Thomson, may
proof of this theorem, by Prof.
be found
in the
Cambridge and
Journal, Jan. ISiS.
a/37 be
ance, then the above equation
and the theorem
p
the electro-motive forces,
is
the electric tension, and
k the
coefficient of resist-
identical with the equation of continuity
da^
db,
dc-i
ax
dy
dz
'
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 electricity, as far as
we now consider them, we may regard 0/^7 as magnetizing forces, p as mafftietic tension, and p
as real magnetic density, k being the coefficient of resistance to magnetic induction.
The
where
proof of this theorem rests on the determination of the
V is
and p has
minimum
got from the equation
to
d'V
d'V
d-7
dor
dy^
dz-
be determined.
The meaning of
this integral in electrical
sence of the media in which k
"quantity" resolved
in
language
may be
thus brought out.
If the pre-
has various values did not affect the distribution of forces, then the
x would be simply
dV
—
dV
and the intensity
-] and a -
and intensity are -fa
—
,
and the parts due
alone are therefore
1 ,
dp\
<iP\
-(a -3dx)
k
dp
dV
dV__^_
--—
anda--T^
dx
k—-
But
the actual quan-
cue
ufi*'
tity
value of
ax
-
dV
k-—.
,
ax
to
the distribution of media
Mn MAXWELL, ON FARADAY'S LINES OF FORCE.
Now
the product
of these represents the
work done on account of
the distribution of sources being determined, and talcing in the
media,
Q
we get the expression
point which
is
due
done by
for the total worlc
to the distribution of
tliat
59
tliis
distribution of
terms
in
y and z
part of the whole effect at any
conducting media, and not directly to the presence
of the sources.
This quantity
which
satisfies
Q
is
minimum by one and
rendered a
Theorem
If
it is
a, b, c
only one value of p, namely, that
the original equation.
be three functions of .r,
V.
y, s satisfying the
da
db
dc
ax
dy
dz
a, /3,
7
always possible to find three functions
equation
wliicli shall satisfy
the equations
d^_dy_
^
Let
of
y,
A
dz
dy
dy
da
dx
dz
da
dd
dy
dx
=fcdy, where the integration
treating
x and «
B = fadz, C = Jhdx,
Let
as constants.
performed upon c considered as a function
to be
is
= jbdz,
A'
i?'
= fcdx,
C
integrated in the same way.
Then
a^A-A'^f,
ax
dy
dz
will satisfy the given equations
d/3
dy
dz
dy
;
=
for
rda
I
— dz
,
rdc
—
-
r
,
dx -
J dz
J dy
•>
db
—
dx
dy
rda
— dy,
+
J
dy
and
=
rda
/
J
dji
dy
dz
dy
—
=
— dx + J
dx
— dx +
r(^^
,
dy
— dx + Jrda
dy
dy
rda
/
J dx
=
,
,
~y~
-^
r^c
—
rda
+
,
dx;
J dz
,
-—dz
J dz
a.
8—2
= fady,
Mr maxwell, ON FARADAY'S LINES OF FORCE.
60
In the same way
it
may be
sliewn that the values of a,
The function \|/ may be considered at present
The method here given is taken from
/3,
7
satisfy the other given equations.
as perfectly indeterminate.
W.
Prof.
Thomson's memoir on Magnetism
(Phil. Trans. 1851, p. 283).
As we
cannot perform the required integrations when
r, y, z, the following
more
method, which
perfectly general
C
discontinuous functions of
be determined from the equations
by the methods of Theorems
I.
drA
d'A
d'A
dx'
ay
dz-
d'B
d-B
d'B
dx'
dy''
d%^
d'C
d^C
d'C
dx^
dy^
dz^
and
II., so that
A, B,
C are
is infinite.
Also
let
_dB^_dC
dz
then
a, b, c are
though more complicated, may indicate
clearly the truth of the proposition.
Let A, B,
or 2
is
dy
dj,
dx
never
infinite,
and vanish when
.r,
y,
Mr maxwell, ON FARADAY'S LINES OF FORCE.
The
function
\|/
is
to
be determined from the condition
da
dfi
dx'^ dy
if
"^
dy _ /d^
ds ~ [dx'
Theorem
a, b, c
be any three functions of
>f,
y, z,
d^\
d'=
^
"*"
the left-hand side of this equation be always zero,
Let
x^^
"*"
d^'J
it is
possible to find three functions a,
dB
dy
ans
dy
dx
a=-dd
dy
dV
,_dy
da
dV
dx
dx
dy
c
Let
and
let
V be
found from the equation
then
satisfy the condition
-
also.
VI.
da
d%
^
must be zero
a fourth V, so that
and
61
^
r +
dy
dx
'
j3,
7 and
Mb maxwell, ON FARADAY'S LINES OF FORCE.
62
where ai6,c„
aifti 7, are
any functions whatsoever,
Q
in
= +
-
ffJl^Trppi
+
(ao«2
capable of transformation into
+
fioh
•y^c.i)}dMydx,
which the quantities are found from the equations
au^o7o^' are determined from
da,
db,
dc,
dx
dy
dz
da,
rf/3,
d.v
dy
by the
a, &, Cj
^7i
^
found from
62 C2 are
a,
71 by
/3i
last
is
J- &c.,
dy
found from the equation
For,
we put
if
a, in the
d^p
d-p
rf'p
da?*
%•
dz'
and
treat h,
bering that
dx
dy
we have by
integration
by parts through
infinity,
remem-
the functions vanish at the limits,
all
Q=-
similarly, then
Cj
'
'
form
dz
and
dw
equations
tlie
dz
p
„
/
theorem, so that
dy
a,=
and
,
ax;
ds
a-j
is
KS ^ I' ^ '^) Q= +
or
""
(f - '^) ^ ^' {^ - '^) ^ - (^ - ^) I-*-.
/// |(47r Fp')
-
(a^a.
+
jSo&2
+
7oC;i) } dcfdyd;?,
and by Theorem III.
ffjVp'dxdydz
=
JJfppdxdydz,
-
+
^A =
so that finally
Q=
If Oi
a.,
functions
(ao«2
7t>f'0
i
dxdydx.
represent the components of magnetic quantity, and a,
6, Ci
intensity, then
tension,
fffl'iTrpp
h,
p
c.-,
will represent
will
the real magnetic density,
be the components of quantity of
deduced from a,h,c,, which
Faraday's Electro-tonic
will
electric
/3i
71 those of magnetic
and p the magnetic potential or
currents, and ao/3o 7o ^^^^ be three
be found to be the mathematical expression for
state.
Let us now consider the bearing of these analytical theorems on the theory of magnetism.
Whenever we
suffix
(
,
).
deal with quantities relating to magnetism,
Thus
a, 61 c,
are the
components resolved
in
we
shall distinguish
them by the
the directions of x, y, z of the
Mr maxwell, ON FARADAY'S LINES OF FORCE.
63
quantity of magnetic induction acting tlirough a given point, and aSiJi are the resolved intenof magnetization at the same point, or, what
sities
is
tlie
same
tiling, tlie
components of the
which would be exerted on a unit south pole of a magnet placed at that point without
force
disturbing the distribution of magnetism.
The
found from the magnetic intensities by the equations
electric currents are
Oo
—
=
When
&c.
dy
dz
there are no electric currents, then
+ y^dz =
a^dx + ^^dy
a perfect differential of a function of
On
y, z.
.?,
f/;;„
the principle of analogy
we may
call p^ the
magnetic tension.
The
forces which act on a mass
m
of south magnetism at any point are
dpi
— m-—
,
m
—
is
,
and — m
dp^
,
dz
and therefore the whole work done during any displacement of a
in the direction of the axes,
magnetic system
—
dp^
dy
d.v
equal to the decrement of the integral
Q=
UlpiPxdxdydz
throughout the system.
Let us now
of
Q
and
will
call
Q
the total potential of the system on itself.
measure the work
lost or
will therefore enable us to
By Theorem
III.
Q may
which
a,
/3i
or
decrease
be put under the form
—
ffj(C\(^\
+
^ifti
+
Ci7i)rf,rrfyd«,
7, are the differential coefficients of pi with respect to
we now assume
If
increase
determine the forces acting on that part of the system.
Q= +
in
The
gained by any displacement of any part of the system,
that this expression for
Q
is
,r,
y,
« respectively.
true whatever be the values of oi
/3i
71,
we
pass from the consideration of the magnetism of permanent magnets to that of the magnetic
effects
of electric currents, and
^^Ifj
So that
1^'^'
~
^ ["""-
in the case of electric currents, the
by the functions
a^fi^yo respectively,
system gives us the part of
We
we have then by Theorem VII.
have now obtained
Q
due
"*"
^°^^
"''
^"^''j r'^'^^y'^^
components of the currents have
and the summations of
all
in the functions a^figyf, the
such products throughout the
means of avoiding the consideration of
Instead of this
method we have the natural one of considering the current with reference
same space with the current
be multiplied
to those currents.
the quantity of magnetic induction which passes through the circuit.
in the
to
itself.
To
or components of the Electro-tonic intensity.
these I give the
name of
artificial
to quantities existing
Electro-tonic functions,
Mr maxwell, ON FARADAY'S LINES OF FORCE.
64
Let us now consider the conditions of the conduction of the electric currents within the
medium diirin<f changes in the electro-tonic state. The method which we shall adopt is an
by Helmholtz
application of that given
mass currents whose quantity
Then
the
amount of work due
measured by a^
is
+
and their intensity by
+
b.fi.^
a.> fi-z y...
is
cry.i)dxdydz
form of resistance overcome, and
dt d
47r dt
in
b>c.,
which would generate in the con-
to this cause in the time dt
dtjjj{a.fii
in the
Conservation of Force*.
tlie
source of electric currents
Let there be some external
ductintr
memoir on
in his
the form of
fjfia^ao
+ ^So + cr/^dxdydz
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
fjf{aMi+
+
6./3.
c.,y.2)d.vdi/dz+
——
where the integrals are taken through any arbitrary space.
a.,a.2
+
b,l3-,
+
C,y.,
=
and
We
^^^ ^°^ ^^ a.b^c^.
variations of Oo/SoYo'
We
must
c,yo)d,vdydz,
tlierefore
have
^'sVo)
at
must be remembered
it
Mo +
— - («.ao + ^So +
47r
for every point of space;
+
///(«2ao
Q
that the variation of
must therefore
is
supposed due to
treat a.^b.^c, as constants,
and
the equation becomes
a, a.>+
•V'
^1
47r
forces
must
due
=
;
A
/3.,
~]
+
^TT
+c.,
7.
+
'V-
dt)
may be independent
In order that this equation
efficients
+6,
dt)
/^ =0dt
4Tr
of the values of
a.,
)
h,
and therefore we have the following expressions
to the action of
magnets and currents
c-,,
for
each of these cothe electro-motive
at a distance in terms of the
electro-tonic
functions,
""*
It
^"
'
47r
dt
'^'
'
47r
dt
appears from experiment that the expression
47r
—° refers to
dt
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 Oj, be expressed as a function of
article,
,r, j/,
z,
and
t,
and
if
,?',
y,
-z
be the co-ordinates of a moving
then the electro-motive force measured in the direction of
Idar,
dx
da^ dy
dua dz
da^\
47r \da;
dt
dy
dt
dz dt
dt
New
Scientific
1
"'~
x
* Translated in Taylor's
Memoirs, Part
is
J
II.
Mr maxwell, ON FARADAY'S LINES OF FORCE.
The
65
The
expressions for the electro-motive forces in y and % are similar.
distribution of
currents due to these forces depends on the form and arrangement of the conducting media
and on the resultant
The
paper
tical
discussion of these functions would involve us in mathematical formulae, of which this
already too
is
It is only on
full.
By
their present form.
help of those
who
which
a more patient consideration of their relations, and with the
are engaged in physical inquiries both in
obviously connected with
in
account of their physical importance as the mathema-
expression of one of Faraday's conjectures that I have been induced to exhibit them at
in
all
any point.
electric tension at
all its relations
it,
I
hope
may be
subject and in others not
this
a form
to exhibit the theory of the electro-tonic state in
distinctly conceived without
reference
to analytical
calcula-
tions.
Summary of
We
in
may
the
Theory of the Electro-tonic State.
conceive of the electro-tonic state at any point of space as a quantity determinate
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
of
state.
artificial notation.
electro-tonic state,
This representation involves no physical theory,
In analytical investigations
and
call
them
we make use of the
electro-tonic functions.
electro-tonic intensity at every point of a closed curve,
call
and
tliree
it is
only a kind
components of the
We
take the resolved part of the
find
by integration what we may
the entire electro-tonic intensity round the curve.
Prop.
divided
I.
itifo
If on any surface a closed curve he drawn, and if the surface within it be
small areas, then the entire intensity round the closed ctirve 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
along twice
Every
effect
in opposite directions,
effect
is
Law
that
I.
and the intensity gained
of passing along the interior divisions
due
The
to the exterior closed
is
in the
one case
is lost in
Prop.
I. it
entire electro-tonic intensify
appears that what
is
therefore neutralized, and the whole
round the boundary of an element of surface
true of elementary surfaces
finite magnitude, and therefore any two surfaces which are
will
passed
curve.
measures the quantity of magnetic induction which passes through that surface,
words, the number of lines of magnetic force which pass through that surface.
By
is
the other.
is
or,
in other
true also of surfaces of
bounded by the same closed curve
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 conduction*.
• See Art. (28
Vol. X.
Part
I.
mk maxwell, on faradays lines of force.
66
Law
The
III.
entire
magnetic
inteyisifii
round the boundary of any surface measures
the quantity of electric current which passes through that surface.
Law
The quantity and
IV.
intensity of electric currents are connected by a system of
equations of conduction.
By
the values of the electro-tonic functions.
I
be deduced from
have not discussed the values of the units, as that
We
be better done with reference to actual experiments.
will
may
these four laws the magnetic and electric quantity and intensity
come next
to the attraction of
conductors of currents, and to the induction of currents within conductors.
Law
The
V.
total electro-magnetic potential
of a closed current
is
measured by the product
of the quaiitity of the current multiplied by the entire electro-tonic intensity estimated in the
same
direction
circuit.
displacement of the conductors which would cause an increase in the potential will be
Any
by
assisted
round the
force
a
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
current might
in
certain cases a displacement in direction or alteration
increase the potential, such an
alteration
would not
itself
of intensity
of the
produce work, and
due
there will be no tendency towards this displacement, for alterations in the current are
to
electro-motive force, not to electro-magnetic attractions, which can only act on the conductor.
Law
The
VI.
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 dirertio7i.
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
tiiese six
laws
in the external circumstances.
have endeavoured to express the idea which
I
believe to be the mathe-
I
matical foundation of the modes of thought indicated in the E.rperimental Researches.
not think that
it
contains even the shadow of a true physical theory; in fact,
temporary instrument of research
There
exists
so mathematical,
is
that
it
however a professedly physical theory of electro-dynamics, wiiich
and
so entirely different
Weber's Electro-dynamic Measurements,
from anything in
this paper, that I
and may be found
Leibnitz Society, and of the Royal Society of Sciences of Saxony
(1)
That two
as
when
particles of electricity
at
rest,
do
does not, even in appearance, account for anything.
axioms, at the risk of repeating what ought to be well-known.
same force
its
I
chief merit as a
when
but that the force
is
in
It
in
*.
is
the
The
is
so elegant,
must
state its
contained in
M. W.
Transactions of the
assumptions are,
motion do not repel each other with the
altered
by a quantity depending on the
relative motion of the two particles, so that the expression for the repulsion at distance r
• When this was written. I was not aware that part of "SI.
Weber's .'Memoir is translated in Taylor's Scientific Memoirs,
\'o\. V. Art. XIV. The value of his researches, both experimen-
1
I
I
tal anil theoretical,
every electrician.
is
renders the study of his theory necessary 10
Mn MAXWELL, ON FARADAY'S LINES OF FORCE.
dr
ee' /
That when
(2)
electricity is
moving
of the conductor
relatively to the matter
is
2
d-r\
,
in a conductor, the velocity of the positive fluid
equal and opposite to that of the negative
Tiie total action of one conducting element on another
(3)
mutual actions of the masses of
electricity of
is
the
fluid.
resultant
of the
both kinds which are in each.
Tiie electro-motive force at any point
(4)
67
is
the difference of the forces acting on the
positive and negative fluids.
of
From these axioms are deducible Ampere's laws of the attraction
Neumann and others, for the induction of currents. Here then is
satisfying the required conditions better perhaps than
of conductors, and those
a really physical theory,
any yet invented, and put forth by a
whose experimental researches form an ample foundation for his mathematical
philosopher
What
investigations.
is
the use then of imagining an electro-tonic state of which
distinctly physical conception, instead of a formula of attraction which
stand
would answer, that
I
.''
it is
a
good thing
to admit that there ai-e two ways of looking at
to
it.
we can
have two ways of looking
Besides,
I
we have no
readily under-
at a subject,
and
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
theory when
is,
that
guide experiment, without impeding the progress of the true
There are
appears.
it
shall
it
also objections to
making any ultimate
on the velocity of the bodies between wiiich they act.
depend
forces in nature
If the forces in nature are to
be reduced to forces acting between particles, the principle of the Conservation of Force
re-
quires that these forces should be in the line joining the particles and functions of the distance
The
only.
experiments of
recently repeated
M. Weber's
With
M. Weber on
by Professor Tyndall,
the reverse polarity of diamagnetics, which have been
establish a fact
which
is
equally a consequence of
theory of electricity and of the theory of lines of force.
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,
aware, original
arose in
;
my mind
from the perusal of Prof.
of Electric,
tation
Part
I. IS.'i],
I
I
"On
a Mechanical Represen-
may
As an instance of the help which may be derived from other
state
that
after I
Diff'raction," Section
paper
to discuss
These are intended merely
till
papers
had investigated the Theorems of
1,
this
paper
Cambridge Transactions, Vol. IX. Part
the theory of these functions, considered with reference to electricity,
rest of this
;
W. Thomson's
Mathematical Theory of Magnetism," Philosophical Transac-
mathematical ideas to be employed
given
am
Stokes pointed out to me the use which he had made of similar expressions in his
"Dynamical Theory of
Whether
his "
Art. 7S, &c.
physical investigations,
Professor
as far as I
Magnetic and Galvanic Forces," Cambridge and Dublin Mathematical
Journal, January, 1847, and
tions,
is,
but the distinct conception of the possibility of the mathematical expressions
in physical research,
remains to be seen.
may
I
lead to
propose
in
i.
new
the
a ievi electrical and magnetic problems with reference to spheres.
as concrete examples of the
methods of which the theory has been
I reserve the detailed investigation of cases chosen with special reference to
have the means of testing their
experiment
results.
9—2
Mr maxwell, ON FARADAY'S LINES OF FORCE.
68
ExAMI
Theory of Electrical Images.
I.
The method
LES.
W. Thomson*,
of Electrical Images, due to Prof.
spherical conductors
simple when we see
has been
its
reduced
to
by which the theory of
great geometrical simplicity, becomes even
connexion with the methods of
We
this paper.
more
have seen that the
pressure at any point in a uniform medium, due to a spherical shell (radius
=
a) giving out
o
fluid at the rate
of A-n-Pa- units in unit of time,
is
kP —
outside the shell, and
kPa
inside
it,
r
where r
is
the distance of the point from the centre of the shell.
If there be two shells, one giving out fluid at a rate 47rPa", and the other absorbing at the
rate 4nrP'a'-, then the expression for the pressure will be, outside the shells,
«
,
a!'
inP —
p = iTrP
; ,
r
r
where r and r are the distances from the centres of the two
to zero
we have,
Now
as the surface of no pressure, that for
/
P'a'-'
r
Pa'
the surface, for which the distances to
of which the centre
O
is
in
Equating
shells.
this expression
which
two fixed points have a given
the line joining the centres of the shells
CC
ratio, is
a sphere
produced, so that
P'd-'f
CC
C'O =
and
its
-=r=
Pa'^-Pa'"]^
radius
If at the centre of this sphere
this source
must be added
to
we place another source of the
that
due
to the other
depends only on the distance from the centre,
where the pressure due to the two other sources
We
it
is
will
two
;
fluid,
and
then the pressure due to
since this additional pressure
be constant
at the surface of the sphere,
zero.
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.
See a series of papers "
ning March, 1848.
On
the Mathematical
Theory of
Klectricity," in the
Cambridge and Dublin Math. Jour., begin-
Mr maxwell, ON FARADAY'S LINES OF FORCE.
69
Let a be the radius of the sphere, and p the given pressure, and let the given sources be
b^ h.^ &c. from the centre, and let their rates of production be iirPi, ^ttP^ &c.
at distances
Then
centre)
—
distances
if at
,
- &c. (measured
we place negative sources whose
the pressure
at
the
=a
surface r
the same direction as 6,62
in
from the
Stc.
rates are
47rPi
will
-
,
-
&c.
r
iirP.
Now
be reduced to zero.
placing a source 47r ~- at
K
the centre, the pressure at the surface will be uniform and equal to p.
The whole amount
of fluid emitted by the
rates of production of the sources within
To
apply
this result to the case of a
The
it.
conducting sphere,
be small electrified bodies, containing
e,
pose that the whole cliarge of the conducting sphere
is
iirP-i, ^tt/'o to
Then
points.
all
that
= a may be found by adding
surface r
let
us suppose the external sources
of positive electricity.
e.2
=
£
Let us
also sup-
previous to the action of the external
required for the complete solution of the problem
is
the
result is
is,
that the surface
of the sphere shall be a surface of equal potential, and that the total charge of the surface shall
he E.
If by any distribution of imaginary sources within the spherical surface
the value of the corresponding potential outside the sphere
potential inside the sphere
We
must therefore
point at
a"
—
and
At
find
really be constant
the
distance b from the
,
,
must
,
.
at that point
the centre
and equal
images of the external
centre
we must
„
.
the true and only one.
electrified points, that
on the same radius
we must place a quantity =—e — of imaginary
,
we must put
effect this,
The
to that at the surface.
find a point
«
.
is
we can
.
is,
for every
at a distance
...
electricity.
a quantity E' such that
E'=E+ei- +e.,~+kc.;
Oj
then if
and
R
be the distance from the centre,
r^r.^
"
bo
&c. the distances from the electrified
points,
r'l/a the distances from their images at any point outside the sphere, the potential at
that point will be
E'
(1
a \\
n
a l\
R
Vi
o^rj
V7-2
62 '•2/
E
e,f a
= o + r p+
R b^\R
b,
a\
r,
--^
rj
e.,1
a
L
+fb.,\R
n\
--+Sjc
rj
6»
+
r^
Mr maxwell, ON FARADAY'S LINES OF FORCE.
70
This
is
At the surface we have
the value of the potential outside the sphere.
R=a
and
63
— = -n
=
-a
&c.
so that at the surface
«=—
and
this
must
also be the value of
p
for
+— +
any point within the sphere.
For the application of the principle of
Thomson's papers
in
V-+&C.
images the reader
electrical
is
referred to Prof.
The
the Cnmhridge and Dublin Mathematical Journal.
which we shall consider
is
-^ =/, and
that in which
6, is infinitely distant
only case
along axis of x,
and £=0.
The
value
p
outside the sphere becomes then
p=Lt
and inside
^ = 0.
force
The
ill
'
the effect of a paramagnetic or diamagnetic sphere in a uniform field of magnetic
On
IL
,3
*.
expression for the potential of a small
the direction of the axis of x
magnet placed
at the origin
of co-ordinates
is
(m\
lm\
X
la
ix \r)
r^
d
'
The
effect
of the sphere in disturbing the lines of force
hypothesis to be similar
be determined.
(We
to
that of a small
magnet
the
at
may be supposed
origin,
as
whose strength
a
first
is
to
shall find this to be accurately true.)
Let the value of the potential undisturbed by the presence of the sphere be
p=
Let the sphere produce an additional
Ix.
potential, whicli for external points is
—x,
p = A ,.3
and
let
the potential within the sphere be
Pi
Let
k'
be
the coefficient
of
resistance
= Bx.
outside,
and
k
inside
the
sphere,
then
the
conditions to be fulfilled are, that the interior and exterior potential should coincide at the
• See Prof.
Thomson, on the Theory of .Magnetic Induction,
Pkit. A/ap. March, 1851.
Theinductivecapaciti/of thesphere^
according to that paper,
the ratio of the quanlity of magnetic
induction (not the intensity) within the sphere to that without.
It is therefore
is
equal to
~.
B -t- = ^7—r.accordmg to our notation.
Mr maxwell, ON FARADAY'S LINES OF FORCE.
71
and that the induction through the surface should be the same whether deduced
surface,
Putting x = r cos
from the external or the internal potential.
d,
we have
for
the external
potential
p=
and
+
Ir
{
A—
\
cos d,
for the internal
Pi- Br
and these must be identical when r =
a,
cos 9,
or
I + A = B.
The
induction through the surface in the external
and that through the interior surface
medium
is
is
and
.-.
-,{I-2A) = - B.
k
k
These equations give
The
moment
effect
outside
the sphere
is
equal to that of a
Suppose
in
north.
this
uniform
field to
—
is I
and
— orI.
2k + k
be that due to terrestrial magnetism, then,
paramagnetic bodies, the marked end of the equivalent magnet
If
k
is
greater than k' as in diamagnetic bodies, the
magnet would be turned
Magnetic field of variable
first
k
is
less
than
be turned to the
unmarked end of
the equivalent
Intensity.
suppose the intensity in the undisturbed magnetic
direction from one point to another, and that
then, if as a
will
if
to the north.
III.
Now
magnet whose length
ml, provided
ml =
k as
little
its
components
field
to
vary in magnitude and
in xyx: are represented
by
a^fi^yi,
approximation we regard the intensity within the sphere as sensibly equal
that at the centre, the change of potential outside the sphere arising from
to
the presence of
Mr maxwell, ON FARADAY'S LINES OF FORCE.
72
the sphere, disturbing
tlie
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
k-k'
,a^a.
k
k-k'
2k +
The
2k +
distribution
to
k-k'
2k + k'«V
-.a^fi.
k'
actual distribution of potential within and without the sphere
result of a
may be
conceived as the
of imaginary magnetic matter on the surface of the sphere
the external effect of this superficial magnetism
magnets
moments equal
is
;
but since
exactly the same as that of the three small
at the centre, the mechanical effect of external
attractions will be the
same as
if
the
three magnets really existed.
Now let three small magnets whose lengths are L l^, and strengths tm, m., m^ exist at the
point xyz with their axes parallel to the axes oi xyz; then, resolving the forces on the three
?,
magnets
in the direction of
X, we have
da lA
Mr maxwell, ON FARADAY'S LINES OF FORCE.
Two
IV.
Spheres in uniform
Let two spheres of radius a be connected together
and
them be suspended
tance
b,
itself
would have been
let
in equilibrium at
any part of the
produce forces tending to make the balls
p — Ir
then, althouo-h each sphere
cos 9,
k-k' a^
,
is
cos Q.
therefore equal to
/,
,
I
r
+
k—
_.
2k
+
k' a^\
,,
-
cos
^
=
p.
k' r^
— k' a^\
k-k'
k
a^
^ (
= /^( 1 - 2 —
-, cos
dr
\
2k + k ry
dp
1
dp
r d9
„
dp
•'''''=
dr
^(
=
-'{
by
the disturbance of the field will
to the sphere is
,
potential
field,
as origin, then the undisturbed potential
P = I 2k v-'—,
+ k r^
The whole
field,
set in a particular direction.
Let the centre of one of the spheres be taken
and the potential due
Jield.
so that their centres are kept at a dis-
uniform magnetic
in a
73
k -
9,
k'
a^\
dp
2k + k
r^J
dtf)
=
0,
is
Mb maxwell, ON FARADAY'S LINES OF FORCE.
74
The
the middle of the line joining
expression for the potential,
/
we
this
y/c^+r'+2 cos 6cr)
'iM-
I
r-
r^
\
c
\
c'
c-
J
6
is
increased
to
c*
turn a pair of spheres (radius a, distance 26) in the direction in which
is
k-Jc APa'V
+k
2k
This
force,
i
-18—-r- sin 20,
I-T^=
dd
and the moment
:
P,
find as the value of
.-.
poles being the
1_
1
\\/cr+r'^-2 cos 6cr
From
the
which tends to turn the
^
.
c^
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 distance of their centres, and inversely as the
sixth
power of the distance of the poles of the magnet, considered
these poles are near each other this action of
the poles will be
as points.
much
As long
as
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
or diamagnetic.
If,
they had been placed in a uniform
by a
magnet according
poles of a
as they are magnetic
between two poles very near to each other,
instead of being placed
such as that of terrestrial magnetism or that produced
field
Ex. VIII.), an elongated body would
spherical electro-magnet (see
set axially
whether
magnetic or diamagnetic.
In
these cases the
all
phenomena depend on t —
magnetically or diamagnetically according as
diamagnetic than the medium in which
VI.
On
the
it is
for the axes of a\ y, z.
and
more or
less
is different
let k'
Let
k^,
k-,,
throughout the sphere, and
be
I,
field into
which the sphere
field
them
is
Let /
introduced, and
m, n.
Let us now take the case of a homogeneous sphere whose
tlie
let
be the coefficients of resistance in these three
be that of the external medium, and a the radius of the sphere.
let its direction-cosines
uniform magnetic
coefficient
in different directions.
parallel
k^,
be the undisturbed magnetic intensity of the
outside
magnetic, or less or more
placed.
Let the axes of magnetic resistance be
directions,
so that the sphere conducts itself
Magnetic Phenomena of a Sphere cut from a substance whose
of resistance
be taken
is
it
k',
whose intensity
is
// in the direction of x.
sphere would be
/
coefficient is
ki-
k' a'\
The
A;,
placed in a
resultant potential
Mr maxwell, ON FARADAY'S LINES OF FORCE.
75
and for internal points
— ^^1
7 7
11
Pi=
2ki
So that
It
is
in
tiie
interior
of
w.
the sphere the magnetization
is
entirely in
the direction of
therefore quite independent of the coefficients of resistance in the directions of
which may be changed from
We
-,
+k
may
^•,
into k^
and
therefore treat the sphere as
but we must use a different
p= I
{Lv +
ks without disturbing this distribution of
homogeneous
We
coefficient for each.
—
my + nz +
+
Ix
K'ih+k'
L
for each of tlie three
w and
.v.
y,
magnetism.
components of
/,
find for external points
Zk^+k'
my + -^
^
Qk,+
,
nss\-).
k'
Jr'j'
and for internal points
- /
3k^
3h,
\2k^+k'
The
external effect
is
the same
2k,
+
''
k'
3ks
\
2k^+k'
I
that which would have been produced if the small
as
magnet whose moments are
2ki + k
had been placed
effect
2k2+
2k^+
at the origin with their directions coinciding with
of the original force / in turning
the axes of x, y, z.
sphere about the axis of w
tlie
The
may be found by
magnets.
The
taking the moments of the components of that force on these equivalent
moment
of the force in the direction of y acting on the third magnet
2*3+ k
and that of the force
in
,
is
mnl'a
« on the second magnet
is
k.,- k'
;
2k,+ V
The whole couple about
the axis of x
is
therefore
3k' (kj
{2k3
+
mnPw'.
-
k,)
k')(2t,
+
mnPa?,
k')
tending to turn the sphere round from the axis of y towards that of
to be suspended so that the axis of
x
is
vertical,
and
let
angle which the axis of y makes with the direction of
expression for the
/ be
/,
m=
z.
Suppose the sphere
horizontal,
cos 0,
then
n = —
if
d be the
sin 9,
and the
moment becomes
/
w,;
1
T-i
2 {2k^+]c)(2k2+
k)
,
tending to increase
9.
The
axis of least resistance
V
sin
29
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,
10—2
Mr maxwell, ON FARADAY'S LINES OF FORCE.
76
the coefficient of this quantity can be but
The
value in space.
medium
independent of the external
The
k
homogeneous
intensity within the
resistance of the shell
is infinite,
shell
is
by supposing
it
would have been
effect
the shell
if
When
be diamagnetic or paramagnetic.
vanishes, the intensity within the shell
is
the
zero.
of the shell on any point, internal or external,
a superficial stratum of
magnetic matter spread over the
by the equation
outer surface, the density being given
= 3/
jo
Suppose the
it
than what
less
shell
and when
In the case of no resistance the entire
represented
shell.
diamagnetic or paramagnetic substance presents no
shell of a
were away, whether the substance of the
may be
k' iok, the
*.
Permanent magnetism in a spherical
VII.
difficulty.
by clianging the value of
t^=/Vsin20,
*^
case of a
altered
expression then becomes
J
The
little
cos B.
now to be converted into a permanent magnet, so that the distribution of
shell
imaginary magnetic matter
invariable, then the external potential
is
p = - I-
due
to the shell will
be
cos 0,
r~
p^= - Ir cos
and the internal potential
Now
let
us investigate the effect of
the resistance
magnetized
is k^
shell
the resistance in the external
may be
9.
up the
filling
with some substance of which
shell
medium being
The
k'.
thickness
of
the
Let the magnetic moment of the permanent magnetism
neglected.
be la^, and that of the imaginary superficial distribution due to the medium k
=
Aa^.
Then
the potentials are
external
The distribution of
medium k, so that
+ A)
—
magnetism
is
p =
real
{I
v
or
cos 9,
internal
;j,
=
(I + A) r cos
9.
the same before and after the introduction of the
A=—
-, I.
2k + k
The
magnetized
effect
of
the
greater or less than
//.
It
external
matter, and diminished by
•
is
shell
is
increased or
therefore increased
filling
it
by
filling
diminished according as k
up the
with paramagnetic matter, such
Taking the more general case of magnetic induction re(2tl), we find, in the expression for the moment
ferred to in Art.
in nature,
we must admit
in the case of electric
those terms which depend on sines and cosines of
through which heat or
is,
The
result
that in every complete revolution in the negative direction
round the axis of T, a certain positive amount of work
is
gained; but, since no inexhaustible source of work can exist
diamagnetic
as iron.
that 7'
=
in all substances, with
This argument does not hold
conduction, or in the case of a body
respect to magnetic induction.
of the magnetic forces, a constant term depending on T, besides
ti.
shell with
is
electricity is passing, for
such states are
maintained by the continual expenditure of work. See Prof.
Thomson,
P/ii/.
Mag.
Blarch, 1851, p. 186.
Mr maxwell, ON FARADAY'S LINES OF FORCE.
VIII.
Electro-magnetic spherical
Let us take as an example of the magnetic
in the
form of a thin spherical
Let
shell.
external effect be that of a magnet whose
may be
the magnetic effect
Let p,
radius be a, and
moment
is
magnetic
be the external and internal potentials,
2>i
a'
•
p^
no permanent magnetism,
is
A=
we draw any
whicli
3la cos
and
d,
6
as
the
the quantity of the current at any point
which flows through the element fdd
shell
referred to unit of area of section
=
a,
the equator,
at
and
at
this
curve
sin 0d9,
some other
will
be
current which flows through
it,
may be found by
- Sla
is
r
magnetic intensity round
total
a measure of the total electric
is
tliis
when
,
-21.
curve cutting
closed
known, then the
is
cos 9,
dr
dr
point for
= Ar
.dp'
— = —dpi-
•
1
and since there
If
thickness
cannot be represented by a potential.
effects
p = I~cos6,
,
an electro-mao-net
t
and let its
Both within and without the shell
/a'.
its
represented by a potential, but within the substance of the shell,
electric currents, the
where there are
shell.
effects of electric currents
its
77
The
differentiation.
so that
quantity
the quantity of the current
is
- 31-
sin e.
t
If the shell be
composed of a wire
round the sphere so that the number of
coiled
to the inch varies as the sine of 9, then the external
made of
the shell had been
effect
we have
the
if
just given.
wound round
If a wire conducting a current of strength /„ be
that
coils
be nearly the same as
uniform conducting substance, and the currents had been
a
distributed according to the law
so
will
a sphere of radius a
measured along the axis of x
distance between successive coils
So
is
—
then
n
there will be
n
coils altogether,
and the value of
/j for the resulting electro-magnet will
be
ba
The
The
potentials, external
interior of the shell
and
internal, will be
p=
/,
is
let
The
coil.
Pi
=-
2/, - - cos 9.
"6a
field.
Effect of the core of the electro-magnet.
P =
external effect
is,
i-esult
n
,
The
0,
us suppose a sphere of diamagnetic or paramagnetic matter introduced into the
electro-magnetic
than k, that
— cos
therefore a uniform masnetic
IX.
Now
-
'or-
Ji
is
-
—
may be
3k'
greater
a"
7
—
or
obtained as in the
^
cos 9,
less
Pi
than
according as the interior of the
=—
last case,
n
sphere
and the potentials become
r
^
cos
2Z,
before,
3k
according as
is
9.
k'
magnetic or
is
greater or less
diamagnetic
with
Mr maxwell, ON FARADAY'S LINES OF FORCE.
78
medium, and the
external
respect to the
beinir sreatest for a
A;
the effect of the core
k'
and the
0,
The
the core
effect
is
may be
of the core
each
=
0,
it
effect of
the electro-magnet
As k has always
has witliout the core.
a finite value,
this.
increased
is
by surrounding
the
with a shell of iron.
coil
on internal points would be
thick, the effect
the
If
tripled.
greater in the case of a cylindric magnet, and greatest of
all
when
Electro-tonic functions in spherical electro-magnet.
to this electro-magnet.
form
oo
=
some function of
r.
and
due
find the electro-tonic functions
will be of the
is
of introducing an iron core into an electro-magnet.
effect
a ring of soft iron.
Let us now
where w
the opposite direction,
electro-magnet we have a uniform field of magnetic force,
shell infinitely
X.
They
than
less
is
the interior of the
intensity of which
=
altered in
were to vanish altogether, the
for the core
would be three times that which
In
is
diamagnetic medium.
This investigation explains the
If the value of
internal effect
Oj
=
fio
Where
7o = -
lo^,
<^yj
there are no electric currents,
we must have
Oj, b^, c,
this implies
d
the solution of which
dw\
[
is
r^
Within
w
the shell
a must vanish at an
cannot become infinite
therefore
;
w =
Q
is
the solution, and outside
infinite distance, so that
C2
is
The magnetic
the solution outside.
_
2
~^'
6a 2k
quantity within the shell
+ k'~ '~
dy ~
dr
is
found by
last article to
be
'
'
therefore within the sphere
0)0=
—
3k
2ar.
Outside the sphere we must determine
value.
The
external value
is
w
k'
so as to coincide at the surface
with the internal
therefore
I^n
"^
where the
+
shell containing the currents is
a'
1
~ ~ 2a 3^ +
k' r'
made up of n
coils of
wire, conducting a current of
total quantity I^-
Let another wire be coiled round the
number of
coils
be
71
;
shell
according to the same law, and
then the total electro-tonic
intensity
found by integrating
EI^ =
/
wa smdds.
EL
let
the total
round the second
coil
is
Mr maxwell, ON FARADAY'S LINES OF FORCE.
The
along the whole length of the wire.
equation of the wire
= -fn TT
cos
where w
is
a large number
;
= a
may be called
£jln=
the
find
The
2,r
3
3
— oia^n =
,1
ann 1--
3^-
-,
+
^
Spherical electro-magnetic Coil-Machine.
action of each coil on itself
is
found by putting n^ or
n"' for nri
.
effects
on both wires, supposing their
/ and
—
N stand for
3
(oft
to
R
be
Let us
and R', and the
then the electro-motive force of the
,
wire on the second
first
is
ft )
- Nnn
Thai of
resistances
/'.
-7-
+
total
on
coil
Let the
be connected with an apparatus producing a variable electro-motive force F.
quantity of the currents
Let
47r
have now obtained the electro-tonic function which defines the action of the one
the other.
first coil
TT ?,\x\^6d$,
the electro-tonic coefficient for the particular wire.
XI.
We
,
sin Qd(p,
= - an
E
is
and therefore
ds
.•.
79
the second on itself
—
dt
is
dt
The
equation of the current in the second wire
-Nnn'~equation of the current in the
- Nn^
Eliminating the differential
therefore
Nn''^-^ = R'I'
dt
The
is
(1)'
dt
first
wire
- Nnn'
-~
dt
^
is
— + F = RI
(2)'
^
dt
coefficients,
we get
nnn
A
and
from which
of
to find
/and T.
AT
iV
— + ^— \
(^
;
m]
\R
For
this
dl ^ F
—
+7= — + JV—
R
R'
n'^
;
dt
dF
—
dt
.
.
(s\
^
'
purpose we require to know the value of
F
in
terms
t.
Let us
case
first
take the case in which
F
is
constant and
of an electro-magnetic coil-machine at the
galvanic trough.
I and
moment when
/' initially
=
the connexion
0.
is
This
is
the
made with
the
Mr maxwell, ON FARADAY'S LINES OF FORCE.
80
Putting i T for iV (-^
+
we
-^.1
find
F
it
The primary
current increases very rapidly from
F n
— and speedily
—
.
.
—
,
,
owing
vanishes,
The whole work done by
is
n
O
to
p
—
t being
to the value of
,
R
and the secondary commences
at
generally very small.
either current in heating the wire or in
any other kind of action
found from the expression
f"
Rdt.
The
total
quantity of current
is
/to
Idt.
For the secondary current we
R n-
Jf,
The work done and
of quantity I'
=
find
4
Rn
v/j
2
the quantity of the current are therefore the same as
if
a current
Fn'
^,
liad
passed through the wire for a time t, where
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
is
/„
and
let
in the secondary
=
0.
equivalent currents are
When
of R.
suddenly cease while the current in the primary wire
Then we
I-Ioe
The
F
the electro-motive force
^
shall
have for the subsequent time
^
,
Ig and
^
-^o
"57
Rf
~
n
>
R'
^^^
will
is
produce a change in the value of t, so that
strength of the secondary current will be increased, and
case in the ordinary coil-machines.
The
quantity
N
'
'
their duration is t.
the communication with the source of the current
This
n
its
cut
if
off,
R
there will be a change
be suddenly increased, the
duration diminished.
This
is
the
depends on the form of the machine,
and may be determined by experiment for a machine of any shape.
Mu MAXWELL, ON FARADAY'S LINES OF FORCE.
XII.
Spherical shell revolving in magnetic
Let us next take the case of a revolving
Experimental Researches, Series
II.,
explained by Faraday in
are
and references are there given
Let the axis of z be the axis of revolution, and
let
be represented in quantity by
the magnetism of the field
field.
of conducting matter under the influence
shell
The phenomena
of a uniform field of magnetic force.
81
the angular velocity be w.
/,
his
to previous experiments.
inclined at an angle
Let
to the
direction of z, in the plane of zx.
Let
R
Oo' /3o, 7oi
be the radius of the spherical
and
shell,
T
the thickness.
be the electro-tonic functions at any point of space;
of magnetic quantity and intensity;
Let P2 ^^ '^^
a^, b^, Cj,
Oj, /Sj,
a^,
6„ e„
Let the quantities
a,, /3i,
7, symbols
72 of electric quantity and intensity.
any point,
electric tension at
dp^
=
02
ka^
V
J
aw
•(0
72=— +^2
The
expressions for oo,
da,,
db,.
dc„
dx
dy
dz
da,
dRo
dyn
dx
dy
dz
aa=
^e
magnetism of*the
<lue to the
70
/3o,
(2);
-^0
+~
= B^ +
y cos
- {z
s,\n
y«= Co-~y sin
Aq, Bq, Co being constants;
and the
dx
~=
-
G,
9
- X cos
d).
0,
velocities of the particles of the revolving sphere are
dy
wy,
dt
We
=
dz
—
=
dt
wx.
dt
have therefore for the electro-motive forces
02
/33
= -
= -
72 =
-
field are
1
doo
47r
dt
1
/
cos
9 wx.
0.
Ma MAXWELL, ON FARADAY'S LINES OF FORCE.
Returning to equations
.
we
(l),
\dz
dyl
idc,
dao\
-^
—
i
i-
/dwj
^
_
db^\
dy
dy,
da^
—
17
=
sin
Ow,
47r 2
dar
^_^
^
dx
^
dxj
V dy
dz
= -f^
dx
dz
Kdx
From which
get
Q
dy
with equation (2) we find
^
*
=
62
0,
11/..
=
Cj
sin ytoa?,
47r 4
A;
——
=
P2
4
47r
7(0
+
(,r^
\
lOTT
cos B
3/^)
- xz sm6\.
These expressions would determine completely the motion of
sphere if we neglect the action of
of circular currents
about
the
axis
proportional to the distance from
of a small
tliese
of
that
magnet whose moment
electricity in a revolving
currents on themselves.
the quantity of
y,
The
axis.
external
They
current
magnetic
express a system
any point
at
effect
will
being
be that
to7 sin 0, with its direction along the axis of y,
is
487rA;
magnetism of the
so that the
The
existence
functions,
and
field
would tend
to turn
it
back to the axis of
a;*.
of these currents will of course alter the distribution of the electro-tonic
so they will react
^n themselves.
Let the
final result
of this action
be a
system of currents about an axis in the plane of xy inclined to the axis of x at an angle
•ind
producing an external
The
effect
equal to that of a magnet whose
magnetic inductive components within the
7i sin
9 —
27' cos
- zT
7, cos
Each of
these
would produce
and these would give
rise
to
new
its
moment
is
I'R^.
shell are
^
in x,
sin (p in y,
6
in z.
own system of
distributions
currents
when
the sphere
of magnetism, which,
is in
when the
motion,
velocity
is
uniform, must be the same as the original distribution,
(/i sin
6 -
2I' cos (p) in
(- 27'
7i
cos
*
it
9
The
in
sin (p) in
x produces 2
y produces
T
2
9 - 27* cos 0)
w
(7, sin
w
(27' sin (p) in x;
in y,
487r/c
z produces no currents.
expression for p^ indicates a variable electric tension in the shell, so that currents might be collected by wires touching
at the equator
and poles.
Mk maxwell, on FARADAY'S LINES OF FORCE.
We
must therefore have the following equations,
since
the state of the shell
83
is
the same at
every instant,
/, sin
9 - 2/' cos
-
=
T
^ =
2/' sin
T
+
/, sin
sin
tt)(Zj
wS/ sin
Q -
9.1'
rf»
cos ^),
whence
T
r=x
..,.
I
V1+
To
understand the meaning of these expressions
Let the axis of the revolving
west.
Let I be the
/cos0
is
The
small angle
of the rotation
T
=
let
vertical,
tan"^
-w
to
is
to
tlie
us take a particular case.
and
total intensity of the terrestrial
the horizontal component in
result
be
shell
-w
247^^•
let the
revolution be from north to
magnetism, and
let
the dip be 9, then
direction of magnetic north.
produce currents in the
the south
shell
of magnetic west, and
about an axis inclined at a
the external effect
of these
247r«;
currents
is
the same as that of a magnet whose
2
loss of
TV
of the couple due to terrestrial magnetism tending to stop the rotation
work due
is
Tu>
247r^
and the
is
^mi cos 9.
2 V'247rA;]-'+
The moment
moment
Tw
1
247rA;|-
+
TV
R'Pcos^9,
to this in unit of time is
Taf"
247r>i
R'rcos^9.
2
This
loss
of work
is
24,rA:]
made up by an
proved by a recent experiment of
+ T'w^
evolution of heat in the substance of the shell, as
M. Foucault,
(see
Comptes Rendus, xli.
p. 450).
11—2
is