James Clerk Maxwell.

The scientific papers of James Clerk Maxwell (Volume 1) online

. (page 16 of 50)
Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 16 of 50)
Font size
QR-code for this ebook


ON Faraday's lines of force. 163

it carries in. A tube which begins within the surface and ends without it
will carry out unity of fluid; and one which enters the surface and terminates
within it will carry in the same quantity. In order therefore to estimate the
amount of fluid which flows out of the closed surface, we must subtract the
number of tubes which end within the surface from the number of tubes which
begin there. If the result is negative the fluid will on the whole flow inwards.

If we call the beginning of a unit tube a unit source, and its termination
a unit sink, then the quantity of fluid produced within the surface is estimated
by the number of unit sources minus the number of unit sinks, and this must
flow out of the surface on account of the incompressibility of the fluid.

In speaking of these imit tubes, sources and sinks, we must remember what
was stated in (5) as to the magnitude of the unit, and how by diminishing
their size and increasing their number we may distribute them according to any
law however complicated.

(9) If we know the direction and velocity of the fluid at any point in
two diSerent cases, and if we conceive a third case in which the direction and
velocity of the fluid at any point is the resultant of the velocities in the two
former cases at corresponding points, then the amount of fluid which passes a
given fixed surface in the third case will be the algebraic sum of the quantities
which pass the same surface in the two former cases. For the rate at which
the fluid crosses any surface is the resolved part of the velocity normal to the
surface, and the resolved part of the resultant is equal to the sum of the
resolved parts of the components.

Hence the number of unit tubes which cross the surface outwards in the
third case must be the algebraical sum of the numbers which cross it in the
two former cases, and the number of sources within any closed surface will be
the sum of the numbers in the two former cases. Since the closed surface may
be taken as small as we please, it is evident that the distribution of sources
and sinks in the third case arises from the simple superposition of the distri-
butions in the two former cases.



n. TTieory of the uniform motion of an imponderable incompressible fluid
through a resisting medium.

(10) The fluid is here supposed to have no inertia, and its motion is opposed
by the action of a force which we may conceive to be due to the resistance of a

21—2



164 ON FARADAY S LINES OF FORCK

medium through which the fluid is supposed to flow. This resistance depends on
the nature of the medium, and will in general depend on the direction in which
the fluid moves, as well as on its velocity. For the present we may restrict
ourselves to the case of a uniform medium, whose resistance is the same in all
directions. The law which we assume is as follows.

Any portion of the fluid moving through the resisting medium is directly
opposed by a retarding force proportional to its velocity.

If the velocity be represented by i', then the resistance will be a force equal
to kv acting on unit of volume of the fluid in a direction contrary to that of
motion. In order, therefore, that the velocity may be kept up, there must be a
greater pressure behind any portion of the fluid than there is in front of it, so
that the difference of pressures may neutrahse the effect of the resistance. Con-
ceive a cubical unit of fluid (which we may make as small as we please, by (5)),
and let it move in a direction perpendicular to two of its faces. Then the resist-
ance will be kv, and therefore the difference of pressures on the first and second
faces is kv, so that the pressure diminishes in the direction of motion at the rate
of kv for every unit of length measured along the line of motion ; so that if w6
measure a length equal to h units, the difference of pressure at its extremities
will be kvh.

(11) Since the pressure is supposed to vary continuously in the fluid, all
the points at which the pressure is equal to a given pressure p will lie on a
certain surface which we may call the surface (p) of equal pressure. If a series
of these surfaces be constructed in the fluid corresponding to the pressures 0, 1,
2, 3 &c., then the number of the surface will indicate the pressure belonging to
it, and the surface may be referred to as the surface 0, 1, 2 or 3. The unit of
pressure is that pressure which is 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 in the same proportion.

(12) It is easy to see that these surfaces of equal pressure must be perpen-
dicular to the lines of fluid motion; for if the fluid were to move in any other
direction, there would be a resistance to its motion which could not be balanced
by any difference of pressures. (We must remember that the fluid here con-
sidered has no inertia or mass, and that its properties are those only which are
formally assigned to it, so that the resistances and pressures are the only things



ON Faraday's lines of force. 165

to be considered.) There are therefore two sets of surfaces which by their inter-
section form the system of unit tubes, and the system of surfaces of equal pres-
sure cuts both the others at right angles. Let h be the distance between two
consecutive surfaces of equal pressure measured along a line of motion, then since
the difference of pressures = 1,

kvh= 1,

which determines the relation of v to h, so that one can be found when the
other is known. Let s be the sectional area of a unit tube measured on a
surface of equal pressure, then since by the definition of a unit tube

vs = \,

we find by the last equation

s = kh.

(13) The surfaces of equal pressure cut the unit tubes into portions whose
length is h and section s. These elementary portions of unit tubes will be called
unit cells. In each of them unity of volume of fluid passes from a pressure p to
a pressure (p — 1) in unit of time, and therefore overcomes unity of resistance in
that time. The work spent in overcoming resistance is therefore unity in every
cell in every unit of time.

(14) If the surfaces of equal pressure are known, the direction and magni-
tude of the velocity of the fluid at any point may be found, after which the
complete system of unit tubes may be constructed, and the beginnings and end-
ings of these tubes ascertained and marked out as the sources whence the fluid
is derived, and the sinks where it disappears. In order to prove the converse of
this, that if the distribution of sources be given, the pressure at every point may
be found, we must lay down certain preliminary propositions.

(15) If we know the pressures at every point in the fluid in two different
cases, and if we take a third case in which the pressure at any point is the
sum of the pressures at corresponding points in the two former cases, then the
velocity at any point in the third case is the resultant of the velocities in the
other two, and the distribution of sources is that due to the simple superposition
of the sources in the two former cases.

For the velocity in any direction is proportional to the rate of decrease of
the pressure in that direction; so that if two systems of pressures be added



166 ON FARADAY S LINES OF FORCE.

together, since the rate of decrease of pressure along any line will be the sum
of the combined rates, the velocity in the new system resolved in the same
direction will be the sum of the resolved parts in the two original systems.
The velocity in the new system will therefore be th€ resultant of the velocities
at corresponding points in the two former systems.

It follows from this, by (9), that the (quantity of fluid which crosses any
fixed surface is, in the new system, the sum of the corresponding quantities in
the old ones, and that the sources of the two original systems are simply
combined to form the third.

It is evident that in the system in which the pressure is the diiBPerence
of pressure in the two given systems the distribution of sources will be got
by changing the sign of all the sources in the second system and adding them
to those in the first.

(16) If the pressure at every point of a closed surface be the same and
equal to p, and if there be no sources or sinks within the surface, then there
will be no motion of the fluid within the surface, and the pressure within it
will be uniform and equal to p.

For if there be motion of the fluid within the surface there will be tubes
of fluid motion, and these tubes must either return into themselves or be
terminated either within the surface or at its boundary. Now since the fluid
always flows from places of greater pressure to places of less pressure, it
cannot flow in a re-entering curve; since there are no sources or sinks within
the surface, the tubes cannot begin or end except on the surface ; and since
the pressure at all points of the surface is the same, there can be no motion
in tubes having both extremities on the surface. Hence there is no motion
within the surface, and therefore no difference of pressure which would cause
motion, and since the pressure at the bounding surface is p, the pressure at
any point within it is also p.

(17) If the pressure at every point of a given closed surface be known,
and the distribution of sources within the surface be also known, then only
one distribution of pressures can exist within the surface.

For if two 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 difference of the pressures in the two former distri-
butions. In this case, since the pressures at the surface and the sources within



ON Faraday's lines of force. 107

it are the same in both distributions, the pressure at the surface in the third
distribution would be zero, and all the sources within the surface would
vanish, by (15).

Then by (16) the pressure at every point in the third distribution must
be zero ; but this is the difference of the pressures in the two former cases,
and therefore these cases are the same, and there is only one distribution of
pressure possible.

(18) Let us next determine the pressure at any point of an infinite body
of fluid in the centre of which a unit source is placed, the pressure at an
infinite distance from the source being supposed to be zero.

The fluid will flow out from the centre symmetrically, and since unity of
volume flows out of every spherical surface surrounding the point in unit of
time, the velocity at a distance r from the source will be

k

The rate of decrease of pressure is therefore hv or — -^, and since the

pressure = when r is infinite, the actual pressure at any point will be

= A

The pressure is therefore inversely proportional to the distance from the
source.

It is evident that the pressure due to a unit sink will be negative and

equal to — - — .

If we have a source formed by the coalition of »S' unit sources, then the

TcS
resulting pressure will be X>=t—,, so that the pressure at a given distance

varies as the resistance and number of sources conjointly.

(19) If a number of sources and sinks coexist in the fluid, then in order
to determine the resultant pressure we have only to add the pressures which
each source or sink produces. For by (15) this will be a solution of the
problem, and by (17) it will be the only one. By this method we can
determine the pressures due to any distribution of sources, as by the method



168 ON Faraday's lines of forck

of (14) we can determine the distribution of sources to which a given distri-
bution of pressures is due.

(20) We have next to shew that if we conceive any imaginary surface
as fixed in space and intersecting the lines of motion of the fluid, we may
substitute for the fluid on one side of this surface a distribution of sources
upon the surface itself without altering in any way the motion of the fluid
on the other side of the surface.

For if we describe the 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.

(21) If the system of pressures and the distribution of sources which pro-
duce them be known in a medium whose resistance is measured by k, then in
order to produce the same system of pressures in a medium whose resistance
is unity, the rate of production at each source must be multiplied by k. For
the pressure at any point due to a given source varies as the rate of produc-
tion and the resistance conjointly; therefore if the pressure be constant, the
rate of production must vary inversely as the resistance.

(22) On the conditions to he fulfilled at a surface which separates two media
whose coefficients of resistance are k and k\

These are found from the consideration, that the quantity of fluid which
flows out of the one medium at any point flows into the other, and that the
pressure varies continuously from one medium to the other. The velocity normal
to the surface is the same in both media, and therefore the rate of diminution
of pressure is proportional to the resistance. The direction of the tubes of
motion and the surfaces of equal pressure will be altered after passing through
the surface, and the law of this refraction will be, that it takes place in the
plane passing through the direction of incidence and the normal to the surface,
and that the tangent of the angle of incidence is to the tangent of the angle
of refraction as k' is to k.

(23) Let the space within a given closed surface be filled with a medium
different from that exterior to it, and let the pressures at any point of this
compound system due to a given distribution of sources within and without



ON fakaday's lines of force. 169

the surface be given ; it is required to determine a distribution of sources which
would produce the same system of pressures in a medium whose 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 place a unit sink. Then
if we make the surface impermeable all will go on as before.

Let the resistance of the exterior medium be measured by k, and that of
the interior by V. Then if we multiply the rate of production of all the sources
in the exterior medium (including those in the surface), by k, and make the
coefficient of resistance unity, the pressures will remain as before, and the same
will be true of the interior medium if we multiply all the sources in it by k',
including those in the surface, and make its resistance unity.

Since the pressures on both sides of the surface are now equal, we may
suppose it permeable if we please.

We have now the original system of pressures produced in a uniform medium
by a combination of three systems of sources. The first of these is the given
external system multipHed by k, the second is the given internal system multi-
plied by k', and the third is the system of sources and sinks on the surface
itself. In the original case every source in the external medium had an equal
sink in the internal medium on the other side of the surface, but now the
source is multiplied by k and the sink by k', so that the result is for every
external unit source on the surface, a source ={k — k'). By means of these three
systems of sources the original system of pressures may be produced in a medium
for which k = \.

(24) Let there be no resistance in the medium within the closed surface,
that is, let /t' = 0, then the pressure within the closed surface is uniform and
equal to p, and the pressure at the surface itself is also p. If by assuming
any distribution of pairs of sources and sinks within the surface in addition to
the given external and internal sources, and by supposing the medium the same
within and without the surface, we can render the pressure at the surface uni-
form, the pressures so found for the external medium, together with the uniform
pressure p in the internal medium, will be the true and only distribution of
pressures which is possible.

For if two such distributions could be found by taking diffijrent imaginary
distributions of pairs of sources and sinks within the medium, then by taking
VOL. I. 22



170 ON Faraday's lines of foece.

the difference of the two for a third distribution, we should have the pressure
of the bounding surface constant in the new system and as many sources as
sinks within it, and therefore whatever fluid flows in at any point of the surface,
an equal quantity must flow out at some other point.

In the external medium all the sources destroy one another, and we have
an infinite medium without sources surrounding the internal medium. The pres-
sure at infinity is zero, that at the surface is constant. If the pressure at the
surface is positive, the motion of the fluid must be outwards from every point
of the surface ; if it be negative, it must flow inwards towards the surface. But
it has been shewn that neither of these cases is possible, because if any fluid
enters the surface an equal quantity must escape, and therefore the pressure at
the surface is zero in the third system.

The pressure at all points in the boundary of the internal medium in the
third case is therefore zero, and there are no sources, and therefore the pressure
is everywhere zero, by (16).

The pressure in the bounding surface of the internal medium is also zero,
and there is no resistance, therefore it is zero throughout; but the pressure in
the third case is the difference of pressures in the two given cases, therefore
these are equal, and there is only one distribution of pressure which is possible,
namely, that due to the imaginary distribution of sources and sinks.

(25) When the resistance is infinite in the internal medium, there can be
no passage of fluid through it or into it. The bounding surface may therefore
be considered as impermeable to the fluid, and the tubes of fluid motion will
run along it without cutting it.

If by assuming any arbitrary distribution of sources within the surface in
addition to the given sources in the outer medium, and by calculating the
resulting pressures and velocities as in the case of a uniform medium, we can
fulfil the condition of there being no velocity across the surface, the system of
pressures in the outer medium will be the true one. For since no fluid passes
through the surface, the tubes in the interior are independent of those outside,
and may be taken away without altering the external motion.

(26) If the extent of the internal medium be small, and if the difference
of resistance in the two media be also small, then the position of the unit tubes
will not be much altered from what it would be if the external medium filled
the whole space.



ON FARADAY S LINES OF FORCE. 171

Oq this supposition we can easily calculate the kind of alteration which
the introduction of the internal medium will produce ; for wherever a unit tube

enters the surface we must conceive a source producing fluid at a rate -^^ ,

and wherever a tube leaves it we must place a sink annihilating fluid at the

k'-k
rate — ^ , then calculating pressures on the supposition that the resistance in

both media is k, the same as in the external medium, we shall obtain the true
distribution of pressures very approximately, and we may get a better result
by repeating the process on the system of pressures thus obtained.

(27) If instead of an abrupt change from one coeflBcient of resistance to
another we take a case in which the resistance varies continuously from point
to point, we may treat the medium as if it were composed of thin shells each
of which has uniform resistance. By properly assuming a distribution of sources
over the surfaces of separation of the shells, we may treat the case as if the
resistance were equal to unity throughout, as in (23). The sources will then
be distributed continuously throughout the whole medium, and will be positive
whenever the motion is from places of less to places of greater resistance, and
negative when in the contrary direction.

(28) Hitherto we have supposed the resistance at a given point of the
medium to be the same in whatever direction the motion of the fluid takes
place ; but we may conceive a case in which the resistance is different in
different directions. In such cases the lines of motion will not in general be
perpendicular to the surfaces of equal pressure. If a, 6, c be the components
of the velocity at any point, and a, yS, y the components of the resistance at
the same point, these quantities will be connected by the following system of
linear equations, which may be called ''equations of conduction" and will be
referred to by that name.

a^P,a + QS + R.y,

h = Fj3+Q,y + EA,

c = P,y+Q,a + JR,l3.

In these equations there are nine independent coefficients of conductivity. In

order to simplify the equations, let us put

Qt + Ji, = 2S„ Q,-B, = 2lT,

&c &c.

22—2



172 ON Faraday's lines of force.

where 4^ = «?,-i2,)' + (^»-^.)' + (^3-^s)',

and I, m, n are direction-cosines of a certain fixed line in space.

The equations then become

a = P,a + SJ3 + S,y + (nfi -my) T,
b=F^ + S,y + S,a + {lY - na) T,
c = P,y + S,a + S^ + {ma~ l^) T.

By the ordinary transformation of co-ordinates we may get rid of the
coeflBcients marked S. The equations then become

a = P(a + (n'^-m'y)T,

b = P:/3 + {ry-n'a)T,

c = P,y+{m'a- Vfi) T,
where I', m, n' are the direction-cosines of the fixed line with reference to the
new axes. If we make



the equation of continuity



becomes



%^-i' -^-|.



da dh c^c _
dx dy dz '



' dx'^ ' dy'^^' dz' ^'



and if we make x = JP^^, y^^fPT^], z = JP^l,

^'^^■^ 3|+^ + ? = °-

the ordinary equation of conduction.

It appears therefore that the distribution of pressures is not altered by
the existence of the coefficient T. Professor Thomson has shewn how to
conceive a substance in which this coefficient determines a property having
reference to an axis, which unlike the axes of P^, P^, P^ is dipolar.

For further information on the equations of conduction, see Professor
Stokes On the Conduction of Heat in Crystals {Cambridge and Dublin Math.
Journ.), and Professor Thomson On the Dynamical Theory of Heat, Part v.
{Transactions of Royal Society of Edinburgh, VoL xxi. Part i.).



ON Faraday's lines of force. 173

It is evident that all that has been proved in (14), (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 is different in different directions. For
il' we examine the proof we shall find it applicable to 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 the most general case of a medium whose resistance is different in
different directions and varies from point to point.

We may by the method of (28), when the distribution of pressures is
known, construct the surfaces of equal pressure, the tubes of fluid motion, and
the sources and sinks. It is evident that since in each cell into which a unit
tube is divided by the surfaces of equal pressure unity of fluid passes from
pressure p to pressure (p — 1) in unit of time, unity of work is done by the
fluid in each cell in overcoming resistance.

The number of cells in each unit tube is determined by the number of
surfaces of equal pressure through which it passes. If the pressure at the
beginning of the tube be p and at the end p\, then the number of cells in
it will be p—p- Now if the tube had extended from the source to a place
where the pressure is zero, the number of cells would have been p, and if
the tube had come from the sink to zero, the number would have been p\
and the true number is the difference of these.

Therefore if we find the pressure at a source S from which S tubes
proceed to be p, Sp \s. the number of cells due to the source S ; but if iS' of
the tubes terminate in a sink at a pressure p\ then we must cut off S p cells
from the number previously obtained. Now if we denote the source of S



Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 16 of 50)