C. L, C
GIFT OF
Professor C. L* Cory
ENGINEERING LIBRARY
C.L..CC,
C. L. C
c. L. com
A TREATISE
ON
ELECTRICITY C A^D MAGNETISM
A TREATISE
ON
ELECTRICITY
AND
MAGNETISM
BY
E. MASCART,
PROFESSOR IN THE COLLEGE DE FRANCE, AND DIRECTOR OF THE CENTRAL
METEOROLOGICAL BUREAU;
AND
J. JOUBERT,
PROFESSOR IN THE COLLEGE ROLLIN.
LANSLAIED BY
E. ATKlWOTMrffiB., F.C.S.,
PROFESSOR OF EXPERIMENTAL SCIENCE IN THE STAFF .COLLEGE.
VOLUME I.
GENERAL PHENOMENA AND THEORY.
LONDON:
THOS. DE LA RUE AND CO.
no, BUNHILL ROW.
i883
4-0
ENGINEERING LIBRARY
PRINTED BY
THOMAS DE LA RUE AND CO., BUNHH.L ROW,
LONDON.
u
PREFACE.
THIS work is based upon a course of Lectures delivered by
one of us in the College de France in the last few years.
It will comprise two parts : the first, which is principally
theoretical, forms the present volume ; the second, which
will be more experimental in character, will be devoted to
the examination of the various phenomena, the methods of
measurement, and the principal applications. %
This method of treatment seemed to us to present great
advantages. The phenomena are, in fact, almost always
very complicated, especially in the case of electricity; and
to understand all the details which these phenomena involve
often requires a more extensive knowledge than that fur-
nished in the chapters with which they are more immediately
connected. The explanation of the experiments will, there-
fore, be materially facilitated by a preliminary account of the
general principles of the science.
After having stated and co-ordinated the facts which
serve to establish the theory, we have investigated the
mechanical consequences. This first volume forms, accord-
ingly, a distinct work ; and we might express the idea
which has guided us by considering it as an Essay on the
Mechanical TJieory of Electricity, if such a title did not
seem too ambitious.
We have endeavoured to bring into prominence the
profound views introduced into science by Faraday, and
842487
VI PREFACE.
so happily developed by Clerk Maxwell, on the conside-
ration of the lines of force, and on the function of the
medium in which electrical and magnetic actions are exerted.
This conception brilliantly elucidates the relations between
various phenomena, and has given rise to a totally unfore-
seen theory of light.
As our principal aim was to be of service to physicists,
we have made all efforts to simplify the demonstrations
without in any way sacrificing the strictness of the reasoning.
Those parts which require a somewhat more advanced
analysis, and which can be readily distinguished, may be
omitted in a first reading; in most cases they are not in-
dispensable for following the development of theory.
In the last few years the science of electricity has under-
gone a real transformation ; we eagerly recognise our nu-
merous obligations to the works of those physicists who
have most contributed to this reform, and particularly to
the memoirs of Sir W. Thomson, and the excellent treatise
of Clerk Maxwell.
In this English edition we have introduced several
corrections, and have given some new proofs of certain
questions. We must express our obligations to Dr. At-
kinson for the care and exactitude with which the trans-
lation has been made.
TABLE OF CONTENTS.
PART I. STATICAL ELECTRICITY.
PAGE
CHAPTER I. INTRODUCTORY i
,, II. ON POTENTIAL 9
,, III. GENERAL THEOREMS 36
,, IV. ELECTRICAL EQUILIBRIUM 51
,, V. WORK OF ELECTRICAL FORCES 77
,, VI. ON DIELECTRICS 87
,, VII. PARTICULAR CASES OF EQUILIBRIUM in
,, VIII. SOURCES OF ELECTRICITY .... .... 175
PART II. ELECTRICAL CURRENTS.
CHAPTER I. PROPAGATION OF ELECTRICITY IN THE PERMANENT
STATE 186
,, II. VARIABLE STATE 213
III. ENERGY OF CURRENTS 236
,, IV. THERMOELECTRICAL CURRENTS 259
PART III. MAGNETISM.
CHAPTER I. PRELIMINARY 280
,, II. CONSTITUTION OF MAGNETS 298
,, III. PARTICULAR CASES 33^
,, IV. MAGNETIC INDUCTION 358
V. ON MAGNETS 385
VI. MAGNETIC CONDITION OF THE GLOBE 407
Vlll TABLE OF CONTENTS.
PART IV. ELECTROMAGNETISM.
PAGE
CHAPTER I. CURRENTS AND MAGNETIC SHELLS 420
II. ELEMENTARY ACTIONS 440
III. PARTICULAR CASES 465
,, IV. INDUCTION 491
,, V. PARTICULAR CASES OF INDUCTION 503
,, VI. PROPERTIES OF THE ELECTROMAGNETIC FIELD . . 546
,, VII. PHENOMENA OF INDUCTION IN NON-LINEAR CON-
DUCTORS 556
,, VIII. OPTICAL PHENOMENA 574
,, IX. ELECTRICAL UNITS 584
, X. GENERAL THEORIES 601
XI. SUPPLEMENTARY , . 628
A TREATISE
C.L.CORY.
ELECTRICITY AND MAGNETISM
PART I. STATICAL ELECTRICITY.
CHAPTER I.
INTRODUCTORY.
1 . MOST bodies, when rubbed, acquire, at any rate temporarily,
the property of attracting light bodies. They are then said to be
electrified. If the attracted body comes in contact with the electrified
body, it is sooner or later repelled, and it is then itself found to be
electrified. Electrical properties can accordingly be transmitted
from one body to another by mere contact.
On the other hand, bodies which show no electrical properties
are said to be in the natural state, or in the neutral state.
2. CONDUCTORS. INSULATORS. On certain bodies, such as
glass, resin, silk, caoutchouc, etc., electricity remains localised for
a shorter or longer time, at the place where it has been produced
by friction, or by contact. Such bodies are said to be bad conductors
of electricity.
On other bodies, on the contrary, such as the metals, electrical
properties imparted to any one part, are almost instantaneously
transmitted to all parts; these are said to be conductors. Most
substances composing the earth belong to this latter class ; air, and
vapours, and generally all gases, belong to the former. Electricity
can only be retained on a conductor by insulating it from the
ground that is, supporting it by a bad conductor, such as a rod
of glass, or sealing-wax, or ebonite, or by silk threads. Hence we
have the term insulators^ applied to bad conductors.
B
INTRODUCTORY.
The distinction between good and bad conductors does not really
correspond to any essential difference of properties. Electricity moves
upon all bodies with greater or less freedom. No bodies are known
which are absolute -insulators that is to say, on which electrical pro-
perties can be retained for an indefinite time without alteration.
In like manner, notwithstanding the rapidity with which elec-
tricity is transmitted on the best conductors, there are none on
which it is propagated instantaneously. Very material differences
are found among them in this respect, and we can determine the
special resistance which each one offers to the motion of electricity.
3. Two ELECTRICITIES. When any two bodies are rubbed
together, a piece of glass and a piece of resin for instance, both
become electrified, but with different characters ; each of them
repels an insulated light body which has been in contact with it,
and which has shared its electricity ; but the resin attracts the body
which has been touched by the glass, and the glass the body which
has been touched by the resin.
The condition of the glass differs, then, from that of the resin,
which is expressed by saying that the electricity of the glass is of
a different kind to that of the resin. Experiment shows, moreover,
that any electrified body behaves either like the glass, or like the
resin of the preceding experiment. It attracts, for instance, the
body electrified by glass, and repels that which has been electrified
by resin, or conversely. There are thus two kinds of electricity, and
only two. This fundamental property may be formulated by saying
that two bodies charged with the same electricity repel each other, and
two bodies charged with opposite electricities attract each other.
4. ELECTRICAL ACTIONS. ELECTRICAL MASSES. The action
exerted between two electrical bodies whose dimensions are small
in comparison with their distance apart, is in the direction of the
straight line joining them.
Coulomb found by direct experiment that this force is inversely
as the square of the distance. It is also a function of the electrical
condition of adjacent bodies, or of their electrification.
If, between two identical bodies of very small dimensions, and
placed at unit distance, the electrical action is equal to unit force,
the quantity of electricity, or the electrical mass of each of them, is
equal to unity.
If, while the condition of one of the bodies remains unchanged,
the distance being also unchanged, the action between them becomes
2, 3 ... times as great, the electrical mass of the other is said to
have become 2, 3 ... times as great
ELECTRICAL FORCE.
The electrical mass of a body, other things being equal, is (pro-
portional to the force which it exerts upon an external body placed
at a distance) considerable in reference to its dimensions ; and the
mutual action of two electrical masses is proportional to the product
of their electrical masses.
5. When two bodies for instance, a glass disc and one of metal,
the latter being insulated are kept in contact after being rubbed
together, the whole system behaves in reference to an external body,
electrified or not, just as if it were in the neutral state. The elec-
trical properties developed by friction have not, however, disappeared ;
for if the two discs are separated, it may at once be shown that each
of them is electrified. The actions of the two bodies in contact are
accordingly equal and of opposite signs. Hence follows this double
conclusion.
By their mutual friction two bodies acquire quantities of electricity^
which are equal and of different kinds.
The law according to which the action varies with the distance is
the same for the two electricities.
6. ELECTRICAL SIGNS. We are thus led to consider electrical
masses of different kinds, as quantities of the same nature and of
opposite signs. When a closed surface contains electrical masses of
different kinds, the action exerted upon an external mass, equal to
unity, and at a great distance, as compared with the surface in
question, is proportional to the difference of the electrical masses of
each kind, and is attractive or repulsive according to the kind of
electricity which.predominates. Affixing to these masses the signs +
and - , we may say that the resultant action is proportional to the
algebraical sum of the electrical masses contained on the surface,
and is repulsive or attractive according to the sign of this sum.
It is usual to apply the term positive to the electricity developed
upon glass when rubbed with resin, and negative to the electricity
acquired by the resin.
7. ELECTRICAL FORCE. The action between two bodies of small
dimensions, charged respectively with the masses m and m', and at
the distance r from each other, is therefore
mm'
f ~r*'
This expression is positive if the two masses are of the same sign,
and the force is then repulsive. In the contrary case it is attractive.
If a mass m is in the presence of any electrified bodies whatever,
it may be considered that the total action which it experiences is the
B 2
INTRODUCTORY.
resultant of all the actions which each of the elementary masses,
considered separately, would exert upon it ; and this whether the
masses belong to separate bodies, or whether they form part of the
charge of one and the same body.
For the sake of abbreviation, we shall apply the term, electrical
force at a point, to the resultant of all the actions which would be
exerted on unit mass of electricity placed at this point.
8. DISTRIBUTION OF ELECTRICITY. Coulomb proved by direct
experiment that when an electrified conducting sphere is put in
contact with an identical sphere in the neutral state, each of them
possesses a mass of electricity equal to half the original mass that
is to say, that each of them acting separately at the same distance
upon an external electrified body, exerts half the action of that of
the sphere in its original state. If the same sphere, instead of being
neutral, is itself electrified before contact, the final charges are also
equal ; each of them is half the algebraical sum of the original
masses, so that it is zero, and the bodies are in the neutral state,
if the initial charges were equal, and of opposite signs.
This would also be the case with two identical conductors of any
given shape, which were made to touch, provided that they were
symmetrical at the point of contact.
If the condition of symmetry be not fulfilled, the charges are no
longer equal ; but their algebraic sum is always equal to that of the
original mass. This is a general fact, and applies to any number of
bodies, however they may be placed in relation to each other ; and
provided that none of the conductors are put, even for a moment, in
contact with the earth, the algebraical sum of the electrical masses of
the system remains the same.
9. ELECTRICITY OF CONTACT. Volta discovered this most impor-
tant fact, that the contact of two different metals, originally in the
neutral state, or more generally, of any two bodies at the same
temperature, is sufficient to place them in two different electrical
states, and to charge them respectively with equal quantities ot
electricity of opposite signs.
Friction is only a particular case of contact. The cause which
produces the electricity seems then to be the same in both cases.
It follows from Volta's discovery, that two conducting spheres of
the same radius would only have equal charges after contact, pro-
vided they were of the same material, and at the same temperature.
But this is no exception to the fundamental proposition, that the
algebraical sum of the charges is the same before and after contact.
10. ELECTRIFICATION BY INFLUENCE. INDUCTION. When a
INDUCTION ON AX CLOSED CONDUCTOR.
body, originally in the neutral state, is placed near electrified bodies
it becomes itself electrified ; the phenomenon is known as electrifica-
tion by influence or induction. If the body under electrical induction
is insulated, its total electrical mass, from what we have seen, must
remain zero. It will then be charged with two masses, equal,
and of opposite signs, distributed according to a certain law. The
phenomenon of induction always precedes the attraction of a neutral
body by an electrified one, and the action which is exerted is simply
that between the electrical masses. We may thus consider it as an
experimental fact that there is never any direct action, except that of
electrical masses on other electrical masses.
11. ELECTRICAL EQUILIBRIUM. The essential characteristic of
induction is that electricity is produced at every point of a conductor
at which electrical force is exerted. Equilibrium can therefore only
exist on a conductor, provided that the electrical force is zero at each
of its points ; the electricity which it possesses exerts, at each point of
its surface, an action equal and of opposite sign to that of the external
masses.
The necessary and sufficient condition for electrical equilibrium
in a system of conductors, insulated or not, is then that the electrical
force be zero at any point whatever of each of them.
12. DIELECTRICS. Electrical force can, therefore, only exist in a
state of equilibrium, on bad conductors, or insulators. For this
reason Faraday gave the name of dielectrics to these bodies, to denote
that they are bodies in which electrical forces may exist or be
transmitted.
13. LOCALISATION OF ELECTRICITY ON THE SURFACE OF CON-
DUCTORS. The experiments of Cavendish and of Coulomb showed
that in any electrical system in equilibrium, conductors have electricity
on their external surface only. The surface of any closed cavity
hollowed in a conductor, and not containing electrical masses, is
destitute of electricity, and the electrical force is null throughout the
above extent of the cavity. We shall find that this fundamental
property is only compatible with the law of inverse squares.
14. INDUCTION ON A CLOSED CONDUCTOR. This localisation
of electricity on the surface of a conductor leads to several important
consequences.
When a conductor is electrified by induction, each of the positive
and negative layers, with which it is charged, forms a mass less than,
or at most equal to, that of the influencing body, or inductor. When
the influenced, or induced, conductor completely surrounds the in-
ductor, the outer electrical layer is of the same kind as that of the
INTRODUCTORY.
inductor, and the charge at each point is independent of the position
of this latter. Nothing is altered, even if the inductor comes in
contact with the internal surface ; but then if it also is a conductor
it only forms with the induced body a single conducting mass, and
the internal surface retains no charge of electricity. There was,
accordingly, in the inside of the induced body, an electrical layer
equal and of opposite sign to that of the inductor, and hence on the
external surface a layer equal and of the same sign.
The quantity of electricity induced by an electrified body on a
conductor which completely surrounds it, is thus equal to the quantity
of inducing electricity. This property also holds if the inducing body
is a bad conductor, and more generally if the electrical masses are
distributed in any manner whatever in the cavity of the conductor.
15. ADDITION OF CHARGES. We have seen above that the
electrical charge of a conductor may be divided into two. In like
manner any electrical masses whatever may be added to a conductor.
It is sufficient for this if the conductor has a cavity almost entirely
closed through which electrified conductors may be introduced, and
which, by contact, transmit the electricities with which they are
charged to the outer surface.
16. We may then increase or diminish at pleasure, the algebraic
sum of the electrical masses contained in the interior of a closed
surface, provided we introduce, or give exit to, positive or negative
masses. It is, however, important to remark that, if no mass tra-
verses the surface in one direction or the other, whatever be the
actions to which the enclosed body is submitted friction, induction,
contact, physical or chemical actions, it is impossible to modify the
total quantity of electricity of the system. We can neither create
nor destroy, on any body, a determinate quantity of electricity, without
at the same time creating or destroying, on the same body, or on
another, an equal quantity of electricity of the opposite sign.
17. HYPOTHESES RESPECTING THE NATURE OF ELECTRICITY.
Electricity, defined and measured as we have explained above, is a
magnitude of a particular kind, perfectly definite from the mechanical
point of view, affected with a sign like a quantity of motion, and the
theory of electrical phenomena may be established from experimental
laws without having recourse to any hypothesis. From the facility
with which electricity is transmitted in conductors, it has often been
compared to a fluid, just as formerly the effects of thermal con-
ductivity were explained by the propagation of a special fluid. The
character of duality, which electrical phenomena present, has been
accounted for in two ways.
ELECTRICAL DENSITY.
According to Franklin, a -body in the natural state contains a
normal quantity of the electrical fluid, and it becomes positively or
negatively electrified, according as its charge of fluid is increased or
diminished by the action of external bodies. The attractions and
repulsions of bodies are explained by the mutual repulsion of the
fluids, and by the attraction which they exert upon ponderable
masses.
The hypothesis of two fluids, devised by Symmer, and adopted,
at any rate provisionally, by Coulomb, assumes that there are two
different fluids, that the molecules of the same fluid repel, and that
different fluids attract; and, finally, that in a body in the natural
state, there are equivalent quantities of the two fluids forming the
neutral fluid. A body is electrified positively or negatively, according
as it contains an excess of one or the other fluid. Attractions and
repulsions are explained in like manner by the actions which they
exert between the fluids and the ponderable matter.
It is the least defect of these hypotheses that they are superfluous.
As, moreover, experiment indicates no limit to the electrification of
a body, we are led to the conclusion that the normal charge of a
body on Franklin's theory, or that the mass of neutral fluid in the
theory of two fluids, is unlimited a conclusion that is manifestly in
contradiction with the notion of a material, fluid.
A certain number of expressions used in the study of electricity
have originated in the idea of fluids ; there is no inconvenience in
retaining them, if we are careful to define them by the mathematical
and experimental properties to which they correspond, with the
object, as Coulomb expressed it, " of presenting the results of calcu-
lation and of experiment with the fewest elements possible, and not
of indicating the true causes of electricity."*
18. ELECTRICAL DENSITY. The idea of a fluid has thus led to
that of electrical density. If the electricity occupies the whole
extent of a body, in the case of a dielectric for instance, and that it
is distributed uniformly, electrical density is the quantity of electricity,
defined as above, which exists in the unit of volume. If the distri-
bution is irregular, the density at a point is the ratio of the electrical
charge of an element of volume at the point to that of the volume
itself.
Conductors only possess electricity on the surface. If the distri-
bution is uniform, the superficial density is the quantity of electricity
which exists upon the unit of surface. In the case of any given
* Histoire de V Academic des Sciences pour 1788, p. 673.
8 INTRODUCTORY.
distribution, the superficial density at this point, is the ratio of the
charge of an element of surface taken about this point to the extent
of the element.
On the hypothesis of fluids, it must be admitted that the layer of
electricity on the surface has a certain thickness, and that it penetrates
to a certain depth, which may be extremely small, in the conductor,
or in the dielectric which surrounds it. As the thickness of this
layer cannot be determined by experiment, holding to the same order
of considerations, we may either suppose the density to vary with
constant thickness, or suppose the density constant with variable
thickness ; in this case, the expressions electric density and electric
thickness at a point are equivalent.
It will be seen that, apart from any idea of fluid, the expressions ot
electric density in volume, or superficial density, have a purely mathe-
matical or experimental meaning quite apart from any hypothesis.
DEFINITION --OF POTENTIAL.
CHAPTER II.
ON POTENTIAL.
19. WE shall assume, in the first place, in conformity with experi-
ment, that the action of two electrified bodies, of small dimensions,
takes place along the straight line joining them, and only varies with
the distance ; that it satisfies, in short, the definition of what are
called central forces ; and, lastly, by definition (4), that the action is
proportional to the product of the quantities of electricity which
the bodies possess.
We shall assume, moreover, that the reciprocal action of two
electrified bodies of finite dimensions is the resultant of the
actions which would be exerted, according to the same function
of the distances, between the elementary masses which make
up the charge.
20. ELECTRICAL FIELD. The term electrical field is applied to
the entire extent of the space throughout which the action of any
given electrical system is exerted. An electrical field is generally
unlimited ; it may be bounded in the case, for instance, in which all
the acting masses are inside an entirely closed conductor. For
masses whose magnitude and position are defined, the electrical force
at each point of the field is merely a function of the co-ordinates of
the point.
The force is zero in all conductors in a state of equilibrium ; the
electrical field does not comprise the volumes of conductors it is
formed of intermediate spaces occupied by an insulating medium or
dielectric.
21. LINES OF FORCE. A line of force in an electrical field is a
line tangential at each point to the direction of the force. Such a
line is obviously continuous, so long as it does not encounter acting
masses.
22. DEFINITION OF POTENTIAL. Consider a system in equi-
librium, and suppose that all the acting masses being fixed in their
several positions, we move unit mass of positive electricity from
10 ON POTENTIAL.
A to B. The work of the electrical forces which corresponds to this
displacement, is independent of the path following in passing from
A to B. This is a necessary consequence of the hypothesis (19) that
the forces are central ; for if it were otherwise, it is obvious that, by
moving an electrical mass on suitable paths between the points
A and B, we might produce an indefinite quantity of work, without a
corresponding expenditure. The work in question only depends,
then, on the co-ordinates of the points A and B ; it is equal to the
difference of the values V A and V B which the same function V has at
these two points, and representing this work by W, we may write
(0 W>V A -V B .
The function V plays a paramount part in the study of electrical
phenomena ; it has been called potential. As this function is only
denned by an integral, its value is only determined to within a
constant, and the variations are measured by the electrical work.
From equation (i), the excess of the potential at a point A over the
potential at B, is equal to the work done by the electrical actions on unit
mass in passing from A to B ; or conversely, it is the work which
must be expended against electrical force to move this mass from B to A.
If unit mass moves along a line of force, the work for an infinitely
small displacement ds is fds t and the total work from A to B is
expressed by
(2) W'
23. EQUIPOTENTIAL SURFACES. ELECTROMOTIVE FORCE. A
level surface, or equipotential surface, is a surface perpendicular at
every point to the direction of the force ; that is to say, a surface
which is perpendicular to all the lines of force which it meets. In
the case of central forces, a surface satisfying this condition can
always be drawn through any given point. If an electrical mass
moves along such a surface, the elementary work is constantly zero,
for the force is always perpendicular to the displacement. The
potential has the same value for all points of the same electrical level.
Let us consider two equipotential surfaces, ! and S 2 , whose
potentials are respectively V 1 and V 2 . The work corresponding to
the displacement of unit mass from a point of the former to a point
of the latter, has the value V x - V 2 ; it is independent of the path
traversed, and even of the position of the point of departure from,
and of the point of arrival at, the two surfaces.
EQUI POTENTIAL SURFACES. II
The work done by a mass x ?# of electricity in passing from the
equipotential surface S x to the surface S 2 , is ;// (V 1 - V 2 ). Electrical
work, like that of gravity on a falling body, appears as a product of
two factors, one m, which corresponds to the weight of the body, and
the other, V l - V 2 , to the height of the fall.
When a mass of positive electricity is left to itself, it tends to move
along a line of force towards the points where the potential is lower ;
negative electricity would move towards high potentials.
If the electrical masses are distributed on dielectrics, they can only
be displaced by carrying along with them the dielectric itself. Con-
ductors, on the contrary, are characterised by the property of allowing
a free passage to electrical masses, which go to the surface, and
distribute themselves there so as to produce equilibrium.
In all cases, the difference of potential V t - V 2 may be considered
as producing the motion of electrical masses ; it is often called the
electromotive force.
24. EXPRESSION OF FORCE AS A FUNCTION OF POTENTIAL.
Consider two infinitely near equipotential surfaces S and S', whose
potentials are V and V (Fig. i). At the point M of the former
surface the force is F ; if dn is the distance of the two surfaces
measured along the perpendicular, the work done by this force on unit
mass in going from M to M' is equal to ~dn. We have then the
equation
which gives
"--
Thus, the force at a point is equal, and of opposite sign, to the
differential of the potential, in reference to the perpendicular to the
equipotential surface which passes through this point.
12 ON POTENTIAL.
The components of the force with reference to any given axis
possess the same property. For let a line MA be drawn through
the point M, making an angle with the perpendicular, and let da
denote the portion of this line between the two surfaces S and S'.
The component F a of the force parallel to the straight line is thus
expressed :
//V
F =Fcos0 = -V-cosl9.
dn
The figure gives moreover,
dn = da cos 6
from which we get,
_^V dn_ = _<)V
a ~ dn da" la'
Thus, the component of the force in any given direction is equal,
and of opposite sign, to the partial differential of the potential along
this direction.
If we consider three rectangular axes, the components X, Y and
Z of the force parallel to the axes are,
(4)
from which we have,
(5)
25. EQUILIBRIUM OF CONDUCTORS. In the interior of a con-
ductor in equilibrium the force is zero (11). Hence for the whole
surface of the conductors we have,
dV dV 3V
- = 0, -- = 0, = 0,
OX oy 02
and consequently,
V = constant.
It follows from this, that the whole volume of a conductor
in equilibrium is at the same potential ; this is what may be
called a level volume. Its surface being then a level surface, the
NUMERICAL VADUE OF POTENTIAL. 13
force is perpendicular at all' points ; hence, the lines of force
proceed perpendicularly from the conductors, or terminate there
perpendicularly.
26. NUMERICAL VALUE OF POTENTIAL. In all these phenomena,
equipotential surfaces are only apparent as differences, and not as
the absolute values of the corresponding potentials. We may
accordingly add to these potentials any given constant.
In the expression
which represents the work corresponding to the displacement of a
unit of electricity from an equipotential surface V 1} to an equi-
potential surface V 2 , let us suppose that the latter is the earth, and
that we agree to take its potential as equal to zero, we shall have
The numerical value of the potential at any given point, is the
number of units of work which corresponds to the displacement of a
Fig. 2.
unit of positive electricity from this point, to the earth, by any path
whatever. The sign of the potential is that of the work of the
electrical forces in this displacement.
In other words, the potential at a point, is the work which must
be done to bring unit mass of electricity from the earth, or from a
body in connection with the earth, to this point.
27. POTENTIAL IN THE CASE OF THE LAW OF THE SQUARE OF
THE DISTANCES. We have hitherto left undetermined the law,
according to which the action of electrical masses varies with the
distance. We shall assume in future that this law is the inverse of
the square of the distance, conformably to the experiments of Coulomb.
In this case, the potential is expressed simply as a function of the
masses and of the distances.
Let us suppose, in the first place, that the electrical system is
reduced to a mass + m placed at a point O. If a mass equal to
unity placed at a point M (Fig. 2) at a distance r from the former, is
14 ON POTENTIAL.
moved through MM' or ds, along any curve whose tangent MT
makes the angle a with the direction of the force, the corresponding
electrical work is
since the force is expressed by
we have
If rj and r 2 represent the distances OA and OB, the work of
displacing unit mass from A to B is
B m m
W. = .
*1 >2
Comparing this equation with equation (i) we see that the two
terms and represent respectively, to within a constant, the value
r \ '2
of the potential at A and at B ; hence the potential of a single mass
m at a point at distance r is equal, except for a constant, to }
that is, to the quotient of the acting mass by its distance from the
point in question.
Let us now suppose that there are several acting masses m, m\
m", . . . , the total work of the displacement of unit of electricity is
equal to the algebraical sum of the partial works corresponding to
each of the masses ; denoting then by ^ the sum of the quotients
of the different masses by their distances from the point of departure
A, and by the analogous sum for the point of arrival B,
r z
If the point B is in connection with the earth, the potential V B is
zero. On the other hand, the expression Jf becomes zero, if the
r 2
point B is at a great distance from the masses in question, whether
in the air or on the ground ; and since the earth, like any conductor
ON THE FLOW OF FORCE.
in equilibrium, has everywhere the same potential, this expression,
which implicitly contains masses relative to induced electricity, is also
zero on the ground. The value of V A is then reduced to ^ .
In general, then, to express the potential V at any point of the
field, we have
(6)
^-<m
47-
Thus, the potential at a point is equal to the sum of the quotients
obtained by dividing each of the acting masses by its distance from
the given point.
28. ON THE FLOW OF FORCE. Let A (Fig. 3) be an equi-
potential surface, and </A an element of this surface at the point P.
Through the contour of this element let lines of force be drawn
Fig. 3-
which cut all the successive equipotential surfaces at right angles.
An orthogonal canal thus bounded by lines of force is called a tube
of force.
The elementary tube of force, bounded by the contour of the
element */A, cuts from any surface S passing through the point P
an element of surface d.
In P the electrical force F is perpendicular to dK; let F n be the
component of this force parallel to the perpendicular to the surface S ;
the angle of the two forces being equal to that of the two elements
of surface, we have
whence
I 6 ON POTENTIAL.
If we suppose that a liquid, in the permanent state, traverses the
element dK at right angles, with the velocity F, the product FdA
represents the volume of liquid which flows through the element dA
n unit time, or, more briefly, the flow of liquid corresponding
to this element. This flow is also expressed by the product F n dS,
obtained by multiplying any section of the tube by the component
of the velocity along the perpendicular to this section.
By analogy, we shall apply the term quantity of force, or flow
of force corresponding to an element of surface, to the product F n dS
of the surface of the element, into the component perpendicular to the
force at this point. The flow of force for unit surface is equal
numerically to the perpendicular component of the force.
The properties which we are about to examine will completely
justify this analogy.
Fig- 4-
The idea of lines of force is due to Faraday, and this eminent
physicist showed all the advantages which may be derived from it in
the study of electrical phenomena. What we have designated a
quantity or flow of force, Faraday called number of lines of force.
It seemed useful to adopt another designation, in the first place for
simplicity of expression, and secondly because the word flow seems
to correspond better with the character of continuity which we want
to determine.
29. GREEN'S THEOREM. Let us consider in the dielectric an
entirely closed convex surface S (Fig. 4), and let + m be an electrical
mass situate at the point O outside this surface An infinitely slender
cone of aperture du, having its apex at O, cuts the elements d$
and ^S' in this surface. Let /and/' be the values of the electrical
force at dS and dS',dA and dA' the corresponding perpendicular
GREEN'S THEOREM.
sections of the cone, and r and r' their distances from the point O.
We have then
/ r i _/>* = ,
and multiplying these equations by each other, we get
but in virtue of the theorem of the flow of force we have also
and, therefore,
f n dS=f n dS'
If we agree to consider as positive, the perpendicular components
Fig. 5-
directed towards the exterior of the surface, and as negative those
directed towards the interior, f n and f' n are of opposite signs, which
gives
If the surface, while still continuous, had concave portions, and
if the cone in question da) cut it in more than two points, it would
meet it an even number of times ; the product f n dS would have the
same numerical value for each of the intercepted elements, but these
products would have to be taken alternately of opposite signs, and
the algebraical sum would still be zero. We have, then, for any
closed surface external to the acting mass m, the equation
1 8 ON POTENTIAL.
that is to say, that the total flow of force which starts from the surface
is equal to zero.
If the acting mass m is within the closed surface S (Fig. 5), the
elements dS and ^S', cut by a cone of aperture d^ starting from the
mass m, always gives the ratio
But in the present case, the perpendicular components f n and/' n
are of the same sign. We have thus for the whole surface,
/ n d?S = w du =
The flow of force which proceeds from a surface S, enclosing an
acting mass, is thereby equal to ^irm. In other words, we may say
that the total flow of force which issues from a mass m t in all
directions^ is equal to
Fig. 6.
It is clear that if each sheet of the cone meets the surface more
than once, it meets it an uneven number of times, for which the
values off n dS should be taken alternately of opposite signs, and the
final result is still the same.
30. Let us now suppose that there are masses m, m', m", . . . com-
prised within the surface S (Fig. 6), and other masses m v m^ m 3 , . . .
on the outside.
At each point of the surface, the perpendicular component F* of
the resultant force F, is equal to the algebraical sum of the perpen-
dicular components of the forces proceeding from all the acting
masses, both internal and external.
Calling 2 fn tne sum J at a P ^ of the perpendicular com-
ponents which arise from the external masses, and fn> tne sum
i
GREEN 'S THEOREM. 1 9
of the components relative to thfe internal masses, the total flow of
force which proceeds from the surface S is expressed by
hvfs= |V/.*s+ p./
Since the flow of force is null for each of the external masses,
we have
For each internal mass, on the contrary, we have
J/ W </S =
and consequently
M being the algebraical sum of all the internal masses.
We have, finally,
(7)
Hence, for any closed surface, drawn in any manner in an electrical
field, the total flow of force, which proceeds from the surface that is, the
excess of the flow of forces which emerge, over the forces which enter, is
equal to the quantity of electricity, comprised within the surface, multiplied
by 47T.
If i be the angle which the direction of the force F at any point
of the surface makes with the perpendicular, the perpendicular com-
ponent is expressed by
F n = F cos i.
If a be the perpendicular to the equipotential surface which
passes through the point in question, n the perpendicular to the
surface S measured towards the exterior, we have, further,
F=-^ and F = - ;
da ^n
c 2
20 ON POTENTIAL.
so that the preceding theorem may be expressed analytically by either
of the equations
Vv
- | cos / dS = 4?rM,
(8)
which are due to Green.
31. EQUATIONS OF LAPLACE AND OF POISSON. Let X, Y and Z
be the components of the force at a point P, whose co-ordinates are
x, y, z, and let us consider the element of volume dx t dy, dz.
If the medium contains the acting masses distributed in a con-
tinuous manner, and if ^M is the total mass contained in the element,
then, denoting the density by p, we have
dM = pdxdydz.
The flow of force which enters by the surface dydz, passing
through the point P, is
Jidydz;
the flow of force which emerges from the opposite face is
/ 3>X \
( X + dx \dydz.
\ ^ x /'
The excess of the flow which emerges is equal to
dX <> 2 V
- dxdydz= --dxdydz.
^x ^x 2
Repeating the same reasoning for the other co-ordinates, it will
be seen that the total flow of force which proceeds from the element
of volume is
/W c) 2 V 3 2 V\
~ ( ^-T + ^TV + TV \dxdydz.
\J)x 2 <Vy 2 c) 2 I
In virtue of the preceding theorem, this excess is equal to the
product of 4?r by the total mass of electricity comprised in the
volume, which gives the equation
DISTRIBUTION OF ELECTRICITY ON CONDUCTORS. 21
Representing by AV the sum 'of the second partial differentials
of the potential in reference to the co-ordinates, we may write :
(9) AV=-47i7>.
If the element of volume is not electrified, p = and
(10) AV = 0.
Thus, the sum at a point of the three second partial differentials
of the potential in reference to three rectangular axes is equal, and
of opposite sign, to the product of 4?r by the density of the mass acting
at this point.
This sum is zero when there is no electricity near the point.
This theorem of the second differentials was first enunciated by
Laplace in the form (10). The more general equation is due to
Poisson.
32. If the equipotential surfaces are concentric spheres, the
force F is inversely as the square of the distance r from the
common centre, and we have
F __^X_A c) 2 V 2 A 2 F
~^7~^' ^*-z--
Taking the z axis along the perpendicular to the surface, the
two others will be in the tangent plane ; if we measure the distance
in the direction of the force, we get
2F
and therefore, by Laplace's theorem,
If the equipotential surface is of any given form, it is readily
seen that the second differential of the potential along the tangent to
a principal section is the same as for the osculating circle. The z axis
being always perpendicular, and the two others along the tangents to
the principal sections, whose radii of curvature are R x and R 2 , we
shall have
W _ F W _ F
^2 = R/ "Sy^R^'
and, therefore,
22 ON POTENTIAL.
33. DISTRIBUTION OF ELECTRICITY ON THE SURFACE OF CON-
DUCTORS. In any conductor in equilibrium, electricity is present
on the surface only.
For we have seen that the force is zero in a conductor in
equilibrium, and that therefore the potential has a constant value
for the whole surface of the conductors. All orders of differentials
of the potential are zero at each point, and we have then
AV = 0, or p = 0.
Hence, in the interior of a conductor in equilibrium, not only
is there no electrical force, but there is no electricity at all. The
distribution is exclusively on the surface.
34. GREEN'S FORMULA. The formula relative to the flow of
force gave, for a closed surface, the equation
- .
I on
J
The mass M, inside the surface, is equal to the sum of the
masses pdv comprised in the different elements of volume ; taking
equation (9) into account, we have then,
(n) </S= - \47rpdv= AWz>.
This equation is a particular case of a more general formula
due to Green.
Let U and V be two finite and continuous functions of x, y,
and z. Let us also put
AV = - + - + -,
and consider the integral
J
IV
UAW?;= | | | U( ^ + ^ + T r ) dxdydz,
ox oy oz I
/
extended to the volume enveloped by a closed surface S. This
integral consists of three terms of the form
U dxdyd*=\ \dydz\V~dx.
GREEN^ FORMULA. 23
Integrating this expression 5y parts, we have
dx = U dydz- \-d
~ " " ~
The first term of the second member should be extended to the
whole surface S, and the second to the volume bounded by this
surface. Repeating this operation for the other co-ordinates, the
sum of the integrals relative to the surface S will be
dydz-\ -- dzdx-\ -- dxdy ).
x "^ "N. "X -^ I
ox oy 02
Now, if we consider V as a potential, which does not restrict the
<)V
general character of the demonstration, the expression -- dydz, or
, represents the flow of force through the surface element dydz;
that is to say, the projection of a surface element d$ on a plane
perpendicular to the axis x. It is the same for other terms, so that,
except for the sign, the parenthesis represents the excess of force
which traverses this element of surface. This parenthesis is thus equal
av
to - F n <afS, or ^S, and we have, finally, Green's formula :
C f
UAW?;=
J J
TTc
(12) UAW?;= U </S-
uav ww wav
. + -- + . I
Making U = i, we obtain the preceding equation (u).
When the functions U and V are identical, we have
fa*- fv
J J ^
If the function V represents the potential of an electrical system,
the force F is determined by the equation,
3*
which gives
(14)
// f*
VAV^= V ^S - PV0,
~bn
/ */
24 ON POTENTIAL.
or, from Poisson's theorem,
(15)
f f 5V f
- Vp<fo= - V a?S+ F 2 <fo.
J J 3 J
35. TUBES OF FORCE. We have seen that a tube of force is an
orthogonal canal with equipotential surfaces, bounded laterally by
lines of force.
Let us consider a tube of force terminated by any two surfaces
S and S' (Fig. 7), and let us apply equation (7). to the volume thus
denned. The lateral surface does not enter into the integral, for
the perpendicular component of the force at each point is zero ;
the integral is reduced therefore to the two terms furnished by the
terminal surfaces.
Let us first suppose that the tube contains no electrical mass.
Fig. 7-
The integral reduced to the two terms of the bases must be zero,
which gives
F^S +
J
or, in absolute values,
J
n </S= f
that is to say that the flow of force is then the same at both ends of
the tube.
It thus appears that the flow of force is the same across any
section of the tube. This flow is kept up like the supply of a
moving liquid, whose velocity at each point is equal and parallel
to the direction of the force.
If the section of this tube is infinitely narrow, and if the
terminal surfaces are perpendicular to the force, the equation
reduces to
COULOMB^ THEOREM. 25
from which it follows that the force at each point of the tube is inversely
as the section.
36. COULOMB'S THEOREM. J^et us consider any elementary
surface </S, on a conductor in equilibrium, and let us draw the
corresponding tube on the outside until it meets an infinitely
near equipotential surface S t . Let us prolong the tube inside the
conductor in any manner, terminate it by any given surface S 2 , and let
us apply the theorem to the volume bounded by the surfaces Si and
S 2 , and the lateral surface of the tube. The force is zero on the
whole surface S 2 which forms part of the conductor, and the
perpendicular component is zero on the lateral surface of the tube ;
there would therefore only be flow of force for the exterior surface
</S. If we denote by a-, the surface density of the electricity on the
element */S, the total mass comprised in the volume in question is
ov/S. We get thus
The two surfaces being infinitely near, and parallel, we have
^S 1 = ^/S, and
F = 47TCT.
Thus, the electrical force at any point infinitely near a conductor
in equilibrium, whatever be the masses in action, is equal to the electrical
density close to this point multiplied by 471-.
Since, moreover
it follows that the density at the surface of a conductor may be
expressed as a function of the external force, or of the potential, by
the ratio,
-__ _L^X
4?r 477 dn
37. CORRESPONDING ELEMENTS. Let us lastly consider an
infinitely narrow tube of force placed between the surfaces of two
conductors, at which it terminates perpendicularly.
The two surface elements ^S and */S', cut by the tube, are called
corresponding elements.
Let us prolong the tube on both sides in the interior of the
conductor, where we will suppose it terminated by any two surfaces.
The flow of force is zero on the whole surface of the volume thus
determined ; for the force is tangential along the sides, and is zero at
26 ON POTENTIAL.
the two ends which form part of the conductors. If cr and a-' are the
densities on the corresponding elements d and */S', their sum is
This sum must be zero, which gives,
Thus two corresponding elements contain quantities of electricity
which are equal and of opposite signs.
If the two surfaces opposite each other are parallel planes, all the
lines of force are perpendicular to the planes ; the corresponding
elements are equal, and we have
cr = cr'.
This is also the case if the two conductors, though not plane,
are infinitely near, for then the surface elements ^S and d$' may be
considered equal.
f
38. UNIFORM FIELD. When the lines of force are straight and
parallel, the equipotential surfaces are parallel planes. The flow of
force being constant in a cylindrical tube, it will be seen that the
value of the force is constant. The field is then said to be uniform.
Conversely, if the force in an electrical field is constant in
magnitude and direction, the equipotential surfaces are necessarily
parallel planes.
39. ELECTRIFIED SURFACE SEPARATING Two DIELECTRICS.
Let us suppose that an electrified surface S (Fig. 8), not belonging
to a conductor, separates two different dielectrics.
Consider an elementary surface ^S, on which the density
is cr. Draw through the contour of this element a straight tube
terminated by two equal bases, and parallel to dS, and of such a
height that the lateral surface is infinitely small as compared with
that of the bases. The flow of force relative to the sides of this
cylinder may be neglected as compared with the flow which traverses
ELECTRIFIED SURFACE SEPARATING TWO DIELECTRICS. 27
the bases. If F and F' are the forces in the two media near this
element, we shall have
= F' n </S - F n </S,
whence
Hence, the difference between the perpendicular components of the
force on the two sides of an electrified surface is equal to the product of
4?r by the density on the surface.
This theorem includes that of Coulomb (36) as a particular case,
in which S is the surface of a conductor.
40. Let us consider two points P and P' in the middle of the
bases of the preceding cylinder. The action F at the point P is
made up of the action of all the masses external to the element, and
of the action - </> of this element. In passing from P to P' the action
of the external masses does not appreciably change, but that of the
element changes its sign, and becomes + <. As this force, moreover,
is perpendicular by symmetry, the perpendicular component of the
force varies by 2<, which gives
whence
(f> = 27TCT.
As the perpendicular component of the force is alone modified,
the tangential components of the forces F and F' are equal on both
sides the surface. If / and /' are the angles of the forces with the
perpendicular on the same side of the surface, we have then
F' sin / ' = F sin i.
The preceding equation gives, moreover,
F' cos /' - F cos / = 47TO- ;
from which we have
tan i' 47TO- 477-0-
r ~ .
tan i P cos i F
The forces undergo therefore a sort of refraction on meeting an
electrified surface. It may be remarked in passing, that as the law of
refraction is determined by the ratio between the tangents of the
angles of the perpendicular and the forces, there could never be any
phenomenon analogous to total reflection.
28 ON POTENTIAL.
41. ELECTROSTATIC PRESSURE. Electricity forms, on the surface
of a conductor, a very thin layer whose thickness it does not seem
possible to determine by experiment, but which is necessarily
limited. It is probable that this layer is restricted to the surface
of the conductor itself, and that it occupies part of the surrounding
dielectrical medium.
Let A B (Fig. 9) be the thickness of this layer. The force is zero
at A on the internal surface S 15 and starting from the point B on the
outer surface S 2 , its value is F = 473-0-. In the interval the force varies
from to F according to an unknown law.
Let p be the density at a point M, and V the potential. The
force at this point is perpendicular to an equipotential surface which
dV
lies between Si and S 2 , and has the value - - . Let e be the
an
Fig. 9.
thickness AB. What has hitherto been called surface density
represents the quantity of electricity on the layer e for unit surface ;
we have then
0- = I pdn.
Let us take three rectangular axes, one of them in the same
direction as the perpendicular at M to the equipotential surface,
that is the perpendicular to the surfaces S t and So, and the
two others of which, x and jy, are in the tangent plane of this
equipotential surface.
7)2y yy
The second differentials and r r being zero (32), the density
ox' 2 oy*
is expressed by,
ELECTROSTATIC PRESSURE. 2 9
Hence, if p be the force on unit surface, the total force exerted
on the electrical layer (n/S of an element of surface is
n -- =- .~dn = -
As -3 is zero at the point A, we have simply,
dn
On the other hand,
F = 47T(7,
which gives
p = - 1 67T 2 CT 2 = 27TO- 2 .
O7T
It is remarkable that this expression might be obtained strictly
without any other hypothesis than that of an extremely small
thickness e, and therefore whatever be the law of distribution along
the perpendicular.
The electricity spread upon each unit of surface is thus impelled
towards the exterior with a force equal to 27r<r 2 , proportional therefore
to the square of the density. This electrostatic pressure, or electrical
tension is counterbalanced by the resistance of the dielectric.
When the conductors are in air, the effect of this force is
to diminish the atmospheric pressure on their surface. If the initial
pressure for unit surface be P in the neutral state, after electrification,
it will be
at each point , of the surface of the conductor.
Thus, an insulated soap bubble will increase in volume when
electrified, and will resume its original volume when it is restored to
the neutral state. Van Marum, for example, found that a balloon
filled with hydrogen became lighter, and that its ascensional force
increased, when it was electrified.
If the density is uniform on a conductor, the same is the case
with the electrostatic pressure, and as this is perpendicular at every
point, its resultant is zero.
If the density is variable, the electrostatic pressure is in general
equivalent to a single force and to a couple. The result thus obtained
30 ON POTENTIAL.
is clearly identical with that arising from a direct consideration of
the action of external masses, supposed to be attached to ponderable
matter, on the various electrical masses of the conductor.
42. CONSEQUENCES OF THE DISTRIBUTION OF ELECTRICITY ON
THE SURFACE OF CONDUCTORS. The preceding theorem is based on
Coulomb's law considered as an experimental fact. We might follow
a different course in taking as starting point this other experimental
fact, which is more easily verified with strictness, that electricity exists
solely on the surface of conductors, and that in the interior there is
neither electricity nor electrical force, even when the conductors contain
closed cavities.
Let us consider an electrified surface S (Fig. 10). Through a
point in the interior P draw a cone du of infinitely small aperture,
Fig. 10.
which intercepts, on the surface, at the distances r and r' two elements,
dS and */S', the densities on which are respectively o- and a-'.
If the electrical forces are proportional to a function of the
distance / (r), the action of the element </S on unit mass at the
point P may be written,
Calling / the angle of the radius vector r with the normal to the
element dS, we have
r 2 do) = d$ cos /,
which gives
The action of the element d$> is, in like manner,
and these two actions are directly opposed.
DISTRIBUTION OF ELECTRICITY ON CONDUCTORS.
If the surface in question is a sphere, the angles / and /' are
equal. If, moreover, the sphere' is insulated, and not exposed to
any extraneous action, the distribution is homogeneous, and the
densities o- and </ are equal.
The force at the point P is zero if the actions of the opposite
elements dS and dS' are equal, and for this it is sufficient if we have
the ratio
= const,
that is to say, that the electrical forces are inversely as the square of
the distance.
The law of the square satisfies this condition, and is the only
one which does. This may be shown in a simple manner by the
following reasoning, which is due to M. Bertrand.
Whatever /(r) may be, we may choose two values, r^ and r^
Fig. ii.
such, that between these two values of the variable the product r 2 f(r)
always either increases or decreases as r increases.
Let us construct a sphere (Fig. n) whose diameter is equal to the
sum r l + r zt and let us consider the point P which divides the
diameter into the two segments r^ and r y The actions of the opposite
elements dS and dS', determined by the same cone of aperture dw on
this point, are : for */S,
cos t
and for
COS I
All the values of r and r' are comprised between the limits r^ and
ON POTENTIAL.
?* 2 , and the value of r is always less than the corresponding value
of r' ; the value of r*f(r) for the elements of the upper portion will
always be smaller or larger than that of r' 2 f(r') for those of the lower
portion ; the action of the zone D AC is therefore smaller or larger
than that of the zone DEC. Equilibrium would thus be impossible
unless r*f(r) is a constant that is, unless the function f(r) were
exactly in the inverse ratio of the square of the distance.
43. ACTIONS OF SPHERICAL LAYERS. The action of a homo-
geneous spherical layer on an external point is the same as if the whole
mass were concentrated at the centre of the sphere.
For let us consider the action which a sphere S (Fig. 12), covered
with a homogeneous layer of density <r, exerts upon an external
point P.
Fig. 12.
The action being obviously directed towards the centre, it will be
sufficient if we take the sum of the components of the elementary
actions in this direction.
This component for the surface element dS at the point A, which
is at a distance p from P, is
<p = COS a.
P 2
Let P' be the conjugate of the point P; that is to say, such that
OP'.OP = R 2 ;
if we draw AP' and AO, the triangles AOP' and AOP are similar,
for the angle at O is common, and we have the ratio
OF R
ACTIONS OF SPHERICAL LAYERS. 33
hence the angles OPA and GAP' are equal ; calling r the
distance P'A, and D the distance OP, we have
P = V
r R*
Lastly, let du be the angle which subtends the element dS, seen
from the point P',
dS cos a.
Replacing dS cos a by this value in the expression for the com-
ponent <, we get
R 2
y = " ^D 2 '
and the total action of the sphere is
IT 47rR 2 o- M
"D 2 D 2 '
It is thus apparent that the action is the same as if the whole mass
M were concentrated at the centre of the sphere.
The potential of the spherical layer on the outside is also the
same as if the whole mass were concentrated at the centre.
If the point P is very near the surface, D = R and the action of
the layer is equal to 4770-, in agreement with what we have already
found (36) for any given conductor.
The component, parallel to OP, of the action exerted by the
surface element dS, only depends on the angle da under which this
element is seen from the point P'. This component is then the
same for an element dS' situate at A', and opposed to the former in
reference to the point P'.
The same is the case for all the elements of the zone CB'C'
compared in pairs with the elements of the layer CBC'.
The plane CC' thus divides the surface of the sphere into two
parts, whose actions on the point P are equal, each of them being
R 2
equal to 2-rra-
If the point P moves to an infinite distance, the two zones tend to
become equal ; if it is infinitely near the surface, the anterior zone
becomes infinitely small, and its action upon an infinitely near point
is reduced to 2770-. We have already obtained this result (40) for
any given surface.
D
34 ON POTENTIAL.
44. ACTION OF A SPHERE CONSISTING OF HOMOGENEOUS
LAYERS. Let us first consider a sphere electrified throughout its
whole mass, and made up of homogeneous concentric layers.
The action of this sphere on an external point is the same as
if the whole mass were concentrated at the centre, or were carried
to the surface so as to form a uniform layer.
On a point in the interior of the sphere the action of the layers
which surround it is null ; that of the layer whose radius is less
than its distance from the centre is still the same as if the mass
were concentrated at the centre.
Proceeding towards the centre the acting mass diminishes then
more and more; the direction of the force is always along the
radius, and depends on the manner in which the density varies.
When the density p of the sphere is constant, the action exerted
on a point in the interior, at the distance r t is equal to
it is thus proportional to the distance from the centre.
If the density at a point is proportional to the n th power of its
distance / from the centre, p = al n ; the action of the layer dl at the
distance r is
and the total action
n
n + s
This action is constant for n = - i ; it increases, on the contrary,
approaching the centre, if n < i.
The total mass of the sphere of radius R is
47T#
which gives
r
M= ^irfial
J
Rn+3
Let us suppose that the density varies according to an arbitrary
law. Let m be the mass external to the sphere which passes through
ACTION OF A SPHERE. 35
the point P v /> the mean density of the whole sphere, and p the
mean density of the external layers, we shall have
M
The force at the surface is F x = , which gives
r, ^
The force may at first be an increasing one on starting from the
surface, then attain a maximum, and then go on decreasing to the
centre. This, for instance, is the case with the variations of gravity
in the interior of the globe. For this we must have
If the thickness h = R r is very small as compared with R, this
condition may be written
(r\^ h p r h p 2
\ <2 , or <-<o'67.
The mean density of the Earth being about equal to 5-5, and
that of the surface to 2-5, we have, in fact,
P 2 '5
= = 0-45.
ft) 5'5
D 2
36 GENERAL THEOREMS.
CHAPTER III.
GENERAL THEOREMS.
45. EMISSION AND ABSORPTION OF FORCE BY ELECTRICAL
MASSES. The flow of force through an orthogonal tube remains
constant, as we have seen (35), so long as the tube does not
encounter any acting mass ; the direction of the transmission is
that in which the potential diminishes.
If the tube encounters a mass of electricity m, the flow of force
experiences an increase qxm at the boundary, which it retains
beyond it as long as the tube does not encounter fresh masses. If
the mass thus encountered is on the surface of a conductor, it is
such as will reduce to zero the flow of force transmitted by the
tube. We may then say that of two corresponding elements dS
and dS', whose densities are a- and o-', the positive electricity of the
element dS emits a flow of force 471-07/8, which at the other end
of the tube is absorbed in the equal quantity of negative electricity
or) the element dS'.
On Faraday's views there are no unlimited tubes of force. A
tube proceeding from an electrified body would always terminate
somewhere on another body, so as to induce on the corresponding
element an equal quantity of electricity of the opposite sign. On
this view, there could nowhere be an absolute and independent
quantity of electricity which has not its complementary quantity at
the other end of the tube.
No line of force can exist between two points charged with the
same electricity. Nor can there be any between two points at the
same potential. In fact, no line of force can correspond on a
conductor to a point not charged with electricity (36).
46. THE POTENTIAL CAN NEITHER HAVE A MAXIMUM NOR A
MINIMUM OUTSIDE THE ACTING MASSES. An insulated mass m
concentrated in a point may be regarded as a layer spread over a
very small conductor. If the mass m is positive, the point A, which
POINTS AND LINES OF EQUILIBRIUM. 37
it occupies, is a centre of the emission of force in all directions ;
if it is negative, it is a centre Of absorption for all directions. In
both cases the adjacent equipotential surfaces are closed surfaces.
In the limit the surfaces are spherical even; for at a small
distance r from the point A, the potential of the external masses
may be neglected as compared with the potential - of the mass m.
The equipotential surfaces being closed about A, the potential
has a maximum or minimum value ; maximum, if the mass m is
positive; minimum, if it is negative.
Conversely, there is electricity wherever the potential is
maximum or minimum.
For, starting from a point A of maximum, the potential decreases
in all directions, the adjacent equipotential surfaces are necessarily
closed, and the flow of force which traverses, for example, a very
small spherical surface comprising the point A, has a finite value
Q. In the interior of this surface there is therefore a quantity
of positive electricity equal to .
4?r
In like manner, a minimum of potential is a centre of absorption
of force, where there is a corresponding mass of negative electricity.
Hence, in any given electrical system, there can neither be absolute
maximum nor absolute minimum df potential outside the acting masses.
47. POINTS AND LINES OF EQUILIBRIUM. Let V be the
potential at a fixed point P . The potential V at an adjacent point
P, whose co-ordinates referred to axes passing through the former
are x, y, z, may be expressed as a function of increasing powers of
the co-ordinates,
H 15 H 2 ---- H n being homogeneous functions of the first, of the
second ---- of the n th degree of the co-ordinates.
As the equation of Laplace should be satisfied separately by
each of these functions, we shall have in general
If the function H } is identical with zero, that is to say, if the
three partial differentials of the potential
p
-
38 GENERAL THEOREMS.
are null at the point P , this point is a singular point of the level
surface V ; the force there is null, it is a point of equilibrium.
For adjacent points we may neglect the powers of the
co-ordinates higher than the second, and the expression of the
potential V reduces to
The equation
represents a cone of the second degree tangential to the equi-
potential surface at the point of equilibrium P .
If the function H 2 is itself identical with null, as well as some
of the following ones, and if H n is the first function of the
development, which does not vanish, the equation
H =0
will represent, in like manner, a cone of the n th degree, tangential
to the equipotential surface at the point P .
This cone will be formed of n sheets, or of a smaller number,
corresponding to an equal number of sheets of the level surfaces.
If the sheets of the cone do not intersect, neither do the
equipotential surfaces, and P is an isolated point of equilibrium.
If the sheets of the cone do intersect, every line of intersection
is tangential to the intersection of the corresponding sheets of the
equipotential surface that is to say, a line of equilibrium which
passes through the point P .
48. If the equipotential surface at the point P consists of two
sheets which intersect^ the intersection of the sheets takes place at a
right angle.
When the equipotential surface consists of two sheets which
intersect, the cone of the second degree, which is tangential at a
point P of the line of intersection, reduces to two planes. If
we take the tangent to this line for the z axis, the equation of the
cone H 2 = will contain no term in z.
In order to satisfy Laplace's equation, which reduces to
<> 2 H
o,
the coefficients of the terms in x 2 and in y 2 must be equal and of
POINTS AND LINES OF EQUILIBRIUM. 39
opposite signs, so that the equation of the cone is of the form
they represent two rectangular planes.
Let us consider, as an example, the surface of a conductor
charged partly with positive electricity and partly with negative;
the line of separation of the two layers is a neutral line. The force
is null in all points of the neutral line (35), and there is in the
dielectric another equipotential surface at the same potential as
the conductor, which cuts it perpendicularly along this line. It
will be remarked that this particular equipotential surface separates
the lines of force, which start from the conductor, from those
which terminate there. It might, therefore, be considered as a
limiting surface of the lines of force.
49. If the equipotential surface consists of n sheets which
intersect along the same line^ the successive intersections take place at
7T
the same angle .
n
Starting from a point P of the line of equilibrium, all the
functions H of the development of the potential, up to that of the
degree n, are identical with zero, since the tangent cone consists
of n sheets. In order that the equation H n = shall represent
n planes passing through the z axis, it must not contain any terms
in z, and Laplace's equation reduces to
If r denotes the distance of a point P from the z axis, and putting
x = r cos 6
y = r sin 0,
Laplace's equation becomes
I*H
~
The function of the degree , which satisfies this equation, is
40 GENERAL THEOREMS.
Making it equal to zero, we get
an equation which represents n planes passing through the z
axis, and the successive angles of which are equal to - .
n
50. THERE is ONLY ONE STATE OF EQUILIBRIUM. It may be
observed, in the first place, that the superposition of two states of
equilibrium is itself a state of equilibrium.
For in each of the two states of equilibrium the potential is
constant on all the conductors. The superposition of the two
systems of electrical layers produces, at each point, a potential
equal to the sum of the potentials relative to the two primitive
states. The potential is constant, therefore, on each of the
conductors, and equilibrium exists.
It follows from this, that if we change the electrical density
at each point in a constant ratio, a new state of equilibrium will
be formed, for the operation amounts to superposing two or more
identical states of equilibrium.
51. A system of conductors A p A 2 , A 3 ..., whose electrical
charges are separately null, is necessarily in the neutral state.
Let Vp V 2 , V 3 denote the potentials of these various con-
ductors, and let Vj be the greatest.
There can be no point in the dielectric where the potential is
higher than V v since there is no maximum of potential outside the
acting masses. The potential sinks, therefore, in all directions from
the conductor A x ; all the lines of force start from this conductor,
and none terminate there. As the sum of the flows of force must
be zero (for by hypothesis the total charge of A x is zero) it is seen
that all the elementary flows of forces are zero. The density is
therefore zero over the whole surface, and therefore the conductor
is not electrified.
The conductor A 1? being in the neutral state, may be suppressed,
and the same reasoning applied to the next conductor ; it may thus
be shown successively that all the conductors are in the neutral
state.
The conductors A v A 2 , A 3 . . . , having charges M 15 M 2 , M 3 . . . ,
differing from zero, let us now suppose that two states of equilibrium
are possible, such that the densities on A 1? A 2 , A 3 are o- 1} o- 2 , o- 3 . . . in
the first case, and <r'j, o-' 2 , </ 3 . . . , in the second.
Changing the signs of all the electrical masses of the second state,
THEOREMS RELATING TO CLOSED SURFACES. 41
:_Si .
there would still be a state of equilibrium, which, superposed on the
first, will give a new state of equilibrium, in which the total charge
will be zero on each of the conductors.
In this case, from the preceding observation, the density must be
everywhere zero. We have, then,
The distribution is therefore the same in both cases, and therefore
the equilibrium is singular. If the proposed system comprised fixed
masses, they might always be supposed to be upon infinitely small
conductors, and the reasoning would not be changed. The theorem
is therefore a general one.
52. THEOREMS RELATING TO CLOSED SURFACES. If the potential
is constant on a dosed surface S, not containing any acting mass, it is
constant throughout the whole interior.
The potential, in fact, could not vary in the interior of the surface
S without attaining at one point a maximum or minimum value,
which is impossible, as there is no electricity (46).
If the surface in question is the external surface of a conductor,
it is seen that the potential is constant not only in the mass of the
conductor, which is a necessary consequence of the conditions of
equilibrium, but also in the cavities which this may contain.
53. If a surface at constant potential comprises a portion of a
dielectric, the potential is constant not only in the interior of this
surface, but also in all the exterior space outside the acting masses.
For, as the potential is constant in the interior, no line of force
can traverse the surface ; all those which could meet it are either
produced there or are absorbed which is impossible, since there is,
no electricity on the surface. Hence no line of force meets the
surface.
As no line of force meets the surface, the potential is constan
on the outside close to it ; hence a new surface S' may be drawn
having the same property, to which the same reasoning may be
applied, and so on indefinitely, provided we always keep outside
the acting masses.
54. This latter theorem was demonstrated by Gauss in a
different manner, by means of the following lemma :
If a spherical surface contains no acting mass, the potential at the
42 GENERAL THEOREMS.
centre is the mean of the values of the potential at the different points
of the surface.
For let R be the radius of the sphere, m one of the masses at
a point A, at a distance d from the centre, and r the distance of
a surface element ^/S from the point A; the mean value, on the
surface, of the potential due to the mass m, is
the sum I is the value at A, of the potential of a homogeneous
layer of density m, which would cover the sphere ; it is equal (43) to
d '
The mean value of the potential on the sphere is then
V --
v nt j)
a
that is to say, equal to the value of the potential at the centre.
This reasoning manifestly extends to any number of masses.
That being admitted, let us suppose that in a portion of the
dielectric bounded by the surface S, the potential has a constant
value V; if the value of the potential were different from V on
the outside, it would always be possible to draw a sphere having
its centre in the inside of S, and on the outside only meeting
points where the potential would be always either greater or less
than V ; but, in that case, the mean value of the potential on the
sphere would be different from its value at the centre. The
potential on the exterior of the surface S cannot, therefore, be different
from V.
55. When a closed surface S surrounds all the acting masses,
and the potential on this surface has a constant value V, at each of
the external points there is a value comprised between V and zero.
Let us suppose V positive. As the potential is zero at an
infinite distance, it cannot have a higher value than V, at a point
P external to the surface, without there is somewhere a maximum
potential, and therefore electrical masses, which is contrary to the
hypothesis. In like manner the potential at P cannot be less than
V 3 for otherwise there would be a minimum. The potential on
THEOREMS RELATING TO CLOSED SURFACES. 43
V
the outside is. therefore between V and zero. It cannot be equal
to V, unless V itself is zero. In fact, it cannot be equal to V
without being a maximum in reference to adjacent points, or without
forming part of a space at constant potential which would extend
to the surface S ; but, from the preceding theorem, the potential
in this latter case would be constant throughout the dielectric, and
could only have the value zero, for it is zero at an infinite distance.
56. A conducting surface S, which contains all the acting masses,
can only have electricity of one kind.
For let V be the potential of the surface, which we will suppose
is positive. If there were negative electricity at a point A, lines
of force would terminate there, and there must be somewhere an
external point where the potential has a higher value than V, which
is impossible from the preceding theorem.
57. When, in a system in equilibrium, a conductor envelopes
diverse masses of electricity, the algebraical sum of the quantities of
Fig. 13-
electricity in the interior, and on the internal surface of the body,
is zero.
Let m, m ', m" ... be the masses comprised within the interior
of the conductor A (Fig. 13), and M the mass of the layer spread
on the internal surface S ; to which we may add that there may
be electricity on the external surface S', and other masses beyond.
On a closed surface S 1} in the conductor, and comprising all the
internal masses, the force is null at each point
The flow of force relative to this surface is therefore null, and
accordingly the algebraical sum of the masses which it contains is
null. We have then
M + m + m' + m" + ..... = 0,
whence
44 GENERAL THEOREMS.
The layer M, developed by induction on the surface S, is equal
and of opposite sign to the algebraical sum of the masses comprised
within the cavity. This layer absorbs the flow of force emanating
from the internal masses. The system of this layer, and of the
masses which it comprises, gives a potential zero, and its action
is zero at every outside point.
If the conductor A is in connection with the earth, its potential
is zero ; if then there are no other acting masses than those con-
tained within the cavity, the external surface S' is in the neutral
state, and the potential is everywhere zero outside the cavity.
An electrical mass M', placed on the insulated surface S', acquires
there a distribution independent of the internal masses. It will
produce a constant potential V throughout the whole extent of the
conductor A, and of the cavities which it encloses ; and this potential
will add itself at each point to the potential already existing. The
mass M' will not exert there any influence on the equilibrium of
the internal masses.
58. If there are no other acting masses than those comprised
within the cavity, and if the conductor A were originally in the
neutral state, its total charge should remain zero ; simultaneously
with the layer M on the inner surface S, it will develop an equal
and opposite layer M' on the exterior surface S', and we shall have
M' = -M =
This layer M', equal to the algebraical sum of the internal masses,
and of the same sign, is of itself in equilibrium whatever be the
position of the internal masses.
We get thus Faraday's law : The quantity of electricity induced by
an electrical system on a conductor, which surrounds it, is equal to the
quantity of inducing electricity.
59. The action which given electrical masses exert on the exterior
of any closed surface, is the same as that of a layer of the same mass
spread on this surface according to a certain law.
Let m, m', m" , .... or JVz, be the given masses, which we will
suppose fixed as if they belonged to non-conducting bodies ; a surface
S of any given form envelopes them. Suppose that, for a moment,
we replace this surface by a material sheet, forming an infinitely
thin conductor in connection with the earth ; the internal surface
of this sheet will become covered with a mass - M = m, whose
potential, for all external points, is equal and of opposite sign to
THEOREMS RELATING TO CLOSED SURFACES. 45
that of ^ m. .A layer + M, distributed in the same way on the
surface S, will have everywhere ' dn the outside a potential equal,
and of the same sign, to that of the masses in question, ^ m.
The layer M will not in general be in equilibrium of itself; that
is to say, that it will not have the distribution which would result
from the form of the surface, and from the action of external
masses; the lines of force will not cut it perpendicularly.
60. The layer +M. will be in equilibrium of itself, if the surface
S is an equipotential surface of the primitive system, taking into account
(he external masses.
Let us denote by ^ m the external masses. The lines of
force of the field are the same at the exterior of S for the system
^m and ^m, and for the system M and ] m' ; these lines
being, by supposition, normal to S, the layer M which covers it
is in equilibrium, and has a constant potential V.
We may then, for external points, replace the system ^m by
an equal mass in equilibrium on an equipotential surface which
surrounds it, the density at each point being determined by the
condition
_ F _ i_JV
4/r 4?r dn
This substitution always modifies the field in the interior of the
surface S ; in the present case the internal potential has become
constant and equal to V, for it is constant on the surface, and the
cavity no longer contains electricity.
If the potential is constant, the force is zero ; the external system
m', and the layer M, exert equal and contrary actions at each
point of the interior of S ; a layer - M would exert actions equal
to, and of the same sign as, that of m'. Thus for all the internal
points of the equipotential surface S, we may replace the action of
external masses by that of a layer in equilibrium, equal to the internal
masses, and of the opposite sign.
61. The following theorems follow from this: If we consider
an equipotential surface S in any electrical system, we may
i st. For all external points replace the internal masses by a mass
M, equal and of the same sign, in equilibrium on this surface ;
2nd. For points in the interior, we may replace the external masses
2? ?n by the same mass M with changed sign; that is to say, by a
mass equal and opposite to the internal masses, this layer being still
in equilibrium.
46 GENERAL THEOREMS.
62. GAUSS' THEOREM. Given two electrical systems, one con-
sisting of the masses m v m^, m B , ..... and producing a potential V,
the other of masses m' v m' 2 , m' 3 , ..... and producing a potential V,
we shall have the identical equation :
that is to say, that the sum of the elementary masses of the first
system, multiplied respectively by the value of the potential of the
second system at the point which they occupy, is equal to the corres-
ponding sum relative to the masses of the second system ; the sum-
mations must be replaced by integrals if the masses occupy a finite
extent.
This proposition is an identity ; to see this, we need only replace
the potentials by their values as functions of the masses, and of the
distances. It then appears that each member of the equation is
equal to the sum of the products obtained by multiplying each
mass of one system by a mass of the other, and dividing the product
by the distance separating them.
It will be sufficient to remark that the sums J5T* m'V and ^ mV'
denote the work which must be expended to bring them respectively
in presence of each other, from an infinite distance to the positions
which they occupy, the two systems ^m and jgm' being supposed
rigid; and in either case the work is evidently the same.
When the two systems we are considering are conductors in
equilibrium, A v A 2 , A 3 , the potential is constant on each conductor,
and if we denote their total masses by M 15 M 2 , M 3 ..... , and M' 1$
M' 2 , M' 3 ..... , the equation becomes
M l V 'l + M 2 V/ 2 + M 3 V ' 3 ..... = M/ 1 V 1 + M/ 2 V 2 + M/ 3 V
63. The following theorems may be considered as corollaries of
that of Gauss, as has been shown by M. Bertrand :
I. If a conductor A in the neutral state, whether insulated or not,
is exposed to the action of an electrical mass m placed successively at two
points P and P' of the dielectric, the potential due to the induced charge
at A, will be the same at the point P' in the first case, as at the point P
in the second.
Let us observe, in the first place, that any point of the dielectric
may always be considered as the centre of an infinitely small con-
ducting sphere, for the charge which this sphere acquires is constantly
zero, and always produces at its centre a potential equal to zero.
EARNSHAV/S THEOREM. 47
If the body is connected with the earth, we shall have to consider
the two follow ng states of equilibrium :
Potentials. Charges.
C Sphere P U m
i st. Sphere P' V
\ Body A x
C Sphere P V
2nd. ] Sphere P' U' m
( Body A x'
Applying Gauss' theorem to these two conditions, we get
v;y,
whence
V = V.
If the body A is insulated, its potential is not equal to zero, but
its charge is null in both states, and the final result is the same.
II. If each of the two conductors A and B is successively put in
connection with a source which raises it to potential V, the other being
in connection with the earth, and therefore at zero potential, the quantity
of electricity developed by induction on the latter is the same in both cases.
We have, in fact, in the first case,
Potential. Charge.
A V x
B -M
and in the second,
Potential. Charge.
A -M'
B V x
Applying the preceding theorem, we get
M'V = MV,
whence
M' = M.
64. EARNSHAW'S THEOREM. An electrified body cannot be in
stable equilibrium in an electrical field.
Let an electrified body A be placed in a field produced by
external masses B, and let us suppose all the masses fixed,
including that of A.
48 GENERAL THEOREMS.
Let m denote the electrical mass of a volume element of A at
a point P, where the potential of the external masses is V. The
energy of the body in the field is
(i) W = mV.
For stable equilibrium, the differential must be null or
positive in any given direction r.
Let x, y, z be the co-ordinates of the point P ; f , 77, f those of a
point P taken in the interior of the body A, and a, b, c that of
the point P with reference to three new axes parallel to the first,
and passing through the point P ; we shall have
The potential V may be considered as a function of a, b, c, and
f > ^ f If the body A is constrained to move parallel to itself,
we shall have
dz=d(,
and therefore,
3 2 V W 3 2 V W D 2 V W
(2) AV = VT + VT + ^ = ^ + ^- + ^79-
cte 2 ty/ 2 c)^ 2 Sf 8 <V ^i 2
As this sum is zero for each of the terms nN of the second
member of the equation (i), we shall also have
^) 2 W t) 2 W _
~ ++ ~
The energy W is therefore a function of the co-ordinates of the
point P , and this function satisfies Laplace's equation, as long as
equation (2) itself is satisfied. We may assume that the point P is
comprised within a sphere of radius r, so small that the body in
question, A, does not meet any external masses. The variation of
energy for any displacement parallel to one of the radii of this
sphere, will be
EARNSHAW'S THEOREM. 49
_ X. ----_... - - - r
Applying Green's equation to the surface of this sphere, we
shall have
pw
As the integral dS becomes null, it follows that the
aw J * r
differential is negative for certain directions, and positive for
others ; hence, the body A is not in equilibrium, and tends to move
towards places for which the energy W diminishes.
aw
There is equilibrium if ^ is always zero ; that is to say, if the
energy is constant, or passes through an absolute maximum or
minimum. We have then
and in like manner,
The components X, Y, Z of the force of the field, produced by
the external masses, may be developed as a function of increasing
powers of the co-ordinates a, b, c, and of the components X , Y , Z ,
relative to the point P , which gives
HP H 2 , ---- H w .. being functions of the i, 2 ---- n.. degrees of
the co-ordinates a, b y and c.
We get then
..... +H n ) = 0.
To satisfy this equation, all the coefficients of the development
must be zero. That necessitates, in the first place, that all the
differentials of X be null ; then, that the position of the point P
be arbitrary; and, lastly, that we have
Hence, either the total mass ^m of the body A must be zero, or
the components X , Y , and Z of the force must themselves be zero.
E
50 GENERAL THEOREMS.
In order, therefore, that an electrified body be in equilibrium
in an electrical field, either the force of the field must be zero, or
the field must be uniform, with the condition that the algebraical
sum of the electrical masses which the body possesses, be zero.
In this demonstration, we have assumed that the electricity was
fixed on the body A, and that the external masses themselves formed
a rigid system. The theorem applies with the more reason to the
case in which the system contained conductors. If stable equilibrium
does not exist when connections are introduced into the system, this
is still less the case when these connections are suppressed; for
instance, when the electrified bodies are in part conductors, which
leaves more play to the displacement of electrical masses.
CONDITIONS OF. THE EQUILIBRIUM OF CONDUCTORS. 51
CHAPTER IV.
ELECTRICAL EQUILIBRIUM.
65. CONDITIONS OF THE EQUILIBRIUM OF CONDUCTORS. The
general problem of electrical equilibrium, when restricted to the case
of conductors, may be thus enunciated :
With conductors of given shape and position, one set insulated, and
the other in connection with the earth, the former being charged with a
definite quantity of electricity, it is required to determine the potential at
each point.
In other words, the problem amounts to determining a function
V of co-ordinates, which satisfies the following conditions :
i st. The function must be zero at an infinite distance, and must
have a constant value on each of the conductors, this value being
zero on all conductors in communication with the earth.
2nd. The sum of the three second partial differentials must be
zero over the whole surface of the dielectrics and in the interior of
conductors, for the electrical density is zero at all these points.
3rd. At every point of the surface of the conductors, the density
is determined by the equation
i </V
4ir dn
so that the total charge upon one of the conductors is expressed by
M= fov/S=- f <JS.
J 4^J dn
These three conditions are sufficient, for they determine a state of
equilibrium which satisfies the data of the question, and only one
state of equilibrium is possible for the system.
This problem frequently presents great difficulties from the
mathematical point of view, and no general solution is known. It
has only been completely solved for a few special cases, the most
E 2
5 2 ELECTRICAL EQUILIBRIUM.
important of which we shall afterwards investigate; but we may
deduce from the general enunciation a certain number of remarkable
properties.
66. GENERAL OBSERVATIONS. Let us first suppose that among
all the conductors which are in presence of each other, A v A 2 . . . A n ,
one of them, A 1? has received a charge M, and that all the others
are in connection with the earth.
When equilibrium has been established, the potential has a
constant value V x on A 15 and it is zero on all other conductors. Let
us suppose V l > 0.
Within the whole dielectric the potential can neither be higher
than Vj nor lower than zero, and it lies between V l and (55).
It follows from this, that those conductors connected with the
earth only possess negative electricity; for if there were positive
spaces on their surface, lines of force would start from them towards
the points where the potential was lower, that is to say negative, and
these points do not exist.
All the lines of force of the field start then exclusively from the
conductor A l ; one set terminates in the conductors in connection
with the earth, the others proceed towards an infinite distance.
It follows from this that the negative charge on these conductors
is only a fraction of that upon A l ; the two charges would only be
equal, provided one of the conductors in connection with the earth
completely surrounded A r
67. Let us now suppose that in the vicinity of Aj there are
other insulated conductors A 2 , A 3 , , at first in the neutral state,
and whose total charge therefore is zero.
The potential is still positive and is less than V a within the
whole dielectric. It has a constant value on each of the other
conductors ; this value is positive, since part of the surface of each
of the conductors is charged with positive electricity, and therefore
lines of force start from them, and these lines of force proceed
towards spaces where the potential is everywhere positive.
Let A 2 be that insulated conductor whose potential V 2 is
highest ; part of its surface is negative, it therefore receives lines of
force. None of these lines of force come to it from the earth, nor, by
hypothesis, from the other conductors whose potential is lower ; they
all proceed then from the conductor A p and therefore V 2 is less than
V r As, moreover, the lines of force received by A 2 do not form the
whole of those proceeding from A v each of the positive and negative
layers which make up the zero charge of A 2 is smaller than the total
charge of A r
RELATION OF CHARGES TO POTENTIALS. 53
The same reasoning applies to all the other conductors; thus
taking them in decreasing magnitude of potential, A 3 , for instance,
receives lines of force from A l and A 2 , and these latter may be con-
sidered as proceeding indirectly from A r On each of the insulated
conductors the negative charge is less than the positive of A 15 provided
that none of them forms a closed surface completely surrounding the
conductor A r
68. RELATION OF CHARGES TO POTENTIALS. Denoting by A 1?
A 2 , . . . A n the conductors, let M 1? M 2 , . . . M n be the respective
charges, and V lf V 2 , . . . V n the corresponding potentials.
Let us first suppose all the conductors insulated, in the neutral
state, and at zero potential. If we give unit positive charge to one of
them, A 15 its potential becomes a n , and those of the other conductors
are respectively a 21 , a sl , . . . a nl . If instead of unit charge, the
charge M x be given to A 15 all the potentials would be multiplied by
M! ; they would be a n M 15 a 21 M 15 . . . a nl M r Let us suppose that
Aj is discharged, and that we give the charge M 2 to A 2 , the
potentials will become a 12 M 2 , a 22 M 2 ..... a w2 M 2 ; and so forth.
Now the final state, when all the conductors receive their respective
charges simultaneously, is that in which all the states, obtained thus
in succession, are superposed ; to express then the potential of each
conductor we shall have an equation of the form
and, therefore, n similar equations for the whole system.
From this the following theorem is deduced.
In any electrical system in equilibrium, the potentials of the several
conductors may be expressed as a linear function of the charges.
Among the n 2 coefficients of the equation (i), the coefficient a pl>
expresses the potential of the conductor A p when it is charged with
unit electricity, all the others being in the neutral state ; a coefficient
such as a qp denotes the potential, which a conductor such as A q
would acquire in the same time.
It is easy to see that these latter coefficients satisfy the relation
(2) o. pq = a qp .
For, let us consider the two successive states in which each of the
conductors A p and A q is alone charged with unit electricity, all the
others being in the neutral state ; applying Gauss' theorem (62) the
relation (2) is at once obtained.
54 ELECTRICAL EQUILIBRIUM.
The remark made above (67) shows that all the coefficients a are
positive, and that a coefficient such as a qp is never greater than a pp
or a qq .
69. If we solve equations (i) in reference to the charges, we
shall have n equations of the form
(3) M p
containing 2 coefficients, the signification of which is at once
manifest.
The coefficient jpp expresses the charge which must be given to
the conductor A p to raise it to unit potential, all the others being at
zero potential. This coefficient, which plays a conspicuous part in
the theory of electricity, is called the capacity of the conductor A p ;
we shall revert to it in a moment. A coefficient such as y qp expresses
the charge acquired by the conductor A q in connection with the
earth ; it might be called the coefficient of electricity induced by
A p upon Ag.
The application of Gauss' theorem in the case of two successive
states, in which each of the conductors A p and A^ is raised to unit po-
tential, the others being in communication with the earth, shows that
these coefficients are also equal in pairs, and that we have the ratio
(4) 7pq
which is only an extension of the theorem demonstrated above (63)
for two conductors.
If we refer to the observation in 66, it is easy to see that while
the coefficients y^, which express the capacities, are all positive, the
coefficients of the induced electricity, such as y pq , are all negative ;
moreover, that the sum of all those which relate to the induction
exerted by the same conductor, is never higher in absolute value
than the capacity of this conductor itself. For instance, we have,
necessarily,
7PP> -[71P + 72P ..... +7np\>
unless one of the conductors in connection with the earth A q , for
instance completely envelopes the conductor A p . In this case, we
should have
7pp= ~7qp>
ANALOGIES OF THE PROBLEM OF ELECTRICAL EQUILIBRIUM. 55
and the n - 2 other coefficients relative to the conductor A p , y^,
7 2 P 7njp W0uld be nul1 '
70. ANALOGIES OF THE PROBLEM OF ELECTRICAL EQUILIBRIUM.
It is interesting to compare with the problem, which we have just
treated, two other problems relating to phenomena which are entirely
different, but which, analytically, present the most complete analogy
that of the uniform propagation of heat in a homogeneous medium,
and that of the steady motion of an incompressible and frictionless
liquid.
In short, the electrical problem is characterised by the existence
of a function of the co-ordinates, which, vanishing at an infinite
distance, has a constant value on each of the conductors, and for
each point of the dielectric satisfies the ratio
AV=0,
the physical signification of which is very simple. X, Y and Z being
the components of the force at a point P, the quantity - AWz;
represents the total flow of force which proceeds from an element of
volume dv taken at this point, and equation AV = expresses that
this flow is nothing in the dielectric, or in the interior of a con-
ductor ; that is to say, where there is no electricity.
Let us now suppose that in a problem of statical electricity the
insulating medium is replaced by a medium which conducts heat,
and which is homogeneous, and isotropic\ that is to say, which has
the same properties in all directions; and let us suppose each of
the electrified conductors replaced by sources which emit or which
absorb heat, so as to maintain constant temperatures on the surfaces
which are respectively equal in numerical value to the initial potentials,
so that for each of the conductors /=V.
When once equilibrium is established, every point of the medium
will be at a definite temperature, and isothermal surfaces can be
traced ; that is to say, surfaces of equal temperature, or of equal
thermal level. It is clear that the temperature of a point P, com-
prised between two isothermal surfaces S and S', is independent of
the situation of the sources, and that it will remain the same when
these sources are suppressed, if the temperatures / and /' of these
two surfaces are kept constant in any other way.
Fourier's hypothesis consists in assuming, what indeed may be
regarded as the simple expression of facts, that heat travels from
layer to layer ; that the thermal effect of a point has no appreciable
influence except on very near points; and that the hotter points
ELECTRICAL EQUILIBRIUM.
tend to raise the temperature of the colder ones. Fourier assumes,
moreover, what is only true for a particular thermometric scale, that
the interchanges of heat only depend on differences of temperature,
and not on their absolute values.
The flow of heat which traverses an element dS of an isothermal
surface S (Fig. 14) is, by symmetry, perpendicular to this surface,
and to all the isothermal surfaces which it meets. The flow of heat
^Q, which passes in unit time from the element dS to the infinitely
near element dS', is proportional to the surface dS v to the infinitely
ds
Fig. 14.
small difference of temperature / - /', to a coefficient h which only
depends on the nature of the medium ; and, lastly, to a function of
the distance of these elements.
We may therefore put
t-t'
If we consider an intermediate temperature t v the flow of heat
from dS to dS' passes first through the element ^S : at the distance
e v which gives
Since the differences of temperature / / x and / /' are, by
continuity, proportional to the perpendicular distances, the function
< is proportional to the distance only. The flow of heat between
two infinitely near corresponding elements, calling dt the variation
of temperature measured in the direction of the flow of heat, and dn
ANALOGIES OF THE PROBLEM OF ELECTRICAL EQUILIBRIUM. 57
the distance of the elements, may be expressed by the following
formula :
</Q = -MS.
an
The coefficient k is the coefficient of conductivity of the medium.
It represents the flow of heat for unit surface between two parallel
planes at unit distance, and whose temperatures differ by i.
In the present case, the value of the flow for unit surface at each
point is
Q--4'
dn
it is proportional to the differential of the potential in reference to
the perpendicular to the corresponding equipotential surface.
The flow of heat is the same across an element dK of any surface
A bounded by the same tube. If 6 is the angle which this element
makes with the equipotential surface, and da the portion of the
perpendicular to the element */A, comprised within the surfaces S
and S', we have
,/Q- -A/A cos B-~ -kdt*-T- -***%--
an an da oa
The flow of heat across an element of any given surface is
therefore proportional to the partial differential of the temperature
in the medium with respect to the perpendicular to this surface.
As equilibrium is established, the total flow of heat corresponding
to any closed surface, not containing sources of heat, must be zero.
Let u, v and w be the components of the flow at a point P, and
dxdydz an element of volume at the same point ; the flow which
enters by one of the surfaces dydz is udydz, that which emerges by
the opposite surface is equal to ( u H -- dx\ dydz, the difference
du \ dx /
is dxdydz ; the total flow corresponding to the entire surface of
the element is then
/"bu cte; <)7>\
dxdydz / + - + \-
\C ty ^Z /
as this flow must be zero, it follows that
+ - + = -M/=0.
02 cy oz
58 ELECTRICAL EQUILIBRIUM.
It is evident, moreover, that the action of the system is not
appreciable at great distances, and that the temperature /, which
it determines, is zero at an infinite distance.
Hence, for every point of the medium and for the limits, the
function / satisfies the same conditions as the function V. It is
seen further that if the constant k is equal to unity, the numerical
values of the flow of electrical force, and of the flow of heat during
unit time, are identical in every point in the two problems.
71. Let us now consider the corresponding hydrodynamical
problem. Let us imagine that the space originally occupied by the
dielectric is filled by a frictionless and incompressible liquid ; let us
imagine, moreover, that -the conductors are replaced by porous
surfaces, so that the liquid has at each point of such surfaces, a
normal velocity, equal to the original value of the electrical force at
this point. The whole of the trajectories of the molecules which
at the same moment have traversed the element dS of the surface
of a conductor, form a liquid thread which issues perpendicularly,
and yields the same supply in all sections. As there is nowhere any
accumulation of liquid, the flow passing through a volume element
dxdydz taken at any point P, is equal to that which emerges ; now
if #, v, w, denote the components of the velocity at the point P, this
condition is expressed by the equation
~+ = 0.
oy cz
The motion is further inappreciable at an infinite distance ; we thus
see that the velocity at each point, depends on a function of the
co-ordinates which satisfies the same conditions as the potential or
the temperature. Lines of flow will coincide everywhere with the
lines of force of the corresponding electrical problem, and at each
point the electrical force and the velocity of the liquid will have the
same numerical value.
The correlation which we have established is of great interest ;
for if it is clear that the analytical difficulties are exactly the same
in the three kinds of problems, it is no less true that certain
consequences present themselves more naturally in one order of
ideas than in another, and it is clear that any result obtained in one
case may be directly transferred, with its special interpretation,
into another. We shall meet with many instances of this in the
sequel
ELECTRICAL CAPACITY. 59
72. ELECTRICAL CAPACITY. We have designated as the capacity
of a conductor the charge which must be given to it to raise it to unit
potential, when all the conductors which surround it are in communi-
cation with the earth.
It follows, from this definition, that the capacity of a conductor
depends not merely on its own shape, but on the shape and position
of the conductors which surround it.
We shall represent this constant by the letter C. If, while the
conditions remain the same, a charge M is imparted to the con-
ductor, in virtue of the principle of superposition of conditions of
equilibrium, its potential will be
from which follows
M = CV.
The problem of determining the capacity of a conductor in a
given case, amounts to investigating the state of equilibrium of the
system formed of the conductor in question, together with those
surrounding it, these latter being in connection with the earth; it
merges then into the general problem of equilibrium.
The word capacity has been borrowed by analogy from the
theory of heat ; but it is important to remark that while the calorific
capacity of a body only depends on the nature and weight of the
body, the electrical capacity of a conductor depends neither on its
nature nor on its weight, but solely on its external shape and on the
shape and position of all the adjacent conductors. The electrical
capacity is not therefore a constant, fixed for the body in question,
as is the thermal capacity.
73. SPHERE. Let us consider a conducting sphere at a great
distance from any other conductor. Let R be its radius, M its
charge. By symmetry this charge forms a uniform layer on the
surface ; it satisfies, moreover, the condition of equilibrium, for its
action on any internal point is null (42). The potential is, therefore,
constant throughout the whole interior; its value at the centre is
M
- ; hence,
V = ^ or M = RV.
-R
The capacity of the sphere is, therefore,
C = R;
60 ELECTRICAL EQUILIBRIUM.
it is equal to the radius. This example shows that the electrostatic
capacity of a conductor is a linear quantity.
74. ELLIPSOID. If a conductor bounded by the surface of an
ellipsoid is covered by a homogeneous electrical layer, bounded itself
by a second ellipsoidal surface concentrical and similarly placed to
the former, the action of the layer on an internal point P is null.
Let us suppose, in fact, that this layer is very thin, and let us
draw through the point P (Fig. 15) an infinitely slender cone */w,
which cuts an element of surface ^S at M at the distance u, and in
the layer, a volume element the height of which along the radius
Fig. 15-
vector is du. The action at P of this volume element is in the
direction of the radius vector, and calling p the density, its value is
The action of the opposite element at M' is also pdudu'. As the
heights du and du' are equal and the forces are directly opposed,
their resultant is zero ; this is also the case for all the elements
of surface two by two, and the action of the entire layer on the
point P is null. An electrical layer distributed on an ellipsoid
according to this law will be then in equilibrium and will have a
constant potential in the interior.
Let (i +a) be the ratio of similitude of the two surfaces supposed
to be very close. The thickness of the layer at a point N is pro-
portional to the distance of the tangent planes from the two
homologous points N and N', and is equal to /a, p denoting the
perpendicular OQ let fall from the common centre on the tangent
plane in N ; the value of the surface density cr is
CAPACITY OF AN ELLIPSOID. 6 1
The ellipsoid being represented by the equation
the total mass of electricity is
M = ^nz&r[(i+a)3-i],3
From this we get
M _ M
7*a / z*
The potential in the interior being constant, it is sufficient to
calculate its value at the centre. We have then, if r be the
radius ON,
fov/S M CpdS
V= = - \- ,
r ^irabc r
/ /
and the capacity of the ellipsoid is given by the equation
C ^.irabc J r
75. For an ellipsoid with three unequal axes the capacity is an
elliptical function, but it may be easily obtained in the case of an
ellipsoid of revolution.
Let us take as element of surface ^S, the zone described by an
element ds of the meridian curve, and let us suppose that the axis a
is the axis of rotation. We have then
2irydx
from which
= 2irb <i dx.
62 ELECTRICAL EQUILIBRIUM.
This ratio shows already that the total charge of electricity is the
same on all zones of equal height. On a very elongated ellipsoid, in
the form of a double point, we may say that the linear density is
constant*
From this we get for the capacity of the ellipsoid
i = i ws i r dx
C qpabcj r 2 a J Jx*+y*'
If it is an ellipsoid of revolution about the major axis, e being
the eccentricity, and Cj the capacity, we have
i +a dx i _ + a i i+e
- [ ex+
If the ellipsoid is one of revolution about the minor axis, the
capacity C 2 is then
dy i I , ey~\ + & arc sin e
arc sin
i i
_L / 02,2
2b\ a *-
J , V ^ 2
~~D .
' -- 2ae\ D I . 0*
Each of these formulae gives
/- x-
when the eccentricity is null; that is, when the ellipsoid becomes
a sphere.
* This remark enables us to explain the power oj points. On a very
elongated ellipsoid in the form of a double point, it will be seen that the surface
density is inversely as the diameter and constantly increases towards the extremity.
If the insulating power of the air were itself without limit, the density and the
tension, which is proportional to the square of the density, might increase without
limit. But, as a matter of fact, when the tension has attained a certain value for a
given pressure of air, the electricity passes from the conductor into the masses of
air surrounding it, aud these being charged with electricity escape along the lines
of force, producing the phenomena known as the electrical aura and of the brush
discharge.
CAPACITY OF CONCENTRIC SPHERES.
76. An elliptical plate may be regarded as a very flattened
ellipsoid. We have then, at the limit,
lira. ff\ =
The density on the plate is expressed by
M i
cr =
I x 2 jy 2
V l ~^~^ 2
and it will be seen that the lines of equal density are concentric
and similarly placed ellipses.
For a circular plate, we have simply, at a distance r from the
centre,
M i M
and the capacity of the plate is reduced to
77. CONCENTRIC SPHERES. Let us suppose that a conducting
sphere is surrounded by a conductor, bounded by two spherical
surfaces concentric to the first.
64 ELECTRICAL EQUILIBRIUM.
Let R be the radius of the sphere A (Fig. 16), Rj and R 2
those of the concentric envelope B, which at first we will suppose is
insulated. If an electric charge M is given to the sphere A, the
envelope B acquires (58) a charge equal to - M upon its inner
surface S 15 and a charge + M upon its outer surface S 2 . The value
of the potential at the centre of the sphere is obviously
M M M |~ i
R~D "O I T)
*i *l L **
The capacity C of the inner sphere being the charge which
corresponds to V = i, we have
c R R, R 2
The two layers +M upon the sphere, A, and -M upon the
surface S x , have a potential equal to zero on the outside. The
potential V x of the envelope B depends, then, solely on the outer
layer +M, and is the same as at the centre of the sphere S 2 ,
supposed to be homogeneous, which gives
If the envelope B is connected with the earth, its potential
becomes zero, the external charge + M disappears, and the capacity
of the sphere is then
iii e
C R Rj RRj
e being the thickness of the dielectric.
If the thickness of the dielectric is small in comparison with the
radius of the sphere, we may neglect the difference between R and
02 RR
RI, and take instead of for the capacity. If this capacity
e e
be expressed as a function of the surface of the sphere, we have
c = R 2 = 4'rR 2= _S_
e 4 4 '
LEYDEN JAR. 65
The charge required to raise the sphere to potential V is ex-
pressed by
It will be seen that in all cases the concentric shell, by increasing
the capacity of the sphere, has the effect of diminishing the potential
relative to a given charge, or conversely, of increasing the charge
relative to a given potential.
78. CONDENSERS. The presence of the envelope makes it
possible to accumulate or condense on the sphere A, a greater
quantity of electricity, for the same potential, than if this envelope did
not exist. The same effect would be produced upon any conductor
A, by the proximity of a second conductor B in connection with
the earth, or insulated, but with a charge null, for this conductor
diminishes the value of the potential for a given charge. The term
condenser is applied to a system of conductors separated by a
dielectric, and arranged so as to increase the capacity of one of
them to a notable extent. In the present case, the sphere and its
envelope constitute what are called the armatures, or coatings of the
condenser, the sphere A being the collector, and the sphere B the
condenser.
The condensing force of a condenser is the ratio between the charge
of the collector when it forms part of a condensing apparatus, and
the charge which it would acquire, for the same potential, if it were
distant from any other conductor. It is therefore the ratio of the
capacities of the collector in these two circumstances. In the
spherical condenser with concentric surfaces, the outer coating of
which is in connection with the earth, the value of the condensing
force is
The application of the condensing force presents no interest ; the
only magnitude which requires to be known is the capacity of a
condenser.
79. LEYDEN JAR. A Ley den jar is a glass vessel coated outside
and inside with metal foil, with the exception of a part near the
opening, so that the coatings may not communicate. A conducting
rod passing through the neck is connected with the internal coating.
The system of these two conducting surfaces constitutes an almost
closed condenser.
66 ELECTRICAL EQUILIBRIUM.
The preceding problem corresponds to the case of a spherical jar
of constant thickness ; the influence of the small zone, which must
be cut off from the outer surface, to allow of communication with
the interior, may be neglected ; the capacity is therefore represented
g
by , and the charge by the formula
It is easily seen that this formula applies equally to a Leyden jar
of any form, of constant and very small thickness, the coatings of
which cover the whole surface outside and inside.
Let Sj and S 2 (Fig. 17) be the two opposite surfaces of the two
coatings, V l and V 2 their potentials. As the outer coating completely
surrounds the inner one, the electrical masses on these two surfaces
are equal, and of opposite signs (57). The value of the force for a
point P of the dielectric, or being the density at A of the inner
layer, is
dV
F = = 47TCT.
dn
As the thickness e is supposed to be very small, the differential
dV V V
is virtually equal to -, which gives
dn c
The charge of a surface element dS is o-dS, and the total charge is
f Vj-Vs/VS
M = U/S = 2 .
4^ I e
CAPACITY OF CONCENTRIC CYLINDERS.
If the thickness is constant, then, putting V = V 1 -V 2 , we have
M =
v i-V
VS
47T
The capacity of the jar in ether words, the charge which corre-
sponds to a difference of unit potential between the two coatings is
expressed in the same way as for spherical condensers; that is to say,
The charge depends thus only on the difference of potentials,
and not at all on their absolute values : this result could be readily
foreseen, as the force itself only depends on this difference.
80. CONCENTRIC CYLINDERS. Given two concentric cylinders
with circular bases of the radii Rj and R 2 , at the potentials V l and V 2 .
Consider the very small angle du formed by two planes passing
Fig. 18.
through the axis, and cut this by two planes perpendicular to the
axis, one of which we shall take for the plane of the figure (Fig.
1 8). All the lines of force being by symmetry perpendicular to
the common axis, the volume thus determined is a tube of force ;
the surfaces which it intercepts on the two cylinders are to each
other as the arcs ^S : and d 2 . If, then, F x and F 2 are the values
of the force at the distances Rj and R 2 , we have
As, moreover,
we obtain, by substitution,
whence
F 2
68 ELECTRICAL EQUILIBRIUM.
The force, therefore, at any point of the dielectric is inversely
as the distance from the axis.
If F be the force, and V the potential at any distance R, we have
A_ _<TV
~~ ~ J
and therefore,
Extending this integral to the volume comprised between the
surfaces S : and S 2 , we have
From which is deduced for the constant A,
V V
A _M _ 12
A-- .
On the other hand, the electrical density o-, at the surface of
the inner cylinder, is expressed by
_ _ __ _
4tr * 47TRJ 47TR/ R 2
Let S be the extent of surface comprised between the two planes
perpendicular to the axis ; the mass M distributed on this surface is
(Vi-V 8 )S
-
If L is the length of the cylinder, thus determined,
S =
which gives, finally,
CAPACITY OF PLANE CONDENSERS. 69
The capacity of a cylindrical condenser for unit length is then
i
C =
This is a problem of great practical importance, as it represents
the case of telegraph cables, which are made of conducting wires
surrounded by an insulator, which in turn is itself protected from
injury by a metallic coating.
81. PLANE CONDENSERS. Consider two conductors bounded
by plane parallel surfaces S x and S 2 , at the distance e, and at the
potentials V x and V 2 (Fig. 19). At a distance from the edges, which
is very great compared with the thickness of the dielectric, the lines
of force are parallel straight lines perpendicular to the surface in
question.
e
Fig. 19.
The electrical field between the two planes is then uniform ; the
force is expressed by
F= l ~ 2 ,
e
and the density on the surface S is
F V\ V
4?r 47T
The electrostatic pressure, that is to say the force exerted upon
unit surface, is
/ = 2 -2 * (TLL^Y;
it is proportional to the square of the difference of potential, and
inversely as the square of the distance.
Suppose that a portion a, of the surface S, at a great distance
from the edges, is movable, and that, being in contact with the
general surface, so as to have the same potential, it is maintained
70 ELECTRICAL EQUILIBRIUM.
there by an antagonistic force; this surface will have a uniform
electrical layer, and the force P, necessary to resist the electrical
attraction, is expressed by
Sir W. Thomson has made use of this property in the construction
of his absolute electrometers ; that portion of the surface S which
surrounds the movable part to keep it at a constant density is called
the guard-plate.
82. Let us now suppose that a conducting plate A, at potential
Vj (Fig. 20), is placed between two conductors B and B', terminated
by surfaces parallel to those of the plate, one at the distance <?, and
potential V 2 , the other at the distance e' and potential V 2 .
to
Fig. 20.
The density on the conductor A, at a great distance from
V V V V
the edges, is -^ 2 for the upper face, and L ^ for the lower
face, so that the charge which corresponds to unit surface of the
plate is
_ /tr ir tr 17' '
Disregarding the variation of density at the edges, the total
charge of the plate, of surface S, is thus
S
M=
If the potentials V 2 and V' 2 are equal, we have simply
M= *"" 2
47T
CAPACITY OF A SYSTEM OF CONDENSERS. 7 1
so that the capacity of this condenser is
S /i i\
c (-+-),
an expression which agrees with that already obtained (79) for closed
conductors.
83. CAPACITY OF A SYSTEM OF CONDUCTORS. Let us con-
sider a number of different conductors whose electrical capacities
are respectively C, C', C", and which are so arranged that their
inductive action on each other is zero. If all these conductors,
being at the same potential, are joined by means of conductors
whose capacity may be neglected, fine wires for instance, no exchange
of electricity will take place, for they were all at the same potential,
and this potential will not change.
They form thus a single conductor, the charge of which is
equal to the sum of the original charges. The electrical capacity
of the system is equal to the sum of the capacities of the separate
conductors.
Let us now suppose that the potentials of the original conductors
are different V, V, V", the corresponding charges are
= VC, M' = V'C', M" = V"C"
All these charges being regularly distributed upon the single
conductor formed by the system, will produce a potential V l given
by the equation
V 1 C 1 = VC+V'C'
whence
1
This expression is frequently used in experimental researches.
It will be remarked how analogous it is to that which represents the
temperature resulting from the mixture of several different bodies at
different temperatures.
84. BATTERIES. This term is applied to the system formed of
several Leyden jars, or condensers of any kind, which are connected
with each other.
If the condensers are virtually closed, as is the case with ordinary
Leyden jars, the external action of each of them is insignificant,
and they can be brought near each other without exerting any
appreciable influence.
72 ELECTRICAL EQUILIBRIUM.
Connection may be made in two ways :
i st. All the inner coatings may be connected with each other, on
the one hand, and all the outer coatings on the other ; the battery is
then said to be arranged for quantity. The whole forms a condenser
whose capacity is equal to that of the capacities of all the jars
separately. If the battery contains p identical jars, each with the
capacity C, the capacity C x of the battery is
2nd. All the jars being insulated, the outer coating of one is
connected with the inner coating of the following one ; the inner
coating of the first jar is charged to potential V p the outer coating
of the last jar being at potential V 2 , and all the intermediate coatings
being insulated ; this arrangement is said to be in series or cascade.
85. CHARGE BY CASCADE. The first jar receives a charge m
on its inner coating, and assumes the potential V 1 an equal and
opposite charge -m is produced on the surface of the dielectric
next the outer coating. The conductor formed of this coating and
the inner coating of the second jar being insulated, will take a
charge + m, which is distributed regularly upon this conductor, as if
the internal charges did not exist, and produces there a potential V.
The greater part of this charge passes to the inner coating of the
second jar, the capacity of which is very great in reference to that
of the conductor in question. Continuing this reasoning, it will be
seen that the inner coatings have continually decreasing charges, but
the diminution is very small, and we may consider all the jars as
having the same internal charge +m, the successive potentials
being V lf V, V", ..... V 2 .
If the battery contains p jars, we may put
m = C (Vj - V) = C'(V - V") = ..... = C<*- x >(VO-'> - V 2 ),
whence
CHARGE BY CASCADE. 73
Adding all these equations, we get
Hence the charge of the first jar, which is the only one that
receives electricity directly, is
v,-v,
I I I
c + c + c 7/
We have thus, for the capacity Cj of the battery,
If the jars are identical, the capacity of the battery has become
/ times less than that of each of the jars.
This arrangement may appear unfavourable, since its effect is to
diminish greatly the capacity of the battery; yet it presents great
advantages for certain experiments. Leyden jars can only sustain a
limited difference of potential, beyond which their coatings discharge
themselves along the surface of the glass, and even sometimes
through the mass of the glass itself, which is then traversed by a
spark. By means of a battery in cascade, the total difference of
potential may be distributed in stages on the successive jars.
This, for instance, is the arrangement adopted in the ordinary
Holtz machines, where the capacity of the conductors is increased
by connecting each of them with the inner coating of a Leyden jar ;
care however is taken to join these in cascade, so as to maintain the
maximum difference of potential, and therefore the greatest striking
distance which the play of the machine allows.
When a large number of jars are available, they may be joined
together for quantity so as to form several batteries, which in turn
are arranged in cascade. In this way the whole of the potential
which a machine can yield may be utilised, and the maximum of
effect obtained with the least expenditure of electricity.
74 ELECTRICAL EQUILIBRIUM.
86. GENERAL PROBLEM OF THE RECIPROCAL INFLUENCE OF
Two INSULATED CONDUCTORS. MURPHY'S METHOD. In order to
determine the distribution of electricity on two insulated conductors
A and B, charged with the total masses M a and M 6 and only sub-
mitted to their reciprocal action, it is sufficient if we know for each
of them :
ist The capacity and the distribution on the surface when it is
insulated and not subject to any external induction ;
2nd. The distribution of the electricity induced on the surface
when it is in connection with the earth, and is subject to the in-
ductive action of an electrical mass placed at any point outside it.
Let m be the capacity of the conductor A alone that is to say,
the charge which would then produce potential unit.
Let us fix this mass, the distribution of which is known, and let
us place in the desired position the conductor B in connection with
the earth. This will be at potential zero, and will become charged
with a known mass of the opposite electricity - m'.
In like manner let us fix the mass m' on B. Let this conductor
be insulated, and let the first one be connected with the earth ; this
latter will acquire a mass m 1 at potential zero.
In like manner let the mass +m 1 be fixed on A, an induced
layer - m" will be obtained on B, and so forth.
Continuing in the same manner, we shall successively obtain the
masses m t m v m z ... on the former, and m\ m", m'" ... on the
latter, each of them tending to verge rapidly towards zero.
The superposition of all the layers m, m v m 2 . . . on A, and of all
the layers m', m", m'" on B will result in a state of equilibrium
with zero potential on B, and potential equal to unity on A. In fact,
the successive layers m and - m' t m l and - m", . . . taken in pairs,
give zero potential on B ; the layers m' and m v m" and m 2 , . . . give,
in like manner, zero potential on A. We have only thus to consider
the mass m on the first conductor, which produces a potential equal
to unity.
Putting Cm+mm + ..... ,
we see that C a represents the capacity of the insulated conductor A
in the presence of the conductor B connected with the earth, and
- C' the coefficient of electricity induced on B (69).
Multiply these two masses by V , the respective charges C a V a
and - C' a V' a correspond to a state of equilibrium with zero potential
on B, and potential equal to V a on A.
RECIPROCAL ACTION OF TWO ELECTRICAL CONDUCTORS. 75
Reversing the functions of the conductors, we shall obtain the
masses C 6 V & on B and - C' b V' b on A, corresponding to a new state
of equilibrium, with zero potential on A, and potential V & on B.
The superposition of these two states of equilibrium gives a new
state of equilibrium with the addition of the potentials on each
of the conductors that is to say, the potential V a on A and V 6 on
B. The total charges of the two conductors are these
These equations enable us to calculate the total masses of the
two conductors when the potentials are known.
In like manner the potentials may be deduced as functions of
the masses, which gives
V =^
a r r
87. RECIPROCAL ACTION OF Two ELECTRIFIED CONDUCTORS.
The preceding method enables us to determine the distribution of
electricity on the two conductors, for the final density at each point
is the sum of the densities relative to the various superposed layers,
and by hypothesis we know the law of distribution for each. We
have then all the elements needed for calculating the action exerted
between the two bodies ; the problem only presents then difficulties
of calculation.
This force consists of the action of each of the two layers C a V a
and - C' 6 V & of the body A, on the two layers C 6 V & and - C' a V a of
the body B. The potentials being supposed positive, the action
f CaYa ls m ade up of two terms one repulsive, proportional to the
product V a V & of the two potentials, and the other attractive pro-
portional to V* .
The action of - C' b V b comprises also two terms, one attractive
proportional to V^ and the other repulsive proportional to the product
v.v 6 .
Calling a, Z>, and c coefficients which depend on the form of the
body and on their distance, the reciprocal action R, considered as
repulsive, has an expression of the form
76 ELECTRICAL EQUILIBRIUM.
If the conductors A and B are identical, and arranged sym-
metrically, the coefficients a and b are equal, and the formula
becomes
We have assumed that the action of the two bodies reduces to a
single resultant. If it were not so, the same reasoning would apply
to the two resultants by which the whole of the forces may be
replaced.
As a matter of fact the calculations required by this method for
determining the coefficients C a , C 6 , C' a and C' 6 , and the resultant R,
are extremely tedious even in the simplest cases. We shall after-
wards explain the application which Sir W. Thomson has made of it
to calculating the reciprocal influence of two spheres.
ELECTRICAL ENERGY. 77
CHAPTER V.
WORK OF ELECTRICAL FORCES.
88. ELECTRICAL ENERGY. When different electrical conductors
are connected with the earth, the system reverts to the neutral state,
and in doing so performs work which is necessarily positive. Any
given system of electrical conductors possesses a store of available
energy corresponding to this work ; it is a potential energy, which we
may simply speak of as electrical energy.
The electrification of a system, requires the expenditure of an
amount of work equal to the potential energy which it possesses in
this new condition.
When two conductors are connected, a change is in general
produced in the distribution of the electrical masses, and this
modification corresponds to a positive work. The electrical energy
of a system of conductors is therefore equal, or superior, to that
of the system obtained by connecting all these conductors in any
way whatever.
When the system contains an electrified insulating body, we may
look upon the several electrified masses, with which the body is
charged, as belonging to infinitely small conductors. If all the
masses are connected together, the energy diminishes. The energy
of a system of bodies, each of which possesses a given mass, is
therefore a minimum when all the bodies are conductors.
The potential energy of a system may be measured either by
the work expended in electrifying it, or by the work which is per-
formed by its discharge.
89. ENERGY OF A SINGLE CONDUCTOR. Let us first consider
a single conductor of capacity C, and let us suppose that a charge
M has been given to it, which raises it to potential V. To increase
the charge by dM, this quantity */M of electricity must be brought
from infinity, or from the earth, to the conductor, and the work
expended in this operation is equal to WM.
78 WORK OF ELECTRICAL FORCES.
The increase dW of the energy of the conductor is therefore
When the mass of electricity changes from M to M I} the in
crease of energy is
As the energy vanishes with the mass, we see that the energy
which corresponds to the mass M is
W = = -CV2 = -MV
2C 2 2
Thus the electrical energy of a single conductor is proportional to the
square of the charge, or to the square of the potential.
90. ENERGY OF A SYSTEM OF CONDUCTORS. Let there be any
number of conductors A v A 2 , A 3 , . . . having charges M 1? M 2 , M 3 , ....
with the potentials V p V 2 , V 8 , . . .
If the density of each point is multiplied by x, a new state of
equilibrium is obtained, in which the potentials are multiplied by the
same factor x. There is the charge xM 1 on A x at the potential
xV v xM% on A 2 at the potential #V 2 , etc.
If we increase x by dx, the masses and the potentials are
multiplied by x + dx, and the increase of charge in the conductor
A 1 is M^x. The corresponding work lies between M 1 dx.xV 1 and
'M. l dx(x + dx)V-^ it is therefore, within an infinitely small expression
of the second order, equal to M-^^dx. This is also the case with
the other conductors, so that the variation of energy of the system
is
dW = (MjVj + M 2 V 2 + ..... )xdx =
Between the two values X Q and x 1 the increase of energy is
If we make ^ = and ^=1, which amounts to supposing that
ENERGY OF A SYSTEM OF CONDUCTORS. 79
to reach the state in question we started from the neutral state, we
have simply
W = -(M 1 V 1 + M 2 V 2 + ) = MV.
We thus see that the energy of a system of conductors is equal to
the half -sum of the products of each mass by the corresponding potential.
91. A conductor which remains insulated during the charge is
merely electrified by induction, and its total charge is zero ; there is
no term, therefore, in the sum of the products, which corresponds to
an insulated conductor.
In like manner, a conductor kept in connection with the earth
remains at zero potential, and does not enter into the expression for
the energy.
It must however be remarked that these two kinds of conductors
affect the value of the energy, by modifying the influence of the
capacities, and therefore the potentials, of the electrified bodies.
Lastly, the same formula holds for the case of insulating bodies,
however electrified. Each of the volume elements of an insulating
body may, in fact, be considered as an infinitely small conductor on
which the corresponding electrical mass is distributed. In this case
the preceding sum becomes an integral; calling p the electrical
density, and V the potential on the volume element dv, the energy of
the system is expressed by
The energy accumulated by electrification on a system of con-
ductors is expended when the system is discharged, and may be
transformed into mechanical work, or into an equivalent effect :
disengagement of heat, chemical action, etc.
92. If electricity were a material substance, the masses consti-
tuting the electrical layers would acquire a certain vis viva during
the discharge, in virtue of which they would, like a pendulum, pass
beyond their position of equilibrium, so as to restore to the system a
fraction of its initial energy ; a succession of discharges alternately
in opposite directions would be produced, until the heat disengaged
upon the conductors had exhausted the whole of the available
energy, and final equilibrium would only be reached after a certain
number of oscillations. Experiment shows, indeed, that under
So WORK OF ELECTRICAL FORCES.
certain conditions the discharges have a distinctly oscillatory cha-
racter ; but we shall see that these oscillations may be explained in
a totally different manner. Hence no conclusion can be drawn in
favour of the hypothesis which assigns a certain inertia to electrical
masses, and in the present state of science no decisive fact can be
claimed for or against this hypothesis.
93. DISCHARGE OF BATTERIES. QUANTITY BATTERY. Total
Discharge. We have seen that the capacity Q of a battery
arranged for quantity is equal to the sum of the capacities of each
of the jars.
If the total energy of the battery is transformed with heat during
the discharge, then, calling J the mechanical equivalent of the unit
of heat, and Q the heat disengaged, we have
W = -MV = -~ = C 1 V 2 = JQ.
2 2 L^^
If the battery consists of p identical jars, of capacity C, the
formula becomes
We thus see that, for a given charge, the energy, or the heat
disengaged, is inversely as the number of jars, and that for a given
potential the energy is proportional to the number of jars.
94. Incomplete Discharge. Let us consider two batteries of the
capacities C x and C 2 , the former charged with a mass M and the
second in the neutral state, the outer coatings being connected with
the earth. Let us suppose that instead of discharging the first, we
join the coatings so as to form a single jar of the capacity C x + C 2 .
The discharge is said to be incomplete; it represents a loss of energy,
and produces a disengagement of heat. Before contact, the potential
energy of the first battery was
After contact, the energy of the system has become
DISCHARGE OF A BATTERY IN CASCADE. 8 1
The energy expended in the discharge is then
2 C, C,+C,
The proportion of the initial energy which has been expended is
Wi-W a= C 2 i
W x ~C 1 + C 2 - Cj'
C 2
Let us suppose that the first battery consists of p l jars of the
surface Sj and thickness e v and the second of A J ars f ^ e surface
S 2 and the thickness * 2 , we shall have
&'iA.S.-.4 ; ;
C 2 A S 2 'i
which gives
l 1+ .l.2
A S 2 ^1
95. DISCHARGE OF A BATTERY IN CASCADE. The capacity C x
of a battery arranged in cascade is connected with the capacities
C, C', C" . . . of the several insulated jars by the expression (85)
_
C C' C"
The expression for the potential energy of the system only comprises
the term relative to the first jar ; for all the other conductors are
insulated, or in connection with the earth, while being charged. We
have thus
If the battery consist of p identical jars,
C =-
which gives for the energy
_i M 2 _iCV 2
~~2 P ~Q,~~2~P~'
For a given charge, the energy of the cascade would be greater
G
82 WORK OF ELECTRICAL FORCES.
than that of a single jar; but for a given potential it would be p
times less. It is the exact opposite of charge by quantity.
All the laws relating to the discharge of batteries have been
experimentally established by M. Riess.
On the whole, then, in working at a constant potential that is
to say, with a constant source of electricity, the best combination
that can be made with a given number of jars, so as to obtain the
maximum energy in the discharge, is to join them in quantity, pro-
vided always that the jars can sustain the maximum potential of the
source. If, on the other hand, only a limited supply of electricity is
available, it is best to arrange them in cascade.
The first is the case most frequently met with in electrical
machines; but as they usually produce very high potentials, it is
often advantageous to select a suitable combination of the jars by
which these high potentials may be used and at the same time the
charge be economized.
96. ELECTRICAL WORK IN THE DISPLACEMENT OF INSULATED
CONDUCTORS. Conductors with a Constant Charge. The value of
the potential energy of a system of conductors is
When the relative position of these conductors is changed, with-
out in any way connecting them, a positive or negative work of the
electrical forces is, in general, produced, and therefore the energy of
the system is altered. If the conductors are left to themselves they
obey the electrical actions which urge them ; the work of these forces
is positive and corresponds to a loss of energy in the system. If,
by any external work, the system experiences a deformation in a
direction contrary to that of the electrical actions, the energy
increases to a corresponding extent.
Hence, calling </T the work of the electrical forces, and dW the
corresponding variation of energy, we have at each moment
(i) </w+dnr=o.
The energy of conductors which are left to their reciprocal actions
tends therefore towards a minimum.
We have, moreover, the general expression,
1 T
2 2
CONDUCTORS AT CONSTANT POTENTIAL. 83
but in the present case the last term is zero, for the charge is con-
stant on each of the conductors ; there simply remains
A conductor originally in the neutral state would be drawn into
the electrical field. Hence the effect of the presence of this con-
ductor is to diminish the energy of the system.
97. Conductors at Constant Potential. Let us now consider the
case of conductors kept at constant potentials by sources of elec-
tricity placed outside the field of action.
We shall suppose that the various conductors A 15 A 2 , A 3 . . . ,
charged with quantities M v M 2 , M 3 . . . , and to the potentials V 1?
V 2 > V 3 . . . , communicate separately with bodies of the capacities
Cj, C 2 , C 3 . . . , withdrawn from any external influence for instance,
closed condensers the outer coating of which is connected with
the earth. This case comes under that which we have been con-
sidering ; if W a is the energy of the conductors and W c that of the
condensers, the energy of the system is
w=w a +w c .
If the system undergoes any deformation without the intervention
of extraneous energy, the theorem (i) applies and gives
(2)
The energy of the conductor is
W a = -
2
and therefore
2 2
For the energy of the condensers, the capacity of which is un-
changed, we shall take the expression
2
G 2
84 WORK OF ELECTRICAL FORCES.
from which is deduced
Lastly, the total charge M + CV for any system consisting of a
conductor and the corresponding condenser is constant; we have
then
and therefore
which gives for the conductor and condenser together
Taking into account this latter relation, equation (2) may be
written
ii ii
22 22
We have then
(3)
^
This equation holds, whatever be the capacities of the condenser.
There is nothing to prevent our considering the capacities as being
infinitely large in reference to those of the conductors, so that the
variations of potential dV v dV 2 . . and the variations of energy
Mj^Vj, M 2 ^V 2 . . . are absolutely negligable. We come then to the
case of conductors kept at constant potentials by external sources,
and equation (3) reduces to
whence
which gives finally, from equation (i),
Thus, when conductors are kept respectively at constant poten-
tentials, the energy of the system, for a given deformation, increases
by a quantity equal to the work of the electrical forces. This
work is positive if the system is left to itself; like the corresponding
CONDUCTORS AT CONSTANT POTENTIAL. 85
increase in . the energy, it is borrowed from the sources which keep
the potentials constant. The sources yield then, at every moment,
a quantity of energy which is divided into two equal parts; one
serves to perform the work dT of the electrical forces, the other
goes to increase by dW a the electrical energy of the system.
In this case the energy of the system tends towards a maximum.
98. We shall proceed to apply these theorems to the following
problem, which may serve as basis of the theory of symmetrical
electrometers.
Let us suppose that a system of conductors is formed of two
fixed unlimited cylinders A and B (Fig. 21) having a common axis,
Fig. 21.
and of a cylinder concentric with the preceding ones, movable
along this axis, the length of the inner cylinder C being, moreover,
so great that the density at each end only depends on that of the
nearest fixed conductor. Let V 15 V 2 , and V be the potentials of
these three bodies, and A , B , and C the charges which they
possess when the movable cylinder is in a position symmetrical with
the two others.
If the cylinder C is displaced by a small quantity x, towards
the right for instance, the distribution of electricity on the various
surfaces near the opening and at the ends is not modified ; we have
merely on this side increased, by a quantity proportional to x, the
surface on which the electrical density is uniform and proportional
to the difference of potentials of the adjacent conductors The right
half of the movable cylinder will have gained a quantity of elec-
tricity proportional to x, and the fixed cylinder B an equal quantity
of the contrary electricity ; the opposite effect will be produced on
the other side.
Thus, calling A , B , C the initial charges on the three cylinders,
A, B, and C the new charges, and a the capacity for unit length of
the inner cylinder at some distance from the middle and from the ends,
= B -cu;(V-V 2 ),
= A + ca;(V-V 1 ).
86 WORK OF ELECTRICAL FORCES.
The variation of energy is then
w - w = I ax ! - v 2 ) v - (v - v 2 )v 2 + (v
- vjvj
The resultant F of the actions of A and B on C is, by symmetry,
parallel to the common axis; the work F#, performed during the
displacement x, is equal to the variation of energy. We get from
this
We may, indeed, express the coefficient a in functions of the
data of the problem. We know, in fact (80), in the case of two
unlimited concentric cylinders, the radii of which are R and Rj and
the potentials V and V 1} that the charge of the inner cylinder for the
length x is
R
From this we get
and therefore
FUNCTION OF THE DIELECTRICAL MEDIUM. 87
CHAPTER VI.
ON DIELECTRICS.
99. FUNCTION OF THE DIELECTRICAL MEDIUM. We have
hitherto reasoned on the hypothesis that the actions between elec-
trified bodies take place at a distance, and have considered the
dielectric as an inert medium, through which the forces act, but
as destitute itself of any active properties.
It appears now to be well proved that heat is a vibratory motion,
the propagation of which takes place through the intervention of an
elastic medium. Now, we have seen that the problem of electrical
equilibrium, and that of heat in the permanent state, are characterized
by the same mathematical properties.
May we not then suppose that the analogy in the two cases is
closer ; that it may be followed into the mechanism of the elementary
actions ; and that there is no other difference in the two orders of
phenomena than that which we ourselves introduce into the physical
interpretation of the laws ? If this is the case, it should be possible
to explain the production of the electrical forces by the action of
the medium only.
Such is the idea which Faraday sought to elucidate, and which
constantly guided him in his researches. This is not the place to
attempt to prove, or to disprove, the exactitude of one or the other
of these points of view, but simply to show their equivalence in
explaining phenomena.
We shall commence by establishing some theorems on the
relations between forces and electrostatic pressure.
100. EXPRESSION OF FORCE AS A PRESSURE. We have already
considered as evident, that the action which is exerted on a conductor
is the resultant of the electrical pressures on the whole of its sur-
face ; but it may be useful to consider this theorem from another
point of view.
88 ON DIELECTRICS.
The pressure, on unit surface, at a point of the conductor where
the density is a-, and the force F, has the value
/=27TO- 2 = F 2 = -F(T,
O7T 2
and this pressure is always directed outwards, whatever be the sign
of the electricity.
For each element of surface, the pressure /^S is the resultant of
the actions exerted, on the mass OY/ S of this element, by all the masses
external to the conductor, and by those on its own surface. The
resultant of all the pressures for the entire surface, is the resultant of
the actions exerted on this conductor, both by the external masses
and by its own electricity. But the resultant of the actions which the
various masses of the conductor exert one upon the other is almost
null; for as there is equilibrium, those masses may be regarded as
fixed on the conductor, and in this case, the elementary forces
taken in pairs neutralise each other ; the resultant of the pressures
is then simply equal to the resultant of the actions of the external
masses.
101. When an electrical system is surrounded by an equipotential
surface S v the action exerted on this system is the resultant of the
pressures which would be exerted on a layer equal to the total charge
of the system, in equilibrium on the surface S r
Let us suppose that an equipotential surface S x divides all the
acting masses into two systems, an internal one M 15 and an external
one M 2 . We have seen that for points external to S 15 the internal
masses may be replaced by a layer of the same total mass M 1 in
equilibrium on the surface. Conversely, the external system M 2 will
act on this layer Mj fixed on the surface S lf as it would act on the
internal masses, supposed to be connected with each other, so as to
form a rigid system.
But, from the foregoing remark, the action of external masses upon
the layer Sj, and therefore on the system M l of the internal masses,
is no more than the resultant of the electrostatic pressures of this
layer.
As the total action of the system M l on all the external bodies
is equal, and of opposite sign to the force which this system
experiences, it is also seen that the action of the system M 15 on
external bodies, is equal to the resultant of the elementary
pressures on the surface S 15 each of them being counted towards
the inside.
EXPRESSION OF ELECTRICAL FORCE AS PRESSURE.
8 9
102. The reciprocal action of two systems M l and M 2 is equal to
the action of two layers + M^ and M l distributed on the two equi-
potential surfaces S x and S 2 , which include M 1 and leave M 2 outside.
For, consider a second equipotential surface S 2 (Fig. 22) which
includes M 15 and again leaves the system M 2 entirely outside.
Let us arrange a layer + Mj in equilibrium on the surface S 1} and
a layer - Mj in equilibrium on S 2 ; the layer on S : may replace the
internal system + M^ for all points external to S x ; and the layer on S 2
is equivalent to the external system M 2 for all points on the surface S 2 .
The system of these two layers gives, moreover, a constant
potential V 1 -V 2 inside Sj, and a zero potential outside S 2 ; and,
finally, a potential varying from Vj V 2 to zero in the intermediate
space. The electrical force is therefore everywhere zero, excepting in
this space, where it retains the same value at all points, either for the
two primitive systems M : and M 2 , or for the equivalent layers
distributed on the surfaces S : and S 2 .
Fig. 22.
The action of the electrified surface Sj on the layer S 2 is thus the
same as on the system M 2 ; that of S 2 , the same on S : as upon M x ;
the reciprocal actions of the electrified surfaces S x and S 2 are thus the
same as those of the two primitive systems M l and M 2 .
But we know from the preceding theorems, that the actions ex-
perienced by the surfaces Sj and S 2 , are merely the resultants of the
electrical pressures /X^i ano - A^2 which are exerted on the elements
of these surfaces. If Fj and F 2 are the electrical forces in the medium,
the values of these pressures near the elements in question are
T
: 8^
i
8^
the first are directed outside the surface S 15 and the second inside
90 ON DIELECTRICS.
the surface S 2 ; and the resultants of these two systems of perpen-
dicular forces /X$>i and / 2 </S 2 are equal, and of opposite signs, as
they represent action and reaction.
103. The real force which acts between the two electrified
surfaces Sj and S 2 may be regarded as arising from the elementary
actions, which are exerted directly and at a distance, between the
different electrical masses which cover them, taken in pairs. This is
the hypothesis which, up to the present, has formed the basis of all
our calculations. But it may also be assumed that this action is
transmitted through the surrounding medium in virtue of a special
elasticity, as Faraday believed. Regarding it from this point of view,
we shall proceed to investigate the mechanical conditions which the
intervening medium ought then to satisfy.
For this purpose, let us consider an orthogonal tube between the
two surfaces S x and S 2 . The flow of forces issues from dS l (Fig. 23),
and is absorbed at ^S 2 , and the two elements dS-^ and </S 2 are exactly
in the same condition as if they were connected by elastic threads
parallel to the lines of force, and pulling the two elements towards
each other, with a force equal to / x for unit surface on </S, and to / 2
on ^S.
Fig. 23.
Let us take in this tube a volume-element bounded by two
infinitely near equipotential surfaces S and Sj at a distance of dn
from each other, and let us suppose this to become solidified. This
element must be regarded as subjected to two tensions pulling its
bases outwards, and the resultant of which is
rfR =/</S' -pd = i (F VS' - FVS).
57T
As from the properties of tubes of force
we have
^ R== 8^
ELECTRICAL PRESSURE. 9 1
The two surfaces being infinitely near, we may write
~~dn H '
which gives
-
dn
Calling R the action relative to unit volume of the dielectric
at a point, we have
R-I#
2 an
The result is accordingly the same as if the forces were exerted
on the dielectric itself, and as if the force for unit volume were
determined by a potential equal at every point to --; hence the
volume element tends to be drawn in the direction towards which
the function p increases.
104. But, under these conditions, the volume-element cannot
be in equilibrium ; it is therefore necessary to bring other forces into
play. It is sufficient if we assume that the element experiences at
each point of its surface a perpendicular pressure p analogous to
hydrostatic pressure, and that the level surfaces corresponding to the
pressure p l coincide with the electrical equipotential surfaces. The
volume element will experience an upward pressure.
-- -,
dn dn
which will balance the resultant pressure R if
,
dn
or
A =IA
The true action on the volume-element ddn consists then of a
pressure p l = -p acting on the whole surface, and of a tension p on the
92 ON DIELECTRICS.
bases. This amounts to saying that the lateral surface experiences
P
a pressure p^ = - , and the bases a tension equal to the difference
/! -/, that is to say equal to - .
105. TENSION AND REPULSION OF LINES OF FORCE. If we
consider a layer bounded by two electrified equipotential surfaces, on
which we assume there are electrical masses capable of replacing the
action of bodies external to the layer, the two surfaces will attract
each other with a force equal to the general resultant of the tensions.
Dividing this layer into two by an orthogonal surface, a repulsion
will come into play, between the two portions, equal to the resultant
of the lateral pressures.
It is easy to extend these considerations to the case in which
the second equipotential surface does not envelope the first.
We may then suppose that conductors are connected with each other
by electric threads stretched along the lines of force ', and which repel
each other. This material representation of the phenomena is a
useful guide in a great number of applications.
106. The foregoing properties are the mathematical translation
of the idea which Faraday formed for himself of the state of dielec-
trics, and which he himself summed up in the two following para-
graphs of his Experimental Researches, Series XL, 1297-1298 :
"The direct inductive force which may be conceived to be
exerted in lines between two limiting and charged conducting
surfaces, is accompanied by a lateral or transverse force equivalent
to a dilatation or repulsion of these representative lines, (1224);
or the attractive force which exists amongst the particles of the
dielectric in the direction of the induction is accompanied by a
repulsive or a diverging force in the transverse direction (1304).
" Induction appears to consist in a certain polarized state of the
particles, into which they are thrown by the electrified body sus-
taining the action, the particles assuming positive and negative points
or parts which are symmetrically arranged with respect to each other
and the inducting surfaces or particles. The state must be a forced
one, for it is originated and sustained only by force, and sinks to the
normal or quiescent state when that force is removed. It can be
continued only in insulators by the same portion of electricity, be-
cause they only can retain this state of the particles."
107. ENERGY OF THE DIELECTRIC MEDIUM. From this point
of view, the whole energy of the electrical system must reside in the
dielectrical medium, and it is easy to calculate its value at any point.
ENERGY OF THE DIELECTRIC MEDIUM. 93
The total energy of a system is (91)
1 if
W = - V m V, or W = - Vpdv.
2 2j
Replacing the density by its value deduced from Poisson's
equation
AV + 47175 = 0,
we get
W= -
Hitherto the energy has been determined as a function of the
electrical masses themselves. In order to vary the signification, we
may apply Green's formula (33)
fvAWz> = fv^^S- (^dv
J *n
to the volume bounded by a sphere of very large radius r which
includes the electrical system we are considering. The first term
of the second member should be extended to the surface of this
sphere. The potential V, as we recede, tends to become inversely
<)V
as r; the factor represents the perpendicular component of the
on
force, and becomes inversely as r 2 . As the surface itself is pro-
portional to r 2 , this integral is inversely as r, and tends towards
zero.
The second member reduces then to the second term, and we
have, for the expression of the energy,
S7T
It appears from this that the energy of the system is the same as
if each volume element of the medium had a quantity of energy
F 2 </z>. The energy w for unit volume is accordingly
OTT
94 ON DIELECTRICS.
Hence the energy for unit volume is equal at every point to the
electrostatic pressure.
108. SPECIFIC INDUCTIVE CAPACITY. If the dielectric does play
this essential part in phenomena, it is not likely that all media
behave in exactly the same manner.
We know, in fact, since Franklin's experiments, that the nature
of the glass is of great importance in the construction of electrical
batteries. Cavendish had already made a great number of ex-
periments to determine directly the comparative effect of various
substances used as insulators in condensers, but his experiments
were unpublished and unknown at the time when Faraday published
his important researches.
Faraday connected the coatings of two spherical Leyden jars
of the same dimensions, in one of which the insulating layer of air
had been replaced by a solid dielectric such as melted sulphur or
resin ; he thus found that when a definite charge of electricity was
imparted to this system of conductors it did not divide equally
between the two jars. That in which the dielectric was solid, took
the larger charge.
This is a general phenomenon, and falls under a very simple law.
The charge, acquired by a closed condenser, with a solid or liquid
dielectric, is in a constant ratio with the charge which it would take,
for the same difference of potential, if the dielectric were replaced by
a layer of air.
Experiment shows, in fact, that air and gases, even when moist,
behave in virtually the same manner, whatever be the pressure and
temperature. If the nature of the gas does exert an appreciable
influence, to which we shall subsequently refer, it may be neglected
in practice.
The ratio thus determined, is what Faraday calls the specific
inductive capacity of the dielectric. It is, as we see, the number by
which the capacity of an air-condenser must be multiplied, to give
that of the same condenser, in which the layer of air has been
replaced by the dielectric in question.
109. ELECTRICAL ABSORPTION. The determination of this
constant offers considerable difficulties for most substances, owing
to the occurrence of a phenomenon, to which Faraday gave the
name of electrical absorption, and which is due to the same cause as
the residual charge of condensers. The capacity of a condenser, in
which the dielectric is solid, appears as a function of the time ; it
increases and seems to tend towards a limit, in proportion as the
duration of the charge increases. Conversely, when the condenser
ELECTRICAL ABSORPTION. 95
is discharged, the disposable electricity which disappears in the
discharge is sometimes far below the whole of that which it
possesses ; it is known moreover that we can successively obtain a
greater or less number of discharges of decreasing intensity.
It appears difficult in the present state of science to account for
this phenomenon. Everything seems to point to its being due to a
progressive change in the structure of the dielectric, to a particular
deformation under the influence of causes which produce polarization;
a deformation which becomes permanent as in an imperfectly elastic
body, and after which the body does not immediately revert to its
original state when the cause has ceased to act.
This view of the matter is confirmed by the facts that all the
circumstances which, in the case of a mechanical deformation,
favour the return of a body to the normal state such as blows,
rapid variations of temperature, and the like, appear also to accele-
rate the disappearance of the residual charge, and its return to the
neutral state.
110. POLARIZATION OF THE DIELECTRIC. Although Faraday's
experiment is incompetent to settle the question of actions at a
distance, it shows unequivocally the part played by the medium in
electrical phenomena. We are thereby led to assume that, in
electrical induction, the medium acquires a state of polarization
analogous to that observed in soft iron when under the influence
of a magnet.
In order to explain magnetism, Poisson made a hypothesis which
was transferred to the study of electrical phenomena by Mossotti,
and then adopted by Faraday. This hypothesis consists in assuming
that the magnetic medium, or the dielectric, is made up of particles,
which may be spherical for instance, which are absolute conductors,
and are disseminated in a non-conducting medium.
" If the space round a charged globe were filled with a mixture
of an insulating dielectric, as oil of turpentine or air, and small
globular conductors as shot, the latter being at a little distance
from each other so as to be insulated, then these would in their
condition and action exactly resemble what I consider to be the
condition and action of the particles of the insulating dielectric
itself. If the globe were charged, these little conductors would all
be polar ; if the globe were discharged, they would all return to their
normal state to be polarized again upon the recharging of the globe."
(Faraday, Experimental Researches , Series xiv., 1679.)
Sir W. Thomson has shown that, without making any hypothesis
as to the constitution of the medium, it is sufficient to assume that
96 ON DIELECTRICS.
each volume-element is changed by induction into a small magnet,
which may, indeed, be considered as an experimental fact. In this
way all the mathematical consequences of Poisson's hypothesis may
be deduced.
111. DEFINITION OF DIELECTRIC. A dielectric placed in a field
becomes polarized, and the algebraical sum of the masses which
form the charge is always null. We know further (59) that, what-
ever be the condition of an electrified body, the action which it
exerts upon an external point is equal to that of a layer of the same
total mass as its own, distributed on the surface according to a certain
law ; in the present case the equivalent layer is formed of two sheets
having equal masses and opposite signs.
According to the theory of magnetic induction, which we shall
afterwards explain, the action of this layer replaces the effect of
polarization, not only for external, but also for internal points. The
distribution is determined by the condition, that at two adjacent
points, one in air or rather in vacuum, and the other inside the
dielectric, the components of the force, perpendicular to the
bounding surface, shall be in a constant ratio />&, so that, if Y n
and F n are the perpendicular components in air and in the dielectric,
taken in the same direction, we have
^ = /,, or F^FV
Without attempting, for the moment, to examine thoroughly the
intimate nature of the phenomenon, we may regard this equation
(i) as defining the function of a certain class of bodies, to which
experiment shows that the dielectrics, such as we know them, must
belong.
We have seen (39) that on both sides of an electrified surface
the components of the forces parallel to the surface are equal, and
that the difference between the perpendicular components is pro-
portional to the density of the layer,
From which, agreeing to count as positive the perpendicular com-
ponents on the side of the dielectric, is deduced
DEFINITION OF DIELECTRIC.
97
Poisson's hypothesis amounts, in short, to supposing that, on the
surface of the dielectric, there is a fictive layer the density of which
cr satisfies this condition.
112. This result may be exhibited under another form. On
both sides of an element PP' or dS of the surface of the dielectric
(Fig. 24) let us draw two tubes of force, and let them terminate
in two orthogonal bases dS l and d$\, one in air and the other in
the dielectric, and just far enough apart to comprise between them
the layer a-dS. The flow of force which enters by the base ^S x is
~F n d -, that which emerges by the base dS\ is
Fig. 24.
the rate of variation of the flow is then equal to (F M - F' n ) dS, or
from equation (i) which defines the dielectric, to ( i -- \ ; it corre-
sponds to a mass of electricity o-d such that
The effect is therefore the same as if a constant fraction of the flow
of force were absorbed or emitted by the fictive layer on the surface ;
the value of this fraction is i - - .
I*
113. REFRACTION OF THE FLOW OF FORCES. The tangential
components being the same in the two media, if / and /' are the
angles which the forces F and F' make with the perpendicular N to
the surface S, the expressions
F cos/=/>tF' cos/',
F sin i F' sin *',
H
98 ON DIELECTRICS.
give the equation
i
tan /= tan /',
which expresses what may be called the law of refraction of the
force, or of the flow at the moment at which the force passes from
air into a liquid or solid dielectric.
114. More generally, let us suppose that the surface S separates
two dielectrics, solids or liquids, whose specific inductive capacities
are respectively equal to /^ and /* 2 . If the surface is replaced by
an infinitely thin layer of air, then if F, F 1? and F 2 are the forces in
air, in the first, and in the second medium, we shall have
11/1
from which is deduced
(F n )
and
The fictive layer is determined by the equation
( F J2 ~ ( F )i =
putting <r = Oj + <r 2 . Lastly, the law of refraction gives for the angles
t\ and /g of the forces, with the perpendicular on both sides of the
surface, the ratio
tan t\ tan /
115. TUBES AND FLOW OF INDUCTION. Let us agree to apply
the term induction at a point, to the product of the force F by the
specific inductive capacity p of the substance, and the term quantity
TUBES AND FLOW OF INDUCTION. 99
or flow of induction across a surface-element, to the product of this
element by the perpendicular component of induction ; the preceding
results may then be expressed in a very simple manner.
We observe, in the first place, that in gaseous media, or at any
rate in a vacuum, //, = i ; the induction and the force have the same
numerical expression, and tubes of force are identical with tubes
of induction just as are the two kinds of flow.
In the case of continuous media whose specific inductive
capacities are /x x and /* 2 , the ratio of the perpendicular components
ft 1 F 1 cos t\ = /* 2 F 2 cos /g
gives
FS cos t = *F^S cos
an equation which signifies that the flow of induction across the
element dS retains the same value in the two media. We are thus
led to the following law :
In a tube of induction the flow of induction retains a constant
value, whatever be the dielectric media which it traverses, so long as
it does not meet a really electrified body. This law merges into that of
the conservation of the flow of force when we are only considering
a single medium.
If the tube encounters a mass of electricity m situate in the
dielectric medium, we may always look upon this mass as sepa-
rated from the dielectric by a layer of air ; in this layer the flow of
induction merges into the flow of force ; as the latter varies by ^irm,
this is also the case with the flow of induction, in virtue of the pre-
ceding theorem.
116. CHARACTERISTIC EQUATIONS OF INDUCTION. If we apply
this theorem to a volume-element dxdydz in a dielectric whose
specific capacity is /*, at a point where the real density of electrifi-
cation is p, we obtain the following equation, analogous to that
of Poisson.
av\ a / av\ a / av\
+ 4^/0 = 0.
As, for the present, we are only considering isotropic media, the
factor fj. is constant, and this expression reduces to
/*AV + 47T/D = 0.
H 2
100 ON DIELECTRICS.
At the bounding surface of two media we shall always have to
distinguish two densities ; the density a- of the fictive layer, which
must be assumed at the bounding surface of the dielectrics in order
that opposite any point, outside this surface, there may be the effect
equivalent to their internal polarization ; there is also the density </
of the true layer, which might have been developed, by friction for
instance, on this same surface.
We shall thus have for the bounding surface of the two media
the equations
Agreeing to count in each medium the perpendiculars from the
surface, and calling V 1 and V 2 the values of the potential on the
two dielectrics respectively, these equations may be written
0.OORY. + S +
3V, SV 2
^ +/+4 =
The inductive capacity //. is always positive and greater than
unity ; in conductors it may be regarded as equal to infinity.
117. In the case in which the dielectrics have not received
electricity either in the interior or on the surface, these equations
reduce to
<>v 2
from which we have
4*7*!
OBSERVATIONS ON THE FICTIVE LAYER. IOI
118. OBSERVATIONS ON THE FICTIVE LAYER. Although the
layer of density o- is a fictive layer, it must be noticed that if,
while the dielectric is under induction, its surface is brought by
any means to the neutral state, by moving along it a flame con-
nected with the earth for instance, and if the sources of induction
are removed, a real layer of density <r will be found on this surface.
This observation enables us to explain the phenomena exhibited
by certain bodies ; for instance, uniaxial pyroelectrical crystals such
as tourmaline. We need only suppose that the normal state of these
bodies is analogous to that which dielectrics acquire under the
influence of electrical forces in other words, that they are naturally
polarized, and that their state of polarization is a function of the tem-
perature.
A tourmaline which in appearance is neutral, is a tourmaline
which, in virtue of its polarization, would exert on the outside the
same forces as a layer of total mass zero, and density a-, distributed
on the surface, but which from any causes for instance, losses by
contact with the surrounding medium has become covered by a
real layer of density - o-, which for any external point neutralises
the effect of internal polarization. If the temperature of the
tourmaline alters, its internal condition may be changed without
modifying the layer developed on the surface; equilibrium is
broken, and could only be restored more or less slowly under the
action of causes which had brought about the previous neutralisa-
tion ; the effect observed under these conditions is the difference
between the actions of the fictive and of the real layer.
119. CHARGES OF Two CORRESPONDING ELEMENTS. The
theorem of corresponding elements (36) also holds when the two
conductors are placed in different media. This will be clear if we
remember that we can always imagine the conductor separated from
the dielectric by an infinitely thin layer of air, between the surface
of the conductor itself and an infinitely near equipotential surface.
Let A and B be the two conductors, /*, and ft 2 the inductive
capacities of the dielectrics with which they are respectively in
contact. If the bounding surface S of the two dielectrics has no
real electrical layer, the flow of induction is the same throughout the
whole extent of an orthogonal tube which cuts, on the conductors
and on this surface, the elements ^S a , ^/S 6 , and ^S.
The force which in air, near the first conductor, would be F a ,
p
becomes F x = in the dielectric.
The apparent density <r' a on the conductor that is to say, that
102 ON DIELECTRICS.
which would give the force F x by the ordinary ratio 4fJ^-' a = F lf is
equal to the algebraical sum of the real density <r a of the conductor
and of the fictive density v l at the surface of the dielectrics ; from
this we get
In like manner, on the conductor B we have
If the surface S has a real layer of density o-', the fictive layer
having the density cr, the perpendicular forces on both sides satisfy
the equation
We have further, in the two media respectively,
Replacing the forces F x and F 2 by their values 47rcr' a and - 47rcr' 6 ,
0.
This equation expresses that the algebraical sum of the apparent
charges of corresponding elements of the two conductors, is equal and
of opposite sign to the total charge of the corresponding element of the
bounding surface of the two dielectrics.
120. ENERGY OF A SYSTEM IN THE CASE OF ANY GIVEN
DIELECTRICS. The general expression of energy is, as we have
seen (107),
i i f
-JVV = -
2^ 2j
- Vpdv.
COMPARISON WITH THERMAL PHENOMENA. 103
The equation
/xAV + 47173=
gives
W
= - f
J
From Green's formula, and the remark already made (107), this
expression reduces to
We have then, for the energy of unit volume,
121. COMPARISON WITH THERMAL PHENOMENA. Let us re-
sume the comparison of the problem of electrical equilibrium
with that of the propagation of heat. We have seen that between
two identical level surfaces, if the coefficient of conductivity is
equal to unity, the flow of heat in the first is numerically equal
to the flow of force in the second ; if the coefficient of conductivity
is k, the flow of heat is k times the flow of electrical force.
Let us now consider two correlative systems, one electrical and
the other thermal, each formed of two media separated by the same
surface S, and such that the equipotential surfaces of the one, coin-
cide with the isothermal surfaces of the other If k- and 2 are the
coefficients of conductivity of the two media, the flow of heat across
an element d$ at the bounding surface, in the first medium, is
and in the second
As thermal equilibrium is supposed to have been attained, these
two flows are equal, and we have
IO4 ON DIELECTRICS.
The electrical system gives, in the same way,
From this we deduce
The flows of induction are therefore proportional to the flows of
heat ; the specific inductive capacity playing the same part in the
electrical problem as the coefficient of conductivity in the thermal
problem.
122. CHANGE OF POTENTIAL PRODUCED BY INTERPOSING A
DIELECTRIC. If we introduce a conductor into an electrical field
due to insulated and electrified conductors, the presence of this new
body has the effect of diminishing the initial energy of the system.
The introduction of a solid or liquid dielectric produces the same
effect to a lesser degree.
Fig. 25.
As an instance of this, let us consider the case of conductor A
(Fig. 25), charged with a quantity M 1 of electricity, and situate
inside a closed conductor B kept at a constant potential V 2 .
Equilibrium being established, let us fix the electrical masses on
A and B and introduce into the interval a dielectric layer C, of
inductive capacity /*, the internal and external surfaces of which,
S and S', are equipotential surfaces belonging to the primitive
system, where the potentials were respectively V and V. It is easy
to see that equilibrium is not disturbed when we distribute on
the surfaces S and S' electrical charges, M and + M, identical with
EFFECT OF INTERPOSING A DIELECTRIC. 105
those which would be produced if this medium were a conductor,
and that the primitive charge of A had been replaced by M.
The charges + M on A, - M and + M on C, and - M on B do,
in fact, establish constant potentials on the bodies A, B, and C ; on
the other hand, the equal and opposite layers, +M X and M 1?
produce constant potentials on the conductors A and B, so that
they are in equilibrium. The form of the equipotential surfaces
intermediate to the conductors A and B is not modified, and the
direction of the force remains everywhere the same.
From the surface S 2 to the surface S' the increase of potential is
the same as if the layer C did not exist ; the variation also remains
the same from S to S r
In order to establish the condition relative to the dielectric, let
us consider an orthogonal tube which cuts on the surfaces Sj and S,
the elements ^/Sj and ^/S, the densities on which have the absolute
values o-j and <r. The flow of force 4Tro- l ^S 1 which issues from the
element ^S x is partly absorbed on the element */S, and the fraction
lost is
As this fraction should be equal to i (112), it follows that
P
the ratio of the charges ov/S and o-^S-^ of the two corresponding
elements is also i . The condition of equilibrium of the dielec-
f
trie is then satisfied if we have
The force having become //, times less between the surfaces
S and S', the fall of potential has diminished in the same ratio, so
that the total increase of potential in going from B to A is now
Calling U 15 the new potential of the conductor A, we have
106 ON DIELECTRICS.
If the body interposed were a conductor, the loss of potential
of the conductor A would be V - V. The effect of introducing the
dielectric has been to lower the potential on the conductor A, and
the fall is a fraction equal to i of what would be produced by a
f*
conductor of the same dimensions as the dielectric. This simple
result is, however, peculiar to the conditions chosen ; it would not
be the same if the dielectric were not bounded by the level surfaces
of the original system.
123. When the dielectric occupies the whole space between the
conductors A and B, so as to form a closed condenser, we have
V = V lt V' = V 2 , and
For the same charge the difference of potentials has become //,
times less, by substituting for the layer of air a dielectric whose
specific inductive capacity is equal to /A. In other words, the
capacity of the system has become /* times as great. This is just
Faraday's experiment.
The above remark (119) gives directly the latter results. By
interposing a dielectric of the specific inductive capacity //., in the
space which separates A and B, the form of the equipotential
surface is not modified, but the apparent density at each point
becomes /* times less than the real density ; the effect is the same
as if the system, retaining its original capacity, had received a
charge //. times smaller.
124. We may represent to ourselves the preceding phenomenon
in still another manner.
Let us suppose that the dielectric comprised between the con-
ductors A and B (Fig. 26) is divided into an odd number of infinitely
thin laminae a, /?, a', /?' by equipotential surfaces of the original
system, so that the variation in potential is the same in all the layers
a, a' ... respectively, as in the layers /?, /3' . .. and that we have,
therefore,
/ S\ 1 Q.1
y _ v" _ v" _ yiv _ _ jj
Let us finally place on each of these surfaces masses equal in abso-
lute value to those on the surfaces S x and S 2 of the conductors
EFFECT OF INTERPOSING A DIELECTRIC.
107
alternately positive and negative, + M on the odd surfaces, and M
on the even surfaces, each of -these layers being in equilibrium under
the action of the original conductors A and B.
It is clear that the system thus obtained is in equilibrium. The
force is not modified in all the odd laminae a, a', a" . . , but it is null
in all the even ones /?, /?', ft" . . . and the potential has a constant
value in each of these laminae. It is as if all the even laminae were
replaced by conducting layers.
s rr
Fig. 26.
This operation has lowered the difference of potential between
A and B. The difference of potential of the surface S : to S", which
was originally Vj - V", has, in fact, become V 1 - V, and we have
In like manner, the original difference of potenial V-V iy , be-
tween S" and S IV is reduced to
V' -V"
The ratios
are the same throughout the whole thickness of the dielectric, and
we may write
a a i
Calculating, in this way, from layer to layer, we see that from the
surface S : to the surface S 2 , the potential varies p times less than in
I08 ON DIELECTRICS.
the original state, so that by interposing the dielectric the difference
of potential of two surfaces has become
which gives the same result as the preceding.
Looked at in this way, the coefficient /* acquires a physical
signification; it is the ratio - - of the sum of the thicknesses
a
of two successive laminae to the thickness of that one which is
odd. Now, experiment shows that many solid dielectrics have a
specific inductive capacity of about 2, from which it follows that
a
the ratio of the thicknesses of two successive laminae would be
sensibly equal to unity.
125. Experiment shows also, and the experiments of Gaugain
on this subject are particularly interesting, that the specific inductive
capacity varies with the time. It has first a minimum value at the
moment of charge ; it then increases rapidly and afterwards more
slowly, tending then towards a limit. In other words, the potential
of the inner coating of a condenser first diminishes rapidly after the
charge, and then more slowly.
The force being zero in each of the ft, /?'..., we see in fact that
the positive layers are all impelled outwards and the others inwards,
and that in consequence of this mutual action the layers which
bound the laminae a tend to come nearer, which more and more
increases the inductive capacity.
Generalising this reasoning, we are led to attribute to conductors
an infinitely great specific inductive capacity.
126. MAXWELL'S THEORY OF DISPLACEMENT. In order to
explain the properties of dielectrics and to account for the phe-
nomena by the intervention of the medium only, Maxwell supposed
that when a dielectric is submitted to induction a phenomenon is
produced equivalent to a displacement or gliding of electricity in
the direction of the induction. For instance, in a Ley den jar
whose inner coating is charged positively, and outer negatively, the
displacement takes place in the substance of the glass from within
outwards.
Any increase of the charge increases the displacement, and
corresponds to a current of positive electricity from the inside
towards the outside; any diminution, to a current going from the
MAXWELL'S THEORY OF DISPLACEMENT. 109
outside towards the inside; the duration of the current is equal to
that of the Variation.
The displacement through any surface is the quantity of electricity
which traverses it. Let o-^S be this quantity ; for an element */S of the
R/S
surface of a conductor the displacement is equal to , it is there-
4?r
fore equal to the corresponding flow of force divided by 477-. In
contact with a dielectric the quantity of electricity has the value
- ; the displacement is accordingly equal to the quotient of the
4 71 "
flow of induction by 477-.
Generally, the displacement ', at any point of a dielectric, is equal to
the quotient of the induction by 477-, and is parallel to this force.
A conductor opposes no obstacle to displacement. In a dielec-
tric the displacement is restricted by the action of antagonistic forces
which the displacement itself develops in other words, by a kind
of elasticity, which may be called the electrical elasticity of the
medium. If, by analogy, we denote the ratio of the force to the
displacement which it produces, by the term coefficient of -electrical
elasticity, and suppose the medium to be perfectly elastic, it will be
seen that the coefficient of electricity is equal to , and that
therefore the specific inductive capacity is inversely proportional to
the coefficient of elasticity of the medium.
The displacement produced by induction across the entire mass
of the dielectric determines the polarization of the medium and the
apparent electrification of the conductors.
Consider a tube of induction between two conductors. Through-
out the whole extent of the tube the displacement is constant ; every
orthogonal section is traversed by the same quantity of electricity.
At one end, the displacement is from the conductor towards the
dielectric, the corresponding element ^/S of the conductor is then
said to be charged with positive electricity of density a- at the other
end the displacement is from the dielectric towards the conductor,
and the corresponding element dS is charged with a density a-'.
Throughout the whole extent of the tube, if the dielectric is
the same, there is no apparent electricity; but this medium is
polarized ; for, conceive for a moment a portion of a tube comprised
between two orthogonal sections : the displacement has taken place in
the contrary direction for the two sections, and they would appear
oppositely electrified if their electrification were not neutralised by
the equal and opposite electrification of the portions of the tube in
110 ON DIELECTRICS.
contact. If the tube traverses the surface of separation of the two
electricities, the displacement is the same in the two media, but the
polarization is not the same, and the surface would have an apparent
electricity equal to the difference of the electrical layers on the
surfaces of the two media in contact.
It is evident that, since the electrification of the conductor is
only apparent, all the energy due to the electrification must reside
in the medium. It is equal to the work expended in effecting
the displacement in a direction opposite that of the elastic forces.
From what we have seen (120), the value of this work for unit
volume is - or . . F : it is therefore equal to half the pro-
STT 2 477
duct of the electrical force with the displacement.
Maxwell's theory of displacement accounts thus for the properties
of the medium in a satisfactory manner. It furnishes a physical
interpretation of Faraday's specific inductive capacity; when multi-
plied by a factor , it is the inverse of the coefficient of electrical
47T
elasticity of the medium.
It explains Faraday's view that it is not possible to impart an
absolute charge of electricity to matter: on this theory, in short,
electricity behaves like an incompressible fluid ; the quantity which
can be contained in a closed surface is invariable, and the production
of two quantities of electricity of equal and opposite signs appears to
be a consequence of one and the same phenomenon.
In conclusion, it is natural to suppose that if the explanation
of electrical phenomena postulates the existence of an incompressible
medium, diffused in space, this medium can be none other than the
ether to which luminous and thermal phenomena are attributed ; this
theory enables us to discern a dependence between the two orders of
phenomena, the confirmation of which would be one of the most
important conquests of physical science.
REPRESENTATION OF THE ELECTRICAL FIELD. Ill
CHAPTER VII.
PARTICULAR CASES OF EQUILIBRIUM.
127. REPRESENTATION OF THE ELECTRICAL FIELD. The con-
dition of an electrical field is defined at every point by the direction
and magnitude of the force. It may be represented either by equi-
potential surfaces or by lines of force.
In the former case, equipotential surfaces are drawn which cor-
respond to the numerical values of the potential i, 2, 3 ....#, and
which, therefore, are such that the transference of unit electricity
from any given surface to the next following one, corresponds to a
unit of work.
The force, at each point, is perpendicular to the equipotential
surface ; its mean value F x between two consecutive surfaces of the
orders n and n + i, at a distance of a from each other, is defined by
the equation
The value of the mean force is therefore inversely as a.
These surfaces may be represented by a graphic method.
Take first the case of a single centre of force, a point charged
with a mass m. The potential at the distance r is
the equation
m
determines the radius of the sphere, the potential of which is V. Let
V have the values i, 2, 3 . . . , and draw the corresponding spheres,
we shall have equipotential surfaces, whose potentials correspond to
the natural series of numbers.
128. Let us now assume that several centres, of masses m, m', m"
act simultaneously ; the resultant potential at a point being the
112 ON DIELECTRICS.
sum of the potentials relative to each of the centres, it is clear that
the points, whose potential is V p , will be obtained by the intersection
of spheres of potentials
such that
n + n' + n"
and that the geometrical locus of all these points will be the level
surface of potential V p .
This is a method of general application, and enables us, in theory
at least, to determine the equipotential surface of any system whatever.
Their representation in a plane could be completely made only
in the case of a system of revolution traced on a meridian plane. The
force will always be in the plane of the figure, perpendicular at each
point to the meridional section of the equipotential surfaces, and
inversely as their distance.
If the system is symmetrical in reference to a plane, we could still
have a complete representation of the state of the field in the plane
of symmetry In any other case the intersection of a system of equi-
potential surfaces by any plane will give a series of curves which are
equipotential curves ; the component of the force along the intersecting
plane is perpendicular to the curves at every point, and is inversely
as their distance ; but the value of the true force is not represented.
129. The lines of force may give an equivalent representation
for the field. Such a line, being perpendicular at every point to the
equipotential surface, indicates the direction of the force ; in order to
represent the strength at the same time, we agree to divide the field
into tubes of force, such that the flow corresponding to each of them
has a constant value, unity for instance.
An equipotential surface being given, it is sufficient to divide it
into elements ^S such that F^/S= i, and to take each of these ele-
ments as the base of an orthogonal tube.
The division is an arbitrary one, and in each case that would be
chosen which leads to the simplest construction.
130. UNIFORM FIELD. In the case of a uniform field all the
equipotential surfaces are equidistant planes, perpendicular to the
direction of the force. The simplest division consists in drawing two
series of planes at right angles to each other, and parallel to the direc-
tion of the force. The equipotential surfaces will then be cut out
in equal rectangles.
Any section by a plane P, parallel to the direction of the field,
will give two systems of equidistant lines of force, which will be the
SYMMETRICAL FIELD.
intersection of the plane of the figure with the two series of planes
perpendicular to the equipoteritial surfaces.
131. FIELD SYMMETRICAL IN REFERENCE TO A PLANE. For
all points of the plane of symmetry, the force is perpendicular to the
plane ; lines of force may be traced such that the product of the
Fig. 27.
force by the distance dl (Fig. 27) of the consecutive lines is constant
since we have
it follows that
dJ L _dl'
dn dri
The curvilinear rectangles dndl, dridl', formed by the two infinitely
near equipotential surfaces L and U, and the two lines of force, are
similar.
The flow of force will only be determined by taking into account
the dimensions of the tube perpendicular to the plane of symmetry.
If we assume that these dimensions are everywhere the same, the flows
will not be equal except in that case in which all the planes parallel
to the plane of the figure are identical. This is the case of a cylin-
drical distribution, all the bodies of the system being parallel cylinders ;
it corresponds to the problem of the propagation in a plane in the
theory of heat. The potential, or the temperature, no longer depend
on only two co-ordinates x and y, and Poisson's equation reduces to
o- being the electrical density on the plane.
i
114 PARTICULAR CASES OF EQUILIBRIUM.
We shall proceed to examine this particular case in some detail,
less for its own importance than as a useful transition to more com-
plicated problems.
132. CYLINDRICAL SYSTEMS. Consider a uniformly electrified
unlimited line of density A, that is to say where the charge is X for
unit length. At each point of the dielectric, the force passes through
the axis, and is perpendicular to it.
The flow of force proceeding from unit length is equal to 4?rA ;
at a distance r^ this flow traverses the lateral surface zirr of the cor-
responding equipotential cylinder, and the force F is defined by the
condition
where
(i) F-.
The force is therefore inversely as the distance, as we have already
seen (80) for cylindrical condensers. The equation
^
gives for the equipotential surfaces
(2) V= -2\t.r + const,
that is to say, a series of concentric cylindrical surfaces. Draw two
planes perpendicular to the axis and at the distance e ; the mass
which they include is m = Xe. The flow of force which will pass
between these two planes will therefore be 477^, and it is evident that
if we draw through the axis 47rw, planes making equal angles with
each other, each of the qicm dihedra thus determined constitutes an
orthogonal tube in which the flow of force is equal to unity.
Take a plane perpendicular to the axis as the plane of the figure.
Let A (Fig. 28) be the trace of the electrified line, and Ax any axis
from which we shall count i, 2, 3. . . , the traces of the %xm planes
drawn through the axis. Lastly, let be the angle which the straight
line, number N, makes with the axis A#, it is manifest that the flow
of force corresponding to the angle is
9
Q = 47TW =
TWO PARALLEL LINES. 115
This flow on the other hand is equal to N units ; we have therefore
N
(3) Q = 2W0 = N, or (9 = .
2M
133. Two PARALLEL LINES. Suppose that the electrical system
consists of two parallel lines A and A', of densities A. and A/, such
that m = t\ /rc' = eA/; take for the x axis the straight line joining
the two lines A and A' (Fig. 28).
Through these two points draw two straight lines An and A'n'
of the orders n and ri respectively in reference to the centres A
and A', and making angles w and o/ with the axis ; join their point
of intersection P to the axis by any given curve PP'. It is evident
that across the cylindrical surface PP', there is a flow , or 2mu from
A, and a flow n' from A', and therefore a total flow equal to n + n' = N.
The same will be the case with all the points of the curve AP, de-
fined by the points of intersection, two by two, of the straight lines
proceeding from A and A', and such that the sum of their numbers
is equal to N ; the locus of all these points is evidently a line of
force of the order N for the resulting system.
The force near one of the acting masses depends only on this
mass, the influence of which predominates. The line of flow of the
order N is therefore tangential at A to the right line of order N
drawn from this point.
The equation of the curve AP is
(4)
from which, replacing these quantities by their values as functions
of the angles,
(5) mot + m'<i>' = mO.
This is the equation of the lines of force drawn from the point
I 2
Il6 PARTICULAR CASES OF EQUILIBRIUM.
A; an analogous equation will give those which proceed from the
point A'. If m and m' are of the same sign, all these lines are
unlimited ; any one of them that of the order N, for instance is
an asymptote to a right line making an angle a with the axis Ax ;
this angle is denned by the condition that the right lines of order
n', connected by the ratio (4), are parallel to each other that is,
that we have
n ri n + n' N m
(6) a = = = - = - = - 0.
2m 2m
All these asymptotes pass through the centre of gravity O of the
masses m and m' t which is evident and easy of verification.
By eliminating the ratio , the equation of a line of force (5)
m
and that of its asymptote give
6-o> 0-a
This is the equation of the line of force as a function of the
angles which the asymptote, and the tangent at the origin make with
the axis Ax.
If r and r' are the distances of a point P to the two lines A and A',
the equation of the equipotential surfaces is
V = const - 2 \M.r + X'l.r'} = const - 2/.(rV A/ ),
from which
r \ r '\' _ C onst.
134. SEVERAL PARALLEL LINES. It is evident that this method
of construction may be applied to any number of electrified lines
A, A', A" ... defined as above by the masses m, m' t m", . . . , on the
condition that these lines are parallel and situate in the same plane.
The general equation of the lines of force starting from the
centre A of the mass m will be in this case
the masses m, m',... may be positive or negative ; that of the cor-
responding asymptote is
(m + m' + m" )a = md
TWO LINES OF OPPOSITE SIGNS.
117
When all the masses are of the same sign, all the lines of force
are unlimited. In the contrary case, part of the flow of force
issuing from positive masses is absorbed by negative masses.
From the method of numbering adopted, the number of un-
limited lines of force is equal to the difference between the number
of positive lines and of negative lines.
If the electrified lines A, A', A" . . . . , while still parallel, are no
longer in the same plane, the construction of lines of force becomes
more complicated. In this case, the value of the potential at a point
P at distances r, r\ r" . . . from the lines A, A', A" ---- , is
V = const - % 2 XI. r = const - 2/. (r
whence
const
135. Two LINES OF OPPOSITE SIGNS. Consider the particular
case of two lines electrified oppositely, defined by the masses +m
Fig. 29.
and - m', situate at two points A and A' (Fig. 29) at the distance 20,
and let m be the greater of these masses. The equation of a line of
force becomes
from which
n ri = N,
mat m'w = mO ;
Il8 PARTICULAR CASES OF EQUILIBRIUM.
that of the corresponding asymptote is
m
m m'
As the angle a cannot become greater than TT, there can only be
unlimited lines of force for values of smaller than
The line of force AP l corresponding to this value separates
the m-m' lines of force proceeding from A, and which are unlimited,
from the m' which are finite and are absorbed at A'. The equation
of this limiting line of force is
ma) - m'<D f = m0 = (m m') TT, or if o/ = -(TT o>).
M
This equation is satisfied, for O> = TT and o>' = 7r; hence the line
meets the axis on the left of the point A', and the point of meeting
O' is symmetrical with the centre of gravity of the system in refer-
ence to the axis AA'. We have, in fact, for any point P x of the
curve
. sin (TT-O))
r sin to sin (TT - o> ) m
r sin sin (TT - <o) sin (TT
If the angle TT - w approaches zero, we get
limY \= , or mxO'A' = m'
\ r'/ m'
As the centre of gravity O of the two masses is determined by
the condition m x OA = m' x O A', it follows that OA = O'A'.
In the case of m = 2m' (Fig. 29) we have
7T
= , and 2w o>' = TT,
from which we get
2 sin
2
TWO EQUAL LINES OF OPPOSITE SIGNS.
119
The limiting line of force is therefore a circumference whose
centre is A', and which passes through the point A.
The equation of the equipotential surfaces is
V = const- 2 1. ( ) = const + 2/. ( -
from which
= const e' 2 .
136. Two EQUAL LINES OF OPPOSITE SIGNS. If we suppose
the two masses equal in absolute values, the equation of the lines of
force reduces to
and that of the equipotential surfaces to
The former represents segments of the circumference such as
ATA' (Fig. 30) passing through the two points A and A' and which
Fig. 30-
may have the angle ; the second represents circumferences
S, S' . . . having their centres on the right line AA', and such that the
two points are conjugate in reference to each of them.
Considering the two equipotential surfaces S and S', a layer + m
120 PARTICULAR CASES OF EQUILIBRIUM.
on each unit of length of the cylinder S, and a layer - m on each unit
of length of the cylinder S', will replace the action of the two
unlimited lines A and A' (61) for all points between the two sur-
faces ; the figure will correspond, in this case, to the problem of a
condenser formed of two unlimited excentric cylinders.
137. Let us suppose that the distance za approximates to zero,
but that the density A varies so that the product 20 A remains con-
stant. The potential at the distance r, in a direction which makes
the angle o> with the straight line, will be
\
-
\ r ]
This equation represents circumferences whose radii vary as
the reciprocals of the series of even numbers.
We have in like manner for the lines of force,
2a sn to
<D to = -
Thus the radii of the circumferences which represent the lines of
force vary also as the reciprocals of the series of even numbers.
138. SYSTEMS OF REVOLUTION. To determine the sections of
elementary tubes of force on an equipotential surface, we shall take
on the one hand equidistant meridian planes, and on the other,
points placed on the meridian section so that in the revolution
about an axis, they divide the surfaces into successive zones, corres-
ponding to the same flow. The surface will thus be divided into
curvilinear rectangles corresponding to the same flow, which will
be taken equal to unity.
139. A uniform field may always be considered as one of
revolution about any line parallel to the direction of the force ; we
may therefore apply to it this mode of representation. An equipo-
tential surface, which is a plane perpendicular to an axis, will be
intersected by a series of circumferences comprising between them
zones of constant surface. The radii, increasing according to the
same law as Newton's rings, will be proportional to the square roots
of consecutive numbers. The lines of force will then be represented
in the meridian plane by right lines, parallel to the axis, and whose
distances from the axis are as the square roots of consecutive whole
numbers.
If F is the strength of the field, and w the angle of the two
CASE OF A SINGLE MASS.
121
meridians, r n and r n+l the distance of two successive lines of force
from the axis, we may take
whence
This method of representation has the inconvenience, as we see, of
not representing a uniform field by equidistant lines of force.
140. CASE OF A SINGLE MASS. A single mass m gives a system
of revolution about any axis passing through the acting mass. By
planes perpendicular to an axis Ax (Fig. 31), the sphere may be
divided into successive concentric zones of the same surface, and
corresponding to the same flow.
The flow corresponding to the circular zone whose semi-angle
Fig. 31. Fig. 32.
at the summit is 0, is proportioned to the cap of semi-aperture 0,
that is to say, to the height PB or to i - cos 0.
Let N be the order of the line of force AN, we shall have
N i - cos 6
471772 2
whence
(7)
COS 0=1-
N
To draw the lines of force, it is therefore sufficient to divide
the diameters BB' (Fig. 32) into 47170 parts, to draw the corresponding
verticals, and to join the points of intersection with the circumference
with the point A.
122 PARTICULAR CASES OF EQUILIBRIUM.
141. ANY Two GIVEN MASSES. Let there now be two masses
m and m' situate at A and A' (Fig. 28) ; they form a system of
revolution in reference to the straight line joining them. The total
flow which traverses any zone of revolution whose semi-arc is PP',
is the sum of the flows which correspond to the angles to and to' for
the two masses separately; that is to say n + n'=N.
For the same reason as above, the sheet which corresponds to
the flow of the value N, and which passes through the point P is
tangential at A to the cone whose angle is 20, which comprises the
same flow for the mass m taken separately.
This sheet is also an asymptote to a cone, the apex of which is
the centre of gravity O of the two masses.
The equation of the line of force AP, that is to say 72 + #'=N,
gives
2irm (i - cos o>) + 2Trm'(i cos a/) = N = 27rm(i cos 0),
whence
(8) m cos to + m' cos a/ = m' + m cos 9.
If the two angles in this equation to and <o' are made equal, we
have the angle of the asymptote with the axis ; we thus get
(9) (m + m')cosa = m' + mcosO.
By eliminating the ratio , between the equations (8) and (9) we
have the equation of the line of force in functions of 6 and of a :
I COS to' I COS a
COS to COS COS a COS
This is still a general method, and may be applied to any number
of centres situate on the same right line.
The equation of a line of force starting from the mass m is
m cos to + m' cos w' + m" cos w" = m' + m" + + m cos 0,
and that of the asymptote
(m + m' + m" . . . . .) cos a = m' + m" + m' + m cos 0.
TWO EQUAL MASSES OF THE SAME SIGN.
123
When the masses are all of "the same sign all the lines of force
are unlimited. If there are masses of contrary signs, the region
which includes the finite lines of force emitted by positive masses,
and absorbed by negative masses, is separated from the region which
contains the unlimited lines of force by a bounding surface, the
meridian section of which is determined by the value of the angle
given by the preceding equation, in which a is made = TT.
142. Two EQUAL MASSES OF THE SAME SIGN. If the system is
made up of two equal masses of the same sign situate at A and A', at
the distance 20, (Fig. 33), the equipotential surfaces are given by the
equation
= + = m ( - + -
r r
The meridian curves are lemniscates. The meridian curve
for the surface corresponding to V = , has two lobes which
intersect at O. The point O is one of unstable equilibrium ; the
force there is equal to zero. At this point the potential has a
minimum relative to the axis AA', and a maximum in reference to
the plane of symmetry PP'.
For all values of V higher than , the equipotential surface
consists of two separate lobes the section of which has the form of an
oval, and which surround each of the two centres. These ovals tend
more and more to merge into circles as we approach the centre.
For lower values of V than , the surface consists of a single
a
sheet the narrowing of which tends to disappear as V diminishes,
124 PARTICULAR CASES OF EQUILIBRIUM.
and which ultimately would be confounded at a great distance with
a sphere the centre of which is the point O.
The equation of the lines of force is
cos o> + cos G>' = i + cos 0,
and that of the asymptote
2 cos a = i+ cos 0,
or
cos a = cos 2 -0.
2
The expression for the force at a point on the transverse
axis OP is
2m . 2my v
F = sin to = - - = 2 m
f T 7 //j2
it is a maximum at points D and D' for which
y-* tt -r-
It is only a maximum in reference to the transverse axis, and on
the contrary is a minimum for the direction parallel to AA'.
The lines of force proceeding from m and m' are separated by
the plane perpendicular to the axis AA', passing through the point O.
143, TWO UNEQUAL MASSES OF THE SAME SlGN. If two maSSCS
of the same sign are unequal, the general form of the equipotential
surfaces is the same as in the preceding case excepting the symmetry.
The point of equilibrium corresponding to the point of intersection
of the surface with two sheets is defined by the ratio
m m 1 r
^ = ^2' or ->'-
Putting r + r' = 20, we get from this
Jm
m
TWO UNEQUAL MASSES OF THE SAME SIGNS. 125
The equation of the lines of force is
m cos o) + m' cos to' = m f + m cos 0.
They always form two distinct systems ; the surface which separates
them corresponds to = 7r, and its equation is
m cos w + m' cos w' = m' m.
If we put = i + e, the equation becomes
COS w + (i + e) COS to' = e,
or in rectangular co-ordinates, the origin being taken in the middle of
the distance 20,
x-a x + a
jy2 + ( x - a y + ( I+ > jjr + ( x + a )2 = '
This equation represents a surface of the sixth degree, which passes
through the point of equilibrium, and which has some analogy with
the sheet of a hyperboloid. Its meridian section, like all the other
lines of force, has an asymptote which passes through the centre of
gravity of the two masses. The equation to this asymptote is
COS a =
2+e
The force makes, with the radius vectors, angles /3 and /?',
defined by the ratio
sin/2 r' 2 m'r 2 w'sin 2 w'
sh^~~m s= mr r * = msm*<a j
from which we get, for the value of the force,
mcos/3 w'cos/5'
144. Two EQUAL MASSES OF OPPOSITE SIGNS. We shall
proceed to examine in greater detail the case of two equal masses
of opposite signs, as it presents several important applications.
126
PARTICULAR CASES OF EQUILIBRIUM.
The equipotential surfaces whose equation is
- (---
\r r
are surfaces of an ovoidal form, with a single sheet, tending to merge
into spheres in proportion as they approach the centres of action.
All of them, which correspond to positive values of V, envelope the
point A, while those which correspond to negative values envelope
the point A'. They are separated by a symmetrical plane at zero
potential.
The equation of the lines of force is
N
cos to cos a>' = i cos 6 =
These lines of force are all limited, proceeding from the point
A, and terminating at the point A' ; they are evidently symmetrical in
reference to the plane of zero potential, which is perpendicular to the
axis AA' at 0, the middle of the distance AA' (Fig. 34).
T'
Fig. 34-
145. The angles ft and ft' which the force makes with the radius
vectors are still determined by the equation (10), which gives
(12)
sin ft
146. The expression of the force is
"cos ft cos ft'
TWO EQUAL MASSES OF OPPOSITE SIGNS.
127
Its value at P a on the axis AA', at a distance d from the centre, is
2d
n _ ^2x2
and on the transverse axis at the same distance d from the centre, or
at the distance p from either of the masses,
The product 2ma, of one of the masses by the distance separating
them, which is called the magnetic moment in the corresponding
problem in magnetism, may be called the electrical moment of the
system.
A'
Fig- 35-
147. When the force is perpendicular to the axis
sn = cos a
equation (12) becomes
COS 0> COS 0)
This is the equation of the curve APX (Fig. 35) which passes
through all the points of the plane where the force is vertical. It
128 PARTICULAR CASES OF EQUILIBRIUM.
consists of two symmetrical branches, starting from A and A' tan-
gentially to the vertical, and which are asymptotes to a straight
line OL.
In order to determine the direction of the asymptote, let us
consider a very distant point ; then, if 8 is the very small difference
<o - co', and observing that the angles co and co' ultimately become
equal,
cos co cos to' cos co' cos co sin co. 8 sin co r8
' 2 r't rt 2r(r' r) 2r 2 r' r
sin to. 2a sin to i sin 2 to
whence
tan 2 co=2.
148, PRINCIPLE OF IMAGES. We have already seen (59) that
we can always replace any mass of electricity by an equal mass
distributed over an equipotential surface which completely sur-
rounds it. This layer is of itself in equilibrium, and its density is
defined by the condition
For all points in the interior, the potential becomes constant and
equal to that of the surface ; but for all external points, nothing is
changed in the state of the field.
Let us consider the plane Oy at potential zero (Fig. 34) in the
preceding problem. For all points on the right we may replace the
mass m on A' by an equal mass in equilibrium on the plane. The
density will be P 2 at each point, the force F 2 being directed towards
the left,
F 2 (21110) i
4?r 477 /o 3
We see thus that it is inversely as the cube of the distance of the
point in question P 2 from the point A.
It may be observed, from this law of distribution, that the charge
of an element of the plane is everywhere proportional to the angle
which it subtends at the point A. In fact, the charge of a surface
element dS (Fig. 36) is
27T
ELECTRICAL IMAGES.
129
Now, -^S is the projection ^S x of the element dS on a plane
P Tq
perpendicular to p, and ^ is the angle dO under which the element
p-
dS is seen from the point A.
We have then
-<rd$ = dO.
27T
If the plane, or the mass m, did not exist, the flow of force
from the mass m in the angle dO would be mdO. In the present case
the flow received by the surface dS is 477 d6 = 2mdO. The dis-
27T
tribution of this flow is the same as if the mass m were alone there,
but the flow is doubled at each point since all the lines of force meet
the plane.
The plane, on which the mass - m is distributed, being at zero
potential, all the space on the left is at zero potential. This is the
A'
Fig. 3 6.
case of an unlimited conducting plane Oy, in connection with the
earth, and under the influence of a mass of electricity + m placed at
a point A. Such a plane completely intercepts the action of the
mass m on points behind it ; it plays the part of an electrical screen.
Thus, the mass +m being placed at A, in the presence of a
conducting plane Oy in connection with the earth, this plane may be
replaced, for all points on the right, by a mass - m at the point A'
symmetrical with A.
Sir W. Thomson looks upon the mass - m at A', considered in
reference to the plane Oy in connection with the earth, as the image
of the mass +m at A. The analogy between the electrical phe-
nomenon and the corresponding optical problem is at once evident.
If the point A is a source of light and the plane Oy a reflecting
K
1 3 o
PARTICULAR CASES OF EQUILIBRIUM.
mirror, the image of A is a virtual one, and is formed at A' ; the
illumination of the space on the right of the plane is the same as
if this plane were replaced by a source of light placed at A', and the
intensity of this virtual source would be equal to A, if the reflecting
power of the plane were equal to unity.
149. INDUCTION IN A MEDIUM CONSISTING OF Two DIELEC-
TRICS SEPARATED BY A PLANE. The principle of images enables us
to determine the condition of two unlimited dielectrics, separated by
a plane surface, in one of which is the acting mass.
Let m be this mass placed at the point A (Fig. 37), /^ and ft 2
the inductive powers of two dielectrics separated by the plane Q, the
acting mass being situate in the former.
Equilibrium may be established by imagining that on the plane Q,
a layer m' is distributed as it would be on an uninsulated conducting
Fig- 37
plane under the influence of a mass m', placed at A, or at the sym-
metrical point B ; in other words, the plane would act on all points
on its left like a mass m' placed at A, and on all points on the right like
the same mass m' placed at B. The potential near the point P,
taken in the plane Q, is, in the first medium, at P p
_^_ m_
1 PA*?! 1
and in the second medium, at
THREE DIELECTRICS SEPARATED BY PARALLEL PLANES. 131
The perpendicular components of the force are, at the same
points,
COS 0)
In order to satisfy the equation of continuity of the dielectrics, the
product of the perpendicular component by the specific inductive
capacity, must be the same on both sides of the surface of separation,
which gives
whence
The density at every point in the plane is
, _
2m a i
47T
2ma i zma i
47T ' p 3
47T ' /O 3 '
150. THREE DIELECTRICS SEPARATED BY PARALLEL PLANES.
Let us imagine three different media, whose specific inductive capa-
Fig. 38.
cities are respectively equal to ^ lf p 2 and yn 3 , separated by parallel
planes Q and Q', and let the acting mass m be situate in the first
medium, at A (Fig. 38).
K 2
132 PARTICULAR CASES OF EQUILIBRIUM.
Let us further take
^-^
The condition of equilibrium on the plane Q is satisfied by a layer
my, which acts on each side as if it were at A or at B.
The mass m at A, and the layer my of the plane Q, produce on
the plane Q' a layer m (i + y)y' = m', which will act as if it were con-
centrated at A or at B r
The layer m reacting on the plane Q, will produce there a layer
- m'y, the image of which is at B x or at A r
In like manner the layer - m'y at Q gives on Q' a layer m'yy
the image of which is A x or B 2 . . . , etc.
The determination of these successive layers is nothing but the
application of Murphy's method.
We shall thus have, from layer to layer :
on the plane Q,
Successive layers. Images.
my A or B
- m'y A : B!
+ m'y*y A 2 B 2
and on the plane Q',
Successive layers. Images.
m(i+y)y' = m' A or B x
-m'yy' A, B 2
A 2 B 3
The algebraical sum of all these layers will give the final state of
equilibrium.
The total charges M and M' of the planes Q and Q' will be
M = my - m'y [i - yy' + (yy') 2 - (yy') 3 + ] = my- m'y ;
r j "
1 + 77 1+77
THREE DIELECTRICS SEPARATED BY PARALLEL PLANES. 133
_. . _ :_V^ .. . . . . . . _. .,__-..-,-, IKJm _
The density at every point is equal to the algebraical sum of the
densities of all the layers superposed.
The potential V, at a point P in the former medium, may be con-
sidered as produced by the mass m, and all the images situate at B,
B lf B 2 ...,etc.
At the different points B I} B 2 . . . , there are two different images
arising from the layers on the two planes Q and Q', and we have :
atBj
m' - m'y = m'(i -y) = m(i- y 2 )/,
atB 2
' = -m'yy'(i-y) = - m(i -y 2 )/.yy',
at B n+1
m'(yy'Ym'(yy'Yy= m'(yy') n (* -?)=
which gives
The potential V 3 in the third medium is produced, in like manner,
by the images situate at points A, A 1 A 2 . . . , on which the masses
are :
m + my + m' = m(i + y) (i + y'),
- m'y - m'yy = - m (i + y) (i + /) yy',
(yy-i + m'y-Y n ) =m(i + y)(i + y) (yy'Y ;
we have then
Lastly, in the second medium, comprised between the planes Q
and Q', the potential is due to the mass m, to the images at A v A 2 . . .
of the layers of the plane Q, and to the images at B lf B 2 . . . of the
layers of the plane Q'. We shall find in like manner
r_L_jzi
|_PA PA X
77' (rrT
h ~PA7"
yy' (yy') 2
PB 2 PB 3
If the third medium is identical with the first, we simply put
then we get y'= y.
134
PARTICULAR CASES OF EQUILIBRIUM.
We thus obtain
1-7* "I-/
I-?*
Vj = m<
+ PB~ (I "
j y'Z y4
PA + PA^ + PAj
...+
fn
]
Denoting by a and /3 the two series containing the distances
PA, PA X . . . , P B 1? PB 2 . . . , which are determinate functions of the
co-ordinates of the point P, we have simply
151. Two EQUAL MASSES OF OPPOSITE SIGNS INFINITELY NEAR
EACH OTHER. Let us suppose that the two equal masses of opposite
signs +m and -m of problem (144) are infinitely near or, what
Fig. 39-
amounts to the same thing, let us consider the condition of the field
at a distance which is very great compared with the distance za of
the two points A and A'. The value of the potential at a point P
(Fig. 39)
r'
rr
reduces to
TWO EQUAL MASSES OF OPPOSITE SIGNS. 135
r r cos (o x
\ m = 2am = 2am ,
r 2 R2 R3
<o being the angle of the direction OP with the axis A' A.
Let cr be a surface, a circle, for instance, traced by the point O
perpendicularly to AA', and let 6 be the solid angle under which this
surface is seen from the point P ; we have
= T COS to,
and therefore
CT
taking CT = 2ma, we get
(16) V = 0.
Thus, the value of the potential at a point, is the solid angle under
which we see from this point, a surface equal to the electrical moment
2ma of the two masses and perpendicular to the middle of the straight
line joining them.
The equation of the equipotential surfaces,
GT cos <o T3x
~~ =
shows that all these surfaces are similar, and that for the same direction
a>, the values of R are inversely as the square roots of the potentials.
152. In the equation of the lines of force,
N
COS to - COS to =
the first member may be transformed in the following manner, de-
noting by 8 the infinitely small difference o> - to' :
, . za .
cos to - cos to = a. cos w = sin <o . 6 = sin o> . - = sin' 5 w.
R R
we have then
sin 2 w i N N N
R
All these curves are similar, and for the same direction, R is
inversely as N. They are tangential to the axis at the origin the
136
PARTICULAR CASES OF EQUILIBRIUM.
loci of the points where the tangent is vertical is evidently the
asymptote found in the preceding problem (144), and the equation
of which is
tan 2 o>=2.
153. Equations (14) and (15) of (146) give for the values of the
force on the axis, and on the transversal, at A and B (Fig. 40),
and therefore
S Q A T
Fig. 40.
From equation (17) we have for a point P at the same distance in
any direction whatever w,
Y= - = -GT
= 3 sin w cos CD.
J
Through the point O as centre, draw a circle of radius R, passing
through the point P, and consider at this point the perpendicular
component F w , and the tangential component F^ of the force ; we
have
(ao)
= X cos o> + Y sin o> = 2 cos w,
R 3
T
. = - X sin o> + Y cos o> = sin w.
'
TWO EQUAL MASSES OF OPPOSITE SIGNS. 137
If / be the inclination of the force F with the tangent, and A
the complement of the angle o>, we have
(21) tan /= -^ = 2 cot w= 2 tan A.
We get lastly, for the force itself,
F 2 = X 2 + Y 2 = F 2 B + F*=/^ -Y( 3 cos 2 <o+i)
(22) ^''-sin'A+i)
154. Prolong the tangent as far as the axis at T, and the direction
of the force to S ; the triangles OPS and OPT give
PS OS
sn <o cos
PS ST
from which we have
cos i sin /
and finally
OS tan /= ST cot o> = ST tan A.
As tan i=2 tan A, we see that
ST = 2OS,
whence
(23) OT = 3 OS.
This theorem is due to Gauss.
The value of the force is easily expressed as a function of the same
lines. We have, in fact,
R = OT cos w = 3 OS cos (o,
= Rcosw,
and, consequently,
,2,.,.
cosw =
OS
138 PARTICULAR CASES OF EQUILIBRIUM.
We obtain then, by substitution,
155. In valuing the force at each point, the electrical masses
only affect the result by their moment 2ma = t3, which may remain
finite for suitable values of m, although the distance 20. is infinitely
small. The total flow of force proceeding from the two infinitely
near centres is not therefore determined, but the flow from a sphere
of given radius R may be easily calculated.
From equation (20) the value of the perpendicular component at
point P, corresponding to the angle w, is
trr
F = 2 cos o>.
The surface of the bow whose angular aperture is 2o>, being equal to
27rR 2 (i cos <o), that of the elementary zone corresponding to the
angle dv> is
sin wtfo).
The flow of force which traverses this zone is therefore
47TS7 .
d Q = sin o> cos w #w,
R
and the total flow corresponding to the angle o> is
O = sin w cos w d(& = sin 2 o>.
R J R
This expression is nothing more than that of the line of force, of
the order N, which terminates at the contour of the zone in question,
and it might have been written directly.
If o) be made equal to , we shall have the total flow on one side
of the transverse plane OB the value of this flow is ; it is seen
to be inversely as R.
To trace on a meridian plane the lines of force which correspond
INDUCTION ON AN INFINITELY SMALL BODY. 139
to the flows represented by the numbers i, 2, 3, 4 . . . , we need only
take, on the transverse axis, lengths corresponding to the numbers
1 >->->- an d by equation (18) draw the lines of force which
cut the axis in these different points.
156. INDUCTION ON AN INFINITELY SMALL BODY. The system
of two equal masses of opposite signs infinitely near each other, repre-
sents the condition of an infinitely small body, conductor or not,
originally in the neutral state, and placed in any given electrical field.
The body is in effect covered with two layers of equal masses, and
of opposite signs, each of which acts as if it were concentrated in its
centre of gravity.
This is also the case with any body, originally in the neutral state,
(that is with a total charge null,) when its action at a great distance is
considered.
157, POLARIZED SPHERE. LAYERS OF GLIDING.* Let us con-
sider two spheres S and S' of the same radius (Fig. 41), of uniform
densities + p and-/), and whose, centres A and A' are at an infinitely
small distance 8. This system is in fact equivalent to that of two
equal layers of opposite signs distributed on the two halves of a
spherical surface.
This particular form of electrification is of great interest, and cor-
responds in magnetism to a very simple method of magnetisation.
For the sake of brevity we may apply the term layers of gliding
* What are here spoken of as layers of gliding (couches de glissement), are the
result of a purely fictitious geometrical operation, which does not aim at repre-
senting a real phenomenon, or a particular constitution of the electrified body.
We shall retain the expression electrical displacement to denote the mechanical
modification of the medium which Maxwell had in view in his theory of dielectrics.
14 PARTICULAR CASES OF EQUILIBRIUM.
to those which are thus produced by two homogeneous masses equal
in density and of opposite signs, one of which has moved through an
infinitely small distance.
The medium may be considered to be polarized, and the axis of
electrical polarization is parallel to the direction along which the
displacement has taken place.
In the present case the density of the layer at each point is pro-
portional to the corresponding thickness P'P of the part which is not
common to the two spheres. Denoting by cr this density on the line
of the centres, we shall have
As the thickness of the layers along the line of the centres is con-
stant, the value of the density, at a point P at the end of radius
which makes the angle o> with this right line, is
o- = (T O cos to = pS cos w.
The action on a point M in the interior, is that of two
homogeneous spheres whose radii are AM and A'M, on a point
of their respective surface. Hence, for the sphere A, it is equal
to TT/a.AM, and is directed along AM; for the sphere A', it is
3
equal to -^Trp.MA', and is directed along MA'. The resultant is
3
therefore proportional to AA' and has the value
4 A, 4 * 4
- 7T/0 . A A = 7T/06 = 7TO-Q ;
j 5 5
it is constant. Let us denote this force by F { , and reckon it posi-
tively from left to right, we shall have
In the interior of the sphere, the equipotential surfaces are planes
perpendicular to the axis AA' and are equidistant ; the potential at a
point varies proportionally to the abscissa x of the point, and as it is
zero at the centre, we have
POLARIZED SPHERE. 141
For the outside, the layer in question may be replaced by two
homogeneous spheres, or by two masses of opposite signs equal
to -7rR 3 /> concentrated at A and A', and the moment of which is
tar = 8 . - 7rR 3 p = - 7rR 3 o- = UO-Q,
<J O
u being the volume of the sphere. We shall have then, for the
distance r> in a direction at an angle o> with the axis,
cos w x
The surface of the elementary zone du being
dS = 27rR 2 sin o> dfo>,
the corresponding mass is
27rR 2 cr sin w cosco du = d. 7rR 2 cr sin 2 o>.
The total mass M of each of the layers is then
o-^S = Y d. 7rR 2 (r sin 2 <o = 7rR 2 (r .
M
This result might have been directly obtained by considering
that at each point of the layer the thickness along the axis being
equal to 8, the total volume is equal to the product of this constant
thickness by the projection of the hemisphere on a plane perpen-
dicular to AA'.
The flow of force from the positive layer is
Q = 4 7rM = (47rR 2 ) 7T(r = STTO-,
0'
S being the whole surface of the sphere.
158. It will be useful to collate here all the preceding results,
and to express each of them by quantities as a function of the
142 PARTICULAR CASES OF EQUILIBRIUM.
maximum density S , or of the internal force F^ ; if a be the radius
of the sphere and u its volume, we have
# _ # 3 _ 3 cos w
V. = wo- = - F, x = - F; -
6 U o to * o
ft
Y e = ~Y 3 sin w cos w = - F { 3 sm w cos w,
2 coso>= -2^ cosw,
.
^smo>= -F { sinw,
o = _ F 2
4 a 4
159. CONDUCTING SPHERE IN A UNIFORM FIELD. Suppose
now that a sphere thus electrified is placed in a uniform field, of
strength <f>, parallel to the axis of x, and let V be the value of the
potential in the plane which passes through the centre. If we have
the resultant force is null in the whole sphere; the potential is
therefore constant, and there is equilibrium if the sphere is a con-
ductor.
CONDUCTING SPHERE IN A UNIFORM FIELD. 143
To obtain the electrical state of a conducting sphere in a uniform
field of strength </>, F^ may be replaced by - < in all the preceding
formulae, which gives
(f>r cos W,
a 3
< cosw,
</> 3 sin to cos w,
-
160. If R be the radius of the circle, drawn on an equipotential
surface, through which would pass the same flow in the original
field that is to say, of the circle which comprises all the lines
of force directed towards the conducting sphere we have
This circle has therefore a surface three times as great as a great
circle of the sphere.
All the lines of force terminate perpendicularly at the surface
of the sphere and proceed perpendicularly from it, always excepting
those which fall upon the equator ; these make with the normal an
angle of 45.
144
PARTICULAR CASES OF EQUILIBRIUM.
In fact, for any point at a distance r in the direction w, the angle
6 of the resulting force with the radius vector r^ is given by the
ratio
tan (9
sin w - ~F t
cos w + F_
tan to
2-
This angle is always null when r = a that is, when the point is
on the sphere. For all points on the equator, however, the angle w
is equal to -, and the expression assumes an indeterminate form.
2 TJ.
Let us suppose that the angle o> is very little different from -, the
angle 6 for a point P (Fig. 42) near the surface is
r a
a tan ( o>
2
r- a
r a
i
tan
Fig. 42.
Multiplying this equation by the preceding, which always holds,
and observing that the difference r - a is very small, we get
a r^ # 3 a 30* (r a) 30?
tan 2 6 = . - = . = = :=i.
r-a
a
r-a
+ 20? r^ + 20?
The lines of force which touch the sphere on the equator make,
therefore, with the surface, an angle of 45.
The equipotential surface at the original potential V of the
centre of the sphere is a plane which terminates at the equator, and
is thus prolonged by the surface of the sphere itself. The equator is
a line of equilibrium.
CONDUCTING SPHERE ,IN A UNIFORM FIELD. 145
161. UNINSULATED CONDUCTING SPHERE IN A UNIFORM
FIELD. It -is easy to pass from the case we have been treating
to that of a sphere situate in a uniform field, and in connection with
the earth. For this, the internal potential which had the constant
value V must be null; this condition is fulfilled by superposing
on the preceding condition a uniform layer capable of producing in
the interior a potential equal and of opposite sign to V . Let - M'
be the mass of this layer and - a-' its density, we shall have
V -
whence
47TO-
The resultant density at any point will be
1 V
(T = (T COS (0 CT' = - < COS 0) -- ,
4?r 471-0
47ro- = 3$ cos a) -- - .
a
For the density to be zero at A, we must have
at the pole of the sphere.
If Vj and V 2 are the original potentials at A' and A, the strength
of the field is
and the potential at the centre
The preceding condition reduces to
whence
146 PARTICULAR CASES OF EQUILIBRIUM.
For the density to be null at A, the original values of the poten-
tial at the two poles of the sphere must be in the ratio of i to 2.
The density at any other point is negative, and therefore the
surface of the sphere is entirely negative as long as V 2 < 2V r
In the contrary case, the greater part of the surface is still
negative, but about the point A there is a more or less extensive
zone of positive electricity.
162. DIELECTRIC SPHERE IN A UNIFORM FIELD. Uniform polari-
zation, or electrification by layers of displacement, also represents,
on Poisson's theory, the condition of a dielectric in a uniform field.
Yet if <f> be the strength of the field, F^ the internal force due to the
fictive layer, the resultant force at each point of the interior, instead
of being zero, will have a constant value equal to < + F^.
We can demonstrate that the condition relative to the equilibrium
of dielectrics is then satisfied that is to say, that there is a constant
ratio over the whole surface between the perpendicular components
on the interior and on the exterior.
This ratio ft, being given by the nature of the dielectric, will
enable us to determine the force F^, and consequently the distri-
bution of the fictive layer.
For a point P on the surface in a direction to, the external
perpendicular component is < cos to + F n = (< 2F^) cos to, and the
internal perpendicular component (</> + F { ) cos to. The ratio of these
two forces
(< - 2F,.) cos to < - 2F,.
cos to
is therefore constant, and we deduce from it
It thus appears that the problem is completely determinate, and that
the state of the sphere is identical with that of a conducting sphere
of the same radius situate in a uniform field, the strength of which
is < ^^ . We deduce from this (159)
p+2
3 A*" 1
DIELECTRIC SPHERE IN A UNIFORM FIELD.
147
163. The flow of force from the sphere is equal to the flow
which traverses it, va 2 (< + F^) increased by the flow - 3?r0 2 F { , which
corresponds to each of the surface layers.
We have therefore
Q =
1 + 2
The equivalent circle of flow on the original equipotential surfaces
would have a radius determined by the equation
!=9/r2
Fig. 43-
so long as ft > i this radius is always greater than that of the sphere.
The force in the interior is
it is equal to - < if /x = 2, which is approximately the case with most
4 3<
dielectrics, and becomes equal to when ft, is very large.
L 2
148
PARTICULAR CASES OF EQUILIBRIUM.
Near the pole A, on the outside, the force is
it is equal to < for //, = 2, and becomes 3 </> when ft is very great.
This force, therefore, is then thrice its primitive value; this is the
case with conductors.
In the present case the external lines of force are no longer per-
pendicular to the surface. It could be easily shown that the tangen-
tial components are equal, and that the ratio of the angles 6 and <o
(Fig. 43) of the perpendicular to the lines of force on the outside and
inside satisfies the law of refraction.
tan o>
For the equator, where co = , this equation also gives 6 =
The lines of force which touch the sphere on the equator are then
tangents to the surface.
Fig. 44.
164. CONCENTRIC SPHERICAL LAYERS IN A UNIFORM FIELD.
It is easy to generalise the preceding problem, and to apply it to a
series of concentric spherical layers. In a uniform field of strength
4> let there be a system of concentric spheres S 15 S 2 , S 3 (Fig. 44)
having the radii a lt a 2t a z . . . , and the specific inductive capacities
CONCENTRIC SPHERICAL LAYER IN A UNIFORM FIELD. 149
Let us consider the inner sphere S r If the medium of specific
inductive capacity fi 2 , which surrounds it, were unlimited and formed
a uniform field of strength </> 2 , this sphere would be covered with a
layer of displacement M 15 giving in the interior a constant force Fj
and a uniform field </> : = < 2 + l ; and for a point P x on the surface
in the direction w, we should have the equation
cos to = * < - 2 cos o>
or
(25)
But the uniform field of strength < 2 , situate on the outside of the
sphere S 19 is that which would be produced for the interior of the
sphere S 2 by an external uniform field of strength < 3 , and by the
internal force F 2 due to the fictive layer distributed on the surface
according to the same law, which would give
For a point P 2 of this surface we have to consider not merely the
action < 3 of the external field, and that of the layer M 2 , but also
the action of the layer M 1 of the internal sphere S r
The law of the conservation of the flow of induction would give
a relation between these quantities analogous to the equation (25)
and which we may write directly in the following manner, suppressing
the common factor cos w :
whence
(26) ^, = ft fa, - 2 F 2 ) + a (ft, -
The same reasoning applies to surface S 3 ; for a point P 3 of its
surface we should then have to take into account the strength ^ of
the external field, together with the actions of the three internal layers
M 3 , M z and M r
We shall thus have
150 PARTICULAR CASES OF EQUILIBRIUM.
Whence
'
The law of the terms is evident. Connecting these equations
with the identities
& -**+*!,
(28) 4> 2
we could determine the values of Fj , F 2 , F 3 . . . The problem is thus
completely solved.
165. Let us suppose that there are two layers bounded by the
surfaces Sj and S 2 , in a uniform external field of strength <, where
the dielectric is air. We may simply put
Equations (28) give then
/XY
Substituting in (25) and (26), and putting /?=( ) we have
F!) = ft (< + F 2 - 2FJ,
+ 2 -
equations which would determine the forces F x and F 2 as functions
of the data of the problem.
166. Suppose, further, that the internal nucleus is also air, which
would amount to determining the state of a spherical layer ; /^ must
also equal i. Let us denote by ft the specific inductive capacity of
the medium previously denoted by /* 2 , equations (29) would become
From this is deduced
POISSON'S HYPOTHESIS. 151
and therefore
2) -2(/X-l) 2 /? '
The force in the interior at the surface S x is
The force is constant inside S p but it is not constant between S x
and S 2 , nor outside S 2 . The value of the force in the interior of S x is
a fraction of the strength of the field, which would be equal to unity
for ft = i, and to zero for ft= oo. With dielectrics whose coefficient
ft does not differ much from 2, the fraction is always very near
unity; if the layer is a conductor, the coefficient /* may be
considered as infinite, and F T becomes zero.
We shall afterwards see the importance of this question in
magnetism.
167. POISSON'S HYPOTHESIS ON THE CONSTITUTION OF DIELEC-
TRICS. Poisson's hypothesis, as revived by Faraday for electricity,
consists, as we have already said (no), in assuming the dielectric to
be formed of small conducting spheres disseminated in an insulating
medium.
The results already obtained enable us to explain the method
adopted by Poisson for calculating, at any rate approximately, the
consequences of his hypothesis.
Consider a sphere of radius a lt and of specific inductive capacity
fij , situate in a field of strength <f> 2 , and of specific inductive capacity
ft 2 ; from equations (25) and (28) the force on the interior of the
electric layer is
and the external potential of this layer on a point at a distance r is
equal to
Let us suppose that a sphere of radius a contains a large number
152 PARTICULAR CASES OF EQUILIBRIUM.
of small spheres of radius a 19 and assume with Poisson that the
electrification of each of them is not influenced by the electrification
of the adjacent spheres, and only depends on the strength of the
field. If n is the number of small spheres contained in the large
one, the value of the potential at a distance, which is very great
compared with a, is
_ r na\
or putting h = , that is to say calling h the ratio of the space, occu-
pied by the small spheres, to the volume of the whole sphere,
cos
If the sphere were homogeneous, and of the specific inductive
capacity /*, the potential V at the same distance would be
The action of the two systems is identical if we have
which gives
this value of ft represents the apparent inductive capacity of the
sphere made up as we have supposed.
If we assume that the small spheres are conductors, we must
make a, = oo ; we have then
I + 2/1
POISSON'S HYPOTHESIS. 153
_jv _
If, finally, the external medium is of air, /* 2 = i, and we get
I +2/1
For those dielectrics, whose specific inductive capacity is near
2, we should have
2 + 2 4
This result may give some idea of the degree of exactitude to
which Poisson's reasoning tends.
In a conducting sphere, the interior force due to the induced
layers is equal to the action of the external field. The external
action of a polarized sphere is very small compared with the internal
AA 3
action, for the ratio of the forces (158) is at most equal to 2( - J
and tends towards zero when the spheres are infinitely small. But if
the volume occupied by the conducting spheres is a quarter of the
total volume, the action which each of them exerts upon the adjacent
ones can no longer be neglected in comparison with the internal
force, and the field is thus modified. The maximum ratio of the
sum of the volumes of the spheres which touch, to the total space is
equal to -= or sensibly = . If this ratio is reduced to - , the
3v/2 v/2 4'
distance of the centres of two adjacent spheres is about equal to the
3 /~T~
diameter multiplied by * / =. The action exerted by the electrical
layer of one of the spheres, at the centre of the nearest one, might
thus attain a fraction of the internal force equal to
It is true that if the reciprocal action of the sphere tends to
increase the electrification parallel to the force of the field, it tends
to diminish it in a perpendicular direction, so that we are not far from
the truth in assuming, with Poisson, that this reciprocal influence may
be neglected.
154
PARTICULAR CASES OF EQUILIBRIUM.
168. Two UNEQUAL MASSES OF OPPOSITE SIGNS. Let +m
and m' be two masses of opposite signs situate at A and A' (Fig. 45)
at a distance of 20, m being greater than m' in absolute value ; let
us put
m
The equation of an equipotential surface is
r r
r r
rr
One of these surfaces corresponds to potential zero, and its
equation is
m m'
7-7=.
whence
It is a sphere to which the point A' is internal ; the two points
A and A' are conjugate in reference to this sphere.
To determine the radius R, and the centre O of the sphere, we
shall use the ratios
BA B'A R OA
TWO UNEQUAL MASSES OF OPPOSITE SIGNS.
155
Remarking that OA - OA' = 2 a, we easily deduce from this
(30)
I
4 -I
The potentials are negative inside the sphere, and positive
outside.
Fig. 46.
All the equipotential surfaces are closed surfaces with one sheet
or with two distinct sheets, except a single one which has two
adjacent sheets S f and S' i5 and which passes through the point I
(Fig. 46) where the force is zero. The position of this point is given
by the equation
m m'
or
156 PARTICULAR CASES OF EQUILIBRIUM.
we have thus
^-k
I A'"*'
and therefore
The value of the potential at I, and on the whole surface with
two sheets, is
2d k
There are evidently two other points C and C on the axis where
the potential has the same value, and which belong to this surface.
For from B to A, the potential increases from zero to infinity, and de-
creases from infinity to zero, from the point A to an infinite distance.
The distances x and x' of the points C and C' from the point A, are
given by the equations
20.
All the surfaces whose potential is greater than V t surround the
point A ; all those whose potential is positive and smaller than V { ,
consist of two sheets, both of them isolated and closed ; one of them
outside the great lobe of the surface S f surrounds the two points A
and A'; the other, which is inside the small lobe S' { , merely surrounds
the point A'.
169. The general equation of the lines of force
m cos to + m' cos o>' = m' + m cos
becomes here
m cos <o m' cos o>' = - m' + m cos 0,
or
(31) cos >' - / 2 cos eo = i - k z cos 6.
TWO UNEQUAL MASSES OF OPPOSITE SIGNS. 157
X _^__^__
The asymptote which corresponds to w = c/, is defined by the
equation
(i / 2 ) cos a = i 2 cos 6]
it passes through the centre of gravity O of the two masses. This
point is given by the ratio
OA w'i
For the asymptote to be real, cos a must be > - i, or
i - & cos e
The condition
i - 2 cos
gives the value of corresponding to the limiting line of force.
The equation of this line of force is then
cos a/ 2 cos to = / 2 i ;
it evidently bounds the flow /^(m - m') from the point A, and corre-
sponds to the value of Q given by the equation
. N 2(m-m')
i - cos = =
m
or
44)
cos = i.
This line of force 2 passes moreover through the point of equi-
librium I, as can be easily shown.
158 PARTICULAR CASES OF EQUILIBRIUM.
We can determine the direction of the tangent as above (145),
which gives
, . sin/3 /' m'r 2 i r 2
/==
170. When the tangent is horizontal, we have
or
sin /3 = sin <o,
sin f}' = sin a/.
Equation (32) becomes then
sin (air 2
sin u>' & r' 2
or replacing the sines by the opposite sides,
r f _ i r 2
r~~&'7*'
which equation may be thus written
The locus of the points where the tangent is horizontal, is there-
fore a sphere comprising the point A'. The centre and the radius
of this sphere may be calculated by formulas (30) in which k is
replaced by $.
171. When the tangent is vertical, we have
or
sin /? = cos a).
sin 3' = cos /.
ELECTRIFICATION OF A SPHERE BY A POINT. \ 159
_ ' X.
In this case equation (32) becomes
COS to
it represents a curve formed of two branches, one proceeding from
the point A, the other from the point A' (Fig. 46).
The branch T proceeding from A is at first vertical at this point.
For points at a considerable distance, the angles co and w' tend
towards equality, and we have
cos <o cos o>' cos W cos to' sin o> . 8r 20 sin 2 o>
from which
sin 2 a>_r(i->E 2 )
COS to 20
The second member increases to infinity with r. The angle o> tends
77
then towards -, and the curve has a vertical asymptote which evi-
dently passes through the point O.
The second branch is a closed curve T'j it passes through the
point A', and through the point I.
172. ELECTRIFICATION OF A SPHERE UNDER THE INFLUENCE OF
A POINT. We know from the theorems already proved (61), that we
can replace the mass - m' by an equal layer in equilibrium on any
one of the equipotential surfaces which surround the point A', com-
prising the sheet S'^ of the surface with two sheets. In like manner
we* may replace the mass m by an equal layer on one of the equi-
potential surfaces which surround A, including the surface S^ The
two masses m and - m' may, lastly, always be replaced, for external
points, by one mass m - m' on one of the surfaces which surround
the two points, including again the surface S^.
If, in particular, we consider the sphere S of potential zero, which
surrounds the point A', we can replace m' by an equal mass in
equilibrium on the sphere. Nothing will be changed for external
points ; but for points in the interior the potential will be constant
and equal to the value which it has on this surface that is to say,
zero. For points inside the surface S, the mass m may be replaced
by a mass + m' in equilibrium on this surface, and thus the potential
will everywhere be zero on the outside.
i6o
PARTICULAR CASES OF EQUILIBRIUM.
The first case corresponds to the electrification of an uninsulated
sphere under the influence of an external mass; the second gives
the influence of an electrified mass on an uninsulated spherical surface
which surrounds it.
The density o- of the layer at each point should satisfy the
ratio
F= 47TCT.
At the point P, on the surface (Fig. 47) the force is directed along
PO ; it is the resultant of the forces /and/', one aeting from A and
Fig. 47-
the other directed towards A'. The triangle formed by the three
forces F,/ and/' is similar to the triangle APA'; we have then
AA' r' r
and, consequently,
/ m m 20, m
I = 20. = 2a = 2ak i = . .
From this we have
m
m
CT =
The force and density at a point on the surface S are therefore
inversely as the cube of the distance, either from the point A, or
from the point A'.
This density is positive if the mass m be replaced by a layer dis-
tributed on the spherical surface S ; this is the case of the inductive
action of a mass - m', placed inside an uninsulated spherical surface.
ELECTRIFICATION OF A SPHERE BY A POINT. l6l
The density is negative if the action of this layer be substituted
for that of the mass - m' t which corresponds to the problem of an
uninsulated sphere S under the influence of the mass m at A.
In these two cases, the sphere is given as well as the position and
magnitude of one of the masses.
Knowing the mass m, the radius of the sphere R, and the dis-
tance AO = </, we get directly
= >
Jx
and, consequently,
a* yJ- m
(T= -
4 7TR
173. If, after having insulated the sphere, we superpose on the
layer - m any uniform layer M, equilibrium still holds, and the
potential of the sphere, which was zero, becomes
If we make M = m', the total charge of the sphere is zero, and
its potential is
This is the case of an insulated sphere, originally in the neutral
state, electrified under the influence of an external point. As the
mass is null, the potential at the centre only depends on the external
mass. We must then have
m '
The density of this new layer being
m m
r -
162 PARTICULAR CASES OF EQUILIBRIUM.
the resultant density is
m - m m
+ =
4 7rR
The density will be zero for all points of the small circle perpen-
dicular to the axis defined by the equation
The plane of this small circle cuts the axis OA on the left of
the point A', since we have r> \/*/ 2 -R 2 . It is the neutral line
which separates the positive from the negative zone. It is a line
of equilibrium the force and the density there are null. It is the
intersection, by the sphere, of the equipotential surface V = ; we
know, moreover, that the two surfaces intersect at a right angle.
The density will be null on the small circle, formed by the con-
tact of the tangent cone to the sphere, and having its apex at the
point A, the plane of which circle passes through the point A', pro-
vided that
M d* - R 2 m
m
4 7rR
and therefore
M R
m
174. The action of the insulated sphere S, electrified by induc-
tion from the mass m^ on all external points, may be replaced by that
of a mass - m' = - , placed at A'. In like manner the uninsulated
k
surface S' acted on by induction from -m', is equivalent for all
internal points to the mass m = km' placed at A.
175. IMAGE OF ANY GIVEN SYSTEM. The principle of images
in reference to a sphere may be extended to any system whatever for
instance, to an electrified layer. For each element of the systems
develops by induction, on the sphere, a layer whose action on external
points is identical with that of the corresponding image. As each of
RECIPROCAL ACTION OF TWO SPHERES. 163
these layers -is in equilibrium, their superposition will be a state of
equilibrium, and the resultant action will be equal to the resultant
action of all the images. The totality of these images will form a
system, which is the image, in reference to the sphere of the given
system. If the given system is a surface 2, the image will be a
surface conjugate to the first.
176. RECIPROCAL ACTION OF Two SPHERES. The principle of
images combined with Murphy's method (86), enables us to solve
completely the very important problem of the reciprocal action of two
spheres. Let S a and S & be the two spheres (Fig. 48), R and R' the
radii. The method consists, as we know, in determining a series of
successive layers in the following manner. On the conductor S a a
layer is placed capable of giving the potential i ; this is a uniform
layer of mass R. This layer acts outwards as if it were concentrated
at A. It is fixed and the induced layer on the surface S & of the
Fig. 48.
second uninsulated sphere is determined, which amounts to deter-
mining the image A' in reference to S 6 of a mass + R at A. The
equivalent layer is next fixed at A', and its inductive action on the
uninsulated sphere S a is determined that is to say, the new image A l
of A', and so on. The same operation will be repeated beginning
with the sphere S 6 , and all the masses thus determined are multiplied
by suitable coefficients. As each of the masses and the densities
can be exactly calculated, the problem of distribution is completely
solved.
The force exerted between the two spheres is the resultant of the
actions exerted by each of the masses comprised within one of the
spheres on all the masses contained in the second.
M 2
164 PARTICULAR CASES OF EQUILIBRIUM.
The calculation does not present any theoretical difficulties, but
it is very tedious. Sir W. Thomson performed it in the case of two
spheres of the same radius when the distance of the centres varies
between 2R and 4R, that is to say, when the distance of the surfaces
is comprised between and the diameter of one of the spheres.
In the present case, if R is the common radius of the spheres
A and B, U and V the potentials, <rR the distance of the centres, M
and N the respective charges, then if I, J, a and b are coefficients
which depend on c, we have
M = R(IU-JV),
N = R(IV-JU),
expressions analogous to those furnished by Murphy's method for any
given bodies.
If we wish to express the force, and the potential as a function
of the masses, we get
E. 2 F = 2/3MN - a(M 2 + N 2 ),
_
(I2 _J 2)2
(
(I 2 -J 2 ) 2
If the charges M and N are equal, we get
M = RV(I-J),
R 2 F = 2 (/3-a)M 2 .
177. As these formulae have only hitherto been calculated for
c = 4, it is useful to see how they may be replaced for greater distances.
RECIPROCAL ACTION OF TWO SPHERES. 165
Suppose that the action of the two spheres is the same as if the
masses were respectively concentrated at the centres, and that the
potential of each of them is equal to that obtained for the centre, by
replacing the adjacent sphere by an equal mass situate at the centre.
We shall have thus
N M
~ +J
MN
<: 2 R 2 '
TJV __ - _ (U 2 + V 2 )
U (U
and for equal charges
F-
If we make <r=4, Sir W. Thomson's formulae give
I-J = 0-80258,
2 (fi- a) = 0-05846,
2(b-a) = 0-03766.
The corresponding values in the approximate formula are
= = 0-0625,
=^ = 0-04.
i) 2 25
Thus, when the distance of the surfaces is equal to the diameter
1 66 PARTICULAR CASES OF EQUILIBRIUM.
of one of the spheres, and the charges are equal, the approximate
formulae give the potentials as a function of the masses, the value
of the force as a function of the masses, and the force as a
function of potentials with a degree of approximation which is
, and for the various forces respectively. As the relative
300 16 17 j
errors amount to , when <r=3'8, it will be seen that they will
diminish very rapidly for greater distances.
We may therefore, as a particular case, regard Coulomb's method
for determining the action of two electrified masses as perfectly
exact; and Coulomb's method will give with the approximate for-
mulae very exact results for the measurement of potentials, if we
take care that the distance of the surfaces of the spheres sensibly
exceeds the diameter of one of them. In determining potentials by
means of the balance, we ought always to take special precautions
for eliminating or for calculating the influence of the sides of the
cage.
178. MOTION OF SMALL BODIES IN THE ELECTRICAL FIELD.
We have seen that (162) a dielectric sphere of radius #, placed in a
uniform field where the force is <, becomes electrified so that the
internal action of the equivalent layer is
This coefficient h is equal to - for the dielectrics whose
4
specific inductive capacity is 2, and it is equal to unity for con-
ductors.
From this we deduce (159)
= u (f> =
47T
The coefficient K is equal to for conductors, and has a
47T
smaller value for dielectrics, since the factor h is always less than
unity.
MOTION OF SMALL BODIES IN THE ELECTRICAL FIELD. 167
Consider now a sphere so small that, placed at any point of a
variable field, it becomes electrified as it would be in a uniform field
where the force was the same ; the electrical moment of this sphere
will be
or = UO-Q = uK<f>,
and the axis of electrification being parallel to the force <, will be
perpendicular to the equipotential surfaces which include it.
Suppose that we fix on the surface the two electrical layers, which
are equivalent to the system of two solid spheres, or of two equal
and opposite masses m at an infinitely small distance 8, such that
7 = m8. If Vj and V 2 are the potentials at the points occupied by
the masses - m and + m, the energy of this sphere in the field is then
W
= f*V = -
At a point P' where the force is equal to <f> + d$> the energy of
the sphere will be
W'= -
The work necessary for bringing the sphere from the point P to
the point P' is then
W'-W= -
In reality, if the layers are not fixed, the electrification changes
with the displacement of the sphere, and the work in question is
between -CT^ and -(cr + ^fer)^; this work only differs therefore
from the value found by an infinitely small expression of the second
degree.
The energy dW expended in effecting the displacement is then
(33) u
dW=
Thus, when an infinitely small sphere passes from a point in the
1 68 PARTICULAR CASES OF EQUILIBRIUM.
field where the force is < , to another where the force is </>, the
increase of energy is
If the sphere is brought from an infinite distance, we shall have
The sphere left to itself tends of course to expend energy, and
therefore, by equation (33), to move in a direction in which the value
of <f> 2 increases most rapidly. It tends then to move to points in
which the force is a maximum in absolute value.
179. Let n be the direction in which < 2 varies most rapidly ; the
expression for the force acting on the sphere is
(34) F-
U'fi x un
and its components parallel to the co-ordinates are
(35)
OZ
180. In a variable electrical field, the force cannot be a maximum
at any point situate outside the acting masses.
This theorem follows directly from the preceding demonstration.
We have seen in fact from Earnshaw's theorem (63), that an
electrified body cannot be in stable equilibrium in a variable field.
Since an infinitely small sphere can only be in stable equilibrium at
points where the value of < 2 is an absolute maximum, that is to say,
where the value of < is a maximum in absolute value, it follows that
this circumstance cannot present itself for any point outside the
acting masses.
DIRECTION OF A DIELECTRIC NEEDLE IN A FIELD. 169
181. The same reasoning applies to the motion of a very small
body of any given form, if we neglect the effects of rotation that is
to say, if we assume that the body always retains the same direction
in reference to the lines of force. This body, in fact, becomes
electrified proportionally to the force of that point of the field in
which it is situated, and the variation of energy is proportional to the
variation of the square of the force.
Independently of this progressive motion, a body which is not
spherical will turn on itself in every point, in such a manner that for
stable equilibrium about its centre of gravity the electrical energy is
a minimum, and the electrification is a maximum.
Such, according to Sir W. Thomson, is the true meaning of the
attraction of light bodies of small dimensions in an electrical field,
so long at any rate as they have not been electrified by direct contact.
These bodies, whether conductors or not, move towards points where
the force is a maximum in absolute value, and they finish by coming
in contact with the electrified surfaces. If they were movable in a
medium in which an extraneous resistance would keep the velocity
very small, they would move towards the electrified body, not along a
line of force, but along a line of maximum variation of the force ; in
certain cases, in which the body is impeded, this motion may even
be perpendicular to the force.
The body is in equilibrium for points in which we have
d<P = 0. or <p#<z) = 0,
This condition may be realised in two ways
</> = 0, or d$ = 0.
Hence there is equilibrium when the force is null, maximum,
minimum, or stationary. There is equilibrium particularly in a
uniform field, which was d priori evident The equilibrium is thus
neutral ; it is stable at the maxima of force, unstable at the minima,
and at the points at which the force is null.
182. DIRECTION OF A DIELECTRIC NEEDLE IN A VARIABLE
FIELD. Let us suppose that on a sphere charged, as we have always
supposed, by layers of displacement, we cause a force F to act which
is constant in magnitude and direction, and which makes the
angle 6 with the axis of electrification ; the moment of the couple
produced by this force will be
F;//.S cos = FCJ cos 0.
1 70 PARTICULAR CASES OF EQUILIBRIUM.
The electrical moment T3 is therefore the moment of the couple
which this sphere would experience in a field equal to unity, the force
of which is perpendicular to the axis of electrification.
If we adopt the conception of Poisson and of Faraday regarding
the constitution of dielectrics, and consider them as formed of con-
ducting spheres disseminated in an insulating medium ; if we admit,
further, that the electrification of each of them is not modified by the
adjacent ones, the electrical moment of a body of any given form in
a uniform field is
U
where U is the volume of the dielectric, n the number of spheres
which it contains, and u the volume of each one of them.
The expression of this moment is therefore the same as for a
homogeneous sphere.
On this hypothesis, a body of any given form in a uniform field
would also be in equilibrium in reference to its centre of gravity,
whatever was its direction. For all the volume-elements become
electrified parallel to the force of the field, and the couple of rotation,
being null on each of them, is null upon the whole.
In a variable field, on the contrary, a very small body, fixed by its
centre of gravity, tends to take a certain direction. As each volume-
element du is only acted on by the force of the field, it tends to
move towards points where the force increases, and the components
of the force which it undergoes are
z = -- .
2 02
183. Consider a short and infinitely thin needle, and let </> be
the force of the field at the centre of gravity O of the needle. Let
us take the direction of the force for the #-axis ; suppose that the
needle can turn about the .s-axis, and that it makes the angle 6
with < .
For the volume element du at a distance a from the centre, and
DIRECTION OF A DIELECTRIC NEEDLE IN A FIELD. 171
X
whose co-ordinates are x and y, the component < of the force, parallel
to the plane, will have an expression of the form
The components X and Y of the force will be
The component tangential to the circle which the volume-element du
describes is
T = X sin 9 - Y cos = (A sin - B cos 0).
The position of equilibrium corresponds to the condition
A sin - B cos = 0, or tan 6 = ,
A
that is to say, to the direction along which the variation of the force
is a maximum.
When the element is turned through an angle dO from its position
of equilibrium, the tangential component is
dT = (A cos B + B sin 0)d0,
the angle being determined by the condition of equilibrium ; from
this is deduced
Acos0 + Bsin<9
We have then
</T =
2
172 PARTICULAR CASES OF EQUILIBRIUM.
For the entire needle, the moment of the resultant in reference
to the axis will be
If p is the density of the substance in question, the time /, of
infinitely small oscillations, is given by the equation
I
cfidu
If the needle is rectilinear, and of the length 2/, we have
cfidu
i
which gives finally
Thus, in a variable field, a dielectric needle able to turn about its
centre of gravity is directed, not along the line of force, but along
the line of greatest variation of the force; the square of the time
of oscillations is not simply in the inverse ratio of the force of the
field, as it depends on the coefficients of the variation, and, other
things being equal, is proportional to the length of the needle.
184. The phenomenon is particularly useful to consider when
the field is symmetrical in reference to the point at which is the
needle. Suppose, for instance, the needle to be placed at the
middle point O of the distance AA' (Fig. 34) between two equal
masses of opposite signs, and that it can oscillate about a right
line perpendicular to AA'. In the plane of oscillation, the force
at the point O is a minimum for the direction A'A or Ox, and
a maximum for the perpendicular direction Oy. For a point
adjacent to O we may therefore write
ACTION OF A FIELD ON A CONDUCTING NEEDLE. 173
The components of the force which are exerted on the element du
are
X = KtluAx = KJuAa cos 0,
Y = - Kduty = - KduBa sin 0,
and the tangential component
T = aKdu(A + B) sin cos 0.
The condition of equilibrium is thus
sin cos = 0,
which gives two directions, = and = -, the first corresponding to
stable, and the second to unstable, equilibrium.
For a very small deviation dO from the position of stable equi-
librium, the tangential component is
If the volume-element is isolated, the duration of the oscillation is
given by the formula
KA + B'
it is thus seen to be independent of a, and to depend only on the
density of the substance, on its electrical susceptibility, and on the
law of the variation of the field.
If we collect on a straight line a series of similar particles, and
if we suppose that they exert no influence on each other, each of
them would act as if it were alone, and the oscillation of the whole
needle would take place in the same time as each of its parts ; the
duration of the oscillation of a needle for the state of the field in
question is therefore independent of its length.
185. ACTION OF A FIELD ON A CONDUCTING NEEDLE. A
conductor behaves in a totally different manner. Consider, for
instance, an infinitely small needle, so that the electrification may
be supposed identical with that which would be produced in a
uniform field.
174 PARTICULAR CASES OF EQUILIBRIUM.
The needle is in equilibrium in a direction perpendicular to the
line of force, but it is evident that then the electrification is a mini-
mum and that equilibrium is unstable.
When the needle is oblique to the force of the field, the two
positive and negative electrical layers are symmetrical in reference
to the centre, and the action of the field evidently produces a couple,
which tends to move the needle in the direction of the force.
The law of distribution is independent of the force of the field,
and each of the layers is proportional to this force. Hence the
moment of the couple which acts upon the needle, for a given
deviation from its position of equilibrium, is proportional to the
square of the force <. Lastly, the duration of the oscillation is
inversely as the force, and we may write
t- A
?'
the constant A only depending on the ratio between the longitudinal
and transverse dimensions of the needle.
ELECTROMOTIVE FORCE OF CONTACT. 175
CHAPTER VIII.
SOURCES OF ELECTRICITY.
186. We have already mentioned Volta's most important dis-
covery that two conductors, and, more generally, any two bodies
placed in contact, assume different electrical conditions on each side
of the surfaces in contact.
In the case of two different conductors in contact and in equi-
librium, the potential is constant on either of them, -but experiences
a sudden change, on passing from one surface to the other.
We need not mention here all the experiments by which this law,
of such fundamental importance, has been established; we shall
merely adduce the following experiment, which will serve to define
the conditions of the phenomenon.
If two plates, one of zinc and the other of copper, in the
neutral state and at the same temperature, while held by insulating
handles, are placed in contact with their parallel faces, and are then
separated from each other, each of them will be found to be elec-
trified, the zinc positively and the copper negatively. The electrical
charge on each of the plates is proportional, other things being equal,
to the extent of the surfaces in contact.
The phenomenon is as if the plates were the coatings of a con-
denser, between which there existed a given difference of potential ;
the corresponding electrical layers being on each side of the surface
of separation in the two metals respectively, and at a very small
distance apart, so that the capacity of the system is simply pro-
portional to the extent of the surfaces facing each other.
187. ELECTROMOTIVE FORCE OF CONTACT. If V 1 and V 2 are
the potentials of the zinc and of the copper, 3V their difference
Vj - V 2 , or the electromotive force of contact, and C the capacity
of the condenser formed when they are in contact, the charge of the
plates will be
176 SOURCES OF ELECTRICITY.
A charge of electricity M may be imparted to the system of the
two plates, which raises the zinc to a potential V, for instance. In
this case the same difference of potential is still maintained at the
surface of contact ; when the plates are separated the charge on each
is made up of the charge m due to the difference of potential
of contact, together with part of the common charge M, which has
been divided between them according to the ordinary laws of dis-
tribution.
The experiment presents great difficulties when it is attempted
to deduce the difference of potential from the magnitude of the
charges ; for the real distance of the plates depends on the degree
of polish of the surfaces ; the plates are often in contact in only a
small number of points, and the capacity of the system may vary
within considerable limits.
More regular results are obtained by keeping the plates parallel
to each other at a certain distance e and connecting them by a
wire of copper or of zinc. The difference of potential SV pro-
duced at the point of contact, is maintained over the whole extent
of the two plates ; the distance of the corresponding layers is sensibly
equal to the thickness of the dielectric which separates them, and
the capacity of the system is inversely as this thickness. If, after
having suppressed the external contact, the plates are moved away
from each other, then, if S is the extent of surface,
This experiment will enable us to determine the absolute value
of 8V. In all cases, if a constant distance e is maintained between
the plates, and if different metals are employed, the ratios of the
electromotive forces of contact may be determined.
As a matter of fact, the experiments are very delicate, owing to
alterations in the results by very slight modifications in the state
of the surfaces. The very nature of the gas which constitutes the
dielectric seems to have a slight influence ; it may be that the layer
of gas, adhering to the metal, changes its physical properties, or that
some particular chemical compound is formed there.
188. VOLTA'S LAWS. LAW OF CONTACT. However this may
be, Volta's ideas have been confirmed, by whatever progress has been
made in electricity, and the following law may be enunciated as the
law of contact:
If two bodies are in contact at the same temperature^ a finite
LAW OF SUCCESSIVE CONTACTS. 177
difference of potential is set up, which depends on their nature, and
which is altogether independent of their dimensions, of their shape,
of the extent of surface in contact, and of the absolute value of the
potential on each of them.
Volta characterized this property by saying that there is a tension
of contact between the two surfaces, but the manner in which he
conceived the phenomenon is quite in agreement with the idea of a
difference of potential.
We shall represent this characteristic difference of potential, or
electromotive force of two metals A and B, by the symbol A|B, the
first letter denoting the metal whose potential is highest. We have
then
We may at once add that this difference is a function of the
temperature, and that the contact of two bodies of the same kind,
but at different temperatures, also gives rise to an electromotive
force.
In all the questions relative to the electrical equilibrium of
conductors, we have hitherto neglected the electromotive forces
due to the contact of heterogeneous conductors. All the calcu-
lations presuppose that the conductors are identical and at the
same temperature, and the results should be modified by allowing
for this new circumstance, unless in the case of very high poten-
tials, where the effects of contact may be neglected.
189. LAW OF SUCCESSIVE CONTACTS. After having confirmed
the fundamental fact of the electromotive force of contact, Volta
compared with each other the results furnished by different metals,
and established experimentally a second law, which may be thus
enunciated :
WJien several metals at the same temperature are soldered to each
other so as to form a continuous chain, the difference of potentials of the
extreme metals is the same as if these two metals are in direct contact.
Let A, B, C ..... L, M be the metals constituting the chain ; this
law, with the symbols adopted above, is represented by the following
formula :
A|B + B|C ..... +L|M = A|M.
We have, further,
A|M= -M|A.
178 SOURCES OF ELECTRICITY.
By bringing all the terms within the first member, the equation
becomes
A]B + B|C + +L|M + M|A = 0,
which amounts to saying that the two ends of any chain which is
terminated by identical metals are at the same potential.
This important proposition is a necessary consequence of the
principle of the conservation of energy. If the terminal metals
A and A', of the same kind, could be kept at different potentials
by intermediate contacts, then if they were joined by a conductor
of the same kind, a continuous discharge would take place both in
the external conductor and in the chain of intermediate metals
that is to say, a permanent flow of electricity. This transference of
electricity, analogous to a succession of discharges, would, as a
necessary consequence, produce calorific phenomena that is to say,
energy which would be a realisation of perpetual motion.
If even we suppose that the disengagement of heat in certain
parts of the circuit, corresponds to an absorption in other parts, there
would be a transport of heat from the colder to the hotter parts,
without any corresponding work. This result is incompatible with
Carnot's principle, which appears as well established in science as the
impossibility of perpetual motion.
190. EXCEPTION TO THE LAW OF SUCCESSIVE CONTACTS.
ELECTRICAL BATTERIES. Volta found that the law of tension
sometimes ceases to apply. We observe, in fact, that it does not
appear necessary as long as there is no source of energy in the
circuit ; but if this contains sources of energy of any kind whatever
for instance, bodies which may give rise to exothermic reactions
correlated to the passage of electricity the energy furnished by
these reactions, might assist in keeping up a permanent current.
Without having very accurate ideas on this point (he was prepared
to find metals which did not satisfy the law), Volta had been led to
divide bodies into two great classes. The first contains those which
obey the law of tensions ; it comprises all the metals and a certain
number of solids. The second contains those which do not obey
this law : it comprises most liquids and solutions. By associating
bodies of the first class with those of the second, a chain can be
constructed the ends of which, though formed of the same metal,
present a finite difference of potential.
Experiment shows that in this case also, Volta's fundamental law
is verified that is, that the electromotive force corresponding to
CONCLUSIONS AS TO THE DISTANCE OF THE ATOMS. 179
each of the contacts, is a constant quantity, independent of the other
bodies constituting the chain ; and the difference of potential between
the two ends of the chain, is the algebraical sum of all the electro-
motive forces due to each of the contacts.
When these two ends are joined by a conductor, no new elec-
tromotive forces of contact are introduced ; a permanent flow of
electricity is set up in the circuit, where it is maintained by the
energy of the chemical actions. This is the principle of electrical
piles or batteries.
191. CONCLUSIONS RELATIVE TO THE DISTANCE OF THE
ATOMS. When two metals are in contact, the existence of an
electromotive force implies the formation of two electrical layers
of opposite signs separated by a finite distance, and these layers
must be localised in the two metals respectively. If we knew the
electromotive force, we could determine the distance between the
layers, by measuring the absolute charge on the two plates when they
are separated after having been in contact ; for if V denotes the
electromotive force we have
= L X
4?r m
It is almost impossible to make the experiment in this form, for
the charges which the plates retain depend solely on the capacity
of the system at the moment the separation is effected ; and this
capacity is in general only a very small fraction of the original
capacity, since contact cannot be broken simultaneously over the
whole extent of the surface.
Sir W. Thomson, however, by a series of ingenious reasonings,
has been able to show what must be the lower limit of the distance
of two electrical layers.
The expression for the electrical energy of the two plates in
contact is
,
S-n-e
and this energy represents the work necessary for separating the two
plates. This conclusion may be verified in another manner. If a-
is the electrical density of each of the layers, we have
V
471-0-=-,
N 2
l8o SOURCES OF ELECTRICITY.
and the force which acts on one of the surfaces is
In order to move the plates to a great distance
expend the work
- fra-
The electrical energy imparted to the system at the moment
of contact, is borrowed from the original potential energy of the
two plates, and it cannot be supposed to be greater than that which
would be available when the two metals are alloyed together.
Suppose that, with zinc and copper, the ratio of the weights
of the plates is that which forms brass. Let p be the total weight,
c the sum of the two thicknesses, and 8 the mean density of the
system, we shall have
or S = ^.
The electrical energy may then be written
This equation holds as long as the metals retain their physical
properties, and for this it is evident that the total thickness of the
two plates must be greater than the normal distance e of the elec-
trical layers.
Suppose now that, without changing the total weight /, we
increase simultaneously the surface of the two plates at the expense
of their thickness, the energy W increases in proportion to the
surface, or inversely as the thickness, so long at any rate as the
thickness e is greater than e. The maximum energy corresponds to
the case in which e = e, and we have then
whence
<= v ''jiS-w-
CONTACT OF DIELECTRICS. l8l
The electrical energy must be less than that which corresponds
to the alloy. As this is known to within about - of its value, we
5
may determine by the preceding equation a lower limit for the
distance e of the electrical layers. If their thickness were less, the
metals would have lost their properties, as they would no longer be
capable of acquiring their characteristic difference of potential on
contact ; we may consider that the molecular constitution of these
bodies would be thus modified, and that the thickness thus calcu-
lated is of the same order as the mean distance of the atoms.
Sir W. Thomson found thus, for copper and for brass,
e =
3-io 8 '
which corresponds to about - - of the wave length of green light.
If we could separate the plates without a partial recombination
of the opposite layers of electricity, each of them would be found at
an extremely high potential. This potential depends on the dimen-
sions of the plates, for the capacity of the system is proportional to
the surface of contact, while the capacity of each of the plates
separately is proportional to its linear dimensions. Assuming that
the distance of the electrical layer is -^ of a millimetre, Helmholtz
has shown that if a disc of zinc 10 centimetres in radius were in
contact with a disc of copper in connection with the earth, the
potential of the zinc when carried to a great distance would be
39.I0 6 times as great as the original potential due to contact.
With Sir W. Thomson's numbers, the final potential would be 30
times as great.
192. CONTACT OF DIELECTRICS. Volta's first law appears to
apply also to the contact either of metals with dielectrics, or of
dielectrics with each other. In these two cases, however, the de-
termination of the electromotive forces of contact presents great
difficulties.
A single point of contact between two conductors is sufficient to
set up equilibrium of the potentials ; the charges only depend then
on the capacity of the system, at the moment at which their sepa-
ration is completed. With bad conductors there is equilibrium at
the points of actual contact only, and only these become charged
with electricity. The total charge will then vary greatly with the
1 82 SOURCES OF ELECTRICITY.
number of points touching. We may further add that the pene-
tration of electricity in the dielectric still further complicates the
experiments.
193. ELECTRIFICATION BY FRICTION. In electrification by fric-
tion, the electricity seems to have no other cause than the contact
of two bodies ; the friction simply multiplies the number of points
of contact.
When the bodies rubbed are the same, they cannot in general be
electrified ; very feeble traces of electricity are sometimes seen, but
in that case the development of the electricity may always be attri-
buted to a more or less visible dissymmetry between the two bodies
rubbed.
194. ELECTRICAL MACHINES. We are thus led to the con-
clusion that there are two modes of producing electricity contact
and induction. All electrical machines bring into play one or other
of these modes, and their only object is to accumulate on a con-
ductor the charges produced.
The general theory of these machines is very simple. Let us
consider a hollow insulated conductor that of Faraday, for instance.
On the other hand, let us take the cake of an electrophorus of resin
or ebonite charged with negative electricity. A metal disc held by
an insulating handle, placed on the cake and connected with the
ground for a moment, will become charged with positive electricity,
and, if it is carried to the cylinder and made to touch the inside, an
equal charge will be produced on the outer surface of the cylinder ;
the disc itself will be neutral on being taken out, and the same pro-
cess may be repeated indefinitely. It is clear that in theory, there
is no limit to the charge of the cylinder, for, whatever be the charge
which it has acquired, a conductor, placed in the inside and put in
communication with it, can retain no electricity. This, reduced to
its simplest expression, is the mode of action of machines based on
induction.
Let us now suppose that instead of the cake of the electrophorus,
we have a disc of copper in connection with the earth, and that we
touch it with a zinc disc held by an insulating handle ; from Volta's
law, the zinc disc would be charged with positive electricity; this
charge, as in the preceding case, could be transferred to the cylinder,
and the experiment could be repeated for an indefinite number of
times.
This would also be the case if we took a piece of cloth, or of
caoutchouc, instead of a copper disc, and glass disc instead of one
of zinc. The insulating nature of the bodies employed must be
ESSENTIAL PARTS OF ELECTRICAL MACHINES. 183
taken into account; contact between the cloth and the different
parts of the surface of the glass, only takes place by friction. When
the glass charged with positive electricity is taken to the inside of
the cylinder, it would cause an equal layer of electricity, but of the
opposite kind, to be formed on the inside of the cylinder; mere
contact, however, with a point of the inner surface would not be
sufficient for neutralisation of the two charges. This result would
be obtained if the inner surface of the cylinder were charged with
points : equilibrium would only be established when the density at
the end of each of the points was null that is, when the electricity
which escaped had neutralised the charge of the glass disc. This is
the arrangement ordinarily used in all frictional machines.
195. ESSENTIAL PARTS OF ELECTRICAL MACHINES. It will be
seen that in all cases the electrical machine is reduced to three essen-
tial parts one which develops electricity, another which transmits it,
and a third which receives it : a producer, a carrier, and a receiver.
The potential energy imparted to the collector is furnished by the
mechanical work performed when the carrier is moved in a direction
opposed to that of the electrical forces, from the producer charged
with the contrary electricity which attracts it, to the receiver charged
with the same electricity which repels it.
In frictional machines, the receiver takes the same quantity
of electricity at each operation, and its charge increases in arith-
metical progression.
Induction machines may be so arranged that the charge increases
in geometrical progression : the two machines must be coupled up in
such a manner that the two inductors develop electricity of opposite
kinds, and that each of them is in metallic connection with the
receiver of the other system. At each operation, the charge of the
inducer increases at the same time as that of the receiver, with
which it is connected, and induces a greater charge in the re-
ceiver at the next operation. This arrangement is made use of in
Varley's machine, and in some of the very ingenious apparatus
invented by Sir W. Thomson, such as the replenisher and the self-
acting reciprocal condenser. Holtz's machine depends on the same
principle, but is somewhat more complicated.
196. LIMIT OF THE CHARGE. Although theory assigns no limit
to the charge of the receiver, a practical limit is soon attained, either
through the losses by air or by the supports, or by the fact of dis-
charges which take place in the form of sparks between the receiver
and the other parts of the machine or the adjacent conductors.
In the latter case, the limit only depends on the shape of the
184 SOURCES OF ELECTRICITY.
machine and on its position in reference to the adjacent conductors ;
in the former case, it depends on atmospheric conditions and on the
rapidity with which the operations succeed each other.
With addition machines, the limit is attained when, at any time,
the gain is equal to the losses, and this limit would always have a
finite value.
In the case of multiplication machines, certain conditions must
be fulfilled if the charge of the receiver is to retain a finite value.
Let C and C' be the capacities of the two receivers, V and V
the potentials in absolute values, c and c' the capacities of the
carriers, and n and ri the number of operations performed in unit
time.
The loss by air of an electrified conductor is sensibly propor-
tional to the charge, or to the potential, for very feeble charges.
If m and m are the coefficients of the proportionality relative to
the two conductors, the losses of the charge for unit time might be
represented by mV and m'V.
During an infinitely small time dt, the increase of charge of the
receiver C is equal to the excess of the electricity which it receives,
over that which it loses ; we shall have then
or
C = V'V'-wV.
at
The other receiver will give, in like manner,
at
Solving these simultaneous differential equations, we could calculate
the values reached by the potentials V and V at the end of a given
time, starting from given initial values ; but the results thus obtained
would only hold within the limits in which Coulomb's law may be
admitted. We know that for somewhat larger charges the loss takes
place according to a more rapid law.
These equations give the conditions necessary for the charge to
go on increasing ; for this it is sufficient if the differentials of the
potential are positive, which gives
n'SV'-mVX),
YIELD OF MACHINES. 185
from this is deduced
m V nc
7 <C~<! ; j
nc V m
or
nn'cc >mm'.
If this latter condition is realised at the outset (and it evidently
does not depend on the original electrification), the charge of
the machine will go on increasing. If the inequality is kept up
notwithstanding the increase of the coefficients m and m', the charge
will have no other limit than that which is determined by the pro-
duction of sparks. If the preceding inequality were in the contrary
direction, the charge would go on diminishing, and would rapidly
become null.
When the apparatus is symmetrical, the condition for the increase
of the charge is simply
nc>m.
The preceding calculation applies particularly to the arrangement
in which the collector of one machine is in metallic connection with
the inductor of the other ; but the same method of reasoning, slightly
modified in details, would also apply to all multipliers of electricity
which act by reciprocal induction.
197. YIELD OF MACHINES. Whatever, moreover, may be the
external cause limiting the charge, it will be seen that all these
machines act like true sources that is to say, as systems which by
the play of their own organs can maintain a conductor at a constant
potential, or maintain a certain difference of potential between two
conductors.
This result is obtained when the quantity of electricity brought
to the conductor is at every instant equal to that taken away from
it, either by loss from contact with air, or by discharges between the
collector and the earth. The yield of the machine is the quantity
of electricity put in motion in each unit of time. It is clear that for
addition machines, the yield, other things being equal, is proportional
to the capacity of the carrier and to the number of operations
performed in each unit of time. If, as with plate machines, the
carrier acts continuously, the yield is proportional to the velocity.
The phenomena in multiplication machines are not quite so
simple ; but experiment shows that the yield is sensibly proportional
to the velocity, although as a general rule it increases a little more
rapidly.
l86 PROPAGATION OF ELECTRICITY.
PART II. ELECTRICAL CURRENTS.
CHAPTER I.
PROPAGATION OF ELECTRICITY IN THE
PERMANENT STATE.
198. PERMANENT CONDITION. When two insulated conductors,
at different potentials V and V, are put in metallic connection,
equilibrium can no longer exist positive electricity flows from the
body at the higher towards the body at the lower potential; a flow
of electricity, an electrical current is produced. If the charges on the
two bodies are limited, equilibrium is established after a time, which
is generally very short, and which depends on the nature and the
dimensions of the intermediate conductor; the current is then
variable with the time. But if by any means the two conductors
are kept at a constant difference of potential, a permanent state
is established, and the intermediate conductor becomes the seat of a
constant current.
199. ANALOGY WITH THERMAL PHENOMENA. The analogy of
these phenomena with those of the propagation of heat between
surfaces, at constant temperatures in a conducting medium, is
obvious, and this analogy is expressed by identical laws in the two
cases.
We have seen, reminding the reader of the principles of Fourier's
theorem (70), that if, in a medium which is a conductor of heat, we
take two near isothermal surfaces at the temperatures t and t + df,
the flow of heat dQ, which in unit time traverses an element of
surface dS, is perpendicular to the element, proportional to the dif-
ference of temperature dt of the two surfaces, and inversely as their
distance dn t and is thus expressed
OHM'S THEORY. 187
k being the coefficient of conductivity for heat ; the sign - signifies
that the flow of heat is in the direction in which the temperatures
decrease. The expression for the flow is the same across an element
dS' of any given surface S', isothermal or not ; it is proportional to
the partial differential , of the temperature in reference to the per-
pendicular ri to the surface S', and we have
200. OHM'S THEORY. Ohm transferred Fourier's method of
reasoning to the study of the propagation of electricity. He assumes
that all points of a conductor in equilibrium are in the same elec-
trical condition, at the same tension. When there is no equilibrium,
interchanges of electricity take place ; the tension at every point is
generally a function of the time and of the co-ordinates but if any
extraneous cause maintains a constant difference between the ten-
sions of the different parts of the conductor, a stationary condition
is established in the system, after a shorter or longer time, in which
the tension at each point becomes independent of the time.
Ohm assumes, further, that between two molecules whose tensions
are U and U', an exchange of electricity is produced in unit time,
proportional to the difference of tensions and to a function of the
distance, such that the adjacent molecules have a preponderating
influence.
This hypothesis is identical with that of Fourier (70). Without
its being necessary to repeat the reasoning, it follows that the
exchanges of electricity take place at right angles to the surfaces
of equal tension, or to the surfaces of electrical level relative to this
new property. The flow of electricity */Q, which traverses an element
</S, of an equipotential surface in unit time, is proportional to the
differential of the tension in respect of the perpendicular to this
surface, and is expressed by
the coefficient c depending on the nature of the medium, and may
be called the coefficient of electrical conductivity. The differential
-- is the electromotive force at the point in question. It will be
an
seen that through an element ^S', of any given surface, there is a flow
1 88 PROPAGATION OF ELECTRICITY.
of electricity which is proportional to the electromotive force along
the normal to this surface, or to the differential coefficient ^-7.
on
On Ohm's theory, as well as on that of Fourier, it is assumed
that the direction of the flow is constantly parallel to the direction
of the force which acts upon it, and is therefore independent of its
previous condition. This hypothesis is incompatible with the notion
of inertia, and therefore with the materiality of that which constitutes
the flow.
Ohm regarded tension as a particular condition in virtue of which
electricity tends to escape, and, when the tensions in any medium are
variable, electricity flows from points at a high towards points at a
lower tension. We see at once the analogies of this function with the
potential, for the tension is also constant in a conductor in equi-
librium. In some of his memoirs Ohm appeared to establish too
close a relation between electrical density and tension ; but he also
points out that the tension at a point, even in the variable state,
could be determined by connecting it with an insulated electroscope,
and the quantity, which is thus measured, is nothing else than the
potential.
201. KIRCHHOFF'S HYPOTHESIS. To express Ohm's theory with
our present views respecting electricity, we must assume with Kirchhoff
that tension and potential are two identical functions.
It follows from this hypothesis that, in a system of conductors
connected together, but not in electrical equilibrium, the flow of
electricity at each point is proportional to the differential of the
potential in reference to the normal to the equipotential surface, or,
in other words, is proportional to the electrical force exerted at this
point that is to say, to the resultant of the actions of all the masses
of the system in their condition for the time being.
Or, more simply, the flow of electricity is parallel and proportional
to the flow of force.
202, SUPERPOSITION OF PERMANENT STATES. In another form
it may be said that the hypothesis amounts to the assumption that
the superposition of two states of electrical equilibrium is itself a
new state of equilibrium, in which the flow across an element of
surface is equal to the sum of the flows relative to the two original
states.
For consider two states in which the flows across an element of
surface dS are AdS and A'dS ; the perpendicular components of the
c)V c)V
force on this element are - ^S and ~-^-dS. By superposing
on on
SUPERPOSITION OF PERMANENT STATES. 189
the two states, .the potentials at each point add together, and the
perpendicular force upon the element dS becomes
- +
By hypothesis, the flow across the element dS has become
AdS +-A'dS.
If we suppose the initial conditions to be identical, the flow becomes
2A^S, and the force is thus doubled ; the flow of electricity is there-
fore proportional to the flow of force.
If the state is permanent that is, if the potential is unchanged
at each point the flow across any surface-element is constant. If
the state is variable, the potentials vary with the time, and the flow
d 2 Q across an element of surface during the time dt is
203. IN THE PERMANENT STATE THE DENSITY IN THE IN-
TERIOR OF A CONDUCTOR is NULL. The flow of electricity cannot
accumulate in the interior of a closed surface, without modifying the
potential, just as an increase in the flow of heat would produce there
a rise of temperature.
If the permanent state has been attained for a system of con-
ductors, the total flow which each element receives is zero. If the
algebraical sum of the flows which enter a volume-element dxdydz is
equal to zero, we find again Laplace's equation :
p being the cubical density of electricity at the point in question,
we have always
AV = - 47173,
from which
/> = ().
The law of the proportionality of the flow of electricity to the
flow of force leads directly to the same conclusion. For, if the state
PROPAGATION OF ELECTRICITY.
is permanent, the flow of electricity for a volume- element is zero : the
flow of force which is proportional to it is zero also ; but the latter is
equal to 477^2, m being the mass in the volume in question ; hence
m = 0. Thus, when a system of conductors has attained a permanent
state, the electrical density is zero at all points of the conductor ; the
electrical masses which produce the potential V, and whose action
determines the current, are therefore entirely on the surface of the
conductors. These masses are not in equilibrium of themselves, and
they produce at each point the electromotive force of the current.
It follows from this that the flow, whatever it may be, if it has a
real existence, is not a flow of free electricity ; on the hypothesis
of two fluids, we must assume that at every instant there is the same
quantity of the two electricities in each volume-element in the
interior of the conductor, and that these move in two equal currents
in opposite directions. On the hypothesis of a single fluid, each
element must be looked upon as containing at each instant the
normal quantity of electricity, while we still assume that this may
be either wholly or partially displaced.
204, LINEAR CONDUCTORS. OHM'S LAW. Imagine a cylin-
drical wire, very long as compared with its diameter, placed in a
perfectly insulating medium, and let us suppose that the permanent
state has been attained.
If there is no loss of electricity, the flow of electricity is parallel
at each point to the generating lines of the cylinder; the equi-
potential surfaces are therefore planes perpendicular to the axis of
the wire. The flow of electricity across any given section in unit
time is the same throughout the whole length ; let us call this flow
the intensity or strength of the current, or more simply the current^
and denote it by /. Let V be the potential at the point P at a
distance x from a fixed plane perpendicular to the wire, and let S be
the section of the wire.
The potential is simply a function of x, and the expression for
the current is
dV
t=-cS- .
dx
As this flow is independent of x, we have
= a, and V = ax + b,
dx
where a and b are constants to be determined.
RESISTANCE THE INVERSE OF A VELOCITY. 19 1
Let Vj and V 2 be the values of the potential at two points A and
B (Fig 49) at a distance / from each other, and let the point A be
the origin of the co-ordinates, we have
(i)
and, consequently,
The quotient = r is called the resistance of the wire between
<:S
the two points A and B, and the inverse of this resistance is the
conductivity of this same wire.
Fig. 49.
Equations (i) and (2) show that :
i st. The potential decreases in arithmetical progression along the
wire, in the direction of the current;
2nd. The current between the two points A and B is equal to the
quotient of the difference of potential of these two points by the
resistance of the intermediate wire.
These two statements form Ohm's law.
It may be noticed that the distribution of potential, and the flow
of electricity in the case we are considering, are identical with the
distribution of temperatures, and with the flow of heat in a homo-
geneous wall bounded by two parallel planes kept respectively at
constant temperatures.
205. THE RESISTANCE OF A CONDUCTOR is THE INVERSE OF A
VELOCITY. The quantity r, which we have called the resistance of
the conductor, has the value ; it is proportional to the length of
o
the conductor, is inversely as its section, and of the coefficient of
conductivity of the medium.
1 92 PROPAGATION OF ELECTRICITY.
The ratio - represents the resistance of a cube equal to unity
parallel to one edge ; it may be called the specific resistance of the
conductor.
The resistance of a conductor is a magnitude of the same kind
as the inverse of a velocity in mechanics.
For we have
V -V
M MJ
The difference of potential V l - V 2 is equal to the quotient of an
electrical mass M by a length a-, the strength of a current, or the
flow during unit time, is equal to the ratio of the quantity of elec-
tricity M', which flows in the time /, to the corresponding time. We
have then
The quotient is an abstract number, and the ratio - is a
M /
velocity. The resistance r is therefore the inverse of a velocity.
206. We may, indeed, imagine an experiment in which this
velocity would have a physical meaning.
Consider a sphere, of radius R, charged with a mass of electricity
M, and let us suppose this sphere connected with the earth by a
conductor of resistance r.
M
The potential of this sphere is equal to ; it diminishes as soon
is.
as it is connected with the earth ; but if the sphere contracts at the
same time as the charge diminishes, it may happen that the potential
remains constant
This condition being realised, then if ^M is the loss of charge of
the sphere, and aTR the diminution of the radius in the time dt,
M_M
~~lf ~ R-
but from Ohm's law,
dlli^-dt.
ANY GIVEN LINEAR CONDUCTORS. 193
In order that the potential may be constant, dR. must also be
proportional to the time ; let us put
We have then
udt ru '
and therefore
i
r=-
u
Thus the resistance r of a given conductor is the inverse of the
velocity u with which the radius of a sphere must decrease for its
potential to remain constant, notwithstanding the loss of electricity,
when it is connected with the earth by the conductor in question.
207. ANY GIVEN LINEAR CONDUCTORS. We have supposed the
conductors to be rectilinear, but the same reasoning evidently applies
to linear conductors bent in any manner whatever, the flow of elec-
tricity being perpendicular at each point to the cross section of the
conductor.
Y*
T7
Fig. 50-
If the circuit consists of two or more cylindrical portions of
different kinds and sections joined end to end, these various parts
may be considered separately.
If Vj and V 2 are the potentials at the points A and B (Fig. 50),
the first in a conductor of section S, and whose coefficient of conduc-
tivity is c, and the second belonging to a conductor in which these
quantities are S' and c'. Let V be the point of contact O of the
two cylinders, at distances / and /' from A and B respectively, and
let us for the present disregard the electromotive force of contact of
the two conductors, to which we shall subsequently return. On each
side of the point O we have
r+r 1
The current is therefore inversely as the sum of the resistances
o
194
PROPAGATION OF ELECTRICITY.
of the two conductors between the points A and B. This is
obviously a general relation.
Thus the resistance of a series of successive cylindrical conductors
is the sum of the resistances of all the conductors.
In conclusion, let us take the case of a conductor of any given
shape terminated at its ends by equipotential surfaces kept at
potentials V 1 and V 2 ; the current is proportional to the difference
v i ~ V 2 f tne potentials, and the number by which this difference
must be divided to give the strength of the current represents the
resistance of the conductor. The number thus obtained is the
resistance of the cylindrical conductor, which for the same difference
of potential would give the same current.
208. KIRCHHOFF'S LAWS. Let us suppose linear conductors, of
various materials and different sections, to be joined to each other
in a complicated manner, the division of the current among these
various conductors must satisfy the two following conditions, which
follow directly from Ohm's law.
Fig. 51.
i st. If several conductors terminate at the same point, the sum of
the currents, counted from this point, is zero.
For, since there can be no accumulation of electricity at the point
in question, the quantity of electricity brought by one set of con-
ductors must be equal to that which passes away by the others in the
same time; so that if we give the positive sign to the currents
proceeding towards the point, and the negative sign to those which
pass away, we must have
(3) 3*/-o.
2nd. If several conductors form a closed polygon, the sum of the
products of the resistance of each conductor, by the current which
traverses it, is zero.
Imagine a series of conductors of resistances r v r^ r^, r n ,
which form the successive sides of a closed polygon (Fig. 51); let
RESISTANCE OF A MULTIPLE CONDUCTOR. 1 95
V 1} V 2 V n , be the potentials at the summits A v A 2 A n , and
let t\, / 2 i n be the currents reckoned positively when the circuit
is traversed in a certain direction. These strengths are not equal,
for at the various summits there may be other conductors which
bring or carry away currents.
We shall have successively :
For the first conductor i^r^ = V x - V 2 ,
For the second conductor 2 2 r 2 = V 2 - V 8 ,
For the n th conductor i n r n = V n -V r
Adding these equations together, all the potentials disappear, and
we have finally
from which
(4) ^i>=0.
The two expressions (3) and (4) are known as Kirchhofs laws.
Fig. 52-
209. RESISTANCE OF A MULTIPLE CONDUCTOR. As an appli-
cation of these theorems, let us consider the case in which the
circuit divides into multiple arcs, between two points A and B
(Fig. 52). Let I be the current in the undivided part in front of
A and beyond B, r v r 2 . . . r n the resistances of the conductors,
*i, I 2 . . i n the respective currents, and lastly let R be the resistance
of the single circuit which would be equivalent to the multiple circuit
between the same two points. We shall have
O 2
196 PROPAGATION OF ELECTRICITY.
The second equation may be written
i-i i t' I
71
R
and we deduce from it
Thus the inverse of the resistance of a number of conductors
terminating at the same points, is the sum of the inverses of the
resistances of the several separate conductors ; in other words, the
conductivity of a multiple conductor is the sum of the conduc-
tivities of the several conductors of which it is made up, which
indeed is evident.
210. HETEROGENEOUS LINEAR CONDUCTORS. The existence of
electromotive forces of contact between metals slightly modifies
Ohm's law. Consider two points P l and P 2 (Fig. 53), separated by
Fig- 53-
two different conductors A and B whose resistances are a and b.
Let V 1 and V 2 be the potentials at the points P l and P 2 , V a and V &
the potentials at the point P on each side of the surface of sepa-
ration of the metals. The current from P : to P 2 is
a + b
Let H a& be the sudden rise which the potential experiences
between the metals A and B going with the current that is to say,
the electromotive force of contact V b - V a = B|A of the conductors
B and A, we get
HETEROGENEOUS LINEAR CONDUCTORS. 197
Let us now suppose that a closed circuit is composed of separate
conductors A, B, C L (Fig. 54) comprising bodies of Volta's
second class that is to say, that the chain of conductors does not
obey the law of tensions; the circuit will be traversed by a per-
manent current.
Fig. 54-
Let r^ r^Tf. *j, be the resistances of the different conductors,
V and V'a, V ft and V' & the successive potentials at the ends of
each going in the direction of the current, and in like manner let
H a6 , H &c Hj a be the successive electromotive forces of contact ;
we have
H..-V.-V,,
The current being the same throughout the whole extent of the
circuit, we have also
from which, reducing, we get
The numerator of this fraction represents the algebraical sum
of the electromotive forces of contact in the chain of con-
ductors; it is the electromotive force E of the circuit The
198 PROPAGATION OF ELECTRICITY.
denominator is the sum of the resistances, or the total resistance of
the circuit. We have thus
211, CASE IN WHICH THE CIRCUIT CONTAINS ELECTROMOTIVE
FORCES. It is easy to see what Kirchhoff's laws (208) become, when
there are electromotive forces at work.
The first theorem is not modified. The sum of the quantities of
electricity which start from, or terminate at, a point is always zero in
the permanent state; for this point can neither be an unlimited
centre for the production of electricity, nor a centre of absorption.
The second theorem must be modified. Suppose that in the
preceding circuit, at the points where the various conductors terminate,
the metals change, or that these places are also points where the
summits of other conductors branch off. The current is not the
same throughout the whole circuit ; let / a , i b ..... t\ be the different
values of the current in the different conductors between any two
successive points of contact or of division. From Ohm's law we
shall have
and consequently
v'.+n+- +ni-(v.- v a )+(v 6 - vy+ . . . +(v,- vy
= ( V 5 - v ') + (V. - v '>) + '-+ (V. - V',)
or
(6) E-^Vf.
Thus, in a closed circuit, the sum of the products of the resistance
of each conductor by the strength of the corresponding current is equal
to the algebraical sum of the electromotive forces of the circuit.
This sum is zero if the circuit is made up of conductors of the
same kind, or of metals at the same temperature, for the latter obey
the law of tensions.
The two relations (5) and (6) give all the equations necessary
CONDUCTORS OF ANY GIVEN FORM. 199
for determining the currents in the various branches of the circuit.
Any modification in the resistances, or in the electromotive forces
which produces no change in the equations, will obviously be without
influence on the currents. For instance :
i st. The resistance of a branch in which the current is null may
be modified at will.
2nd. In all conductors which terminate in the same point, we
may introduce equal electromotive forces tending to produce currents
which all approach or recede from the point in question, these elec-
tromotive forces neutralising each other in pairs in all the closed
contours which pass through that point
212, CONDUCTORS OF ANY GIVEN FORM. ELECTRODES. The
analogy of electrical conductivity with thermal conductivity, and
of this latter with the phenomena of statical electricity, enables us
to establish directly some theorems relative to the propagation of
electricity.
Let us consider, in the first place, a single isotropic unlimited
medium. Let us suppose that, for the three different orders of phe-
nomena, the temperature on the one hand and the potential on the
other are kept constant on different closed surfaces Sj, S 2 , S n ,
and that on each of the surfaces the temperatures and the potentials
/! and V p / 2 and V 2 are represented by the same numbers, or
by proportional numbers. In the thermal problem, these surfaces
will represent sources of heat ; in the problem of statical electricity,
the conductors; in the problem of the propagation of electricity,
they are called the electrodes.
The temperature and potential of any point of the medium are
functions of the co-ordinates defined by the condition that these
functions acquire determinate values on the bounding surfaces, and
in the interval of these surfaces satisfy the condition
A/=0, or AV = 0.
The temperature and the potential at every point will therefore have
values which are either equal or are in a constant ratio. The iso-
thermal and the equipotential surfaces will be identical throughout
the whole extent of the medium, and therefore the tubes of flow are
identical.
Across an element dS of any given surface, the flow of heat (70)
is - kd , the flow of electrostatic force is - d$ , and the elec-
9* av ^ n
trical current is - cd$ - , k and c being the coefficients of thermal
200 PROPAGATION OF ELECTRICITY.
and electrical conductivity. These three flows are therefore pro-
portional to each other.
According to this, whenever a problem of the uniform propa-
gation of heat or of statical electricity has been solved, the
corresponding problem of the propagation of electricity in the
permanent state, will be found to be solved in the same way.
As a particular case, we have seen that if we assign the potentials
V lf V 2 . . , V n to fixed surfaces S 1} S 2 . . , S n that is to say, the poten-
tials of different conductors in air the potential at every point of the
medium is defined, and the problem of equilibrium has only one
solution. In like manner, if the dielectric medium is replaced by a
conductor, and the potentials are kept constant on the same surfaces
Sj, S 2 . . . S n , the flow of electricity is determined for each point, and
there is only a single state of equilibrium.
213. Suppose now that two dielectrics, whose inductive capacities
are /^ and /x 2 , are separated by a surface S. The flow of force is not
maintained on each side of the surface, but the flow of induction is
maintained, and counting the normal n to the surface in the same
direction for the two media, we have (121)
C.L.COHY. 3V w .
It is the condition of continuity on the surface. If we replace the
dielectrics by conductors whose coefficients of conductivity are ^ and
c^ the flow of electricity is then the same on each side of the surface
S, which gives
W <>V 2
It will be seen from this that if, in a problem of electrostatics
containing dielectrics whose inductive capacities are yu 15 //. 2 , . . //, 3 , we
replace 4he dielectrics by conductors whose coefficients of conduc-
tivity are respectively proportional to the corresponding inductive
capacities, such, that is to say, that we have
the flow of electricity at each point will be proportional to the flow
of induction of the correlated electrostatic system.
HETEROGENEOUS CONDUCTORS. 2OI
Thus, all electrostatical problems which have been solved for a
system of dielectrics, furnish also the solution of the corresponding
problems in the propagation of electricity. Such, for instance, are
the following cases :
Concentric spheres (77).
Concentric cylinders (80), or eccentric cylinders one of which is
inside the other (136).
Parallel planes (81).
Closed condensers of constant thickness (79).
Successive concentric cylinders formed of different media (164).
214. HETEROGENEOUS CONDUCTORS. We have also seen (167)
that, by comparing a dielectric to a medium whose specific inductive
capacity is /* 2 , and in which is disseminated little spheres of the
specific inductive capacity /*j, the medium thus constituted behaves
like a homogeneous dielectric, the specific inductive capacity of which
would be represented by the expression
in which h represents the ratio of the sum of the volumes of the
small spheres to the total volume of the space in which they are
disseminated.
In analogous conditions, the mean specific conductivity of a
medium of conductivity r 2 , containing small spheres of conductivity
<r 1? is expressed by the formula
If the ratio of the conductivities of the spheres and the sur-
*1
rounding media is very great, the formula reduces to
In like manner, the problem of 150 will give the conductivity
of a system formed of three different media separated by two parallel
planes.
202 PROPAGATION OF ELECTRICITY.
215. ANISOTROPIC CONDUCTORS. We have hitherto only con-
sidered the case of isotropic bodies that is to say, bodies which
have the same properties in all directions. If the medium is
anisotropic, but homogeneous like crystallized bodies, the physical
phenomena depend on the direction in which they are regarded.
The expansion, for instance, may be very unequal. There are
then three principal directions, rectangular to each other, and such
that the expansion of an infinitely thin cylinder considered in the
medium parallel to one of the principal directions takes place along
the axis of the cylinder. Each of these directions is denned by a
particular coefficient, which gives for the medium three principal
coefficients of expansion, /, /' and /". If we suppose in the medium
an infinitely thin cylinder in any given direction, making angles with
the principal directions, the cosines of which are a, a! and a", this
cylinder turns at the same time that it dilates, but remains rectilinear
if the medium is homogeneous. The expansion parallel to the axis
of the cylinder is equal to the sum of the projections on this axis
of the three principal expansions, and the value of the coefficient L
relative to this direction is
The same considerations apply to the propagation of heat, to the
propagation of electricity, and to electrostatic induction.
In an anisotropic medium, the flow of heat at a point is no longer
perpendicular to the corresponding isothermal surface ; but, just as in
the case of the expansion, and generally for all properties which are
linear functions of the causes on which they depend, there are again
three rectangular directions along which the flow of heat is perpen-
dicular to the isothermal surfaces, and to which correspond the three
principal coefficients of conductivity , k' and k".
Across an element of surface dS, the perpendicular to which
makes angles with the axes the cosines of which are a, a' and a", the
flow of heat is equal to the sum of the flows which correspond to
the projections a</S, aWS, a'WS of the element perpendicular to the
three principal axes ; taking, then, these three directions for axes
of the co-ordinates,
In like manner again, if <:, c', c" are the coefficients of electrical
ANISOTROPJC CONDUCTORS. 203
conductivity of the medium along the principal axes, the flow of
electricity dM across an element of surface dS will have an analogous
expression as a function of potentials. If l n is the strength of the
current for unit surface along the perpendicular to the current, we
shall have
or
av
r ay "sv av~i
? /S= -</S i + ^o' + ^a f/ .
~dx ty 1)2 J
^y ~dz I
This latter expression is the sum of the projections on the per-
pendicular of the currents I, I', I" in the three principal directions,
or the projection of the resultant current I .
If 9 is the angle made by the perpendicular to the element dS
with the direction of the current, which makes with the axes angles
whose cosines are A, A', A", we have then
I n = (al + aT + a"I") = (aA + a' A' + a"A")I = I Q COS 0,
and
A A' A"
216. We may now without difficulty extend the same kind ot
reasoning to phenomena of induction in anisotropic dielectrics.
Here again are three principal directions of induction, such that the
flow of force is perpendicular to the equipotential surfaces, and
which we may characterise by the specific inductive capacities /*, //
and /A". If (f> is the flow for unit surface, the flow of induction
across an element dS is
f t>V dV <>V~1
= -^S ;ua + /a' + /t"a"
~bx ty ~bz J
If /?, ft' and ft" are the cosines of the angles formed by the
electrical force F at the point in question, which is not perpendicular
to the equipotential surface, we may write
</> = (pap + /* 'a' ft' + p "a" ft") F.
2O4 PROPAGATION OF ELECTRICITY.
In this case the electrical displacement is no longer parallel to
the electrical force.
217. CONDUCTORS IN Two DIMENSIONS. The preceding con-
siderations apply to unlimited media. It is clear that nothing is
altered if we limit the conducting medium by a surface formed
entirely by the lines of flow of the unlimited system. For a limited
conductor placed in an insulating medium, the external surface,
whatever it may be, is always parallel to the lines of flow, and
therefore, if the medium is isotropic, it is perpendicular to the equi-
potential surfaces.
This is the case of the propagation of electricity in a thin plate,
which may be regarded as a conductor of two dimensions. We may
determine by experiment the locus of points which have the same
potential, by the condition that no current flows through a conducting
wire one end of which is in connection with a fixed point in the
plane. The results obtained by experiment are in complete accor-
dance with those deduced from Fourier's formula, and furnish
a fresh confirmation of the analogy between the two orders of
phenomena.
In both cases we may suppose the propagation to take place
either with or without loss in the surrounding medium.
If there is no loss in the external medium, Poisson's equation
for any point outside the electrodes reduces to
and, for any point in the outside of the plate, we have
It is easily seen that this problem merges into that of the problem
of equilibrium in the case of a cylindrical distribution (132 et seq.)
We have seen there that in a plane traversed perpendicularly at points
Aj, A 2 , A 3 ..... by parallel lines of the densities A p A 2 , A 3 . . . , the
potential at a point P, at distances r v r^ r z ... from these lines, has
the value
The flows of force which, in a layer of the thickness e comprised
RESISTANCE OF ANY GIVEN CONDUCTOR. 205
between two parallel planes, proceed from the different lines are
If, in the problem of propagation, we regard the same portions of
the lines as sources of electricity, as electrodes, the flows of elec-
tricity, or the strengths of the currents, are
from which we have
A i =
and the expression of the potential at P becomes
V = const- V =const -- L/.r, + I/.r 2 + ..... .
^ 27JV6 27TT6 \_ l J
A particularly interesting case is that of two electrodes A l and A 2
furnishing equal flows of opposite signs. In this case
I 1 + I 2 = 0,
and
V = const -- /. .
The lines of flow are segments of circles passing through the two
points A x and A 2 (Fig. 30); the equipotential lines are circum-
ferences having their centres on the line A T A 2 and such that these
two points are conjugate in reference to each of them. From the
remark made above, it is clear that the problem will remain the same
2O6 PROPAGATION OF ELECTRICITY.
if, instead of an unlimited plate, we consider a circular plate
having two electrodes on its circumference, or again, any plate
comprised between two circular segments passing through the
points AJ and A 2 .
218. RESISTANCE OF A CONDUCTOR OF ANY GIVEN FORM.
Whatever be the conductor, it may be always supposed to be divided
by two series of surfaces parallel to the lines of flow into infinitely
slender tubes, each of which is a tube of flow. Each of these tubes
may itself be compared to a conducting wire of varying section, the
resistance of which is at each point inversely as the section. The
total resistance can be deduced from the resistance of this complex,
by the ordinary laws of multiple conductors ; the reciprocal of the
total resistance, or the conductivity, will be the sum of the reciprocals
of the resistances of all these tubes.
The calculation will in general be very complicated ; but if the
value of the potential on the two electrodes is known, as well as the
corresponding flow of electricity, it is easy to determine the total
resistance of the conducting medium by Ohm's formula (207).
Let us take as an example the case of two electrodes A l and A 2
in an unlimited medium. We may suppose these electrodes to be
small spheres of radius p. Let V x and V 2 be their potentials, and I
the absolute value of the flow of electricity corresponding to each of
them. If the radius p can be neglected in comparison with the
distance A 1 A 2 , we may assume that the potential close to each of
the electrodes is inversely as the distance r, and is represented by -,
which, on the spheres themselves, will give
The current is then
'/'
//v
-
dn
As, on the surface of the sphere, we have
DISTRIBUTION OF ELECTRICITY ON LINEAR CONDUCTORS. 207
we get
or
According to this, the total resistance R of the medium is ex-
pressed by
The same reasoning would apply to the case of a medium
unlimited on one side, and bounded on the other by a plane on
which two hemispherical electrodes A x and A 2 are placed. The
resistance would then be the double of the preceding, and we should
have
TTCp
It is remarkable that the resistance is independent of the dis-
tance of the two electrodes, and only depends on their dimensions
and on the conductivity of the medium. This case may be regarded
as corresponding to that of the earth when two points of the soil are
connected with electrodes kept at potentials of equal values and
opposite signs.
219. DISTRIBUTION OF ELECTRICITY ON LINEAR CONDUCTORS.
When the state is permanent, as the density is zero in the interior of
the conductor (203), the potential is simply due to the electricity
which exists on the surface; this electrical layer is distributed ac-
cording to a law which can be determined in a few simple cases.
Let us consider a rectilinear cylindrical wire, the diameter ot
which is very small as compared with its length, and placed in its
whole extent in conditions which are the same in reference to neigh-
bouring conductors. If this wire were electrified and in equilibrium,
the distribution of the superficial layer at some distance from its
ends would be uniform that is, that any portion of a surface
comprised between two planes perpendicular to the axis, and at
unit distance, would have the same quantity of electricity: let
A be this quantity, which may be called the linear density of
the wire.
208 PROPAGATION OF ELECTRICITY.
The potential V of the wire is, moreover, proportional to the
total charge, and therefore to the charge of each unit length.
We have then
A being a constant which depends on the section of the wire, and on
its position in reference to external conductors.
If the charge of the wire varies from one point to the other of
the length, the linear density at a point, is the limit of the ratio of
the charge to the corresponding lengths.
When equilibrium does not hold, it is not generally speaking
exact that the potential at each point is proportional to the
density; but this proportionality is evidently in particular true for
cables, in which the conducting wire is surrounded by a dielectric
layer of constant thickness, which in turn is surrounded by a con-
ductor in connection with the earth. The various parts of the wire
are then without appreciable action on each other, and the potential
at each point is that due solely to the nearest electrical masses. If 7
be the capacity of unit length of the wire that is to say, the charge
which would correspond to unit potential the charge of a length dx
at potential V would be equal to yVdx.
220. PROPAGATION IN A WIRE WHEN THERE is A Loss ON
THE SURFACE. Let us still consider a cylindrical wire traversed by
a current, and let us suppose that the permanent state has been
attained, but that there is a loss of electricity on the surface. The
flow is no longer parallel to the axis throughout the entire extent of a
perpendicular section ; it tends to become perpendicular to the wire
close to the external surface. The equipotential surfaces S, S'
(Fig. 55) are still planes throughout the greater part of their extent,
S
.
T <Lc y
Fig. 55-
but they bend just near the edges and then join with the outer
surface of the wire.
PROPAGATION OF ELECTRICITY IN A WIRE. 209
The loss, which takes place at the surface is no more than a flow
of electricity in the external medium ; it is therefore proportional to
the flow of electrical force (201). For a length dx of the wire, the
charge is yVdx and the flow of electrostatic force is ^iryVdx. If c'
is the coefficient of conductivity of the medium, the strength of the
lateral current would therefore be c'^TryVdx. As this strength is
also equal to the quotient of the potential V by the resistance of the
medium, from the lateral surface in question, to the points where the
potential is zero, the resistance relative to the- length dx is ;
T & 4*fyi*
and the resistance p' for unit length is equal to .
fgpy
The permanent state being established, the total flow of electricity
through the surface S of the volume-element dx should be equal to
the sum of the flows by the opposite surface S' and by the lateral
surface, which gives
dx \ dx dx*
that is to say
dx* cSp' ' '
The product <:S represents also the reciprocal of the resistance p
of the wire for unit length ; putting ft 2 = = , we get
cSp p
This is Fourier's equation for the propagation of heat in a
cylindrical bar. The integral of this equation may be put under
the form
To determine the constants A and B, we must know the poten-
tials V and V x at two points P and P l at a distance of / from each
other. The value of the potential at a point P, situate at a distance
p
210 PROPAGATION OF ELECTRICITY.
x from P and of l-x, or 7, from P 1? is
w v-v.i?^;*
If the point P 1 is on the ground, we have V x = 0, which gives
e py _ e -y
(9) V = V ^T^'
If the wire is unlimited, the constant A is null, and we have
v=v *-0*
221. The current in the wire is given by the expression
VI 1
~~ P '~d^~
if the point P l is connected with the earth, it becomes
V
=
and, if the wire is unlimited,
I being the current at the origin of the wire.
222. RESISTANCE OF A CONDUCTOR WHERE THERE is A Loss
BY THE SIDES. From equation (n), the total resistance offered to
the electrical current from the point P to the ground, allowing for
the branches, is
RESISTANCE OF A CONDUCTOR. 211
Let us consider, as a more general case, a wire the different points
in which P lf Pg, P 3 , are connected with the earth by conductors
of resistance p v p# /> 3 , Let R x be the total resistance from
the point P x to the ground, R 2 the resistance starting from the point
P 2 , , ; finally r v r 2 , r 3 , . . . . the resistances of the wire between
the successive points of contact P^, P 2 ^3' > etc - From the point
P! the total conductivity is equal to the sum of the conductivities
which the various paths offer, which gives the equation
We should have a series of analogous equations, and ultimately
I
P2
I
Pn r n
This is the case of overhead telegraph wires when we take
into account the leakage by the insulators. If the leakage is
continuous, and R is the resistance starting from the point P,
R + </R the resistance from the adjacent point P' at the distance
dx, the coefficients p and p having the same meaning as above,
equation (12) becomes
I I
T? '
dx
or
</R dx
212 PROPAGATION OF ELECTRICITY.
The integral of this equation is
The constant C is determined by the limiting conditions. If the
resistance in the external medium starting from the end of the wire
P! is equal to R 15 then, making # = /, we get
APPLICATION OF FOURIER'S FORMULAS. 213
CHAPTER II.
VARIABLE STATE.
223. APPLICATION OF FOURIER'S FORMULAS. The problem of
the propagation of electricity in a conductor when the permanent
state has not been obtained as, for instance, when a battery is
discharged through a wire offers great difficulties.
The flow of electricity which penetrates into a volume-element is
not zero, since the charge varies with the time; but it cannot be
asserted, a priori, whether the internal density varies, or, indeed,
whether it still remains zero, and the increase of charge takes place
only on the surface.
In the absence of adequate experimental data, the simplest idea
is to pursue the analogy between the propagation of heat and that of
electricity, and to try to apply Fourier's formula to the variable state
of conductors. This is to assume implicitly that the flow of electricity
at each point, is proportional to the electrical force at this point, or to
the differential of the potential of all the acting masses. This
proposition seems natural enough, if it is the case that the electrical
forces really act at a distance and in an instantaneous manner, as is
readily admitted in the case of universal attraction ; but if, on the
contrary, electrical actions are transmitted through the intervention
of a medium, in virtue of what we have called the electrical elasticity
of this medium, it is necessary to assume that the state of electrical
tension (99, 126) is set up from layer to layer. A physical effort of
this kind must necessarily require a finite time, however small this
may be. This question of time, which does not affect problems of
equilibrium in the permanent state, may have a preponderating
influence in the phenomena of the variable state.
In other words, we may assume that the electrical force is
propagated with an extremely great, but not infinite velocity, or else
that the potential of an electrical mass is itself propagated with a
finite velocity.
214 VARIABLE STATE.
In this case it is still possible that the flow of electricity at each
point is proportional to the actual electrical force, but this force will
not depend solely on the position of the acting masses it will
depend also on the velocity of these masses, and the effects may
be very different, according as the velocity of displacement of the
acting masses is, or is not, of the same order of magnitude as the
velocity of propagation of the potential.
We shall see lastly, in connection with the phenomena of electro-
dynamic induction, that the displacement of electrical currents and
their changes of strength, produces new electromotive forces, which
can be calculated in a certain number of cases, and which may greatly
modify the results relative to the variable state.
The two effects which we have mentioned are perhaps produced
by the same mechanism; we shall not for the present take them
into account.
With this reservation we can again apply Fourier's formulas.
In any case, the results to which they lead must be so much
nearer the truth the slower are the changes in the variable state ;
in fact, these results represent very approximately the propagation
of electricity in submarine cables, and apply rigorously to Gaugain's
experiments on the propagation of electricity in bodies of great
resistance, such as cotton threads, or columns of oil.
224. VARIABLE STATE IN A CYLINDRICAL CONDUCTOR. Let
us consider, then, in a cylindrical conductor the volume-element dx
comprised between two infinitely near sections S and S' (Fig. 55).
The potential at a point P is no longer a simple function of x that
is to say, of the position of this point but it is also a function of the
time /. During the time dt, the amount of electricity which this
volume-element gains, is equal to the excess of the flow through the
section S, over the flow which issues by the section S', together with
the loss by the external surface that is to say :
-V dxdt.
I- -vl.
I p*bx 2 p
/ 1 <) 2 V I \
The increase of charge ( -^~2~~^ )& for unit length, will
\pDx* p )
produce a variation of potential dV or dt\ if then we assume
that the ratio of the charge to the potential, remains equal to the
VARIABLE STATE IN A CYLINDRICAL CONDUCTOR. 215
capacity, as in the phenomena of statical electricity, or of the
permanent state that is to say, that the new charge is entirely on
the surface,
If a 2 = 7/0, the equation becomes
If the loss by the surface may be neglected, the coefficient fP is
zero, and we have
We may observe, moreover, that if the loss is not zero, we
may put
equation (i) takes the form of equation (2) and becomes
In both cases, the potential V is a function of x and of *,
which tends to become a simple function of # when the time
increases.
The general integral of this equation has been given by Fourier
under several forms.
If we consider a wire of length /, originally in the neutral state,
one of whose ends is in connection with the earth, while the other
end is suddenly raised to potential V , and then kept at this electrical
level, the general value of the potential at the distance x from the
2l6 VARIABLE STATE.
origin of the wire, and at the time / from the establishment of
contact, is given by the expression
V _ew -e-> __L_ = n '*', . TZTT
o =i
which, if the loss may be neglected, becomes
V _t-X ^ M=C I 27r % . W7T
225. DURATION OF THE RELATIVE PROPAGATION. The second
equation shows that if there is no loss by the sides, the ratio of the
potential at a distance x, to the potential at the origin, is the same for
two different wires, at two points whose distances from the origin are
proportional to the total lengths of the wires, when the ratio - has
or/- 1
the same value.
The time / necessary for the potential at any given point (in the
middle of the wire, for instance) to attain a definite fraction of the
initial potential or of the final potential, is therefore proportional to
a 2 / 2 or y/>/ 2 that is to say, to the square of the length of the wire, to
the capacity, and to the resistance of unit length. This condition
gives what may be called the time of relative propagation of
electricity.
There is not, therefore, in the preceding conditions, a deter-
minate velocity for the propagation of electricity as there is for sound,
or for light. The apparent velocity which it has sometimes been
attempted to estimate, by supposing the propagation uniform, and
determining the time necessary for the electrification produced at
one end of a wire to have a sensible effect at a certain distance,
depends on constants characteristic of the wire, and on the
sensitiveness of the means by which these electrical effects are
made evident.
226. UNLIMITED WIRE. Fourier's general integral lends itself
with difficulty to numerical applications; but we may choose simpler
conditions, which really correspond to several of the observed
phenomena, and enable us to find again the principal results
obtained by Sir W. Thomson,
DURATION OF THE RELATIVE PROPAGATION. 217
Consider an insulated wire, whose loss by the surface may be
neglected, originally in the neutral state, and of an unlimited length,
or at any rate of a length such that the condition at a point is not
appreciably modified by that of the most distant end. At the end
of the wire a constant potential V is maintained. At the end of a
time /, the potential at a distance x is defined by the equation
For a second wire placed in the same conditions as the first, and
the material of which is defined by another coefficient a', we shall
have in like manner
Let us put x' = mx^ t' = nt, m and n being constants ; we may
then consider the potential V as a function of the variables x and t,
and equation (5) will become
If the coefficients m and n are chosen so that
that is to say
/'
the potentials V and V satisfy the same differential equation (2),
and the same limiting conditions; they represent then the same
function of x and of /.
227. Hence, for unlimited wires, which is practically equivalent
to wires so long that the duration of the propagation has an appre-
a?x 2
ciable value, the potential V does not change when the ratio
has the same value ; it is, therefore, a function of this ratio.
2l8 VARIABLE STATE.
Hence follows the conclusion which has been established above
(226), that the time required to produce a definite potential at the
distance x, or more precisely, a definite fraction of the potential at
the origin, is proportional to the square of the distance, and to the
coefficient a 2 , which is special to the wire.
In these conditions, equation (2) really only contains one inde-
pendent variable, and putting
IT
it becomes
20^ = 0;
dz* dz
from this we easily deduce
The constants A and B are determined by the initial conditions.
For = 0, that is to say x = or /= oo , we have V = V ; for = oo ,
that is to say x = oo , or / = 0, we have V = 0.
We get then
There is no simple expression for the integral contained in this
formula, but it is met with in a great number of problems ; for
instance, in the theory of probabilities, and tables of it have been
calculated, so that its numerical values are well known.
JTT
Between the limits and oo , for instance, it is equal to >
which gives
DURATION OF THE RELATIVE PROPAGATION.
219
The curve A (Fig. 56) represents the values of the ratio of the
V a?x 2
potentials , as a function of /, by taking a = = z 2 t. The
VG 4
ordinate remains zero for some time near the origin, and only com-
mences to acquire an appreciable value from the period in which
&
/ = -. The asymptote to the curve is parallel to the /-axis at a
4
distance from this axis equal to unity.
0,5
\
\
\
\
\
TV
\
sin
O.I
Fig. 56.
228. The expression for the current at the distance x, is
__ _oi_ _
L " ~ j o
p dz ' ttor p JIT 2 Jt '
replacing z 2 by - , we get
(7)
T "0 / "
=A/ 7
220
VARIABLE STATE.
This strength is a function of the time; it is zero for /=0, and
<)I i
becomes a maximum when = 0, which gives z 2 = - . If T is the
ot 2
time of the maximum we have
*JK*
2. 2
The curve A (Fig. 57) represents the value of the expression
\ -e~~ t which is proportional to the strength of the current. The
Fig. 57-
time of the maximum being proportional to a 2 ,* 2 , we see that the
curve is the more bent, the greater is the distance x from the point
taken as the origin.
229. MOMENTARY CONTACTS. Let us suppose that the end
of the wire is only put in momentary -contact with a source at
constant potential, that is to say, that this end is only raised to
potential V for a very brief period T, and is then connected with the
earth.
The potential at any given point will be obtained by superposing
MOMENTARY CONTACTS. 221
two states, the first due to the permanent potential V , established at
the origin of the wire at the beginning of the time, the second to
the permanent potential - V , set up after the time r. The value of
the potential at the distance x, corresponding to each of those states,
is the same function of the time which elapses from establishing at
the origin of the wire the corresponding potential; the resultant poten-
tial U is therefore equal to V(/) - V(/-r). If we suppose that the
time r is infinitely short, we get
From this we deduce
The value of U is no longer then a simple function of z*. If <
stands for the function
z fix _
- e r** = e t
t 2/1
we may write
(8)
230. We may, moreover, determine graphically the value of ,
v o
by taking the difference of the ordinates of the curve A, and of
another identical curve which has been displaced towards the right
by a quantity T. The curves I, II, III, IV, V, represent the result
of this superposition for values of r equal respectively to a, 20, 30,
40, $a.
This construction and the formula will show that the momentary
connection of the end of the wire with a source of constant poten-
tial, gives rise to a sort of electrical wave, which is propagated
according to a somewhat complex law, and which spreads itself out
as it travels.
222
VARIABLE STATE.
We obtain in the same way the strengths corresponding to
momentary contacts; curves I, II, III (Fig. 57), represent the law
of the strengths at a point, when contacts are made for durations
respectively equal to a, 20, 30.
0,1
Fig. 58.
231. The period T v at which the maximum potential of the
wave is obtained for an infinitely short contact, is determined by
the condition
^ = Q, or = 0.
Since we have
lit
the maximum takes place when 2z 2 = 3, or
2d
3
HP
- X 2 = .
6 3
DURATION OF PROPAGATION OF AN ELECTRICAL WAVE. 223
This time T l corresponds to the point of inflexion of the curve
W 2 2 V
A (Fig. 56), for the condition = 0, is equivalent to ^7 = ; it is
a third of the time T, required to attain maximum current at this
point, with a constant potential at the end of the wire.
The time T, may therefore be considered as expressing the
duration of propagation of an electrical wave.
232. We may, in like manner, determine either graphically or by
calculation, the wave which would result when the origin of the wire
is put alternately in contact with sources at potentials + V and - V
during equal or unequal times. The curves in Fig. 58 correspond
thus to alternating contacts which may be collated in the following
table :
Curves. Duration of Contacts.
I a a
II a a a
III 2.a 2a
IV 2.a a
V 30 2a
VI 30 a
VII 3# 2a a
It will be seen that, by choosing the duration of these contacts,
we may obtain a far shorter wave than by a single contact. The
wire is then quickly restored to the neutral state after the passage
of the wave ; this is the problem which it has been attempted to
solve in certain cases of telegraphic communication.
The curve of contact T may be simply constructed by adding
algebraically the ordinates of the curve A, and of another curve
- AT, the origin of which has been displaced through T. The curve
corresponding to the succeeding contact r of the opposite sign, is
obtained in like manner by the ordinates of the curves AT and
+ AT+T/. The curves corresponding to the contacts r and r' of
opposite signs will therefore be obtained by the sum of the ordinates
of the three curves A-2A T + A T+T /; we thus avoid the separate
construction of curves relative to the different contacts.
233. In the most general case, the potential V at the origin of
the wire does not pass suddenly from zero to a constant value ; it is a
continuous function F (9) of the time 6, counted from the moment
224 VARIABLE STATE.
the electrification begins. The element ^U of the potential at the dis-
tance x and at the time /, which corresponds to the potential V set up
during the time dO and to the time at the origin of the wire is equal
(229) to M </>(/-#) dO. If the total duration of the electrification
VTT
is r, we shall have for the resultant potential
U = -=
It may, however, be remarked, that this expression has no meaning
except for values of /, greater than T.
If the potential at the origin of the wire varies periodically
according to a simple law if it is represented, for instance, by
V sin 2nt, the ultimate electrical state of the wire at each point
varies obviously according to the same period. The potential at
the distance x may be expressed by the formula
in which b is a constant, and A a function of x. Substituting this,
in equation (2) we get finally
(9) V = V Q e- ax ^ n sm(2nt-xa f Jn).
Hence a definite phase of the potential at the origin is trans-
mitted along the wire with a constant velocity equal to - . In
this case it may be said that there is a regular velocity of propa-
gation, but this velocity depends on the period of electrical
oscillation; the time necessary to traverse a definite length is
proportional to a, and not to a 2 , as has been found for the relative
length of propagation considered above (225).
If the electrification at the origin, comes under a more or less
complex law, and if the expression for the initial potential is de-
composable into a series of simple periodic terms, the potential
in the wire will be represented by a series of corresponding ele-
mentary waves; but these waves will be propagated with different
velocities, and a kind of electrical dispersion will be produced
analogous to the phenomenon of the dispersion of light in a re-
fracting medium.
ELECTRICAL WAVE. 225
234. Let us suppose that the potential at the origin V has
alternately constant values which are equal, and of opposite signs
during the very short and equal times r, and that the operation is
repeated an uneven number of times, 2n+ i for instance that is
to say,
+ V from to T,
- V from T to 2T,
+ V from 2T to 3T,
+ V from 2m to (272+1)1-.
We have then
TT T 2nr
The different values of the function <f> (t - 0) are :
For the first contact - + <(/- 0) =+</>(/),
For the second contact - -</>(/- T) = - <j>(t)
For the third contact - + <(/- 2r) = + <.(/) - 2r</>'(/).
For the ( 2 n + i ) th contact +</>(/- 2 nr) =+</,(/)_
Adding these equations, we have
and, therefore,
The maximum value of U at the distance x is produced after a
time T n defined by the condition
which gives
/ \ r~ /
Q
226 VARIABLE STATE.
If we replace /, in all the terms containing r as a factor, by the
2/7
approximate value , which corresponds to the maximum T l of the
first wave, we get
or finally
The time at which the maximum is attained diminishes,
therefore, as the number of contacts increases ; the waves are thus
shortened, and the duration of the phenomenon is thus materially
diminished.
A series of uneven numbers of equal and alternate contacts
of very short duration would thus produce along the wire an im-
pulse in the same direction as that of a single contact, but it would
be far shorter.
235. For a contact of an infinitely short duration T, the potential
at a point at the time t is
TT V -^-rs-z*
j*i
The value of the current is
or
V T a /*\!/ a \ a
(10) I = . =. ( ) (2 i }e~t .
P Jir 2ai\tJ \ t )
It is easily proved that the maximum determined by the con-
dition = is attained when
CURRENT IN A WIRE OF FINITE LENGTH.
227
The curve I (Fig. 59) represents the values of the expression
i/W a \ n
-( - ) ( 2--I W-7,
4V/ V ' /
which is proportional to the current.
o,3
\
Fig- 59-
236. WIRE OF FINITE LENGTH. To pass, to the case of a wire
OE (Fig. 60) of finite length /, the end of which is in connection
with the earth, let us consider an unlimited wire X'X, and imagine
x- o,.
0'
o"
0'" X
Fig. 60.
on this wire two series of sources O^ O 2 . . . , O', O", O"' . . at
successive distances each equal to 2/5 the first O v O 2 , . . . are
identical with the given sources which exists at the point O, and
the others O', O", O'" , ... are of the same numerical value but of
opposite signs.
All these sources, O and O', Oj and O", . . . being symmetrical
in pairs in reference to the point E, and of opposite signs, the
potential at E will always be zero. In like manner, all the
sources added, O l and O', O 2 and O" . . . are symmetrical in pairs
in reference to the point O and of contrary signs: the potential
Q 2
228 VARIABLE STATE.
at this point will only depend on the source found there. The
portion OE of the unlimited wire is therefore in the same condition
as if it were alone.
The current at the point P, on the wire OE at a distance x from
the origin O, is the algebraical sum of the currents which would be
produced at this point if we suppose that all the sources were on
an unlimited wire.
If all the sources are raised to the constant potential V , then,
for a portion of the potential due to the source O, we shall have
(228)
V a _a"* V a
* = 4=* ir
pjirt
For sources on the left it will be sufficient if we successively
assign to x the values x + 2/, x + 4!, . . . x + 2#/, . . . The sources
O', O", O"' will produce currents in opposite directions to the
preceding if they were at the potential V ; but, as their sign has
been changed, the flows of electricity which they produce are still in
the same direction; we ought accordingly to replace x by 2/-^,
4/- #,..., znl-x, . . . , which gives for the current I,
When we have made x = /, that is to say, when we consider the
phenomenon at the point E, at the end of the wire in connection
with the earth, the expression becomes simplified, and we see
directly that the intensity is double that which all the sources on
the left would give. We have then
2V, a F 2 ' 2 a2 ( 3? > 2 2 (50 2 -I
Putting e u = v, we get
The current is at first zero, since v is zero when t is equal to
zero ; it then increases towards the limiting value .
pl
CURRENT IN A WIRE OF FINITE LENGTH. 229
The curve represented by this series would be very easily calcu-
lated, for the terms rapidly decrease when v differs appreciably from
unity.
237. Sir W. Thomson has, however, solved the problem by the
help of another series, which is more easily discussed, and which
follows directly from Fourier's equation (4).
According to this formula, the expression for the current at the
distance x from the origin, is
1 ^ v v of vV= - t ** 1 *
1= --. -2 1 + 2 V e c^ 2 cos * .
p ^x pl\_ - 1 /
For the end of the wire which is in connection with the earth,
x = /, and we get
Giving to n the successive values i, 2, 3 . . . , the cosine takes
alternately values equal to - i and + i. If for the sake of brevity
we put
IT"*
we get
(12) I = l -
For very small values of t, u tends toward unity, the series
in the parenthesis is equal to - , and the current null. As the
time increases, u diminishes, the series tends to zero, and the
y
current diminishes up to a limiting value 7.
pi
The series can, moreover, be easily calculated; according to
Sir W. Thomson, it does not differ appreciably from its maximum
value, until u is greater than - If a' is the time at which this
4
value is attained, we have
3 -^' " 2 / 2 7 /4
- = e a/, or a --rjr-/, ( -
4 ^ \3
230
VARIABLE STATE.
We may then write
3V
- \a'
4y
The curve A (Fig. 61) represents as a function of the time and
taking the final strength as unity, the curve of the current produced
at that end of a wire which is in connection with the earth, when
a constant potential is established at the other end.
Til
\
\
Fig. 61.
238. MOMENTARY CONTACTS. In order to obtain the strength
corresponding to the case in which the wire is connected with a
source of constant potential V , for the time r, it is sufficient, if, as
in the case of the insulated wire, we calculate the expression
or construct geometrically the curve, the ordinate of which is equal
at each point to the difference of the ordinates of the two curves
F (/)andF (/-T).
Curves II, III, IV, V, VI, VII (Fig. 61) represent thus the
currents arising from contacts whose durations are respectively
equal to 20', 30', ..... , 70'. The phenomenon appears as an
electrical wave, or a momentary impulse at the end of the wire.
If the time of contact is infinitely short, the arrival curve of the
current / is represented by the equation
d d du
MOMENTARY CONTACTS.
231
which gives
^
pi
7T 2
^T 2
This current is represented by the curve B (Fig. 61). It is a
di
maximum when = 0, that is to say, when
u-
an equation which gives sensibly
/3\s
=(-), or /=3.
W
239. Finally, in order to shorten the arrival waves, and to
discharge the wire, the origin of the wire may be put alternately at
equal potentials and opposite signs, during equal or unequal times,
by connecting it with one of the poles of a battery.
Fig. 62.
Curves I, II, and III of Fig. 62, represent the arrival waves of
the alternative successive contacts :
Curves.
I
II
III
Duration of contact.
4*
3*'
232 VARIABLE STATE.
Without, for the present, dwelling further on this important
question, it will be seen what is the nature of the problem, and
what methods may be utilised for accelerating the transmission of
signals in electrical wires.
240. USE OF CONDENSERS. We may add that in practice it
has been found very useful to keep the cable constantly insulated by
joining each of its ends with a condenser. The battery electrifies
one of the coatings of the condenser at the sending station ; the
other coating, which is connected with the cable, is electrified with
the opposite kind, and a flow of the same kind as that which the
battery would have given passes to the first coating of the condenser
at the other end. The second coating of this condenser is con-
nected with an electrometer, or is in communication with the earth
by a galvanometer.
If the contact at the origin is continuous, the electrometer tends
towards a maximum deviation ; the galvanometer gives a deviation
which increases at first and then reverts to zero, so that even for a
permanent contact, the phenomenon appears as an electrical wave.
It is easily understood from this, that momentary alternate con-
tacts suitably chosen, may produce waves which are materially
shorter than if the wire had been directly charged by the
battery.
241. PROPAGATION IN DIELECTRICS. The conclusions from
Fourier's formula applied to the variable state, are verified, at any
rate approximately, for good conductors in the phenomena presented
by transatlantic cables, and, for imperfect semi-conductors, by the
experiments of Gaugain. The formula appears general therefore,
and we are led to apply it to dielectrics, which are never absolutely
destitute of conductivity.
A dielectric submitted to the action of an electromotive force,
may be considered as being at once the seat of a phenomenon of
polarisation, and of a phenomenon of conduction subject to the
ordinary laws.
Let us suppose that the dielectric is isotropic and let /* be its
specific inductive capacity, and c its coefficient of conductivity.
The general equation of induction (116) applied to a volume
element dv situate at a point in which the density is p, gives
/xAV + 4717) = 0.
On the other hand, the variation of the charge ckdvdt of the
element, during the time dt, produces a corresponding increase of
RESIDUAL CHARGE OF CONDENSERS. 233
density, which gives the equation
from which we deduce
p i dp
A V = ATT =
p c dt
and therefore
putting T = .
This equation shows that the density p constantly decreases, and
that if for any reason the dielectric has received a charge in the
interior, it will not retain it indefinitely ; this charge will always
finish by being altogether on the surface, like that of a good con-
ductor evidently an a priori conclusion.
242. RESIDUAL CHARGE OF CONDENSERS. The phenomena
of absorption and of residual charge to which dielectrics give rise
should not be considered as effects of their own conductivity. Let
us examine, from this point of view, the series of phenomena to
which the charge or the discharge of a condenser gives rise.
Let C be the capacity of a condenser, R the resistance of the
dielectric, E the difference of potential of the two coatings at the
moment /, r the resistance of the circuit which joins the two coatings
on the outside ; let E be the electromotive force of a source inter-
posed in the circuit. The increase of charge G/E of the condenser
during the time dt, is equal to the excess of the flow of electricity
"F "F "P
dt furnished by the source, over the flow dt which traverses
r R
the dielectric. From this we have the equation
ETP TT //TT
- & H, tf-C'
and therefore
putting Tj =
234 VARIABLE STATE.
Suppose that at the moment / x we open the circuit, and leave the
apparatus to itself during a time / 2 , equation (13) reduces to
denoting by E : the difference of potential at the time / : between
the armatures, by E 2 that which exists at the moment / 1 + / 2 , and
putting T 2 = CR, we have
Suppose, lastly, that we discharge the condenser by connecting
the two coatings by a conductor of small resistance />, we shall have
the equation
E E_ E
+ ~ ^~*
and, at the end of a time / 3 that is to say, at the period ^ + / 2 + / 3
E = E-| with T
The total loss of the condenser during the time / 3 , is C (E 2 - E 3 ) ;
the portion which traverses the outer circuit and constitutes the
p
discharge Q, is equal to C(E 2 -E 3 )- - , which gives finally
JK. + p
CR 2
To have a complete discharge, we must make / 3 = oo ; we see
that we attain this complete discharge in a continuous manner, and
without any of the alternatives to which condensers give rise.
Maxwell has shown that a system formed of parallel dielectric
layers, and even of different dielectrical elements mixed in any way
whatever, may give rise to residual charges, although each of the
constituent dielectrics is destitute of this property. But the want
RESIDUAL CHARGE OE CONDENSERS. 235
of homogeneity does not seem to be the sole cause of the phe-
nomenon, and experiment shows that the existence of residual
charges must in most cases be ascribed to a kind of elastic de-
formation which is caused by the polarization of the dielectric.
It must be observed that all actions, such as repeated shocks,
vibrations, sudden variations of temperature in opposite directions,
which facilitate the return to the normal state, of a body which
has undergone any permanent elastic deformation, also facilitate
the appearance of residual charges and their return to the natural
state.
The propagation of heat gives rise to no phenomenon which
could be compared to the residual charge of dielectrics, and in this
respect the analogy, which in so many respects is so close between
the two orders of phenomena, ceases to hold.
236 ENERGY OF CURRENTS.
CHAPTER III.
ENERGY OF CURRENTS.
243. DISENGAGEMENT OF HEAT. When a system of electrified
conductors undergoes any modification whatever, without the inter-
vention of any external force, the electrical energy in the second state
is necessarily less than in the first. The energy lost during the trans-
formation may be utilised in an equivalent form, such as a mechanical
work, the raising of a weight, increase of the vis viva of the system,
a change of physical state, or finally a disengagement of heat.
For any infinitely small transformation of the system in
question, the loss of energy is equal to the sum of the products
of each of the electrical masses, into the difference of the values
of potentials at the points in which they were placed before and
after the transformation.
Let us consider two points A and B kept respectively at the
potentials V : and V 2 , and on equipotential surfaces which are
traversed at A and B by two corresponding portions S x and S 2 that
is to say, cut by the same tube of flow. The quantity of electricity
which traverses the two surfaces is the same ; the energy lost by the
current in this interval in unit time, is equal to the product of the
mass of electricity which issues that is to say, of the strength of
the current I by the difference of potentials V l - V 2 , if the current
goes from A to B that is to say, by the electromotive force between
these points. Hence, as a measure of the energy lost, we have
W = I(V 1 -V 2 ) = IE.
We shall assume as an experimental fact, that no part of this
energy is employed in changing the vis viva of the electrical masses.
The fact is obvious if the surfaces Sj and S 2 are equal, for then the
velocities are the same on entering and on leaving the tube. For
the general case, we have already observed that the flow is parallel
to the force at each point, and that therefore no effect attributable to
JOULE'S LAW. 237
electrical inertia seems to intervene in the phenomena of the perma-
nent state.
If, on the other hand, the conductor is rigid, at any rate as a
mechanical whole, and if, finally, the current produces no external
work, the energy is necessarily spent in the conductor itself.
244. JOULE'S LAW. Two cases may present themselves : either
the fall of potential between the points A and B is continuous, and
takes place in accordance with Ohm's law ; or there are, somewhere
between these two points, two adjacent surfaces between which there
is a sudden fall of potential, constant and independent of the strength
of the current that is to say, a constant electromotive force H.
The manner in which the electrical energy is distributed along the
conductor, depends on the law according to which the potential
varies, and is not identical in the two cases. Wherever the variation
of potential is continuous, energy is expended in a continuous
manner ; it is transformed into thermal energy, and gives rise to a
disengagement of heat along the conductor. Wherever there is a
sudden fall of potential, there is a sudden change of electrical
energy, which reveals itself either by some thermal phenomenon or
by some other equivalent physical effect.
Let us first consider the former case, and let us suppose that
there are no variations of potential independently of the current.
If R is the resistance of the conductor between two points A and B,
Ohm's law gives
The expression for the energy expended between the two points
is therefore
W = IE = I 2 R = .
R
Accordingly, the thermal energy developed in a conductor during
unit time, is equal to the product of the square of the current strength
into the resistance of the conductor. If Q be the quantity of heat,
such as is measured by calorimetrical methods, and J is the
mechanical equivalent of heat, we have
The quantity of heat disengaged is proportional to the resistance of
the conductor, and to the square of the strength of the current.
This is Joule's law.
238 ENERGY OF CURRENTS.
245. CONNECTION BETWEEN OHM'S AND JOULE'S LAWS. This
result can be arrived at in another way :
Let us consider a conductor of capacity C, a battery for instance,
electrified to potential V ; the value of the potential energy is (89)
-CV 2 .
2
Let us now suppose this battery connected to earth by a wire
whose resistance R is so great that its discharge has an appreciable
duration. During the time dt, a mass of electricity dlA flows out,
and the potential diminishes by^V; we have
and the loss of energy in the same time is
= C WV = WM = I Vdt.
Ohm's law applies if the current remains sensibly constant during
the time dt ; from this it follows that
-i-
and
V 2
that is to say that the energy expended in unit time is expressed by
Joule's law.
We have here deduced Joule's law from the principle of the
conservation of energy together with Ohm's law. Ohm's law might
conversely be deduced from the same principle combined with
Joule's law. For Joule's law gives
W = I 2 R.
We have further
W-Elj
from which follows
E = IR,
that is to say Ohm's law.
246. We may here observe that in a multiple circuit, which does
not contain localised electromotive forces, the quantity of heat
developed is a minimum, when the currents come under Ohm's law.
PELTIER'S PHENOMENON. 239
Suppose, for instance, that between two points A and B, at
potentials Vj and V 2 , there is a series of conducting arcs (Fig. 52).
Let R be the resistance of one of them, and I the strength deduced
from Ohm's law that is to say, such that IR = V 1 -V 2 = E, and
suppose that by a change of conditions, the strength in this con-
ductor becomes ! + /. The expression for the total quantity of
heat developed in the new system will be
but the product RI is a constant for each of the arcs, and on the
other hand i is necessarily zero, if the current which terminates
at the point A is not modified; the quantity of heat reduces
therefore to
and it is obviously a minimum, for z = that is to say when the
strength divides in the branches according to Ohm's law.
247. PELTIER'S PHENOMENON. Let us now suppose that
between two points A and B, always kept at the same potentials
Vj and V 2 , the value of the potential, instead of varying in proportion
to the resistances, undergoes a sudden fall Uj - U 2 = H, at a point
P between two adjacent surfaces, which is independent of the
current ; the expression for this strength will no longer be the
same as in the preceding case.
If Rj and R 2 are the resistances of the two portions AP and PB,
we have thus (210)
V 1 -U 1 U 2 -V 2 V 1 -V 2 -(U 1 -U 2 )E-H
_
R 2 Ri + R 2 R
The total energy expended between the points A and B is
W = I(V 1 -V 2 ) = IE,
which gives
This energy consists then of two parts ; one which is propor-
tional to the square of the strength of the current, and which heats
the conductor throughout its entire length corresponding to Joule's
law ; and another, which is proportional to the current, is localised
240 ENERGY OF CURRENTS.
at the point P. This latter is positive if the fall is in the direction
of the current, and negative in the contrary case. If there is no
other work than that corresponding to changes of temperature, this
energy will appear as a disengagement of heat at P in the first case,
and by an absorption in the second that is, by a cooling. This is
the effect which is known as Peltier's phenomenon, produced at the
contact of the two metals.
It may be that the localised energy IH is correlated to a chemical
reaction, which expends heat if H is positive, and on the other
hand produces heat if H is negative, so that the changes of tempera-
ture are then merely due to the heat disengaged in accordance with
Joule's law.
248. The' converse of the conclusions which we have established
is evident. If, at any point of the circuit, a thermal or chemical
phenomenon is produced, the energy of which is proportional to
the strength of the current, it may be affirmed that at this point
there will be a sudden variation of potential positive or negative,
according to the sign of the work, and that the variation is indepen-
dent of the current. If, further, the work changes with the direction
of the current, we conclude from this that the corresponding variation
of potential is fixed, and is independent of the current.
Let us consider this latter case ; let r be the resistance of the
region in which the fall of potential is manifested, and let us suppose
that only thermal phenomena are produced at that place. The
quantity of heat disengaged is made up of two parts ; one defined
by Joule's law is expressed by IV, and is independent of the
direction of the current; the other, due to the Peltier effect, has
the value IH, and changes its sign with the direction of the current.
If the current passes in one direction, the total quantity of heat
disengaged is
iBfi+I^
V
and if it passes in the opposite directicfh
i r
VTT
"
In proportion as the current is diminished, the term I will
ri
become smaller and smaller, the Peltier effect will predominate, and
the reversal of the current will more and more tend to produce
equal effects and contrary signs.
CHEMICAL DECOMPOSITION. 241
A question presents itself here in reference to Peltier's phenomenon.
The thermal effect observed during the passage of the current at the
soldering of the two metals, measures the sudden fall of potential at
this point, and it would seem as if it should measure the electro-
motive force of contact between them on Yalta's theory. Does the
result thus obtained agree with that given by other methods the use
of electrometers, for instance ?
Experiment answers this question in the negative ; not merely do
the series of numbers obtained by the two methods disagree with
each other, but the bodies themselves are not arranged in the same
order ; the numbers of the two series are not of the same order of
magnitude ; they are even sometimes of opposite signs. It is certain
therefore that we are not measuring the same phenomenon in the
two cases. The most plausible explanation of this discrepancy is that,
in the electrostatic measurements, we are dealing with a complicated
phenomenon in which the nature of the medium, necessarily inter-
posed between the metals in contact, plays a considerable part.
249. CHEMICAL DECOMPOSITION. Whenever a compound
liquid is traversed by a current it splits up ; one of the elements
appears at the conductor by which the current arrives, the other at
that by which it leaves. Faraday gave to this phenomenon the
name electrolysis ; the body submitted to decomposition he called an
electrolyte, and applied the term electrodes to the two conductors by
which the current enters and leaves ; the former being the positive
electrode, and the latter the negative electrode.*
Two conditions are necessary for the occurrence of electrolysis ;
the current must traverse the compound, and the compound itself
must be liquid, or at any rate in the pasty state. Thus, glass at a
red heat gives evident signs of decomposition, for it becomes at
once a conductor, and pasty.
It is extremely remarkable that the products of decomposition
only appear on the electrodes. Clausius, developing a theory which
was originally propounded by Grotthiis, explains this phenomenon
in a very ingenious manner. On his view the molecules of which
the body is made up are in a constant state of agitation ; but while
the excursions of each molecule are restricted in the case of solids,
these excursions may take place to any extent and in any directions
in liquids. Thus the molecules of hydrogen which form part of the
* Faraday called the electrode by which the current enters the anode, and that by
which it leaves the cathode ; he applied the term ions to the elements decomposed.
The anion is that which is liberated on the anode, the cation that on the cathode;
these terms have not however, like the former, been generally adopted
R
242 ENERGY OF CURRENTS.
molecules of water are not invariably united to the corresponding
molecules of oxygen ; but, carried along in an incessant eddying, they
may quit the first molecule of oxygen, to become combined with
adjacent ones ; and thus by a series of successive interchanges they
may be carried to distances which are infinitely great in comparison
with their radius of activity. In the ordinary condition, the directions
of these motions are perfectly irregular ; the passage of electricity
imparts to them a systematic tendency, owing to which the molecules
of hydrogen moving with the current are impelled towards the
negative electrode ; those of oxygen, on the contrary, going in the
opposite direction move towards the positive electrode.
250. FARADAY'S FIRST LAW. The first experiments on the
decomposition of water by electricity appear to have been due to
Troostwyk and Diemann in 1795. They employed the spark of the
battery passing between two gold or platinum wires. The experiment
was repeated in 1800 by Carlisle and Nicholson by means of the
current of the voltaic pile. In working with sparks it is advantageous
to use what are called Wollastoris electrodes, which consist of a
platinum wire passed into a glass tube in such a way that only the
mere section of the wire is in contact with the liquid. Wollaston,
Faraday, Armstrong, have shown that the effect of the spark is
identical with that of the battery. Whatever be the origin of the
electricity, the quantity of water decomposed is proportional to the
quantity of electricity which passes.
This law, which was enunciated by Faraday, has been more
particularly verified by the electromagnetic measurement of currents ;
but the direct determination of the quantity of electricity by elec-
trostatic methods also allows of a very exact demonstration. In
some recent experiments Dr. Warren De La Rue discharged a
condenser which had been charged to potentials i, 2, 3, through
water, and verified the exact proportionality between the quantity of
electricity and the quantity of water decomposed. This propor-
tionality enables us to regard electrolytes as measurers of electricity ;
the term voltameter is applied to an apparatus arranged so that the
gases arising from the decomposition of water may be collected.
251. The work of chemical decomposition being proportional to
the strength of the current, it follows, from the remark made above,
that there must be somewhere in the voltameter a sudden fall of
potential H, independent of the strength. The energy made avail-
able by the fall of the current at this point, is used in decomposing
the water, and may be calculated in absolute value.
Let M be the quantity of electricity which has passed through
POLARIZATION OF THE ELECTRODES. 243
the voltameter in unit time, and P be the weight of water decom-
p
posed. These two quantities being proportional, the quotient =/
expresses the weight of water decomposed by unit electricity. If a
is the heat of combination of unit weight of water at constant
pressure, JaP represents the energy necessary to decompose a weight
of water equal to P. This energy being furnished by the fall of the
current, we must have
W
from which is deduced
Hence, between the two electrodes of a voltameter traversed by
a current there is, besides the difference of potential due to the
resistance of the intermediate conducting liquid, a sudden fall, the
exact seat of which is indeterminate, and which may be produced
either wholly upon one electrode, or partially on both, and which is
numerically expressed by the mechanical work corresponding to the
energy absorbed by that quantity of water which a unit of electricity
decomposes.
252. POLARIZATION OF THE ELECTRODES. By what mechanism
is this difference of potential produced ? It is clear that before the
current passes, the two electrodes, if they are of the same metal,
(both of platinum, for instance,) are, by Volta's law, at the same
potential, which probably differs from that of water ; but the sudden
and opposite changes which then take place at each of the electrodes
would produce in the voltameter an amount of work which is ob-
viously zero. When the current is started, the two falls are unequal
and their difference is equal to H. Following Volta's ideas, we are
led to the conclusion that the surfaces in contact are modified. A
deposition of the elements of the electrolyte on the electrodes gives
a sufficient explanation of this modification.
For if a plate, which has been used as an electrode, or which has
been immersed in a gas, is placed in water in presence of a plate of
the same kind, but clean, or recently heated to redness, a difference
of potential is set up between the two plates.
Let us consider, as a particular case, the decomposition of water.
The first portions of gas which come in contact with the platinum,
if they do not form with it a true combination, seem at any rate to
be deposited there in a state of condensation in which the gas has
far less potential energy than it has in the free state. This effect
of condensation of the gas takes place particularly at the outset,
R 2
244 ENERGY OF CURRENTS.
and then goes on progressively diminishing until the thickness of
the layer becomes so great that the fresh bubbles no longer expe-
rience an action on the part of the plate, and can then escape freely.
The work of the decomposition of water only attains its
normal value from this period. Hitherto the normal value has
been diminished by the work of condensation in question; experi-
ment shows that at the outset the value of this difference may
be very small.
The modification which the surface of the plates thus undergoes is
the cause of the phenomenon known as polarization of the electrodes,
and which manifests itself by the development of an electromotive
force opposed to that which produces the current. We can thus
understand how it is that this polarization is not instantaneous, that
it may increase continuously from zero to a maximum limit; and,
finally, how the quantity of electricity required to produce a given
state of polarization depends on the condition and dimensions of
the plates. This quantity is often called the capacity of polarization
relative to the given system.
By taking electrodes of very unequal surfaces and passing a given
quantity of electricity at a given potential through the voltameter,
we can produce polarization of either electrode at will; recent
experiments by M. Blondlot show that the phenomenon follows the
same law whatever be the direction of the current, and that for a
given electrode and given electrolyte the capacity does not depend
on the direction of the polarization.
253. SECONDARY CURRENTS. When once polarization is set
up, if the original current is opened and the two electrodes are
themselves joined by a wire, the electromotive force of polarization
H, produces a current in a direction opposite to that of the original
current, but the current rapidly diminishes and finally disappears
more or less completely ; this current is called the secondary current.
It is easy to account for this phenomenon ; when the two elec-
trodes are connected by a conductor, the layer of gas gradually
disappears, reforming water ; the electromotive force diminishes and
disappears with it; and lastly it is clear that the total quantity of
electricity set in motion while the secondary current lasts, must be
equal to that expended in effecting the polarization of the electrodes.
It is manifest that the current would remain constant provided
the electromotive force H could be kept constant; it would be
sufficient for this if the layer of gas necessary for complete polariza-
tion were maintained at the surface of the electrode. This is
precisely what takes place in Grove's gas battery.
FARADAY'S SECOND LAW. 245
254. SUCCESSIVE CHEMICAL ACTIONS OF THE CURRENT.
FARADAY'S SECOND LAW. Let us suppose that several Grove's
cells and voltameters are arranged in series in one and the same
circuit. Let n be the number of cells, ri the number of voltameters,
R the total resistance of the circuit, and I the strength of the
current which flows through it. In each unit of time the work
done by the whole of the cells is n]apl' } that expended by the
voltameters is ri]ap\. Lastly, a quantity of work RI 2 is converted
into heat in accordance with Joule's law. If there is neither
positive nor negative external work, the sum of the positive works
must be equal to the sum of the negative works, which gives
from which
The product IR is necessarily positive ; the current can only exist
therefore provided that
n>n'.
The numbers n and n' are whole numbers if the polarization is a
maximum in all the cells ; if the polarization was incomplete in
one of them, the corresponding electromotive force would only be a
fraction of H, and n should then be considered as a fractional
number. In all cases, the necessary and sufficient condition for
the existence of the current is that n shall be greater than *ri.
When the permanent state has been attained, the polarization
being supposed complete in the cells as well as in the volta-
meters, the same work is done during the same time, positive in the
one, and negative in the others. In other words, for each unit of
electricity which traverses the system, the same quantity of water is
found in the couples, and is decomposed in the voltameters.
255. Faraday's second law holds even when the polarization is
not complete at all points of the circuit in question. Suppose, for
instance, that in one of the couples the thickness of the layer of
gas has fallen below its limiting value, and that at a given moment
the electromotive force has only the value H', which is less than H ;
the transport of a unit of electricity no longer represents the same
work as in the others, but the relation H' = ]a'p is still satisfied, if by
ct we represent the heat of formation of unit weight of water with
246 ENERGY OF CURRENTS.
the oxygen and hydrogen in that state of partial combination in
which they exist on the platinum, and the couple thus altered gives
rise to the same quantity of water as all the others.
The law, moreover, is general ; the weight of elements combined
or decomposed in any electrolyte is proportional to the quantity of
electricity which passes ; and this whether the operation is positive or
negative ; whether it takes place with polarization of the electrodes,
as in the decomposition of water, or of cupric sulphate with
platinum electrodes ; or whether the polarization can be neglected,
as in the electrolysis of cupric sulphate by two copper electrodes.
This statement includes as a necessary consequence that the electro-
lyte never acts as a mere conductor, and never allows any fraction of
the current to pass without correlative decomposition.
In the electrolysis of cupric sulphate by two copper plates, if the
two plates are really in the same condition, the electromotive force
of contact of the metal with the liquid is the same on both sides,
and since just as much copper is dissolved at the positive electrode
as is deposited at the negative electrode, the heat produced must be
equal to the heat expended. On the other hand, any difference in
the state of the two plates would be shown by thermal work.
256. We may, however, state here an important restriction in
the principle of the equivalence between chemical energy and
electrical work. It is assumed that, at the place where the chemical
action takes place, no external work, and no change of temperature
is produced independently of the resistances. If this is not so, we
must take into account all the physical or chemical secondary work
to which the electrolysis may give rise.
In the decomposition of water, for instance, the energy of the
current first brings about the separation of hydrogen and oxygen,
and then does the work required by the gases in occupying a certain
volume at the external pressure.
When the current arises from a Grove's battery, each element
performs the same work. So long as Mariotte's law holds, the
external work is always the same for the same weight of water
decomposed, and therefore for the same expenditure of electricity.
Within these limits the condition of equilibrium of the cells and of
the electrolytes is independent of the pressure.
Mariotte's law is far from holding at very high pressures ; the heat
of combination of oxygen and hydrogen is thus modified, and it is
known that decomposition by the battery requires the employment of
far greater electromotive forces. The heat of formation of water is,
moreover, a function of the temperature, and the condition of
ELECTROCHEMICAL EQUIVALENTS. 247
equilibrium in a circuit may be modified, if the cells and the
electrolytes are at the same temperature.
It may happen, on the other hand, that certain of the elements
decomposed, experience secondary reactions which are independent
of the action of the current, and give rise to an absorption or a
disengagement of heat. The final result of the electrolysis would
no longer be in a simple ratio with the electromotive force, and
this latter could no longer be calculated from the heat of com-
bination of the elements, taken in the condition in which they
appear after the electrical operation.
257. ELECTROCHEMICAL EQUIVALENTS. Let A, A', A", . . .
be various electrolytes, /, /', /", . . . the weight of each decomposed
by unit electricity. These numbers are called the electrochemical
equivalents of the various bodies, and experiment shows that they
are proportional to their ordinary chemical equivalents.
If a, a', a", . . . are the heats of combination for unit weight of
each of the compounds, the elements of the combination being in
the condition due to the passage of the current (that is to say,
without taking into account the secondary reactions), the products
ap, a'p', a"p", . . . will be the heats of combination of the equivalents.
By analogous reasoning to that in the case of water, we see that the
electromotive forces relative to these various electrolytes are deter-
mined by the ratio
H =Jaf,
H' -*
which give
H H' H"
ap dp 1 d'p"
It follows from this that the electromotive force of an electrolyte is
equal to the mechanical equivalent of the heat of combination of its
electrochemical equivalent.
258. E. BECQUEREL'S LAW. The application of this law of
Faraday presents no ambiguity in the case of analogous chemical
compounds. If, by one and the same current, we effect the electro-
lysis of water, and of a series of neutral sulphates of the protoxides,
for instance, the electrochemical equivalent of each metal is the
weight which is deposited for the disengagement of a gramme of
hydrogen ; but there may be some doubt when the compounds have
not the same formula. With two neutral sulphates, one of the pro-
toxide, and the other of the sesquioxide of iron, decomposed by the
248 ENERGY OF CURRENTS.
same current, it may be asked whether it is the same weight of metal,
or the same weight of oxygen, which is liberated in the two electro-
lytes. M. E. Becquerel showed that the metalloid determines the
law -, consequently the weights of iron for the two electrolytes will
be in the ratio of 3 : 2. This is also the case with the salts of other
acids, the chlorides, sulphurets, etc.
259. ELECTRICAL COUPLES. Let us now consider a compound
circuit made up of various electrolytes, one set giving rise to positive
actions and the other to negative actions. If a denotes the heat of
combination for unit weight of those of the first kind, and b for
those of the second, R the total resistance, and I the strength of the
current, we shall have
or
The product IR, which corresponds to the heat liberated in the
circuit owing to the resistances, being essentially positive, the current
could only exist provided that
If this condition is not fulfilled, and all the electrolytes are at
first in the natural state, the current is established the moment the
circuit is closed. An incomplete decomposition polarizes the elec-
trodes, and the current ceases as soon as the sum of the electro-
motive forces of polarization attains the value ^ap ; the system
remains then in equilibrium. This is the case with a circuit formed
of a DanielFs cell (263) and a voltameter; the replacement of
copper by zinc in Dani ell's cells gives 24-2 thermal units, while the
decomposition of water requires 34*5.
260. DEPOLARIZATION BY DIFFUSION. It may, however, happen
that an extremely feeble current is then observed. This current is
due to the following cause : the polarization of the electrodes of the
voltameter is gradually dissipated in consequence of the diffusion
of the gas ; it can be seen that this diffusion will be more or less
rapid according to the conditions of the experiment, but especially
according to the value of the polarization itself, and its deviation in
reference to the maximum polarization. The current observed in
these circumstances will be that necessary to re-establish the losses
due to diffusion, and to maintain the state of equilibrium which
VOLTA'S COUPLE. 249
corresponds to the maximum of polarization for the conditions of
the experiment In this way are explained the various peculiarities
to which the phenomena of polarization give rise.
When we connect, with a voltameter, a source of electromotive
force insufficient to produce a continuous disengagement of gas,
experiment shows that the electromotive force of polarization in-
creases with the strength of the permanent current in question, but
less rapidly ; that for a given value of this current, the electromotive
force diminishes when the surface of the electrodes is increased ;
and finally, that the electromotive force is constant if the current,
and the surface of the electrodes, increase in the same ratio.
261. VOLTA'S COUPLE. A few words only are now needed to
complete the theory of the battery. Volta's couple, in the strict
sense of the word, consists of a plate of zinc and a plate of copper
placed in water, to which a small quantity of sulphuric acid, or of
any salt has been added, to make it conduct ; the plate of copper
being soldered to a plate of zinc which forms part of the next couple.
Thus, between two terminals of the same kind there are three
contacts, zinc-copper, copper-water, and water-zinc. The electro-
motive force may be expressed by the ordinary symbols
E = Zn|Cu + Cu|Aq + Aq|Zn.
Volta assumed that water only played the part of a conductor, and
thus we shall have
Cu|Aq + Aq|Zn = 0,
and therefore
E = Zn|Cu.
On this point of view, the electromotive force of a Voltaic couple
only depends on the contact zinc-copper, and these two metals joined
by a layer of water are at the same potential. The alteration of the
surface of the metal when in contact with the liquid or the gas, makes
it very difficult to establish Volta's hypothesis in a rigorous manner.
However this may be, this alteration is so rapid, and produces
such changes in the electromotive force, that the electromotive force
of Volta's couple must practically be considered as depending, to a
considerable extent, on the medium which forms the third element.
When the couple is closed by a conductor whose resistance is R,
a current is produced the strength of which is given by the ratio
4
250 ENERGY OF CURRENTS.
but the water is soon decomposed, oxygen goes against the current
and oxidizes the zinc plate, while hydrogen goes along with the
current and polarizes the copper plate ; from this follows an inverse
electromotive force. When the stationary condition is established,
the electromotive force of polarization is E', and the strength V
satisfies the ratio
(E-E')r = r 2 R,
or
R
If the couple is allowed to rest, the polarization disappears slowly,
owing to diffusion. When it is again closed after the lapse of some
time, the current at first reappears with its original strength I (if the
influence of the layer of zinc oxide may be neglected), to regain the
intensity I' after a lapse of time which is usually very short, but which
may be very long if the surfaces of the electrodes are very large and
the resistance of the circuit is considerable. As long as the couple
is open, the difference of potential of the extremities is equal to E.
In a closed circuit the available electromotive force is E E'. In
each couple, oxide of zinc and hydrogen are produced at the expense
of the zinc and of the water. As we may assume that the oxygen has
passed through the gaseous state in going from the water to the zinc,
it will be seen that the disposable energy of the couple, corresponds
to the excess of the heat of formation of the oxide of zinc over that
of the formation of water for the same weight of oxygen. If the
water is acidulated, the difference corresponds to the substitution
of zinc for hydrogen in sulphuric acid: this difference is about 177
thermal units.
The layer of hydrogen which covers the copper has also the
effect of greatly increasing the resistance of the couple, which is a
fresh cause for the enfeeblement of the current.
262. UNPOLARIZABLE CELLS. Mechanical means, such as the
agitation of the liquid, or rubbing the copper plate with a foreign
body, greatly diminish the resistance, and even the polarization, by
getting rid of the greater part of the gas ; the layer of gas may be
completely removed by chemical action, and thus non-polarizable
couples or cells be obtained.
A liquid which merely dissolved the hydrogen without calorific
action, would increase the electromotive force by the whole amount
of the work which the gas performs in filling a given volume at the
external pressure ; but if the hydrogen enters into a new chemical
UNPOLARIZABLE CELLS. 251
combination, or even if we allow for the heat of solution, the
electromotive force is equal to the algebraical sum of the energies
produced at the two electrodes, or at the two poles of the cells.
Such, for instance, is the couple employed by Joule, in which
the copper plate is covered by a layer of oxide, which the hydrogen
gradually reduces. The electromotive force is equal to the difference
between the heats of oxidation of the copper and of the zinc for the
same weight of oxygen.
In other cases, a salt of the metal which forms the positive
electrode is dissolved in the liquid ; for instance, a solution of
cadmium sulphate, in which is placed a plate of zinc and a plate
of cadmium. The dissolved sulphate undergoes electrolysis when
the circuit is closed, and a weight of cadmium is deposited on the
cadmium plate which is equivalent to the zinc dissolved. The
electromotive force corresponds to the heat of substitution of zinc
for cadmium in the sulphate that isj about 8-3 thermal units.
This condition lasts as long as the weight of zinc dissolved is not
so great that the salt itself takes part in the electrolysis. From this
time the polarization of the cell is again produced.
The electromotive force of this cell may be expressed, in terms
of the electromotive forces of contact, by the following symbols :
E = Zn|Cd + Cd|CdO.SO 3 + CdO.SO 3 |Zn.
263. CELLS WITH Two LIQUIDS. In DanielPs cell two liquids
are used : a concentrated solution of copper sulphate surrounding
the copper plate, and a solution of zinc sulphate in which is the
plate of zinc. The two liquids are separated by a membrane, such
as bladder, or a vessel of porous porcelain, so as to hinder the
liquids from mixing, without destroying the conductivity.
The electromotive force is
,SO 3 + CuO,SO 3 |ZnO,SO 3 + ZnO,S0 3 |Zn.
While the zinc plate dissolves, the copper arising from the elec-
trolysis of copper sulphate is deposited on the copper plate. The
electromotive force corresponds to the difference between the heats
of formation of the zinc sulphate and of the copper sulphate that
is to say, to the heat of substitution of the zinc for the copper in the
sulphate, or 24*2 thermal units.
This cell is remarkably constant, and is one of those which
undergo least change from variations of temperature.
In Grove's cell the copper is replaced by platinum : the hydrogen
252 ENERGY OF CURRENTS.
is absorbed by nitric acid, and forms nitro-compoimds of a lower
degree of oxidation. The zinc is placed in a solution of sulphuric
acid or of zinc sulphate. By substituting carbon for platinum, we
get Bunsen's element.
The energy available in Grove's and Bunsen's cells represents a
quantity of heat of about 47 thermal units; they have therefore
almost twice as great an electromotive force as that of DanielPs cell ;
the liquids have, moreover, a far smaller resistance. Accordingly
they are usually employed whenever very powerful currents are
wanted ; but the liquids change rapidly, the resistance increases, the
electromotive force diminishes, and the strength of the current is
soon lessened.
264. ELECTROSTATIC PHENOMENA IN PILES OR BATTERIES.*
The name of pile, frequently given to the association of several
couples in connection with each other, arises from the form originally
devised by Volta. Volta's pile consists of a series of double plates
of zinc and copper arranged one upon the other in the same order,
and separated from each other by discs of moistened cloth.
A couple consists of the whole of the bodies which exist between
two zincs that is to say, zinc, copper, water, zinc. It may be sup-
posed that each of the zinc plates is the half of two successive couples.
If the battery commences at the bottom by a copper and ends at
the top in a zinc, it will be seen that the first copper plate does not
come into play. The difference of potential being equal to e for
each couple, the potential will go on increasing from the bottom
upwards ; and if there are n couples, the electromotive force of the
battery is
E = en.
265. UNINSULATED BATTERY. If the bottom of the battery is
connected with the earth by conductors whose influence may be
neglected, the top disc A has a potential V a = E = ne which is propor-
tional to the number of couples. This is easily verified, either by
means of an electrometer, or by measuring charges given to a
condenser.
266. INSULATED BATTERY. If the battery, which we will suppose
is formed of identical and equidistant couples, has not been connected
with the ground, or at any rate after such a length of time that it has
attained equilibrium, its total charge will be zero, and the distribution
of potentials will be symmetrical in reference to the middle. We
* The term battery is more generally used in this country and will be here
adopted. TRANS.
INSULATED BATTERY. 253
shall have therefore for the ends A and B, or the two poles,
and accordingly
V.-V.
Suppose that we give an extra charge M' to the battery; this
charge will distribute itself as it would on an ordinary conductor
of the same shape, and will produce a constant potential V in the
interior, such that if P is the capacity of the battery,
The potential V being added everywhere to the original potential
will not affect the law of contacts. Hence, at the top A we shall
have
and, on the mth couple from the bottom,
V
Let us now suppose that we connect the mih couple of an
insulated battery, whose total charge is zero, with a conductor whose
capacity is C. This will take a charge M ; there will be a fall of
potential V in every point of the battery, so that if V m is the new
potential of the couple in question, we shall have
from which we get
v =_ v __ --
a 2 "~2 P 2 P
We have, moreover,
254 ENERGY OF CURRENTS.
Eliminating the intermediate potential V m from these two equa-
tions, we get
P n
If the mth couple is connected with the earth, we clearly have
The distribution of potentials on any given battery, symmetrical
or not, would be determined in the same way. In the latter case,
the neutral point of the insulated and uncharged battery is no longer
in the middle.
267. REPRESENTATION OF POTENTIALS IN THE INTERIOR OF
THE BATTERY. Let us represent the battery by a straight line such
that each portion of the length is proportional to the resistance
of the part which it represents, and at each point draw an ordinate
proportional to the potential at this point. Let us suppose the case
to be that of a battery of Volta's couples, the potential increases
by a constant quantity at each zinc-copper contact ; the curve will
show then at the corresponding points a sudden change of the
ordinate, which is always the same. If the battery is open, the
potential is constant in the battery from one contact to the
following; the curve representing the potentials will be formed of
a series of equidistant steps like those of a ladder. The line of
zero potential passes through the middle, if the battery is insulated,
or it passes through any given point which is connected with a
conductor of some capacity or with the ground.
Three cases may present themselves in the case of a closed
battery :
i. The interpolar conductor has a resistance which may be
neglected in comparison with that of the battery. The two poles
are sensibly at the same potential, and each contact produces the
same variation of potential ; but from one contact to the following
there is a progressive fall of precisely equal amount. If n is the
number of couples, and r the resistance of each of them, Ohm's law
gives
the current is the same as with a single couple.
BATTERY PLACED IN A CONDUCTING MEDIUM. 255
2. The' resistance of the battery may be neglected in comparison
with that of the interpolar conductor. The variation of potentials
in the interior of the battery is almost exactly the same as if it were
open. On the outside the fall of potential is continuous; and if
R is the resistance of the interpolar conductor, the current is
it is then proportional to the number of couples.
3. Finally, if the resistance of the interpolar is of the same order
as that of the battery, the potential rises by a constant quantity at
each contact, and sinks continuously, but to a less extent, from one
contact to the next ; the difference of potentials has a finite value,
less than in the case of an insulated battery, but which is greater
the greater the resistance of the interpolar, and by Ohm's law,
the current is
268. BATTERY PLACED IN A CONDUCTING MEDIUM. We have
hitherto supposed that there is no loss of electricity by the lateral
surface of the battery. Imagine that a battery, of Volta's original
construction, made up with infinitely thin plates, is placed in a
conducting medium, and that electricity flows both from the sides
and from the ends ; this would be the case of a battery immersed in
water, if the effects of polarization are neglected. Let < be the
electromotive force of the battery for unit length, p the internal and
p the external resistance for unit length (220).
The flow of electricity is still parallel to the generating surfaces
for the greater extent of each normal section of the battery, and part
escapes at each point, so that the equipotential surfaces are plane
and agree with the lateral surface, as in Fig. 55.
Between the infinitely near points P and P', whose potentials are
V and V, the strength I of the current in the interior satisfies the
equation
The current at each point is therefore given by the equation
'-*-)
256 ENERGY OF CURRENTS.
Let us assume that the permanent state has been attained,
and consider two successive layers. The flow of electricity which
traverses the first is equal to the sum of the flow I' which traverses
the second, and of the flow / which escapes by the lateral surface,
that is
or
/=!_!'= -dl.
As we have
._ i _Vdx
?L p>
dx
it follows from equation (i) that
Making /3 2 = , this equation becomes
the same as for the permanent state of a wire when there is an escape
at the surface (220). To determine the constants of the integral
let us suppose that the lengths are calculated from the middle of the
battery, and that, the whole being symmetrical, the potential is zero
when x = ; it follows that
V- A (<*-*-*).
If Vi is the potential at the ends of the battery and / its
length, we have
BATTERY PLACED IN A CONDUCTING MEDIUM. 257
269. The expressions for the current in the battery, and that
for the lateral current /, are
(3)
V
t-^ ">]
e-e
In order to determine the potential V l of the ends, we must
estimate the current which flows through each of them.
We have then, if R x is the resistance of the medium measured
from the ends,
v, ir M- ? -fn
''-sh; -- s-^' +e i
e* -e *
which gives
I
(5)
If the external medium is an insulator, />' = oo and /3 = 0. The
second term of the current appears then in an indeterminate form,
but we get finally the ordinary expression,
i=
The total resistance of the battery, and of the medium,
s
258 ENERGY OF CURRENTS.
measuring from the point P, is given (222) by the expression
R-
We can determine the constant Cj by the condition that this
resistance becomes equal to R x , for x = - , which gives
270. ELECTROCAPILLARY PHENOMENA. The preceding experi-
ments have shown that any modification of the surface of contact of
two bodies brings with it a variation in the electromotive force.
This may be considered as a general law, and we must assume
a priori that there is a relation between the electromotive forces of
contact of two bodies, and any other property dependent on the
state of the surfaces.
If, for instance, we use a surface of mercury as negative electrode
to decompose water, the mercury becomes polarized that is to say
that the difference of potential at the contact of the two liquids
increases with the external electromotive force until the disengage-
ment of bubbles of gas begins. The capillary properties of mercury
(that is to say its surface tension), depend simply on the state of the
surface, and must therefore change with the polarization.
M. Lippman's experiments have shown that this is the case.
The capillary tension of mercury in contact with acidulated water,
increases at first with the electromotive force of polarization until
this reaches 0.9 of the electromotive force of a Daniell's cell, and
then diminishes in proportion as the polarization increases.
Reasoning and experiment alike show that the converse of this
is true. If by any mechanical process whatever, the surface of the
mercury is deformed, and therefore the surface tension of contact of
the two liquids is made to vary, the difference of potential changes at
the same time ; during the deformation the potential varies in such a
way that the surface tension which corresponds to it tends to oppose
the motion produced.
SEEBECK'S DISCOVERY. 259
CHAPTER IV.
THERMOELECTRIC CURRENTS.
271. SEEBECK'S DISCOVERY. We have seen that a closed circuit
consisting of several metals at the same temperature, cannot give rise
to a current ; but this law no longer holds if the different parts of
the circuit, and particularly the solderings of the metals, are not
at the same temperature. The circuit is then traversed by what is
called a thermoelectrical current.
This important discovery was made by Seebeck in 1821.
In a circuit formed of a bar of bismuth, the ends of which are
joined by a strip of copper, the current goes from the bismuth to the
copper through the heated soldering ; the copper is then said to be
negative to the bismuth. With a couple antimony-copper, the current
is reversed it goes from copper to antimony through the heated
junction; the antimony is accordingly negative in reference to
copper.
It is natural to suppose that the metals could be classed in a
regular series based on this new property, and that antimony, which
is negative to copper, is much more negative to bismuth. This, in
fact, is what experiment shows, and the electromotive force for the
same temperature at the junctions, is greater with the couple
bismuth-antimony, than with either of the two couples bismuth-
copper or copper-antimony.
The electromotive force of a thermoelectrical couple may
be obtained by breaking the circuit at a point outside the
junctions, and determining the difference of potential at the two
ends.
In a circuit consisting of a single homogeneous metal, it is
impossible to set up an electrical current by variations of tempera-
ture, whatever may be the shape and section of the conductors near
the heated points. Currents might, however, be produced if the
s 2
260 THERMOELECTRIC CURRENTS.
metal has either a temporary or permanent dissymmetry in its physical
properties, on either side of the heated part.
272. LAWS OF THERMOELECTRICAL CURRENTS. Without dis-
cussing the experiments which demonstrate these special points, and
which have served to establish the laws of the phenomenon, we shall
confine ourselves to giving the laws themselves.
I. LAW OF VOLTA. There is never a current in any metallic
circuit all of whose points are at the same temperature.
For the algebraical sum of all the electromotive forces of con-
tact is necessarily zero since the metals obey the law of successive
contacts (189).
II. LAW OF MAGNUS. In any homogeneous circuit there is never
a permanent current, whatever may be the shape of the conductor, and
whatever the variations of temperature which exist between the different
points of the circuit.
This law leads to the conclusion, either that the variation of
temperature from one point to another determines no difference of
potential between these two points, or that this difference, if it
exists, only depends on the temperatures themselves, and not at all
on the law of variation.
From the hottest part of the circuit to the coldest, we find,
in fact, by two different paths, the same fall of temperature,
but with variations entirely independent on either side. If there
are variations of potential in the circuit, the sum of these
variations is null; hence between the two temperatures t and /',
the total variation of the potential must be the same on each
side.
It follows from the law of Magnus that the electromotive force
only depends on the temperature of the two junctions, and not
at all on the distribution of temperatures in the conductors which
separate them.
We shall represent by Ef(AB) the electromotive force of the
two metals A and B when the junctions are at the temperatures /
and /', the current going from A to B across the hottest junction at
the temperature /'. This electromotive force is a function of the
two temperatures t and /'.
III. LAW OF SUCCESSIVE TEMPERATURES (BECQUEREL). For a
given couple the electromotive force corresponding to any two tempera-
tures t and t' of the two junctions, is equal to the sum of the electro-
motive forces, which correspond to the temperatures t and on the one
hand, and B and t' on the other, being a temperature between the
two former.
LAW OF INTERMEDIATE METALS. 261
This law may be expressed as follows :
We have already learnt that the electromotive force only depends
on the temperature of the two junctions ; this latter law shows that
the electromotive force may be expressed by the difference of two
terms, one of which only contains the temperature t and the other /',
these two terms being the values of the same function of the tem-
perature.
We may then write
IV. LAW OF INTERMEDIATE METALS (BECQUEREL). If two
metals A and B are separated in a circuit by one or more inter-
mediate metals, with all intermediate junctions kept at the same
temperature /, the electromotive force is the same as if the metals
were directly connected, and the junction raised to the same tem-
perature t.
The law of intermediate metals may be expressed by the
equation
For if two metals A and B are connected at the hot junction by
an intermediate metal C, from the law of Magnus we may suppose
that a point P of this third metal is at the lower temperature /, and
interpose, in like manner, at the cold junction, a piece of the metal
C kept at the temperature of this junction. We have then the two
couples AC and CB in the circuit between the same limits of tem-
perature; the electromotive force is that which would be directly
produced between the metals A and B.
This law is of great practical importance; it shows that the
soldering at the junction of two metals has no influence on the
phenomena to which they give rise.
V. PHENOMENA OF INVERSION. In the case of some thermo-
electric couples, the strength of the current increases continuously as
the temperature of the heated junction is raised, that of the cold
junction remaining unchanged. The couple is said to work uni-
formly when the electromotive force is proportional to the difference
of the temperatures of the two junctions.
In most cases, on the contrary, the electromotive force of the
couple, after having passed through a maximum, becomes null, and
then changes its sign.
262
THERMOELECTRIC CURRENTS.
Hence, at a certain temperature, there is an inversion of the
current, and the strength then increases continuously without showing
a fresh inflection. This phenomenon was discovered by Gumming
in 1823.
Gaugain found that the temperature of inversion depends on
that of the cold junction, and that for a given couple the mean of
the temperatures of the two junctions at the moment of inversion is
constant and always equal to the temperature of the maximum strength.
273. GRAPHICAL REPRESENTATION OF THE PHENOMENA.
Gaugain, in a remarkable research on thermoelectrical phenomena,
represents their course by a graphical method by which the pre-
ceding laws may be readily verified. Taking for the abscissa the
difference t - / of the temperatures of the two junctions (the cold
one having a constant temperature of 20), he erects at each point
an ordinate proportional to the corresponding electromotive force.
o p x p
The following properties are observed in these curves (Fig. 63) :
i. They are symmetrical in reference to the maximum ordinate,
which verifies the law relative to the temperature of inversion ; for
if t m is the temperature of the maximum, and t { that of inversion,
'o + 'i
These curves are calculated by Gaugain to be branches of hyper-
bolas with a vertical axis, but they may be replaced by parabolas ;
the difference of the ordinates calculated for the two curves are
of the same order as experimental errors ; both represent equally
CONCLUSIONS FROM VOLTA's LAW. 263
well the results of experiment. Theory indicates, as we shall see
later, that the curve which represents electromotive forces as a
function of temperature must, in effect, be a parabola.
2. If a horizontal line is drawn through a point M 1? which
corresponds to the temperature / 15 the ordinates, counted from this
straight line, will represent electromotive forces relative to the tem-
perature /j for the cold junction. The law of successive temperatures
is thus found to be verified, for we have
that is to say,
The temperature of inversion corresponds to the point where
the new line of the abscissa meets the curve. If OP represents the
temperature of the cold junction, OP' will be that of inversion ; it
will be seen that it depends on the temperature of the cold junction.
3. If the curves AB and AC represent electromotive forces for
couples formed of a metal A associated respectively with two metals
B and C, the difference MN of the ordinates of the two curves
represents the electromotive force of the couple formed by the two
metals B and C. The relation MP = PN + NM is therefore equivalent
to the equation
E(AB) = E(AC) + E(CB),
which expresses the law of intermediate metals.
274. CONCLUSIONS FROM VOLTA'S LAW. Disregarding the
principle of inversion, we may look upon the preceding laws as
consequences of the principle of Volta that is to say, that there is
an electromotive force at the contact of two metals^ and that this elec-
tromotive force is a function of the temperature.
On this view, the electromotive force of a couple is the algebraical
sum of the two electromotive forces in contrary directions which
exist at the two junctions.
Let us agree to represent by the symbol the electromotive
force H of contact of two metals A and B, at the temperature /, we
shall have
264 THERMOELECTRIC CURRENTS.
Let I be the current which traverses the circuit whose total
resistance is R. In unit time, the work withdrawn from the heated
junction is IH 2 , and the work expended at the cold junction is
IHj ; the difference of these two works is transformed into thermal
energy, which is disengaged in the circuit in accordance with Joule's
law, and we have
whence
T _H 2 -H 1
R
The system may therefore be looked upon as a heat engine, the
boiler of which yields a quantity of heat Q 2 given by the equation
JQ 2 = IH 2 , while the condenser absorbs a smaller quantity of heat
Qj, defined in like manner by the equation JQ 1 = IH 1 , the difference
of these two quantities being employed to heat the circuit, from
which it follows that
The law of Magnus is contained in the hypothesis that there is
no electromotive force at the junctions.
The law of successive temperatures follows from the identity
B|A
Lastly, the law of intermediate metals is also evident, for, by
definition, we have
On the other hand, Volta's law of tensions gives, for any given
temperature,
B|C C|A_B|A
~~ " ~>
the preceding equation thus becomes
CONSEQUENCES OF INVERSION. 265
275. CONSEQUENCES OF INVERSION. The principle of Volta,
restricted to the contact of bodies of different kinds, is not sufficient
to explain the phenomena of inversion.
Let us consider, in fact, a circuit consisting of two metals A and
B. In order to account for inversion as a mere effect of contact,
we must assume that the difference of potential of the junction at
first increases with the temperature, passes through a maximum, then
diminishes, and, at the temperature of inversion, becomes equal to
the difference of potential at the cold junction. The value of H 2
would then continue to decrease; and next, the current having
changed its sign, the play of the electrical forces would produce a
disengagement of heat at the hot junction, and an absorption at the
cold one, besides the heating of the circuit in virtue of Joule's law.
We may imagine that the causes of the cooling of the circuit are so
diminished that it is possible to dispense with the source of heat, and
that the mere passage of the current would be sufficient not merely
to keep up the temperature of the hot junction, but even to increase
it, and to diminish that of the cold one, the effect of which would
be to intensify the current. In this way we should have realised a
metallic circuit possessing the remarkable property of transferring
heat from the colder to the hotter parts without any expenditure
of energy. And although such a result is not so obviously im-
possible as that of the impossibility of perpetual motion, it is
incompatible with the general course of thermal phenomena ; it is,
moreover, in direct contradiction with Carnot's principle.
If, on the other hand, thermoelectrical currents were merely
due to the electromotive forces at the junctions, Carnot's principle
would necessitate that all couples had a uniform course.
Let us imagine, for example, that a thermoelectrical couple
working between the temperatures / x and / 2 is connected with an
electrolyte whose electromotive force of decomposition is E; we
shall have the ratio
or
If the current I is very small, and the resistance R moderate,
the term I 2 R may be neglected, the opposing electromotive force E
is very little less than H 2 - Hj , and the excess of heat furnished
by the hot source is employed in performing the external work IE.
Let us suppose that, by any means, E is made to increase to the
266 THERMOELECTRIC CURRENTS.
value E', which is little more than H 2 -H 15 the direction of the
current would change ; if the absolute value of the strength remains
the same, the same quantities of heat would be put in play at
each junction, but in opposite directions, and the electrolyte would
produce heat instead of absorbing it. In the case of a very
feeble current, the thermoelectrical pile would behave as a re-
versible calorific engine, and we may apply the principle of Carnot.
If T x and T 2 are the absolute temperatures of the two junctions,
the quantities of heat Q 1 and Q 2 , absorbed or furnished by the two
sources according to the working of the machine, must be proportional
to the absolute temperatures T l and T 2 , and we should have
0,0,
>
or, A being a constant,
T T
1 2 1 l
From this would follow
1 - = = A,
T 2 -T X TJ-T!
and therefore
Hence the electromotive force of all couples should be pro-
portional to the difference in temperature of the two junctions ; all
couples would have a" uniform course, and the phenomena of in-
version could never be met with.
276. SIR W. THOMSON'S THEORY. Volta's principle is therefore
incapable of giving a complete explanation of thermoelectrical phe-
nomena ; we must accordingly assume the existence of electromotive
forces other than those of contact, and capable, like them, of pro-
ducing reversible thermal phenomena.
The least changes in the physical condition of metals, such as
tempering, torsion, or traction, etc., modify their electrical proper-
ties ; it is accordingly natural to assume that the contact of two
parts of the same metal at different temperatures also gives rise to
a difference of potential.
SIR w. THOMSON'S THEORY. 267
The electromotive force resulting from variations of temperature
is null in a homogeneous wire (law of Magnus), for the total fall
of potential on either side of the maximum is of the same value ;
but this compensation no longer holds on each side of the junction
of two different metals, and we must take into account the continual
change of potential which variations of temperature determine along
conductors.
To give greater definiteness to these conceptions, let us consider
a copper-iron pair, for example, working between the temperatures
/! and / 2 , and let Hj and H 2 (Fig. 64) be the electromotive forces
of contact at these two temperatures; suppose, further, that the
potential has increased along the copper C M , in consequence of a
rise of temperature from ^ to / 2 , by a quantity c independent of the
strength of the current ; and that conversely there is a fall of potential
!H2
1
Fig. 64.
fj on the iron Y e for the same excess of temperature ; the potential
near the hot junction will be higher by a quantity f+c=h, and the
electromotive force of the couple will now be
We have implicitly assumed that the temperature / 2 is lower than the
temperature of inversion. The current goes from copper to iron
through the hot junction; the thermal energy absorbed at the hot
junction, as well as on the adjacent points, is equal to (H 2 + ^)I, and
that which is expended at the cold junction H 1 I.
The lower temperature / x being fixed, the electromotive force
of the couple will increase as long as H 2 + h increases that is, so
long as
dh
dt dt
268
THERMOELECTRIC CURRENTS.
and the maximum will take place at the temperature / m , which is
evidently independent of f lt and is denned by the condition
H 2 dh A
- + =0,
dt dt
We shall see that at this instant the value of H 2 is zero, and
that it then becomes negative. The difference of potential near
the junction is then simply due to the variations of temperature
on the two metals (Fig. 65).
Cu
Fig. 65.
As the temperature continues to rise, H 2 changes its sign ;
the iron which was positive to the copper becomes negative; the
Fig. 66.
distribution of potential is represented by Fig. 66, and heat is
disengaged at the two junctions.
Inversion takes place at the moment at which
The electromotive force changes its sign at a higher temperature
at the heated junction, and we have
THOMSON EFFECT.
269
In this case, which is represented by Fig. 67, the current absorbs
thermal energy at the two junctions, IH 2 at the hot one, IH 1 at the
cold one, and a quantity \h is liberated at those points where the
temperature varies.
Such is a general idea of Sir W. Thomson's theory, the
mathematical consequences of which we shall proceed to develop.
We shall apply the term Thomson effect to the difference of
potential due to the differences of temperature which form the
basis of this theory.
;
1
F
Cu
Fc
Fig. 67.
277. THERMOELECTRICAL POWERS. We have seen that, by
the law of successive temperatures, the electromotive force of a
couple is the difference of the values of one and the same function
for the temperatures of the two junctions. If these temperatures
/ and t + dt are infinitely near, the electromotive force is infinitely
small, and is expressed by
= dt;
dt
we may therefore write
dt
Sir W. Thomson calls the function <J>(t) the thermoelectrical
power of the two metals at the temperature t. This function is
nothing but the angular coefficient of the tangent to Gaugain's
curves. We can deduce from it the electromotive force of the
couple for the temperatures ^ and / 2 of the two junctions by the
formula
278. This function possesses a remarkable property, in virtue
of which thermoelectrical phenomena may be very simply expressed.
270
THERMOELECTRIC CURRENTS.
The thermoelectrical power of two metals A and B at a temperature
t is equal to the difference of the thermoelectrical powers of the same
metals A and B in reference to any third metal C.
For the law of intermediate metals gives the equation
E(AC) = E(AB) + E(BC).
From which we deduce
^E(BC)
dt
dt
dt
or
and, therefore,
(2)
If, then, we know the thermoelectrical power of different metals
in reference to a standard metal X, it will be easy to deduce from
this the thermoelectrical power of any two metals by the formula
= </>(AX)-<HBX).
Fig. 68.
Let AX (Fig. 68) be the curve which represents the value of ^>
as a function of / for the two metals A and X, and BX the analogous
curve for the two metals B and X ; from equation (2) we shall have
THERMOELECTRIC POWER. 271
279. The expression for the electromotive force of the couple
AB between the temperatures / and f 19 is
E! (AB) = f (/)<//= f
jij A
it is therefore represented by the area of the quadrilateral comprised
between the curves AX and BX, and the ordinates corresponding to
the two temperatures ^ and /.
If, while the cold junction remains at a constant temperature t lt
the temperature / of the hot junction is increased, the electromotive
force increases with the corresponding area, until the temperature
attains the value t n1 which corresponds to the point of meeting of
the two curves.
At this temperature t n the thermoelectrical power of the two
metals is zero ; this is called the neutral point. When the tem-
perature exceeds that of the neutral point, the electromotive force
decreases, for it is now represented by the difference of the two
triangular areas which have their apex at I; it becomes zero, as does
the strength, at a temperature / 2 such that
area M 2 IN 2 = area MjINp
Lastly, as soon as the temperature of the hot junction exceeds
/ 2 > the electromotive force becomes negative and inversion takes
place. Hence the temperature of inversion depends on the tem-
perature of the cold junction.
280. The previous results become very simple when the curves
AX and BX are straight lines. The figure M^NN^ is then a
trapezium, the surface of which has the value
We have further
2
= const. = #,
and, therefore,
272 THERMOELECTRIC CURRENTS.
This expression is in conformity with the laws of Gaugain (272).
If the straight lines AX and BX were parallel, we should have
and the couple would have a uniform course.
281. SPECIFIC HEAT OF ELECTRICITY. Suppose now that the
variations of potential, to which electromotive force is due, are of
two kinds; sudden variations, resulting from Volta's principle, and
continuous variations connected with variations of temperature, and,
like the former, capable of producing reversible calorific phenomena.
It is clear that, if we designate the variations of the former kind by
H, and the sum of the continuous variations which exist between the
two points A and B of a conductor by I dh t the value of the whole
electromotive force will be
E =
The variations of the second kind between two points M and M'
of the same metal, according to the law of Magnus, only depend
on the temperatures / and /', and not at all on the intermediate
resistance. We may then put
If the current is so small that the heating of the circuit on
Joule's law may be neglected, the quantity of heat absorbed or
developed in unit time in that portion of the circuit in which is
produced the heat in question, by the passage of a current I, will
clearly be expressed by
Idh = If(t)dt=I<rdt.
The quantity a- is the variation of potential, and therefore the
thermal work for unit current which corresponds to a variation
of temperature equal to unity ; it is a characteristic function of the
nature of the conductor, but which varies with the temperature.
Sir W. Thomson has given the name specific heat of electricity to this
new physical quantity.
ELECTROMOTIVE FORCE OF A THERMOELECTRIC COUPLE. 273
282. ELECTROMOTIVE FORCE OF A THERMOELECTRIC COUPLE.
This being admitted, let us consider a circuit (Fig. 69) formed
of any number of metals A lf A 2 , . . . A n . Let H 15 H 2 , . . . H n be the
sudden variations corresponding to the electromotive forces of con-
tact; o- 15 cr 2 , . . . o- n the specific heats of electricity of the metals A lf
A! H! A., H. ? A B H tt A 1
t o-i t t <r a t 2 ar n t n o-j. t
Fig. 69.
A 2 , . . . A n ; finally, let f lt / 2 , . . . t n be the temperatures of the junc-
tions, and f Q the constant temperature of the external wire. The
expression for the electromotive force of the circuit is
f*i f a rtn rt
[ 2 ..+H n + 0y#+ <r. 2 dt. . + <r n dt + cr^df,
Jt Jti Jtn-l Jt n
or, combining the two extreme integrals,
ft. rt a rt n
/ _ \ "p ^/ T-T -I- I rr /// I I /// | _\__ I JA
\O/ <4M * 2 ^^ t \^ l .**'
}u ft J^.v
Suppose that the circuit only consists of two metals A and A',
the electromotive forces of contact H x and H 2 are in general of
opposite signs. Taking these signs into account, we shall have
(4)
PV-
A
If the difference of temperature of the two junctions is infinitely
small, we have / 2 - / t = dt> and the equation becomes
(5) -+cr'-<r=
at at
We have seen that for an infinitely weak current the circuit may
be regarded as a reversible heat engine; we may therefore apply
Carnot's theorem, and state that the algebraical sum of the quotients
obtained by dividing the quantity of heat absorbed at a point, by the
corresponding absolute temperature, is equal to zero.
274 THERMOELECTRIC CURRENTS.
Let T 2 and Tj be the absolute temperatures between which the
couple works, and T the absolute temperature of any given point
of the conductor. At the hot junction the calorific work is H 2 I ; at
the cold junction H-J ; on an element of the conductor taken between
the temperatures T and T + ^T, this work is IvdT. Hence, sup-
pressing the common factor, we shall have
a.a + r*
T 2 T, J Tl T
If the difference T 2 - T x is infinitely small and equal to *?T, we may
write
d
or
i </H H o-'-o-
---- + - = 0,
T *rr T 2 T
and, lastly,
H </H ,
= -- \-<r or.
T </T
From equation (5), the second member of this equation is
nothing but the thermoelectrical power < (/) of the two metals ;
we have then
(8) H = T*(/).
Thus, the electromotive force of contact of two metals, and
therefore the Peltier effect at any given temperature, is equal to the
product of the absolute temperature by their thermoelectrical power
at the same temperature.
We deduce from this same equation
and, therefore,
(9) E- TV/- p ?-
JT, JT, T
TAIT'S HYPOTHESIS. 275
283. The discussion of this formula leads to the different cases
the examination of which we have anticipated (276). Let T n be
the temperature corresponding to the neutral point ; for this point
the thermoelectric power is zero, we have then
<(/) -0 and H n = 0.
So long as the temperature T 2 of the hot junction is lower than
T n , the function H 2 is positive, and the current cools the hot
junction.
When T 2 = T n , the heating effect is null at the hot junction. If
the temperature T 2 is between the maximum temperature and that
of inversion T f , H 2 is negative ; the current heats the hot and the
cold junction at the same time.
Finally, if the hot junction is at a higher temperature than that
of inversion, the current cools the two junctions.
According to Sir W. Thomson, and in conformity with Gaugain's
experiments, the electromotive force of a couple may be empirically
represented by the formula
It follows from this that, for an infinitely small difference
between the temperatures of the two junctions,
and, therefore,
The electromotive force of the couple and the electromotive
force of contact between the two metals, expressed as a function
of the temperature, will therefore both be represented by parabolas.
284. TAIT'S HYPOTHESIS. Professor Tait arrived at the same
result by assuming that the specific heat of electricity o-, characteristic
of each metal, is proportional to the absolute temperature. Hence,
denoting by k and k' the constants for each metal,
T 2
276 THERMOELECTRIC CURRENTS.
In this case equation (7) becomes
and we deduce from it
T
We have further, for the neutral point, H n = 0, or
which gives, finally, putting k' k = a and replacing the absolute
temperatures by ordinary temperatures,
H = (K - )T(T n - T) = aT(t n - t),
In this way we rediscover the empirical formulae of Sir W.
Thomson and of Gaugain.
The thermoelectrical power of the two metals is then
it will therefore be represented by a straight line as a function of the
temperature.
Suppose we take the thermoelectrical powers in reference to the
same metal for which k is zero, which, according to Le Roux's
experiments, seems to be the case with lead, the equation reduces
to
ELECTRICAL CONVECTION OF HEAT.
277
The straight lines, which represent the thermo-electric powers
of the different metals, are unequally inclined to the axis of tem-
peratures; they cut this axis at the point corresponding to the
temperature of the neutral point with the metal compared, and their
inclination to the axis is the specific heat of electricity corresponding
to each of the metals.
285. ELECTRICAL CONVECTION OF HEAT. It was important to
verify experimentally the hypothesis which serves as the basis for this
theory that is to say, the existence of changes of potential due to
changes of temperature. The method employed by Sir W. Thomson
consisted in establishing that, by the passage of a current, reversible
calorific effects are produced analogous to the Peltier effect.
Let us consider a bar of metal (iron, for instance), the central
part of which AA' (Fig. 70), is kept at a constant temperature T,
while the ends B and B' are kept at o.
Fig. 70.
A' M' B'
The distribution of temperatures is represented by the curves
BDD'B'; so long as the current does not pass, the distribution is
obviously symmetrical, and is represented by curves such as BPD
and D'P'B'. The passage of a current produces two effects at each
point ; firstly, a heating regulated by the law of Joule ; secondly, a
disengagement or an absorption of heat produced by the fixed fall of
potential corresponding to the difference of the temperature of two
adjacent points.
If we only take into account Joule's law, the distribution of
temperatures is still symmetrical, and may be represented by the
dotted lines BQD, D'Q'B'.
The second effect is reversible with the direction of the current.
If the current goes from left to right with the arrow, there will be a
fresh heating in the anterior part of the wire BA, where the tem-
peratures increase, and a cooling in the posterior part where they
decrease. The distribution of temperatures is then dissymmetrical,
and may be represented by the dotted curves BRD, and D'R'B'.
278 THERMOELECTRIC CURRENTS. ,
At two symmetrical points, M and M', the difference of the
final temperatures t and /', corresponds exactly to the Thomson
effect. As the heating is greater in the part in front of the median
region AA', where the temperature is a maximum, there is a kind
of electrical convection of heat in a direction opposite to that of the
current, and this convection is proportional to the strength of the
current.
Sir W. Thomson found in this way that for iron the electrical
convection of heat is negative (that is to say, in the direction
opposite to the current), and that the convection is positive^ but far
feebler, for copper.
M. le Roux extended these same observations to a great number
of metals ; he proved that the effect is proportional to the strength
of the current, and ascertained that it is almost zero for lead, so that
from this point of view lead is sensibly neutral.
286. NATURE OF THE PELTIER PHENOMENON. We may now
discuss, with more accuracy, the Peltier phenomenon.
The electromotive force of contact between two metals is
expressed in Sir W. Thomson's theory by the general formula
If the course of the couple is uniform, we have, for the two
temperatures T T and T,
E = A(T-T 1 ),
from which follows
H = AT.
The electromotive force of contact between two metals is then
proportional to the absolute temperature, and the Peltier effect
should follow the same law. At the temperatures of 25 and 100,
for instance, we shall have
=i ( 75 _ T , T
'
H 25 273 + 25 298 4
According to M. le Roux's experiments, the bismuth-copper
couple exactly satisfies this condition.
NATURE OF THE PELTIER PHENOMENON. 279
A current which traverses a junction, heats it when it passes in
one direction, and cools it when it passes in the contrary one.
M. Becquerel showed that the direction in which there is cooling,
is that of the current which would produce the artificial heating of
the same junction. When a thermoelectrical current traverses a
circuit, the variations of temperature produced at the junction by the
current itself, tend then to become weaker, and we may say that their
effect is to develop an electromotive force opposed to that which
the current produces. That is a necessary condition ; if it did not
take place, an accidental current in a metallic circuit would produce
a difference of temperatures between the junctions, which would go
on increasing, and the current would maintain itself for an indefinite
time.
In a circuit of two metals the hot junction of which is. at a lower
temperature than the neutral point, the electromotive force increases
with the temperature, the Peltier effect tends then to diminish the
temperature of this junction. Beyond the neutral point, on the
contrary, the electromotive force diminishes when the temperature
increases, and the Peltier effect would tend to increase the tem-
perature of the hot junction.
The Peltier effect at the hot junction has, therefore, a different
sign according as the temperature of this junction is lower or higher
than that of the neutral point ; it follows from this that the electro-
motive force of contact must have changed its sign at the neutral
point. It is by analogous reasoning that Sir W. Thomson first
showed this property of the neutral point, and deduced from it the
necessity of the existence of electromotive forces in a homogeneous
conductor at variable temperatures.
280 PRELIMINARY.
PART III. MAGNETISM.
CHAPTER I.
\
PRELIMINARY.
287. ON MAGNETS. From the earliest times the name of load-
stone has been given to certain natural ores which have the property
of attracting iron filings ; they consist of an oxide of iron whose
chemical formula is Fe 3 O 4 . The various parts of a loadstone possess
these properties of attraction to unequal extents : the filings attach
themselves in preference to certain parts of the surface in the form
of tufts.
These phenomena have an evident resemblance to those of
statical electricity. The analogy, however, is not complete, and
observation indicates essential differences between them; thus, the
loadstone does not act indiscriminately upon all substances ; the
filings when attracted are not repelled after contact, and when
once detached are not found to possess any new property. At
each step in the course of this new study, we shall have to point
out analogies and differences of this kind between the two orders
of phenomena.
The loadstone, by mere rubbing, can magnetise steel that is to
say, can impart to it the property of attracting iron, and this without
losing any of its own power. As steel bars, magnetised artificially,
have a more regular form than loadstones, they are more convenient
for investigation ; experiment shows, moreover, that the phenomena
are of exactly the same nature in the two cases.
288. MAGNETS NATURAL AND ARTIFICIAL, PERMANENT AND
TEMPORARY. The general term magnet is given to all substances
which have the property of attracting iron filings. Natural magnets
are the pieces of magnetic ore found in nature; artificial magnets
are pieces of steel, or specimens of iron more or less pure, to which
the same properties have been imparted.
MAGNETIC AND DIAMAGNETIC SUBSTANCES. 281
Some artificial magnets retain this new property when the cause
which produced it has ceased to act ; these are permanent magnets.
Tempered steel is the body best suited for preparing permanent mag-
nets, and is ordinarly employed in the form of long rods or bars.
Different kinds of cast and of wrought iron may also be power-
fully magnetised by natural or by artificial magnets, but they lose
most of their properties when the magnetising agent has ceased to
act. In this way temporary magnets are obtained ; and the term
residual magnetism is applied to the comparatively feeble mag-
netisation which persists, at any rate for some time, in bodies
which have been temporarily magnetised.
289. MAGNETIC AND DIAMAGNETIC SUBSTANCES. Until the
present century, iron was the only substance which was known to
be attracted by magnets ; it was afterwards found that certain metals,
such as nickel and cobalt, whose chemical analogies with iron are so
remarkable, also possess these properties, but to a smaller extent.
It has further been found, by means of very powerful magnets, that
a great number of substances are also attracted by magnets, but the
actions are incomparably weaker. The term magnetic is applied to
all bodies which can be attracted by a magnet, and the term mag-
netism is applied to the whole of the phenomena to which magnets
give rise, and, by extension, to the cause of these phenomena.
In 1778, Bruginans observed that a piece of bismuth is repelled
by a magnet. The importance of this observation was disregarded,
until Faraday's discovery that a certain number of other substances
also possess this property. From the special way in which the
experiment was made, Faraday applied the term diamagnetic to
bodies which are repelled by magnets.
We may say, in short, that all bodies in nature are more or less
susceptible of the action of magnets. They have been divided into
two groups magnetic, paramagnetic, or positive substances, which are
attracted, like iron ; and diamagnelic or negative substances, which are
repelled by magnets, like bismuth.
290. DISTRIBUTION OF MAGNETISM IN MAGNETS. POLES.
We may obtain a bar magnetised regularly a needle, for example
by rubbing it several times, and always in the same direction, with
a piece of natural magnet, or with the same end of any artificial
magnet.
When such a needle is placed in iron filings, the particles attach
themselves more particularly to the ends of the needle, to a certain
distance from them, and they stick end to end to each other, forming
more or less abundant tufts.
282 PRELIMINARY.
Magnetic actions appear, therefore, to be concentrated at the
ends of regular magnets. We shall call these ends the poles of the
magnet, and shall afterwards define this term with more precision.
291. THE Two KINDS OF MAGNETISM. The two ends of the
magnet are not of the same kind; any magnet free to turn in a
horizontal plane takes up a fixed direction in space. This direction
is nearly north and south. When a magnet is displaced from this
position it reverts to it when left to itself, and it is always the same
end which points to the north. That end of the magnet which
points to the geographical north is called the north pole, and that
end which points to the south is the south pole. These poles may
be marked once for all on permanent magnets.
292. LAW OF MAGNETIC ACTIONS. Magnets act on each other'
The north pole of a magnet when presented to the north pole of
another magnet repels it, but on the contrary attracts a south pole.
In like manner two south poles repel each other. The phenomena
are therefore analogous to those of electrical action : Two poles of
the same kind repel, and two poles of opposite kinds attract, each other.
Hence there are four actions between two magnets which are
near each other : two, which are repulsive, between poles of the same
kind, and two attractive between different poles. If the magnets are
very long as compared with their transverse dimensions, and are
situated at a considerable distance in reference to these same dimen-
sions, the action of each end may be considered as concentrated in
a point. The reciprocal action of these two magnets consists, then,
of four forces directed along the straight line which join in pairs the
centres of action or the poles of the two magnets, and it is impossible
in experiments to reduce them to a smaller number.
Yet, by causing the poles of two very long magnets to act on
each other at distances, and in positions, such that the actions of the
two other poles may be neglected, Coulomb experimentally estab-
lished the law that the attractive or repulsive actions between two poles
are inversely as the square of their distance.
It may, however, be remarked that nothing proves the existence
of these elementary forces. Experiment shows, indeed, that the
reciprocal action of the systems which constitute the magnets, and
which are probably very complicated, may be reduced to attractive
or repulsive forces directed along the right line joining the poles ;
but as one pole can never be separated from its fellow pole, the
action of poles is a purely mental conception, advantageous no
doubt in representing and calculating phenomena, but without any
real existence as demonstrated by experiment. If it should happen
MAGNETIC MASSES. 283
that other views as to the nature of elementary actions lead to the
same conclusions, as regards the effects which we can measure
(if, for example, we abandon the idea of action at a distance),
we may consider these new views to be just as legitimate as
previous ones.
293. MAGNETIC MASSES. The action of two poles at a given
distance depends on the special power of each of the magnets.
Experiment shows that the actions exerted on the poles of two
magnets by a given system are in a constant ratio ; we may consider
this ratio as being that of the magnetic masses of the two poles. It
follows from this definition, that the action of any given system on a
pole, is proportional to its magnetic mass; hence the reciprocal
action of two poles is separately proportional to the mass of each
of them that is, to the product of their magnetic masses.
Calling these two masses m and m\ the action f, of the two poles
at the distance d, is
mm'
4> being a coefficient which depends on the choice of unit mass. In
order that this coefficient may be unity, we must take as unit mass
that of a pole, which, acting on an identical pole at unit distance,
exerts a repulsion equal to unit force. We have then
mm '
f = ~j?~
and the action is repulsive or attractive according as the poles are
of the same or of opposite kinds.
If two poles of masses m and m', are connected with each other,
the action of the system thus formed on a third pole M placed at
distance d, which is very great in comparison with that of the two
poles, is equal to
m'M
it is proportional to the sum m + m of the two masses if the poles
are of the same kind, and to the difference m-m' if they are
different Magnetic masses can be added like algebraical quantities,
and we may affix to them the signs + and - as we can to electrical
PRELIMINARY.
masses ; we shall agree to give the sign + to the magnetic mass of a
north pole, and the sign - to that of a south pole.
The action of two poles, expressed by the formula /= - , will
be positive in the case of repulsion, the masses being of the same
sign, and negative in the case of an attraction.
The law of elementary actions being the same as that for elec-
trical phenomena, we may apply all the theorems relative to electrical
potential, at any rate as regards fixed masses, and disregard, for
the present, phenomena relative to conductors. The consideration
relative to lines of force, to tubes and flows of force, are more
particularly directly applicable to magnetism.
294. MAGNETIC FIELD.- A magnetic field is a space in which
magnetic phenomena are produced. The direction and strength of
the field at a point, are the direction and intensity of the force
which would act on a positive magnetic mass equal to unity placed
at this point.
295. DEFINITION OF POLES. MAGNETIC Axis OF A MAGNET.
We have assumed in the foregoing remarks, that the actions of a
magnet reduce to that of two magnetic centres situate at the
ends; this is the case, but then only approximately, when we are
dealing with long cylindrical magnets, at a very great distance in
reference to their transverse dimensions. Magnetic properties are
really perceptible throughout the whole extent of the magnet, and
only exhibit a very marked maximum near the ends. This is readily
seen from the manner in which filings attach themselves to the
magnet. We must admit, therefore, that in the magnet there is a
series of magnetic masses, some positive and others negative, which
are distributed according to a certain law, and the whole of which
constitutes the total magnetic mass.
This being assumed, we may define more precisely what are
called the poles of a magnet.
Let us suppose the magnet placed in a uniform magnetic field.
The actions exerted by the field on the different points of the
magnet are parallel to each other, and for each volume-element
are proportional to the mass present there. All those which act on
the positive masses are in the same direction : they have a resultant
equal to their sum, and parallel to their direction, which is applied
at the centre of mass, or the centre of gravity of the positive masses.
The same is the case for negative masses, on which the field pro-
duces actions parallel to the preceding, but in the opposite direction.
The magnet is submitted to the action of two parallel and opposite
MAGNETIC MOMENTS. 285
forces, one applied at the centre of gravity of positive masses, and
the other at the centre of gravity of negative masses. These two
points of application are the poles of the magnet; the magnetic
axis of the magnet is the line joining the two poles, and the
direction of the magnetic axis is reckoned from the negative pole
towards the positive one.
The magnet is evidently in stable equilibrium when its magnetic
axis is parallel to the direction of the field, and pointing in the same
way ; equilibrium is unstable if these two directions are parallel but
in contrary directions.
296. THE MAGNETIC MASS OF A MAGNET is ZERO. The
vicinity of the Earth may be considered as a uniform magnetic
field.
Experiment shows, in fact, that throughout a region whose extent
is considerable in reference to the dimensions of the magnet, but
small compared with the radius of the Earth, all magnets, when
under the influence of the Earth alone, tend to assume the same
direction.
Coulomb showed, moreover, that the action of the terrestrial
field on any magnetised bar is purely directive that it has neither
vertical nor horizontal component ; it has no vertical component, for
the weight of a bar of steel is exactly the same before and after
magnetisation ; the horizontal component is also zero, for any
magnet which can move in a horizontal plane has no tendency
towards a motion of translation. The two forces of opposite directions
applied at the two poles are therefore equal, and constitute a couple.
From this follows this important conclusion that in any magnet
the sum of the positive masses is equal to the sum of the negative
masses ; in other words, the total sum of the magnetic masses is zero.
We have then always ^m = 0.
From this point of view the state of a magnet is comparable with
that which a dielectric, or an insulated conductor, acquires by
induction.
297. MAGNETIC MOMENTS. Let m be the absolute value of the
mass of each pole, and / the distance of the two poles ; the product
ml, of the mass by this distance, is called the magnetic moment M of
the magnet.
This magnet may be represented by a straight line OA (Fig. 71)
having for direction, the magnetic axis, and for length the numerical
value of the magnetic moment M.
This mode of representation amounts to supposing that all the
poles are identical, that their mass is equal to unity, for instance,
286 PRELIMINARY.
and to their being placed on the magnetic axis at a distance pro-
portional to the magnetic moment of the magnet in question.
When a system formed of several magnets connected with each
other is placed in a uniform field, the action of the magnet is
reduced to a couple; accordingly, in order to estimate the total
action, we may move all the magnets parallel to themselves, for in-
stance, in such a manner that all the negative poles are superposed.
Consider two magnets represented by the right lines OA and OA'
(Fig. 71), and let G be the middle of the line AA' that is to say,
the centre of gravity of two masses equal to unity, placed at A and A'.
The system is equivalent to a single magnet whose length is equal to
OG and the masses equal to 2, or to a magnet of double length OB
with masses equal to i. The resultant magnet is thus represented
by the diagonal of the parallelograms constructed on the right lines
OA and OA'.
Magnetic moments of magnets may therefore be compounded as
can forces. For any given system of magnets connected with each
other, the resultant moment is represented by the straight line which
closes the polygon constructed by adding, end to end, the moments
of all the magnets.
The projection of this line on any given axis being equal to the
sum of the projections of all the others, we see that the magnetic
axis of any given system, is the right line on which the sum of the
projections of the separate moments of the magnets constituting the
system is a maximum.
In like manner, a magnet may be replaced by any given number
of magnets, the resultant magnetic moment of which is equal to the
moment of the proposed magnet for instance, by the three pro-
jections of this magnetic moment on three rectangular axes.
When two magnetic systems are very distant from each other,
their reciprocal action is equal to that of the resultant magnets, for
each of the systems may be regarded as being situated in a uniform
field produced by the other magnet.
ASTATIC SYSTEMS. 287
298. ACTION OF A UNIFORM FIELD ON A MAGNET. If we con-
sider a magnet whose moment M equals ml, situated in a uniform field
whose strength F makes the angle 6 with the axis of the magnet,
the moment of the couple produced by the action of the field is
equal to Yml sin 6 or FM sin 6. This is the moment of the couple
which would tend to turn the magnet about a straight line perpen-
dicular to the magnetic axis and to the force of the field.
If the magnet is movable about a given axis, the couple of
rotation only depends on the projections M x and F x of the magnetic
moment and of the strength of the field, on a plane perpendicular to
the axis, for the projections on the axis are without influence. If B 1
is the angle of the directions of M x and of F 1? the moment of the
couple is equal to FjMj sin # r
Let, generally, #, , c and a, /?, y denote respectively the cosines
of the angles which the directions of F and M make with three
rectangular axes, and let us replace these magnitudes by their pro-
jections on the three axes. The moment of the couple which tends
to turn the magnet about the axis of z is
Z = F. Ma -F
We shall have, similarly, for the other axes,
The product FM is sometimes called the moment of the action
of the field upon the magnet ; it is the moment of the couple which
would be produced if the magnet were perpendicular to the direction
of the field. As a particular case, if T is the intensity of the
terrestrial field, the product TM will be the moment of the terrestrial
action on the magnet.
299. ASTATIC SYSTEMS. Take the particular case of two mag-
nets (Fig. 72) whose magnetic moments are OA and OA', the
Fig. 72.
resultant moment is the diagonal OB of the parallelogram drawn on
OA and OA'. If the moments OA and OA' are equal and exactly
288 PRELIMINARY.
opposite, the resultant moment is null, and the equilibrium is neutral
in any position whatever in a uniform field ; such a system is said to
be astatic.
If the moments OA and OA' are almost equal and make an
angle nearly equal to 180, the resultant OB is very small and is
directed sensibly in the direction of the line bisecting the angle
AOA'; it is therefore perpendicular to each of the needles. Thus,
when two magnetised needles, forming a quasi astatic system, are
in a uniform magnetic field, they are in stable equilibrium in a
direction at right angles to the force of the field.
This is exactly the case of magnetic needles used for certain
galvanometers. The system is so much the more nearly astatic the
more nearly the direction of the free needles is to being perpen-
dicular to the magnetic meridian.
300. MAGNETIC POLARITY. RUPTURE OF A MAGNET. When
a magnetic needle is broken, each of the portions becomes a com-
plete magnet having two equal poles of opposite kinds, and the
phenomenon can be repeated indefinitely as far as we can carry the
division by mechanical means. This is a fact of prime importance
in the theory of magnetism : it proves in the first place that it is
impossible to get an independent mass of negative or positive
magnetism which is not associated with an equal mass of the
opposite kind ; and further that magnetism is an essentially molecular
phenomenon. We are led to admit that magnetism is due to a
kind of polarization of ponderable molecules, each of which is a
small magnet with its two poles exactly on the terminal faces.
301. INDUCED MAGNETISATION. The tufts of iron filings which
remain adhering to a magnet prove that each particle of filing has
itself become converted into a small magnet. The number of grains
in direct contact with the magnet is relatively very small ; the others,
attached in succession to each other and to the first, form chains
where the particles are united by their poles of contrary names.
The magnetisation acquired by these filings is transient; as soon
as they are detached from the magnet, they resume their original
neutrality. In like manner, a bar of soft iron is magnetised when
placed in the prolongation of a magnet, and acquires two poles
similarly placed to those of the magnet that is to say, the two
adjacent ends of the magnet and the soft iron have magnetisms
of opposite signs. This magnetised soft iron may in turn act
similarly on a second piece, and so on. As soon as the original
magnet is removed-, the magnetisation of the first bar of soft iron
and of all those which follow it disappears more or less, and all the
COERCIVE FORCE. 289
actions, which they exert on each other, disappear at the same
time.
More generally, when any magnetic body is placed in a mag-
netic field, it becomes itself a magnet. This is a magnetisation by
influence or induced magnetisation.
The axis of magnetisation at each point is parallel to the direction
of the resultant force. This resultant arises from the action of the
field and of that which is produced by the induced magnetism itself.
If the body in question is infinitely small, the magnetisation is exactly
parallel to the force of the field at the point in question.
This conclusion also follows, that the action of a magnet is
null on a neutral body, and that any action exerted by magnets
on magnetic bodies, is preceded by a magnetic induction on
the latter.
Here again we see the analogy of this phenomenon with that of
electrostatic induction, and particularly the induction in dielectrics.
The magnetism thus induced does not depend merely on the
strength of the field, but also on the nature of the substance in
question; magnetisation, which is very powerful with pure iron
and nickel, is far feebler with all other magnetic substances.
302. SOFT IRON. COERCIVE FORCE. Iron is said to be abso-
lutely soft if, after having been placed in a very powerful magnetic
field, it loses its magnetisation when it is withdrawn from it. Soft
iron, in the magnetic sense, is also soft in the ordinary meaning
of the word ; it may be easily bent, worked, and it has but little
elasticity. Conversely, ordinary iron is not soft in the magnetic
sense of the word ; when it is impure, or has undergone mechanical
changes, it remains more or less magnetised, and this property is
designated by the somewhat barbarous term coercive force. A speci-
men of iron has so much the greater coercive force the greater is its
quantity of residual magnetism; at the same time, this kind of
iron is more readily magnetised by induction. Coercive force is
therefore a property analogous to friction. Within certain limits it
opposes the changes which external forces tend to produce in mag-
netisation, and hinders any one single state of equilibrium from
corresponding to given external conditions.
The coercive force in steel is very great ; it becomes magnetised
by induction with more difficulty than soft iron, but it retains the
magnetisation, once acquired, so much the better. The magnetic
qualities of steel vary with the composition of the metal and with its
mode of preparation ; they depend greatly on the manner in which
the tempering has been effected, as well as on the degree of annealing
2QO PRELIMINARY.
to which the bar has been afterwards subjected. The harder the
steel, and the more brittle the temper, the greater is the coercive
force.
We have hitherto implicitly assumed that the magnetisation of a
magnet is invariable, and is independent of the forces to which the
magnet is subjected ; but this kind of magnetic rigidity is a limiting
case which is never realised with perfect completeness. When a
magnet placed in a strong magnetic field is in its normal position of
equilibrium, its magnetisation slightly increases ; it diminishes on the
contrary if it is in the opposite direction. The variations thus pro-
duced are generally feeble, and usually transient like those of soft
iron in the same circumstances ; these variations may ordinarily be
neglected in the case of powerfully magnetised steel bars placed
in a magnetic field of no great strength like that of the earth, for
instance.
303. INFLUENCE OF TEMPERATURE. Heat acts also on the
magnetism of magnets. A moderate increase of temperature
diminishes the magnetisation, but only temporarily, and the magnet
resumes its original magnetisation with its original temperature.
Within the ordinary variations of the surrounding temperature, the
effects produced are sensibly proportional to these variations, so that
if M and M t are the magrietic moments of a magnet at the tempera-
ture of zero, and of / degrees, we have the ratio
the coefficient a depending on the nature of the steel.
A greater degree of heating, above 100 for instance, produces a
definite enfeeblement of the magnetisation, and a bar of steel heated
to bright redness has usually lost all traces of magnetisation when it
returns to the ordinary temperature.
A rise of temperature produces analogous effects on the magnetic
properties of soft iron. At the ordinary temperature the magnetisa-
tion induced in iron by a given field, changes but little with variation
of temperature, but beyond 100 the diminution of induced mag-
netism becomes very rapid. At a temperature beyond red heat, iron
no longer possesses the power of being attracted by magnets ; at that
temperature it is not even magnetic.
304. ON MAGNETIC FLUIDS. The physicists of last century,
more especially yEpinus and Coulomb, attempted to explain
magnetic phenomena by a hypothesis analogous to that of electrical
fluids.
TERRESTRIAL MAGNETIC ELEMENTS. 2QI
From this point of view we must attribute to the fluids, to the
magnets, and to magnetic substances, a certain number of properties
by which all the experiments may be explained.
We assume, then, the existence of two imponderable magnetic
fluids, consisting, like the electrical fluids, of molecules which act by
repulsion on the molecules of the same fluid, and by attraction on
molecules of a different kind, these reciprocal actions being inversely
as the square of the distance. The combination of these two fluids
in equal quantities has no action on external bodies, and constitutes
what may be called the neutral fluid.
In virtue of the phenomena of induced magnetisation, we must
assume that the neutral fluid exists in almost unlimited quantity in
magnetic bodies, and is divided into two distinct fluids under the
influence of the magnet.
Since permanent or temporary magnets are always complete,
whatever may be their dimensions, we must assume also that the
fluids present in an element of volume never quit it to pass to an
adjacent element, so that the separation of these fluids is confined to
the extent of each molecule.
Finally, no internal force opposed to the directive actions of the
magnetic fluids, hinders their separation or their reunion in soft iron.
In cast iron and in steel, on the contrary, there is a special resistance
(a kind of friction called coercive force), which restricts the magnetisa-
tion by induction, and then hinders the recombination of the fluids
when the external force has disappeared.
It is not surprising that the theory of fluids, with all the acces-
sories which are only arbitrarily connected with it, can explain the
phenomena ; in these conditions the agreement of experiment with
theory affords no argument in favour of the exactitude of the
hypothesis ; and we shall make no further use of it.
305. DEFINITION OF TERRESTRIAL MAGNETIC ELEMENTS.
The magnetic field which surrounds the earth, and which may be
called the terrestrial field, is sensibly uniform throughout a space of
small dimensions as compared with that of the terrestrial radius ; but
the direction and the intensity of the force vary from one point to
another. In fact the force in any one place changes in magnitude
and direction in the course of time ; we shall disregard for the
moment these variations which are feeble, and shall suppose we are
considering the magnetic state of the globe at a definite time.
The magnetic axis of a magnet suspended freely by its centre of
gravity, and withdrawn from any other action than that of the
terrestrial magnetic field, would, when in equilibrium, assume the
u 2
PRELIMINARY.
direction of the terrestrial forces. In our country this direction
is almost north and south, and it makes a considerable angle with
the horizontal line, the north pole pointing downwards.
The magnetic meridian in any place is the vertical plane passing
through the direction of the earth's magnetic force.
The declination is the angle which the magnetic meridian makes
with the astronomical meridian ; the declination is said to be west
when the north pole of a free magnet turns to the west of the
magnetic meridian which passes through its centre ; it is east if this
north pole is to the east of the meridian.
The inclination is the angle which the earth's force makes with
its projection on a horizontal plane.
Let
D be the declination,
I the inclination,
T the strength of the earth's field,
H the horizontal component = T cos I,
Z the vertical component = T sin I.
A magnetised needle, movable about a vertical axis, will only
obey the horizontal component of the earth's force, and will place
itself so that its axis of magnetisation is in the magnetic meridian.
If we move it out by an angle 8, the moment of the couple which
tends to bring it back, has the value
HMsinS;
M being the magnetic moment of the needle. It is proportional to
the sine of the angle of deviation. This result has been verified by
the very accurate experiments of Coulomb by means of his torsion
balance.
If the needle is suspended freely by its centre of gravity, or is
movable about a horizontal axis passing through this point, and
perpendicular to the magnetic meridian, the direction of the mag-
netic axis, when the needle is in equilibrium, is the direction of the
earth's force itself; the angle which its magnetic axis then makes
with the horizontal measures the inclination.
Let us now suppose that the horizontal axis of rotation makes
an angle a with the perpendicular to the magnetic meridian. We
may replace the horizontal component H by its two projections, the
one H sin a parallel to the axis of rotation, and other H cos a per-
pendicular to this axis. The needle only obeys the two forces Z and
TERRESTRIAL MAGNETIC INDUCTION. 293
Hcosa situate in the plane which it describes; the value of the
resultant of these two forces is
and the needle, in its position of equilibrium, makes with the hori-
zontal an angle /, defined by the equation
Hcosa
cot i = - = cotlcosa.
The angle * is the apparent inclination in the vertical plane which
is at an angle a with the magnetic meridian. If we have a = - , it
7T 2
follows that /=- and the needle is vertical. If the angle a changes
by - , we have cos ( a- J = +sina, and the value of the new in-
clination is
cot** = + cot I sin a.
From these two equations we have,
COt 2 / + COt 2 /' = COt 2 !,
a formula frequently used in determining the inclination.
If the needle is loaded with an accessory weight, or, what is the
same thing, if the axis of rotation does not pass through the centre
of gravity, the direction of the equilibrium is modified.
Suppose, as a particular case, that a weight /, at a distance d
from the axis of rotation, keeps the needle horizontal in a certain
plane ; the moment of the magnetic couple is reduced then to the
moment of the vertical component, and the condition of equilibrium
is
This condition is independent of the azimuth of the vertical plane in
which the needle moves ; the counterpoise which makes the needle
horizontal in one plane, would make it horizontal in all planes.
Hence, placing a needle on a vertical pivot, we may counterpoise it
so that it is always horizontal ; but the weight of the counterpoise
depends on the vertical component, and this ought to be modified
if we wish to use the needle in other latitudes.
2Q4 PRELIMINARY.
306. DISTRIBUTION OF TERRESTRIAL MAGNETISM. The ele-
ments of terrestrial magnetism, strength, declination, and inclination
are not the same at the different points of the earth. These elements
vary as a function of the geographical co-ordinates according to very
complicated laws ; but if we are content with a first approximation,
the variations may be formulated in a very simple manner.
The magnetic meridian at any one place cuts the surface of the
globe along a great circle; all points of this great circle have the
same plane for magnetic meridian.
All magnetic meridians intersect along the same diameter ; this
diameter is the magnetic axis of the earth ; the points where it cuts
the surface have been named, though incorrectly, magnetic poles.
The magnetic axis makes an angle of about 15 with the axis of
rotation of the earth.
It is evident that the declination varies from one point to
another on the same magnetic meridian. The only exception
is the meridian which, passing both through the magnetic axis
and the terrestrial axis, is identical with the geographical meridian ;
for all corresponding points the declination is null. On one side
of this great circle, the north pole turns to the west and the
declination is west; on the other, it turns to the east and the
declination is east.
The great circle perpendicular to the magnetic axis is called the
magnetic equator. In all points of the magnetic equator, the earth's
force is horizontal and the inclination zero. On either side the
inclination increases to the magnetic poles where it is 90: in the
northern hemisphere, the north pole dips downwards ; and in the
southern hemisphere, the south pole.
307. HYPOTHESIS OF A TERRESTRIAL MAGNET. Biot tried if
it were possible to represent the magnetic condition of the globe,
and the variation of the magnetic elements on its surface, by the
hypothesis of a central magnet in the direction of the magnetic axis ;
he found that the results of calculation agreed the better with the
observations the smaller was the distance of the poles of this
imaginary magnet.
If we thus replace the earth by a magnet which is infinitely small
as compared with the radius that is to say, by two equal masses
of opposite signs which are very near each other, we know (153)
that at the latitude A, counting from the magnetic equator, the
inclination is given by the equation
tan 1 = 2 tan A.
HYPOTHESIS OF A TERRESTRIAL MAGNET. 295
The intensity of the force T at any point of the surface may be
expressed as a function of the force at the equator T e by the formula
At the magnetic pole the intensity is
it is therefore twice as great as at the equator.
These two formulae are at any rate approximately in agreement
with observations made at a given moment over the whole surface
of the earth.
The absolute magnetic moment of the earth rs may be obtained
simply by means of the equation
TR 3
in which R represents the earth's radius.
The hypothesis of a terrestrial magnet was introduced into
science by Gilbert. The pole in the southern hemisphere received
the name of austral pole, and is evidently of the same kind as the
north pole of magnets ; the pole situated in the northern hemisphere
is of the same kind as the south pole of magnets.
This conception of a terrestrial magnet has also led to the
designation of austral pole being applied to that pole of the needle
which turns towards the north, and of boreal pole to that which
turns towards the south; in the theory of fluids, in spite of the
contradiction, we may say that a north pole contains austral fluid
and a south pole boreal fluid. But it is better to abandon the
expressions austral and boreal, which may give rise to misconcep-
tions, and to call, as we have done, positive magnetism that which
corresponds to the north pole of magnets, and negative magnetism
that of the south pole.
308. The hypothesis of an infinitely small central magnet is
only one of the forms under which the earth's magnetism may be
represented ; it is even that which is least probable, seeing that the
undoubtedly very high temperature of the centre of the earth is
incompatible with the existence of bodies strongly magnetised.
We know, for instance, that two superficial hemispherical layers
equal and of opposite signs, distributed so as to produce a constant
296 PRELIMINARY.
force at any point in the interior, and which we have called layers
of gliding (157), will produce on the exterior the same effects as
two infinitely near masses. The values of the densities of these
layers at the poles will be
and, at a point of the magnetic latitude A,
o- = cr sinA.
From this point of view, the earth must be considered as covered
with two magnetic layers, the one negative in the northern hemi-
sphere and the other positive in the southern one, the density at
each point being proportional to the sine of the magnetic latitude.
The total mass of each of the layers is expressed by
it may therefore be easily calculated if we know the absolute value
of the force at the equator.
We shall see, in the sequel, that there are other modes of repre-
senting terrestrial magnetism; the infinitely small central magnet,
which of itself is inadmissible, is really the very simple mathematical
expression of several equivalent states, which are quite compatible
with the known properties of magnetic substances.
309. VARIATIONS OF TERRESTRIAL MAGNETISM. The elements
of the earth's magnetism also undergo changes with the time ; one
kind are purely accidental, while others have a well-marked periodical
character. The variations of long periods, which are called secular
variations, may be represented, as a first approximation, by a rotation
of the magnetic axis about the earth's axis, a rotation in virtue of
which the magnetic axis should describe from east to west a cir-
cular cone of about 30.
As the earth's magnetic pole, which at present is in New South
Wales in 100 W. longitude, was in 1660 near the North Cape in 20
E. longitude, we see that the period of complete revolution is about
800 years. The declination at Paris, which at first was east, was
null in 1666; since this time it has been west, and went on in-
creasing until 1824; it is now decreasing and will be null in 2050,
VARIATIONS OF TERRESTRIAL MAGNETISM. 297
if the phenomenon continues to follow the same course ; the mag-
netic pole will then be on the other side of the north pole in
reference to us. Since 1666 the inclination in Paris has been
continually decreasing : it will attain a minimum when the declination
is null.
The variations with a short period seem to be connected with
the apparent motion of the sun, of the moon, etc., and are governed
by laws which at present are not well known.
The mean values of the declination, for instance, have in one
and the same place a well-marked daily oscillation with two maxima
and two minima. The amplitude of the excursion of the magnet is
far greater during the day than during the night, and the time of the
extreme variations is very different according to the positions of the
stations. Thus, while the average maximum westerly deviation
over a whole year is at 9 a.m. at Hobart Town (Tasmania), Batavia,
the Cape, and St. Helena, this time corresponds to the maximum
easterly variations in the northern hemisphere. The maximum
westerly excursion is at i p.m. at Toronto (Canada), London, and
Paris; at 2 o'clock at St. Petersburg; at 3 at Nertchinsk and
Pekin. The hours of these maxima and minima vary, moreover,
with the seasons. The other magnetic elements, inclination and
components of the force, present analogous oscillations.
By eliminating the mean daily variation from observations rela-
tive to the various magnetic elements, we may refer them to the
lunar day, and we thus find a regular variation in the residual effects.
There is, further, an annual periodical variation.
Finally, the accidental variations themselves, which seem to
be produced simultaneously over a great extent, if not over the
whole surface of the globe, and which are ordinarily known as
perturbations or magnetic storms, appear also to occur in certain
annual or secular periods as regards their main effects/ These
perturbations are directly related to the phenomenon of the aurora
borealis, and are accompanied by accidental currents in telegraph
wires.
298 CONSTITUTION OF MAGNETS.
CHAPTER II.
CONSTITUTION OF MAGNETS.
310. MAGNETIC FILAMENTS. The experiment of breaking a
magnetised bar demonstrates this central fact that any volume
element of a magnet is itself a complete magnet, having in its then
state a magnetic axis and a definite moment. We say in its then
state, for it is clear that if the volume element instead of being
conceived as separated from the surrounding medium, were really
detached, it would no longer retain the same state as when it formed
part of the general mass. Let us consider two molecules placed
end to end, and only touching with their opposite poles ; if they
are equally magnetised, the action for any external point would
reduce itself to that of its two free ends. In like manner, if a
series of molecules equally magnetised are placed end to end, all
the magnetic axes being arranged on the same line, the external
action of the linear magnet thus constructed still reduces to that
of its two ends, each intermediate point giving rise to equal and
contrary actions which neutralise each other. Such a system of
magnetised particles constitutes a uniform magnetic filament.
311." FREE MAGNETISM. But if the magnetisation in this line
of molecules is variable, at each point there will be a certain quantity
of apparent or free magnetism, equal to the differences of the mag-
netic masses of two adjacent molecules in contact. If we suppose,
for instance, that the magnetisation diminishes from the middle of
the filament to the end, we see that on one of the halves of the
linear magnet, there will be an excess of positive magnetism distri-
buted according to a certain law, and on the other half an equal
excess of negative magnetism. The magnetic filament thus con-
structed is no longer uniform, but it is evident that we can regard it
as the resultant of the juxtaposition of uniform filaments of different
lengths.
POTENTIAL OF A MAGNET. 299
312. UNIFORM MAGNET. A magnet of finite dimensions, which
is formed of identical filaments placed parallel to each other,
might be called a uniform magnet; the poles of the elementary
filaments being placed at the ends, on the surface of the body, it
will be seen that the action of the whole magnet would reduce to
that of two magnetic layers, distributed on the surface according to
a simple law.
313. ANY GIVEN MAGNET. At a point P within any given
magnet the magnetic axis has a determinate direction, and this
direction varies continuously; hence, inside a magnet we may
draw lines tangential at every point to the magnetic axis, and
we may imagine magnetic filaments directed along these lines of
magnetisation.
The magnet would thus be subdivided either into non-uniform
magnetic filaments closed or terminating at the surface, or into
uniform filaments, some closed, and others terminating on the
surface, and others, lastly, terminating in the interior. So long as
no hypothesis is made as to the form of the filaments, this con-
ception is a pure and simple translation of facts, and has nothing
hypothetical. It leads to considering any given magnet as formed of
a magnetic layer distributed on the surface, and of magnetic masses
disseminated throughout the interior. We may accordingly consider
a surface density of magnetisation and a volume density.
The density of the free magnetism at a point is the limit of the
ratio of the magnetic mass contained in a volume element taken
about this point to the volume itself; the surface density is the
quotient of the quantity of magnetism which exists on an element
of surface about this point, by the area of the element
314. POTENTIAL OF A MAGNET. This being admitted, it is
clear that the value of the potential of the magnet at any external
point P will be
(i) V
=/>/'
In the first integral, which will extend to the entire surface of
the magnet, o- denotes the surface density on the surface element
</S, whose distance from the point P is equal to r. The second
integral must be extended to the whole volume of the magnet, p
denoting the volume density of magnetism in the element dv at a
distance rv from the point P. These densities p and a- may be
considered as those of a particular fluid.
300 CONSTITUTION OF MAGNETS.
As the sum of the magnetic masses is always zero for any given
magnet, we have the equation of condition
\pdv.
The components of the magnetic force at the point P are
W
C. L, CORY. z Jj ;
and the value of the force itself is
These formulae are general, and apply not only to points on the
exterior but also to points in the interior of the magnet.
It can be shown that, as in electricity, the sum AV of the three
partial secondary differentials of the potential is equal to zero for any
external point, and to -4^ for any point inside the magnetised
bodies.
The fundamental equations are the same as for statical electricity.
Hence, without a fresh demonstration we may apply the theorems
already established, provided that these theorems do not depend on
the properties of conductors, and that, on the other hand, the
coercive force does not come into play.
315. A MAGNET is EQUIVALENT TO A MAGNETIC SURFACE.
We may demonstrate directly Poisson's theorem, that the action of
a magnet on an external point is equivalent to that of a fictive layer
of a total mass equal to zero, distributed along the surface according
to a certain law.
Suppose that, as a matter of fact, the masses in question are fixed
electrical masses, and that the magnet is covered with an infinitely
thin conducting surface in contact with the ground. On the inner
surface of the conductor a layer of opposite sign to the internal
POISSON'S THEORY. 301
masses will be developed by induction; as the force and the potential
are now everywhere null on the exterior, the layer induced has, on
any external point, a potential equal and of opposite sign to that of
the original masses. A layer equal and of the opposite sign dis-
tributed on the external surface according to the same law, will
produce a potential equal and of the same sign as those of the
system in question, and will form therefore a system equivalent for
external points. This conclusion obviously applies to magnetism,
the two kinds of masses obey the same elementary law.
The external actions exerted by the magnet enable us to calculate
the density of this fictive surface layer, but will teach nothing respect-
ing the real distribution of magnetism. In order to investigate this
distribution, it would be necessary to determine experimentally the
forces which act on the interior of the magnet, and make cavities
into which test needles could be introduced; but the withdrawal of
a mass, however small, modifies the force in the cavity, for the
adjacent masses are suppressed, whose effect cannot then be
neglected. The magnet is equivalent then to two fictive layers,
one on the outer and the other on the inner surface of the
cavity; the sum AV of the three secondary differentials of the
potential is become null in this region, while it was originally
equal to -47rp.
316, POISSON'S THEORY. Hitherto we have made no hypothesis
as to the manner in which magnetic masses are distributed in the
magnetised substance. To establish the theory of magnetisation by
induction, Poisson considers a magnetised body as made up of
magnetic particles disseminated in a medium impervious to magnet-
ism. These particles are spherical and equidistant if the body is
isotropic and homogeneous ; each of them contains equal quantities
of positive and negative fluids, part in the neutral state in the interior,
and part in the free state on the surface. The magnetic moment of
each particle, of volume u, may be represented by uq, the factor q
depending on the degree of magnetisation. If we consider a volume
dv> which is very great compared with the dimensions of the particles,
but infinitely small in reference to the dimensions of the magnet, all
the particles which it contains will have their magnetic axes sensibly
parallel, and the magnetic moment of the volume element will be the
sum of the moments of the particles. Calling ^, as we have already
done (167), the ratio of the space occupied by the particles to the
total volume dv, the total volume of the particles contained in this
element is proportional to hdv> and its magnetic moment will be
hqdv. This element will act on any point at a finite distance like
302 CONSTITUTION OF MAGNETS.
an infinitely small magnet, or like the system of two infinitely near,
equal, and contrary masses (151). The magnetic moment of the
magnet for unit volume is equal to hq.
The value of the ratio h varies with different magnetic bodies,
and for the same body the value of q at each point depends on the
degree of magnetisation ; external actions increase or diminish with
the product hq. In bodies which have no coercive force, nothing
prevents the movement of fluids in the interior of a magnetic particle ;
equilibrium can only exist when the resultant of all the forces,
internal as well as external, is zero for every point of the molecule ;
on the contrary, in a body endowed with a certain coercive force,
which acts like friction, it is sufficient if this resultant be less than
the value given for the coercive force.
Poisson's theory is not bound up with the hypothesis of two
fluids, but it is more difficult to free it from this particular concep-
tion of the structure of magnetic media.
317. SIR W. THOMSON'S THEORY. We shall prefer to explain
the theory of magnetism in the form given to it by Sir W. Thomson.
This theory agrees with that of Poisson in its essential results, but
it has the advantage of being independent of the idea of fluid, and
of any hypothesis on the constitution of the medium, so that it seems
to be in closer agreement with experimental facts. The fundamental
notion is to consider any given portion of a magnet as being a
complete magnet, defined by the direction of the axis and by its
magnetic moment that is to say, as an infinitely small magnet
having masses + m and - m at its ends, a length <&, and therefore a
magnetic moment equal to mds.
318. INTENSITY OF MAGNETISATION. That being admitted, the
term intensity of magnetisation I at a point, is the quotient of the
magnetic moment of a volume element by the volume itself in
other words, the value of the moment for unit of volume.
We shall have thus
mds
This intensity of magnetisation I, represents the product hq in
Poisson's theory.
The intensity of magnetisation is a geometrical magnitude defined,
like a force, by its direction, which is the magnetic axis of the
volume element, and by its numerical value ; it will therefore be
represented at every point by a straight line of given direction and
magnitude.
EXPRESSION FOR POTENTIAL. 303
All magnetic phenomena may be expressed as a function of this
quantity alone.
319. EXPRESSION FOR POTENTIAL. Let I be the intensity of
magnetisation at a point M of the magnet whose co-ordinates are x,
y, and z. If the intensity of magnetisation makes, with the axes,
angles whose cosines are A, ^ v, its components A, B, C, along the
axes will be expressed by
A-U,
The magnetic moment of a volume element is mds = Idv. Its
potential at a point P at a distance r along a right line, making an
angle B with the direction of the magnetic axis (that is to say, with
the direction of the strength of magnetisation), is equal to (151)
This potential may be regarded as the sum of the potentials
dVtf dV b , dV c , due to the three components A, B, C, of the
magnetisation. If we denote by 8 the angle which the right line
MP makes with the axis of x, and by f, yu, , the co-ordinates of the
point P, we have,
On the other hand, the equation
gives
and, therefore, _ I
> > * ~
-x i^r r
From these we deduce
304 CONSTITUTION OF MAGNETS.
In like manner we shall have
c
Of
The potential of the entire magnet will be obtained by extending
these expressions to the whole volume, which gives
The potential at the point P is thus expressed as a function of
the distance r of this point from the different elements of volume of
the magnet, and of the intensity of the magnetisation.
Each of the terms of which the second member of the equation
(3) is composed contains a factor which is an exact differential, and
may be integrated by parts ; we then obtain
the former integral should be extended to the whole surface, and tfce
second to the volume of the magnet.
Let I be the intensity of magnetisation at a point of the surface
S (Fig. 73), and the angle which its direction makes with the
perpendicular ; a, /?, y, the cosines of the angles of the perpendicular
Fig. 73-
with the axes; lastly dS, an element of surface at the point in
question, we have
L/S cos = L/S (a A + /?/* + yv)
= U. cw/S + . J&/S + Iv.
UNIFORM MAGNETS. 305
The products ly, I/*, Iv are the components of the magnetisation,
and adS, /3dS, ydS, the projections of the element of surface on the
co-ordinate planes. We have then
Ids cos 6 = Kdydz + Edzdx + Cdxdy,
and the expression for the potential becomes
i/DA 3B 3C\
(4) V= -</S- - + + \dxdydz.
ty ^ /
The formulae (i) and (4) represent the same potential ; if they
are identified it will be seen that the surface density, and the volume
density of magnetism, may be expressed as a function of the intensity
of the magnetisation in the following manner.
(5)
(6) .
Hence, the surface density is the resolved part of the intensity
of magnetisation in the direction of the perpendicular to the surface
drawn outward.
The volume density is equal and of opposite sign to the sum of the
partial derivatives of the components of magnetisation referred to three
axes.
The quantities p and o-, which represent the densities of a fluid
on Coulomb's hypothesis, may be regarded as purely mathematical
quantities. They are two symbols defined by the equations (5) and
(6) ; for the sake of brevity the name of densities will be retained,
without attaching to this word a literal meaning.
We may observe that Poisson's equation, relative to secondary
differentials, becomes in the present case
For an external point, the second member is identical with zero
since the strength of magnetisation is constant and equal to zero.
320. UNIFORM MAGNETS. Let us consider the particular case
in which the magnetisation is uniform, that is to say, in which the
x
306 CONSTITUTION OF MAGNETS.
strength of magnetisation is constant in magnitude and direction
throughout the whole extent of the magnet ; the differentials of the
components A, B, C are null, and the equation (6) gives
there is magnettem therefore on the surface only.
The potential reduces then to
or, if ^Sj_ is' the projection of ^S, on a plane perpendicular to the
magnetisation,
All the elements of volume* being magnetised parallel to each other,
the magnetic moment of the whole, is equal to the sum of the
moments of all the elementary volumes ; we have, then,
or = \Idv = I \dv = vl.
Thus, the magnetic moment of a uniform magnet is equal to the
product of the volume by the strength of the magnetisation.
The expression for the surface density
or = I cos Q
shows that the external action of a body magnetised uniformly is
equivalent to that of two layers of gliding (157) that is to say,
of two layers which would result from the superposition of two
homogeneous magnetic masses of densities p equal and of opposite
signs, of which the positive part has been displaced parallel to the
magnetisation by a quantity 8 such that p8 = I.
We have said (308) that we may explain the action of the
earth by an infinitely small magnet placed at the centre, or by
two layers of gliding ; we see that we may also suppose the earth
uniformly magnetised; this latter condition being equivalent to
the two others.
FORCE IN THE INTERIOR OF A MAGNET. 307
321. FORCE IN THE INTERIOR OF A MAGNET. We cannot
determine the magnetic action in the body of a magnet itself without
making a cavity in which a small test magnet may be placed ; but
the creation of a free surface in the interior of a magnet is equivalent
to the formation of a surface layer having in general a finite action
on 'the points which it contains, and this action depends on the
form of the cavity.
A mass placed in the cavity is then outside the acting masses,
and the force which it undergoes may be determined in the usual
manner. Tfris force is the resultant of two others, one due to
external masses and the other to the surface layer of the cavity ; the
second force depends on the shape and orientation of the cavity
while the former is independent of it.
The strength of magnetisation may, moreover, be regarded as
constant in magnitude and in direction, throughout the whole extent
of the infinitely small magnet which is removed ; it would therefore
be possible to determine the second force for certain simple forms
of the cavity.
322. Let us consider, in the first place, a cylindrical cavity
the generating lines of which are parallel, and the bases at right
angles, to the strength of magnetisation. The density of the fictive
layer will be null on the lateral "walls, since the perpendicular com-
ponent of the magnetisation is zero at every point ; on the two bases
which are perpendicular to the magnetisation, the density will
be uniform, equal to + I on one and - I on the other. If the
extent of the base is equal to a, there will be equal and contrary
magnetic masses + al and - al at the two ends of the cylinder.
Let us imagine the cylinder to be circular : let r be the radius
of the base and zh the height ; the action of two layers on a point
in the middle of the axis is double that of a homogeneous disc on a
point of the perpendicular raised at its centre. In order to calculate
this action, let us suppose the density equal to unity and the disc
divided into concentric elementary zones ; the component along the
axis of the action of one of these zones at a distance p from the
point in question is
h
or, taking into account the relation p 2 = r 2 + /^ 2 ,
#w*,
X 2
308 CONSTITUTION OF MAGNETS.
Integrating between the limits p = h and p = *Jr 2 + /z 2 , we get
/ h \
/= 27T ( I ) :
we shall have therefore, for the action of the two layers,
h
Two cases are particularly interesting those in which one of the
two qualities h and r is very great in comparison with the other.
When the cylinder is very long, the ratio - is very small, and the
h
value of R tends to zero. The real force which then acts in the
cavity is reduced to the action of external masses. An infinitely
thin cylinder with any base will clearly give the same result ; this
will also be the case if we imagine in the magnet a section made by
any surface parallel at each point to the lines of magnetisation, and
if we suppose an infinitely thin interval between the two separated
parts. h
When the cylinder is reduced to a flat disc, the ratio - is very
small, and in the limit the value of R becomes
R = 4 7Tl.
The components of this force parallel to the axes are
These values will agree also with the case of an infinitely
thin section made in the magnet perpendicular to the lines of
magnetisation.
If the cavity is in the form of a sphere, the sides will be covered
with two equal and contrary layers distributed like layers of gliding ;
the action R of this layer on any point in the interior is constant,
parallel to the magnetisation, and is expressed by
MAGNETIC INDUCTION. 309
For a narrow slit, the perpendicular of which makes an angle
with the direction of magnetisation, the internal force of the surface
layers is perpendicular to the slit, and has the value
477-0- = 471-1 cos 6.
The influence of the shape of the cavity is well marked in these
various examples.
323. MAGNETIC FORCE. In the case of an elongated cylindrical
cavity, or of a slit parallel to the lines of magnetisation, the force
at a point only depends on the potential V of masses external to
the volume-element which has been removed. The components
of this force are
5V 3V 3>V
.A , x , Z/= .
ox oy 02
The expression magnetic force > or resultant force at a point of the
magnetised mass, is more especially assigned to the force thus denned.
324. MAGNETIC INDUCTION. If the cavity is a very flattened
cylinder, or an infinitely thin slit perpendicular to the lines of mag-
netisation, the components of the true force F x have the values
(7)
The force F x plays an important part in the study of magnetisation
by influence ; it is called magnetic induction.
The sum of the three partial differentials of the function F x
gives the equation
r +-r- + ^
ox oy oz
= 47T/0 4?T/3 = .
Hence magnetic induction satisfies Laplace's equations both for
points inside and outside the magnetised media. It is, moreover,
identical with the magnetic force for all external points, since the
magnetisation I, and its components A, B, and C, are then equal to
zero. Magnetic induction has therefore the same properties as
electrostatic induction (116).
310 CONSTITUTION OF MAGNETS.
A line of induction is a line to which the force of induction is
tangential at every point ; a tube of induction is a channel bounded
laterally by lines of induction ; lastly, flow of induction across an
element of surface is the product of the surface of the element by
the perpendicular component of induction. Since induction satis-
fies Laplace's equation for all internal and external points, it follows
that the flow of induction is a constant quantity throughout the
whole extent of a tube of induction.
325. DIFFERENT KINDS OF MAGNETS. We may divide mag-
nets into distinct categories, according to the manner in which the
intensity of magnetisation varies.
326. MAGNETIC SOLENOIDS. A simple solenoid is a magnet in
the form of a filament with an infinitely small constant section, at
each point of which the intensity of magnetisation is itself constant
and tangential to the direction of the filament.
The magnetic density is zero throughout the whole mass of the
filament and on its lateral surface (310) ; at the ends only are two
equal and opposite magnetic masses ; if I is the strength of mag-
netisation and a the section of the filament, the absolute value of
these two masses is
This product a\ maybe called the magnetic power vt the solenoid.
If, while the section of the filament, and the intensity of the
magnetisation are variable, the product a\ remains constant, the
system will still form a simple magnetic solenoid.
A simple solenoid acts on all external points as would a magnet
whose poles were exactly at the ends.
The potential at a point P (Fig. 74), at a distance r z from the
positive pole A 2 , and at a distance r from the magnetic pole A 15 is
thus expressed,
MAGNETIC SOLENOIDS. 311
If such^ a solenoid is closed, the potential is everywhere zero on
the outside ; the force is therefore zero, and we can only discover
the magnetism in the system by breaking it at a point and separating
the ends.
327. A magnetic filament with a constant or variable section
tangential at every point to the direction of magnetisation, and
in which the magnetic power is not constant, constitutes a complex
solenoid. Such a system may be regarded as several simple sole-
noids of unequal lengths united to form one bundle.
Fig. 75-
The potential P at an external point of an element of length ds
(Fig. 75), whose magnetic power m = a\ is
mds dr dr
dN = - cos = -m = - a I .
r z r i r i
The potential at P of the whole filament is
' A * mdr
V =
Integrating the second member by parts, and calling -m^ and
2 the masses of th.e extremities Aj_ and A 2 , we get
'A, rdS
The potential is the same as if the linear density at each point of
312 CONSTITUTION OF MAGNETS.
the filament were defined by the ratio
dm _ d(al]
~~~~ ~
and we may write
y ^S*p*
'i. J /
328. SOLENOIDAL MAGNETS. A magnet is said to be solenoidal
when it may be divided into simple solenoids terminating at the
surface or closed upon themselves. There is no free magnetisation
in the interior of the magnet ; the distribution is entirely superficial.
The volume density p being zero, we have
(8) + + = 0.
Conversely, if the condition (8) is satisfied, the density is zero at any
point in the interior, and the magnet is solenoidal.
329. MAGNETIC SHELLS. A simple magnetic shell is a magnet
formed of two infinitely near equidistant surfaces, charged with equal
and opposite uniform magnetic layers ; or is a magnet formed of
two infinitely near but not equidistant layers, always equal and of
opposite signs, and such that the density at each point is inversely
as their distance.
If h be the thickness of the shell at a point, and o- the density of
the layer, the product Jvr must be constant ; it is called the magnetic
power of the shell.
We may also define a simple magnetic shell as being an infinitely
thin plate, the magnetisation of which is perpendicular at every point
to the surface, and its intensity inversely proportional to the thickness.
If < be the magnetic power of the shell we have
That portion of the shell which corresponds to an element dS,
may be regarded as an infinitely small magnet, the moment of
which is
MAGNETIC SHELLS. 313
The potential at the point P (Fig. 76) of this element of the
shell is expressed by
cos0
6 being the angle formed by the perpendicular N drawn externally to
the positive surface, with the right line r, which joins the point P to
the element ^S.
Fig. 76.
The solid angle du, under which the element d is seen from the
point P, is given by the equation
d$ cos = rVw or du> = .
From this it follows that
As the factor <> is constant, the potential of the shell at P is
expressed by
(9) V = $a>.
It is important to define with precision the significance of the
solid angle o>. The potential dV is positive or negative according as
the point P views the positive or the negative surface of the element
d$ of the shell that is to say, according as the angle is acute or
obtuse. The angle </<o, considered itself as positive or negative in
314 CONSTITUTION OF MAGNETS.
the same conditions, is the surface cut on a sphere of radius equal to
unity, whose centre is the point P, by a cone described with vertex
at P on the surface of this element as base ; it is the apparent surface
of this element. The angle to, or the apparent surface of the
whole shell, is therefore defined by a limited cone on the contour
of this shell; it is positive or negative according as the surface
of the shell which the point P views throughout the contour is itself
positive or negative. The potential of the shell is therefore inde-
pendent of its form, and only depends on its magnetic power and on
its contour. From this follows the important theorem of Gauss :
The potential of a simple magnetic shell at an external point is
equal to the magnetic power of the shell by its apparent surface seen
from this point.
In order that the potential shall be zero at this point, the appa-
rent surface of the shell must be zero.
The apparent surface of 'the shell is null if, the contour being a
plane, the point in question is situate in this plane.
It is zero, whatever may be the shell, when it consists of parts
of opposite signs whose algebraical sum is zero.
As a particular case, if the shell forms a closed surface, the
potential for any external point is zero. For a point inside the
shell, the angle o> is equal to 477, the potential is therefore constant
and equal to 4^ ; it is of the same sign as the internal surface.
The value of this potential being constant both inside and outside,
the action of the closed shell on any given point is zero.
330. If two equally strong magnetic shells S and S' (Fig. 77)
have the same contour, and if their surfaces which face each other
are of opposite signs, their potentials are equal for all points outside
the space which they comprise ; these potentials differ, on the con-
trary, by 47r ( i > for all points between the two surfaces. For the
potential of one of the shells S is positive and equal to 3>w, that
of the other shell S' is -<a>'; the difference is therefore
In like manner, for two infinitely near points situate on each
side of a magnetic shell at a finite distance from the contour, the
difference of potentials is equal to 4^, for it is 3>(o for the one
and < (4?r co) for the other. Hence, when the point in question
traverses a shell in the direction of the magnetisation that is to
say, from the negative to the positive face the potential suddenly
increases by
MAGNETIC SHELLS. 315
If, while the point was fixed, the shell altered its shape so as to
pass from the position S' to the position S (Fig. 77), the potential at
P would undergo the same increase of
Fig. 77-
As a matter of fact, the change of potential does not take place
suddenly on a geometrical surface, since the shell has necessarily a
finite thickness, and it is easy to see that the potential at P varies
continuously while the point traverses the magnetised layer.
For, let the shell SS' (Fig. 78) be divided into two parallel layers
Fig. 78.
of thicknesses x and h - x, and of power ^ and $ 2 , and consider
the point P at the surface of separation of these two shells. The
value of the potential at P is
V = o>3> 2 - (477 - co)*!
We have further
and, consequently,
x x
V = (0< - 47T^> - = <( CO- 47T - 1.
h n
The perpendicular action of the shell at the point P is
This expression, as might have been foreseen, is the perpen-
dicular component of the induction at the point P. For we have
316 CONSTITUTION OF MAGNETS.
implicitly assumed that we placed the point P in an infinitely thin
slit perpendicular to the lines of magnetisation : the term 473-! is the
force which must be added to the external actions in order to have
the value of the true force in the interior of the cavity.
We may further observe that if the intensity of magnetisation is
finite, the magnetic power ^> of the shell is an infinitely small
quantity ; for any external point at a finite distance from the contour
of the shell the value of the force is infinitely small, while in the
interior of the shell the force has a finite value 473-!, directed along
the perpendicular and in an opposite direction to that of the mag-
netisation.
331. LAMELLAR MAGNETS. A magnet is said to be lamellar
when it may be divided into simple closed magnetic shells or into
open shells with their edges on the surface of the magnet.
Let < be the sum of the magnetic powers of the shells which we
meet in going from a given point to a point whose co-ordinates are
x,y, z, along a line of force drawn in the interior of the magnet.
This quantity 3? is a function of the co-ordinates independent of the
line joining the two points ; it has a constant value on the whole
surface of a shell, but varies from one shell to another.
The lines of magnetisation are, by definition, at right angles to
the surfaces of the elementary shells, and the strength of the mag-
netisation at each point is inversely as the perpendicular distance dn
of two consecutive shells. We have then
332. POTENTIAL OF MAGNETISATION. The function < has
therefore, taking into account the sign, the same properties in
reference to magnetisation as the potential in reference to external
forces. Hence, by analogy, we may call the function -& the
potential of magnetisation. The components of magnetisation along
the axes of the co-ordinates, are respectively equal to the corre-
sponding partial differentials of the function < :
From this we deduce
(i i) MX + >dy + Cdz =
POTENTIAL OF MAGNETISATION. 317
_ V _ _
The first member of this equation is thus an exact differential.
Conversely, if the expression Adx + 'Bdy + Cdz is the exact dif-
ferential of a function of the co-ordinates, the components of
magnetisation are respectively equal to the partial differentials of
this function, and the magnetisation is lamellar.
The condition of lamellar magnetisation may be expressed by
equations in which the function < does not appear.
We have, in fact,
which gives the three equations
3A SB
N -\ '
dy ox
(12) as ac
_ .-, __ _ y ?
oz oy
333. A magnetic shell is said to be complex when, the mag-
netisation being always perpendicular at each point, the magnetic
strength is not constant throughout the whole extent of the shell.
The potential at the point P of the element d of the shell is
still
and the potential of the entire shell
the integral being extended to the whole surface of the shell.
When a magnet can be divided into complex magnetic shells, the
strength of magnetisation is no longer inversely as the distance of
two infinitely near shells, but the lines of magnetisation are still
orthogonal to the surfaces of these shells, which gives the condition
A_JB __(:_
(13) ~~*
318 CONSTITUTION OF MAGNETS.
In this case, the expression A.dx + >dy + Cdz is no longer an
exact differential. We may again eliminate the function 3> between
these equations, and we get
This is the condition which must be satisfied to have a complex
lamellar magnetisation.
Conversely, if equation (14) is satisfied, the magnet is formed
of complex magnetic shells, for the lines of magnetisation are
orthogonal to a system of surfaces ; unless each of the expressions
in the parenthesis is separately zero, in which case the magnetisation
would be lamellar, from equations (12).
334. POTENTIAL OF A SOLENOIDAL MAGNET. The general value
of the potential of a magnet is
' P -dv.
If the magnet is solenoidal, the density p is everywhere zero, and
the potential is reduced to
The potential of a solenoidal magnet at any internal or external
point only depends then on the surface density, or on the per-
pendicular component of the strength of magnetisation at every
point of the surface. This potential is independent of the manner
in which the internal magnetisation varies, or in other words, on the
internal form of the solenoidal filaments which terminate at the
surface, as well as of the existence of closed filaments.
We may suppose, for instance, that the magnetism of the earth
is produced by solenoidal filaments, maintained in the surface
rocks at a low temperature, and terminating on the surface in such
a way as to produce a distribution equivalent to that of a uniform
magnetisation.
335. POTENTIAL OF A LAMELLAR MAGNET. If the magnet is
lamellar, it consists of closed magnetic shells, and of open shells with
their contour on the surface. The force outside only depends then
POTENTIAL OF A LAMELLAR MAGNET. 319
on the form and position of the edge of the open shells that is
to say, of the infinitely thin zones cut on the surface by two adjacent
shells, and not at all on the form of the shells.
For a point in the interior, the force in a slit between two shells,
or the magnetic induction, will be obtained by combining the action
determined by these successive zones, with a force in the opposite
direction to the magnetisation at the point in question, and equal to
4 TT I. The potentials by means of which these forces may be expressed,
are directly obtained from the following considerations.
Let us first of all disregard the closed shells, and suppose that
after having removed all the open shells which the magnet contains,
we replace them by shells respectively of the same power terminated
by the same edge, but applied on the surface itself; this operation
would be realised physically if each of the shells were formed of
Fig. 79.
an elastic membrane, fixed by its edge, which could be stretched so
as to be applied on the surface of the magnet without modifying its
magnetic strength. Let us assume, for instance, that in Fig. 79 all
these shells have their positive faces turned upwards, and that they
are made to cover the point A of the surface of the magnet
where the function <& has its maximum value.
The entire surface will then be occupied by a series of shells, the
superposition of which forms a complex shell, and produces at every
point outside, the same potential as the magnet itself.
Let us now consider a point P in the interior. The potential
has not changed by the fact of the transformation of those shells
which passed between A and P ; but for each of the other shells
which have been traversed by the point P, the potential is less by
Let then-^p be the potential of magnetisation at P, and
320 CONSTITUTION OF MAGNETS.
- $0 the value of this potential at the point O of the surface where
the function * is a minimum ; during the transformation the potential
at P will have diminished by the product of 477 by the sum of the
magnetic powers of all the shells between the points P and O that
is by 4 Tr^p- <>()), and this quantity must be added to the new
potential at the point P to give it the value which it originally had.
At any point M of the resultant superficial shell thus formed, the
magnetic strength is equal to the sum / d& of that of the shells which
have been superposed there ; it is therefore equal to * - $ , calling
- < the value at this point of the original potential of magnetisation.
Consequently, the potential of all the layers on the point P is equal
to
If the point P is not surrounded by closed shells, the potential at
this point has diminished by 4?r (& p - <1> ) during the transformation
the original value of this potential was therefore
Let us now suppose that there are closed shells ; only those which
comprise the point P need to be taken into account. Let $ x be the
value of 3> on the largest of them. The sum of the magnetic powers
of the open shells from the point O to the point P is 3^ 3> ; that
of the closed shells which comprise the point P, and which have not
been displaced by the preceding transformation, is equal to & p - $ r
The potential at the point P is then
or
It will be seen that the closed shells do not modify the expression
of the internal potential.
The external potential is not changed by the transfer of the shells
to the surface ; it is expressed by
(16)
POTENTIAL OF A LAMELLAR MAGNET. 321
The two formulae (15) and (16) may be simplified if we observe
that the integral I d& is equal to zero for external points, and to
-47T for internal points.
We get then
(15)'
(16)'
Denoting by 12 a function defined by the ratio
(17)
we might put the potential in the form
(15)" V. = fl + 4T($-$ )-
(16)" V, = fi.
336. It is easy to show that, notwithstanding the difference in
form of the expressions for V e and V\, the potential varies in a
continuous manner when the surface of the magnet is traversed.
For consider two infinitely near points M. e and M i? one without and
the other within the surface S. In passing from M e to M^ the
function fi diminishes by
Hence, on both sides of the surface, we have
(18) fl t = fi i + 4 ,r(*-* ).
The magnetic potentials at M e and M^ are
the two values are therefore equal.
Y
322 CONSTITUTION OF MAGNETS.
337. POTENTIAL OF INDUCTION. The function plays, in
reference to the induction F 15 the same part as the function V in
reference to the magnetic resultant F. For the values of the com-
ponents of the force F 1 are (326)
3V
Xl =--
ay
Yl =--
**
We know, on the other hand, that we have (332)
, 13 ~, Lx .
ox oy 02
From the equation
we deduce
W c)fi ^ ^0
---= __ 47r __= - - 4?rA,
d^ ojc o^: ox
"SV M ^ ttt
-^ = - -47T~-= - -47TB,
oy oy oy oy
_ = _ -47T = - T --
OZ 02 02 02
and, therefore,
an
The components of the induction Fj are therefore equal and of
opposite sign to the partial differentials of the function ft.
POTENTIAL ENERGY OF MAGNETS. 323
On the other hand, the functions V and tt are identical for all
points external to the magnetised media points for which the induc-
tion and the magnetic force are themselves identical. Hence the
function
= f($_
may be considered as the potential of magnetic induction of a lamellar
magnet.
338. POTENTIAL ENERGY OF MAGNETS. The general expression
for the energy of a permanent magnet in a magnetic field produced
by an invariable system, where m is the magnetic mass situate at the
point where the potential of the field is V, is
or again, as a function of the surface density and of the volume
density of the magnetism,
= /
This energy is the work which must be expended to bring the
magnet in question from an infinite distance to the position which
it occupies, or conversely the work done in moving it to an infinite
distance.
In order to express the energy as a function of the intensity
of magnetisation, we must replace the densities by their known
values ; but it is simpler to consider the problem directly. A volume
element dv, the magnetic moment of which is Idv, is equivalent to a
small magnet of mass m, and length ds y parallel to the direction of
magnetisation. If V and V are the potentials of the field at the
points at which are the masses -m and +;;z, the energy of this
element of volume is
dW = m(V - V) = mds^-^-=ldv .
ds &
If 8 be the angle which the direction of magnetisation makes
with the direction of the field, and dn the perpendicular distances
of the two equipotential surfaces V and V at the point in question,
we have
dV dV
y- = cos 8= -Fcos 8= -
Y 2
324 CONSTITUTION OF MAGNETS.
X, Y and Z being the components of the force of the field, A, //, and
v the direction cosines of the directions of magnetisation. The
expression for the elementary energy is therefore
and hence the energy of the whole magnet is
(20) W= -
If the field is uniform, the components X, Y and Z are constant.
If a, ft and 7 are the cosines of the angles of the force F with the
axes, we get
W= -
If K be the magnetic moment of the magnet, /, m and n the
cosines of the angles which the magnetic axis makes with the axes
of the co-ordinates, we have
(MV
= K/, 'Rdv = Km , cdv = Kn ,
and the energy becomes
(21) W= -FK(al+ftm + yn)= - FK cos 8,
8 being the angle which the magnetic axis makes with the direction
of the field. This result may be written directly.
The energy is a minimum and equal to - FK, and therefore the
equilibrium is stable when the angle 8 is zero that is to say, when
the magnetic axis is parallel to the direction of the field. The
equilibrium is unstable if these two directions are opposite ; the
energy is then a maximum and equal to FK. The energy, lastly,
is zero if the two directions are at right angles.
339. ENERGY OF A MAGNETIC SHELL. If the system is a
simple magnetic shell S, the magnetic moment of a surface-element
of the shell is <I?*/S and the value of its potential energy in the field
is
the energy of the shell is therefore
W= -</"( AX + ^Y
ENERGY OF A MAGNETIC SHELL. 325
The quantity in brackets (XA + Y/^ + Zv) represents the projection
F n of the force of the field on the perpendicular to the shell ; the
product F n ^S is the flow of force of the field corresponding to the
element dS ; this flow is positive when it traverses the shell from the
negative to the positive face, and negative when in the opposite
direction. Hence the integral of the second member simply ex-
presses the value of the flow limited to the edge, and therefore
is independent of the form of the surface to which it is attached.
Let Q be the value of this flow, the expression for the potential
energy of the shell is
(22) W=-$Q.
Consequently, the potential energy of a shell is equal to the product,
with the contrary sign, of the power of the shell by the floiv of force
which penetrates its negative surface.
340. This result may be directly obtained. For the energy
of a mass m, in the field of a simple magnetic shell, is expressed by
But the product ma* is the flow of force which starts from the
point in the angle w, and which therefore traverses the shell entering
by the positive surface. The flow dQ which enters by the negative
surface has the same value with the contrary sign - mu. We have
thus
But the energy of a magnetic system in the field of the shell is
the sum of the energies of different masses; it is therefore the
product, taken with the opposite sign, of the magnetic strength < of
the shell by the sum of the flows of force which traverse it that is
to say, by the flow of force which starts from the system and enters
the shell by the negative face.
341. If this system is a second shell S', the flow of force Q is
proportional to the magnetic strength <' of this second shell, and we
may write Q = M3*', the coefficient M being the flow of force which
the former shell would receive, if the power of the second were
equal to unity. The energy of the first shell, in the field of the
second, is therefore
(23) W=-<M>'M.
326
CONSTITUTION OF MAGNETS.
The energy of the second shell in the field of the first has the
same value, and is expressed in the same way,
as a function of the flow of force which proceeding from the first
would traverse the second ; from it we infer
(24) M = M'.
Thus when two magnetic shells of equal strengths are in presence
of each other, the flow of force which starts from one and traverses
the other, entering by the negative face, is the same for both.
It will be observed how analogous this property is with the
theorem demonstrated above (63) relative to the electrostatic in-
duction between two conductors.
342. Equation (22) shows that the energy of a shell in a mag-
netic field only depends on the flow of force which crosses the
surface bounded by the contour of the shell, and that it is inde-
pendent of the form of this surface. This energy, and therefore the
force exerted on the shell, may be expressed then as a function
of the curve of the edge.
In like manner the reciprocal energy of two shells given by
equation (23) only depends on the two edges; this energy and
the reciprocal force may then be expressed as a function of the
two curves which bound the shells.
343. ACTION OF A FIELD ON A SHELL. Consider a shell S
(Fig. 80) placed in any given magnetic field. When the shell ex-
periences an infinitely small displacement, the increase of the
potential energy is
ACTION OF A FIELD ON A MAGNETIC SHELL.
327
</Q being the increase of the flow of force which traverses the
negative face of the shell. The work dT of the magnetic forces
being equal and of opposite sign to dW, we have
As the form of the shell is a matter of indifference, we may suppose
that it forms part of a continuous surface S, passing through the
positions C and C\ which the edge occupies, and that this only
makes it glide on the surface.
The work of the magnetic forces is proportional to the excess
of the flow of force which traverses the surface bounded by the
edge G! over that which traverses the surface bounded by the
edge C. The flow of force relative to the portion common to
the two shells disappears by difference, so that calling q and q' the
flows which traverse the spindles AMB and AM'B, we have
Let ab be an element of the first contour, a^ its new position
after the displacement, F the force of the field at this point. To
obtain the part dq of the flow relative to the displacement, we must
multiply the force F, by the projection of the parallelogram, abb^
which this element has described, on a plane perpendicular to this
force.
Fig. 81.
Finally, in order the better to see the geometrical signification
of this product, let us imagine an observer laying along the curve C
so that, looking at the shell, he has the negative face on his right
hand. The positive direction of the arcs is that of a moving body
which goes from the feet to the head of the observer. Let us take
328 CONSTITUTION OF MAGNETS.
the plane Fds for that of yz, and the direction of the force F as
axis of y (Fig. 81). Let a be the angle which the element ds,
calculated in the positive direction, makes with the force F. The
projection of the parallelogram abb^ on the plane of the xz per-
pendicular to the force F is a new parallelogram ab'b'^. We may
consider this latter as having for base ab' = ds sin a, and for height ac
that is to say, the abscissa of the point a lt or the projection e of the
displacement aa l on the axis ax perpendicular to the plane fds. We
have then
(25) dq = ds sin a x e.
The corresponding work is
?Fds sin axe.
This work is the same as if the element ds were subjected to the
action of a force
sin a
parallel to the axis of x that is, perpendicular to the plane Fds.
We are thus led to this important theorem :
The action of a magnetic field on a shell is equivalent to that of a
system of forces applied at the different elements of the edge.
The force, which we must suppose applied at each element, is
perpendicular to the plane which passes through the element, and to
the direction of the field, and is on the left of an observer placed in
the element along the positive direction, and looking at the direction
of the force F.
344. If we consider these as real actions, we may enunciate
the following theorem :
The action of a magnetic field on an element of the edge of a
shell) is equal to the product of the magnetic strength of the shell, by the
force of the field, the length of the element, and the sine of the included
angle in other words, by the surface dA. = Yds sin a of the parallelo-
gram constructed on the force F and the element ds.
We have then simply
As a particular case, if the magnetic system is reduced to a single
mass m at a distance r from the element ds, the force F is equal
to , and the elementary action becomes
r*
d<b = < ds sin a .
ACTION OF A FIELD ON A MAGNETIC SHELL. 329
Hence, 'the action of a magnetic pole on an element of the edge
of a shell is proportional to the magnetic strength of the shell, to the
sine of the angle which the element makes with right line joining the
pole to the element, and inversely as the square of the distance.
345. We may further remark that dq represents the flow of
force cut by an element ds during the displacement aa 19 the flow
of force thus cut being counted positive or negative, according as the
displacement is to the left or right of the observer whose position has
been defined as above. From this follows the theorem :
The work of magnetic forces during the displacement is equal to the
product of the shell by the sum of the flows of force cut by each of the
elements of the contour.
346. Let us suppose that the external system reduces to a
magnetic mass equal to unity, and placed at the origin O of the
co-ordinates. Let C be the edge of the shell, and ds an element at
the distance r, at a point M whose co-ordinates are x, y, and z.
Let A, fj,j and v be the cosines of the angles which the force d$
makes with the axes. This force being perpendicular to the element
ds and to the straight line OM along which is directed the action
proceeding from the point O, we have the ratios
= ,
\dx + pdy + vdz = ;
from which we deduce
ydz - zdy zdx xdz xdy -ydx rds sin a '
a being the angle which the right line OM makes with the element
* *
As the force d$ is equal to ds sin a, its components d^ dy, and
(zdx xdz) ,
= (xdy -ydx}.
33
CONSTITUTION OF MAGNETS.
The action of the point O on the shell will be obtained by ex-
tending these expressions to the whole surface. Lastly, the action F
of the shell on the point O, passes through this point, and the com-
ponents X, Y, and Z of this force, are equal and of opposite sign to
those of the action of the point on the shell ; we have then
(26)
= <i> j* z z x = 3> f_ d r\
J r Jr \J
__ c|) I *_ = . = ^) I "^_
J r a
d *
y
347. RECIPROCAL ACTION OF Two SHELLS. We may now de-
termine the reciprocal action of two shells S and S'. The action
of S on S' may be considered as the resultant of actions, determined
by the previous rule, which the shell S would exert on each of the
elements ds' of the contour C' of the second shell.
Fig. 82.
Let us suppose that one of these elements ds is at O (Fig. 82)
and is directed along the axis of x. The action d$ which is exerted
on this, element is equal to &Fds' sin a, and is situated in the plane
of zy ; the components of this force are
^' = <!>' Fds' sin a' . cos /5 = '
^' = - $>'F<&' sin a' . sin )8 = -
RECIPROCAL ACTION OF TWO SHELLS.
33*
which, expressing the forces Z and Y as a function of the co-ordinates
x, y and z of the point M where the element ds is situate, gives
(27)
>' ds'(
Ix-xdy
zdx - xdz
We may also consider the action of the edge C on the element
ds' t as the resultant of the direct actions which each of the ele-
ments ds would exert on the element ds'. The only condition
imposed on this elementary action is, that the integral of the
partial components extended to the edge C shall reproduce the
preceding expressions.
348. In accordance with this, the simplest solution for the action
of ds on ds' is a force/, the components of which parallel to the axes
f-x>fy>fz are > representing by a the product
(28)
xdy-yd X= _ ( ,y_
r 6 r 3 \x
xdz - zdx
j /
= -ad ( -
349. To each of the components of the elementary action an
exact differential of the co-ordinates x, y, and z may be added, since
the integrals extended to the contour C will give values of zero for
these terms. There is therefore an infinite number of expressions
by which the actions of the elements of two magnetic shells may be
expressed.
Let X, Y, and Z be functions of the co-ordinates x, y, and z ;
the problem will be satisfied if we take as components of the action
(29)
332
CONSTITUTION OF MAGNETS.
350. Let us, for instance, impose the condition on this force, that
it shall be directed along the right line which joins the elements, so
that we have
x y z
it follows that
or
Q " \ I '
X X
In order that the second members of these two latter equations
shall be exact differentials of a function of the co-ordinates, we must
have
and, therefore,
-
r 3
x r
3 / '
The components of the elementary force will then be
(30)
The force itself may be determined by the ratio
which gives
r ./x-\ 2a\ . 3-^rl 2a\"6x $xl>r~\
f=a-d( \= \ dx---dr\^ \ - ---- \
x \^ 3 / ^ 2 L 2 r r L 2 r J
ds .
RECIPROCAL ACTION OF TWO SHELLS.
333
If 6 and 6' are the angles which the elements ds and ds' make
respectively with the right line OM which joins them, and e the angle
of these two elements, we have
- = cos<9',
and we get
r = I cos - - cos cos 0' I ds
2
If we consider the action of ds upon ds' and take the distance
r as positive in the direction MO, we must change the sign of the
force and replace the angles and 0' by TT - 6 and TT - B' t which
does not change the sign of the product of the cosines.
Let us represent by d^ the action of ds upon ds' , which is an
infinitely small quantity of the second order, and consider this force
to be repulsive ; we shall have finally
351. We may give another form to this expression, which is more
convenient for estimating the work.
Fig. 83.
Let C and C' (Fig. 83) be the edges of two shells, ds and ds'
the elements at P and P', and let us count the arcs s and s' respec-
tively from the fixed points O and O'.
334 CONSTITUTION OF MAGNETS.
From the figure we have
co.*--*
From the extremities P' and P" of the elements ds' let perpen-
diculars P'A and P"A' be drawn to the tangent to the curve ; we get
PA =rcos<9,
from which we deduce
OS
On the other hand, the distance AA' is the projection of the
element ds' on the tangent at P to the curve j, which gives
A , jt
AA =- v ^ , ds = ds cos e ,
OS
and, therefore,
M r~
COS = ,
V ^f 05
The elementary action may then be written
~&,r 3
'^y^y'F <) 2 ^ i ^r ^r
* ?>s~ds' 2^s^)s'
We have further
ayr
_ _^ ^ _ i g^gr __ ^_ r
' 2 N /rt)j()j / ^r^r^s^s' 2r^r^
RECIPROCAL ACTION OF TWO SHELLS. 335
which gives finally
(32 )
352. To determine the relative energy of the system, let us
suppose that the shell S' moves away, and that during the time dt
the distance r of two elements varies by dtor 2>Jr--dt. The
Ql Of
corresponding elementary work of the force d^ is equal to
d^ dt t so that the total work ^ 2 T relative to the element ds
for the time dt is
Integrating by parts, we have
rv^^v? rv^v^i _ ryray; d ,
} It ^to' ^ [_ to to J J to to'to
The first term of the second member is zero for the closed
surface C', which gives
to
The elementary work relative to the actions of the two circuits
in the time dt is therefore
to
This work being symmetrical in reference to the edges C and
C', we have also
to
336 CONSTITUTION OF MAGNETS.
From which follows, taking the half sum of these expressions
to
or
The relative potential energy W of the two shells is equal to the
work which the forces can perform when one of the shells C moves
to an infinite distance. We have then
(33)
353. This expression for the energy may be put under several
different forms.
We have, in fact,
3-'
W/>__!**;_ _ cosecosff = _:* '
os os ^rosos ^r 405 os
We may then write
'o-
117 ,^'f f COS(9cOS(9 '^,7> AA'ff * r r jJ'
W = <M>' - ^y = $$ r -^-, ^f^f '
JJ r JJ ^^
Integrating this latter expression by parts, we have
NEUMANN'S FORMULA. 337
The first term of the second member is null, being extended to
the closed contour C ; we have then
f4V>= - fi-\**- f?LV
J ^s V J r W J r
and therefore
( 34 )
This remarkable formula is due to F. E. Neumann. We deduce
from it for the value of the coefficient M (341), which expresses the
flow of force common to the two sheets, each supposed equal to unity,
(35)
33$ PARTICULAR CASES.
CHAPTER III.
PARTICULAR CASES.
354. POTENTIAL OF A UNIFORM MAGNET. The magnetic
action of a body uniformly magnetised being equivalent to that
of two layers of gliding (320), the potential V may be readily
deduced from that of a homogeneous mass which would fill the
volume. Let P be the value of this potential at a point M, when the
density of the mass is equal to unity, its value will be pP if the
density is p.
The potential of the system of the two layers is evidently the sum
of the potential /oP of the positive mass, and of the potential - pP' of
an identical negative mass which has been displaced in the opposite
direction to that of the magnetisation, by an infinitely small quantity
dx = 8. The potential pP' is that of the positive mass at the point M',
whose co-ordinates are the same as that of the point M, except the
abscissa parallel to the magnetisation, which has increased by dx.
We thus obtain
,, ,--,(,***).-,
Consequently the potential of a uniform magnet is equal, and of
opposite sign, to the product of the intensity of magnetisation by the
partial differential, referred to the direction of the magnetisation,
of the potential, which a uniform mass, of density equal to unity
occupying the whole volume of the body, would have at the point
in question.
The components X, Y, and Z of the magnetic force are equal,
and of opposite sign, to the partial differential of the potential,
which gives
SPHERE. 339
355. SPHERE. For a sphere of volume , for instance, the
value of P at an external point at a distance r from the centre, is
P---
~ r'
from which follows
and therefore
ux
r 3
as we have previously seen (157).
In the interior of the sphere the action of a mass of unit density
will be equal to - irr (44), which, if a is the radius of the sphere,
O
gives for the potential
from which we get
and consequently
DP 4
= --TTX
^x 3
4 T Y-0
7T1, 1 U,
that is to say, that the internal action of a uniformly magnetised
sphere is constant, parallel, and in a contrary direction to the direc-
tion of magnetisation, a result which has already been established
directly (157).
The magnetic induction in the interior of the sphere is constant
and equal (324) to
4 8
~3* ~3^ '
Z 2
34 PARTICULAR CASES.
so that the total flow of induction which traverses the great circle,
perpendicular to the magnetisation, is expressed by
356. ELLIPSOID. Consider a homogeneous ellipsoid, the axes of
which 2#, 2$, zc, are taken as axes of the co-ordinates. Denoting by
L, M, N, known functions of the axes, the potential of this ellipsoid
at a point in the interior, the co-ordinates of which are x, y, z, is
P = - - (L* 2 + M/ + N* 2 ) + const.
s
If the ellipsoid is uniformly magnetised in a direction which
makes, with the axes, angles whose cosines are /, m, , the com-
ponents of the magnetisations are
A_B_C
1 }
/ m n
and the state of the ellipsoid may be considered as being produced
by the superposition of these three magnetisations, respectively
parallel to the axes. The potential in the interior is
and the values of the components of the force parallel to the axes are
X=-AL, Y=-BM, Z=-CN.
The interior magnetic force of a uniformly magnetised ellipsoid,
is therefore constant in magnitude and direction, and makes, with
the axes, angles whose cosines are respectively proportional to AL,
BM, and CN.
The components of the induction parallel to the axes are
ELLIPSOID. 341
Induction is therefore a constant force, which makes, with
the axes, angles whose cosines are respectively proportional to
( 4 7T- L) A, (477- M) B, and (471-- N) C.
Lastly, the values of 'the flows of induction across the three
principal sections are respectively irbc (477 - L) A, irca (477 - M) B,
and nab (477 - N) C.
357. If the magnetisation is parallel to one of the axes the axis
a, for instance we have simply
-IL.
From the manner in which the layer is formed, the quantity of
magnetism M a , distributed on each of the halves of the ellipsoid, is
equal to the total charge which would exist on the principal section,
parallel to the two other axes, if the density were uniform and equal
to I ; we have then
The magnetic moment S7 a of the magnet thus formed, is equal to
the product of the volume by the intensity, which gives
4 4
n = - irabc\ = irabc .
3 3 L
The poles of the magnet, or the centres of gravity of each of the
two layers, are at a distance a' from the centre determined by the
equation
which gives
, ICT 2
a = --=-a.
2M 3
Thus the pole of an ellipsoid uniformly magnetised in a direction
parallel to one of the axes, is at a distance from the centre equal to
2
- of the length of the corresponding half-axis.
342 PARTICULAR CASES.
The density at a point of the surface is given by the equation
-. o
y* g* a* t
/ being the perpendicular let fall from the centre, on the tangent
plane at the point whose co-ordinates are x, y, and z.
The total charge of a zone determined by two planes perpendicular
to the axis a and at a distance dx, is equal to the product of the
intensity I by the difference dS of the sections of the ellipsoid corres-
ponding to these two planes. At a distance x the section is bounded
by the ellipse
the surface of which is
\
we have then
and, consequently, the charge of the zone is
2Trbcl
= xax.
The ratio ^ of the charge of the zone to its height, which may
be defined as the linear density, in reference to the axis of magneti-
sation, is therefore proportional to the distance of this zone from
the centre of the ellipsoid.
358. When the axes of the ellipsoid are unequal, the coefficients
L, M, N are given by the partial differentials of a definite elliptic
integral; for the complete calculation we must refer to special
treatises, and must limit ourselves to giving the results relative to
ellipsoids of revolution. In this case, in fact, the problem is more
simple, and the coefficients are expressed by means of the ordinary
functions.
ELLIPSOID. 343
If the ellipsoid is one of revolution about the minor axis c, we
have, if e is the eccentricity of the meridional ellipse,
^^ i-* 2 ~|
--- ,
sin e e* J
For an ellipsoid of revolution about the major axis,
i->p s i+, -i
= 47T -/. I
e* \2e i-e
Making * = in these formulas, we find again the results already
obtained for the sphere ; that is to say,
M = N = -
*J
V 4 T
F= -- TT!.
For a very flat ellipsoid, in which the eccentricity e is near unity,
we have, at the limit,
If the ellipsoid is very elongated, we have approximately
M = N = 27T,
and the coefficient L tends towards zero, when the eccentricity
tends towards unity.
344 PARTICULAR CASES.
For a very flattened ellipsoid, which at the limit might be con-
founded with a very thin disc, the force in the interior is given by
the equations
F=-47rI, or F= -Tr 2 ^/! -e? 2 I,
according as the magnetisation is perpendicular or parallel to the
plane of the disc.
For a very elongated ellipsoid we have, in like manner,
F-- 3 rf, or F-v*- t (/.^-i I,
a A \ o
according as the magnetisation is perpendicular or parallel to the
major axis.
359. CYLINDER MAGNETISED TRANSVERSELY. The case of a
cylinder might be deduced from that of an ellipsoid, but it can be
easily treated directly. If we consider an unlimited circular cylinder
of radius #, and density equal to unity, the mass of unit length will
be X = ira\
The external action of this cylinder, at a distance r from the axis,
\ 2
is equal (132) to - , which gives for the potential
p = - 27T# 2 /. r + const.
If the cylinder has a uniform transverse magnetisation, and if we
take the axis of #, parallel to the magnetisation, the external potential
will then be
V= -I = 1 - = 1-
ox r ox r 2 -
On a point in the interior, the action of a homogeneous circular
cylinder reduces to that of a cylindrical core passing through the
point. This can be easily seen by reasoning analogous to that which
has been applied to the sphere (42). The action of a cylinder on
271"?^
a point in the interior is therefore equal to - = 27rr, and the
potential becomes
P= -7rr 2 + const.
POTENTIAL OF MAGNETIC SHELLS. 345
The potential of the uniformly magnetised cylinder is then
\
<)P ~br
V = I = \2irr = \2Trx.
ox ox
Consequently the force in the interior is constant and equal to - 2?rl ;
its direction is opposite that of the magnetisation.
The induction is also constant and has the value
F! = 477! - 2?rl = 27rl.
360. POTENTIAL OF MAGNETIC SHELLS. We have seen that the
potential of a uniform magnetic shell is equal to the product of its
magnetic strength < by its apparent surface o> at the point in question.
If the shell is not uniform the expression for the potential is
By a method like that of the preceding, the calculation of the
potential may be reduced to the potential of a magnetic layer, so as
to avoid the determination of solid angles.
The potential at a point M, of an element of the shell d, is
equal to that of two magnetic layers o*/S, equal and of opposite
signs, the perpendicular distance of which satisfies the condition
vdn = <.
Let us denote by n the distance of the element from a fixed
point on the perpendicular, on the same side as the negative face,
the potential of the layer <nS may be regarded as the product of
ds by a function of this distance n, so that the potential of the
element of the shell will be
d$f(n) - d$f(n - dn) = d$dn .
on
As the distance dn of the two surfaces may be supposed to be
constant, the potential of the whole shell is
^)n ^n
J J
34^ PARTICULAR CASES.
But the expression I/^S represents the potential U of the
positive surface of the shell ; we have then
(3)
The value of the potential Q of a layer whose density at each
3?
point is equal to the magnetic strength <, is U- or ~(3dn ; from this
follows
-g-
If - be the distance of the point M from the element dS, the
f r
potential Q is equal to I j&ds, which further gives
(5) V
The factor p represents the potential of unit mass placed at the point
M on the element ^S.
361. If the shell is uniform the factor ^ is a constant. If P is
the potential of a layer of density equal to unity, we shall have
Q = 3>P, and expressions (4) and (5) become
> "-
362. Let us consider, for instance, a shell bounded by a plane
surface ; this may be replaced by a plane shell of the same strength,
bounded by the same surface. Let us place this shell in the plane
of yz, the positive face on the side of the #-axis. The abscissa of
the point M being x, we have evidently dx = - dn, and therefore
(7)
POTENTIAL OF MAGNETIC SHELLS. 347
If the shell is uniform we have
v * *
an expression which might have been obtained by the consideration
of layers of gliding (354).
363. For a shell on a sphere of radius a, the point M being at
the outside, on the positive face of the shell, we have, in like manner,
(8) V = -
and, if the shell is uniform,
(8') V = *
On the contrary, if the face turned away from the side of the
point M is negative, we must take the expressions
<>
364. In the case of a sphere, the potential p is a homogeneous
function of the degree - i of the radius 0, and of the distance r
from the point M at the centre, which gives the condition
or
348 PARTICULAR CASES.
from which follows
The distances r and being constant during the integration, we
may write
(9)
365. POTENTIAL OF A CIRCULAR LAYER. The potential of a
uniform shell with a circular edge may be calculated from the
potential of a plane circular layer, or of any given layer spherical,
for instance bounded by the same edge.
Consider, in the first case generally, a layer of revolution about
the axis of x. For a point M the abscissa of which is x 9 and which
is at a distance /> from the axis, the potential P is a function of x and
of p. If this potential be developed in increasing powers of p or
of - , the series, from symmetry, will only contain even powers of
P
the variable. We may therefore write
- I +
the coefficients A , A 2 ..... , B , B 2 ..... being functions of x.
When x and p are taken as independent variables, Laplace's
equation AV = becomes
V~2 -^~ T = -
ox 2 p op Op z
This condition gives for the first series, a new series developed
in increasing powers of p, in which the coefficients of all the terms
must be separately null, from which follows the general condition
POTENTIAL OF A CIRCULAR LAYER. 349
We have thus successively
A i . ______
4 ~~ ~ A*' 7W2 ~ "'"/- / ,\2 < "7wT'
(2.4)
A , , .
6 2 ' ()^ 2 (2 . 4 . 6)2 '
^ 2n ~ 2 _ _j_
If we know the first coefficient A , all the others can be deduced.
This coefficient A is given by the expression of the potential on the
axis, which depends on the form of the layer and on the law of
distribution.
The potential outside the axis is thus
(2. 4 ) 2 ' (to* (2.4.6) 2
For the second series, Laplace's equation would have given the
general condition
which does not enable us to determine the successive coefficients in
the same way.
366. In the case of a homogeneous circular layer of radius a
and of density equal to unity, the value of the potential P on the
axis, taking the centre as origin of the abscissae, is
P
Putting
u = v /fl 2 + x 2 ,
which gives
350 PARTICULAR CASES.
we have thus, for the first development of the potential as a function
of powers of /a,
A =2Tr(u-x),
<>A flu \ (* \
= 27T( - I )=27r( I ),
^ V x J \ u
"^ = 2Ir w ' etc '
The potential at P outside the axis is
= 2
The successive differentials would be easily calculated.
367. When the layer is circular, it is often more advantageous to
carry out the development in another manner.
Let a be the radius of the circle which bounds the layer, r the
distance of the point M from the centre of the circle, and 6 the
angle which the direction of this right line makes with the axis. We
may express the potential by one of two series
V <$/
according as r is smaller or larger than a that is to say, that the
point M is inside or outside the sphere of radius a. The coefficients
are functions of the angle 0, and as the two expressions should have
the same value on the sphere, they satisfy the condition
AO+AJ+ ..... = B O + B I + ......
The potential being considered as a function of r and of 6,
Laplace's equation becomes
OT DP W VP
r 2 -+ 2 r + cotan 6 4 = .
or 2 or ov o# 2
POTENTIAL OF A CIRCULAR LAYER. 351
We find thus that the coefficients A and B satisfy the general con-
ditions
If we develop the potential on the axis
as a function of increasing powers of - or - , we obtain the two
series
- TT f _* I / :r \ 2 _lli/^\ 4 i i 3 /-A 6
2\a) 2.4\aj 2.4.6\aJ
(14)
In order to have the expression for the potential outside the axis
as a function of the ratio - or of - , we need only remark that if the
a r
density of a spherical layer is symmetrical in reference to a diameter
taken about the axis of x, the potential of this layer at a point M
only depends on the distance r of this point from the centre O of
the sphere, and of the angle which the right line OM makes with
the axis.
From a well-known theorem of Legendre, this potential may be
expressed by the general formulas
/>\ 2
- \
352 PARTICULAR CASES.
in which A , A] . . . B , Bj . . . are constants, and X 1? X 2 . . . functions
of the angles known as Legendre's polynomials, and which are
denned by the series
- 2x cos e + x *\ =
all these functions become equal to unity when the angle is equal
to zero.
As we know the development of the potential of a homogeneous
circular layer for a point of the axis that is to say, when is zero
and r equal to x, the coefficients are known. It follows that the
potential P for a point outside the axis is expressed by
[r i />\* i i /r\ 4 i i -\ /A 6 "1
I -x l -+-xJ-}- x 4 (-}+- -|x 6 (-) -.. ,
1 a 2 2 \aJ 2.2 *\aj 2.4.6 *\a J
A 7
- ) + . .
r J
P
P = 27T<2 ----
2
2.4 r 2.4.6 r 2.4.6.8
368. POTENTIAL OF A UNIFORM CIRCULAR SHELL. The
potential of a uniform circular shell may now be obtained by the
expression
We thus find, with the first form (12),
The first terms of the development are then
(16)
V =
and the series is convergent whenever p< u.
POTENTIAL OF A SPHERICAL LAYER. 353
Taking the expression
the second form of the development will give
_j_ T -3 i a \ % T '3*5 f a \ x 4...
369. POTENTIAL OP A SPHERICAL LAYER. Let us finally con-
sider any given spherical layer of radius a. The potential at an
external point M, at a distance r from the centre, may still be
expressed by the series
(18)
in which the coefficients depend on the law of distribution and on
the direction of the right line r.
Let u be the angle of the right line r with the axis of z, / the
angle of the plane rz with the plane of yz ; the co-ordinates of the
point M are then
z = r cos u ,
(19) y = rsin u cos /,
x = r sin u sin /.
Taking r, u and / as independent variables, Laplace's equation
gives
3*(rV) W "SV i -&V
r + - + coten u - + --. - = 0;
A A
354 PARTICULAR CASES.
from which follows the general condition for the coefficients
n(n + i)A n + -^ + cotan u ^ + -^- . -2 = 0.
' n 2 2
The general integral of this equation was given by Laplace ; if we
put
_f n-m (n-m}(n-m-i} n-^-2
L 2( 2 -i)
Isin^
(\ / \
2H I/ (271 2)
the coefficient A n , expressed by means of the new symbols A w . m ,
consists of 2 + 1 terms developed according to the sines and
cosines of multiples of the angles /, and its value is
The factors denoted by g, h> with the different indices, are
numerical coefficients which must be determined in each special
case.
370. If we consider a sphere magnetised in any given way, its
external action is equal to that of two layers of equal mass and
contrary signs, distributed on the surface according to a certain law.
The coefficient A of the first term is null. For, in fact, at a
great distance, the potential simply becomes equal to the quotient of
the total -mass by the distance. The product A # 2 which forms the
numerator of the first term represents in this case the total mass
and we know that in every magnet the total mass is null.
The value of the coefficient A x of the next term is
or, taking equations (19) into account,
SOLENOIDAL MAGNETS. 355
This term becomes predominant at a great distance, and the potential
reduces then to
it follows from this (151) that the three products
represent respectively the magnetic moments of the sphere in refer-
ence to the axes of x, of jy, and of z. Denoting by 3 K the resultant
magnetic moment, and by a, /? and y the cosines of the angles which
its direction makes with the axes, we have
K = ^ =
371. SOLENOIDAL MAGNETS. The potential of a solenoidal
magnet (330) only depends on the surfaces formed by the ends
of the elementary solenoids which constitute it.
If all the solenoids are closed, the potential of the magnet is
everywhere null, and the magnetic force null. In this case the
induction is reduced at each point to 477!, and it is parallel to the
magnetisation.
372. Suppose that a solenoidal magnet is bounded by a channel
surface, the magnetisation being everywhere normal to the perpen-
dicular section of the channel. The flow of induction across an
element dS of the right section is equal to 47rL/S, and the value of
the flow of induction is
Each of the solenoidal filaments forms a closed curve of length /,
perpendicular at every point to the right section of the channel. If
the structure of the magnet is such that the product of the strength
of magnetisation I of a filament, by its length /, is a constant quantity,
examples of which we shall see afterwards, the flow of induction
could be expressed by the formula
(20)
'/
A A 2
356 PARTICULAR CASES.
If the magnet is a ring of revolution, and x is the radius of an
elementary solenoid, we shall have
A (Vs A (Vs
Q = 4?rA = 2 A .
J 2TTX J X
Consider a torus or anchor ring, for instance. Let a be the
radius of the section and R the distance of its centre from the axis
of rotation taken as axis of z; we have then
and the value of the total flow of induction is
() Q
373. CYLINDER. A cylinder uniformly magnetised and termi-
nated by right sections, is equivalent to two equal and opposite
magnetic layers 1, which cover the two bases A and B. The
potential of any such magnet at a given point is equal to the
sum of the potentials V a and V & , of the two terminal layers.
If the right section of the cylinder is circular, the potentials
V a and V 6 may be expressed by the formulae found previously
(365 and 366).
The expression for the magnetic force on a point M of the axis
on the outside and on the side of the positive face A, is
F=27rl(i -COS a)- 27rl(i - COS fi) = 27rl(cos ft - COS a),
a and /? being the angles under which the radii of the two bases are
seen from the point M, and it is in the same direction as that of the
magnetisation.
For a point in the interior, the actions of the two bases are of the
same sign, which gives a force
F = 4?rl - 2?rl (cos a + cos ft),
in the opposite direction of that of the magnetisation.
CYLINDER. 357
Lastly, the induction on the axis in the interior is
Fj = 47rl - F = 27J-I (cos a + COS ft)
it is parallel to the magnetisation and varies very slowly so long as
the point in question is at a considerable distance from the bases.
If / is the length of the cylinder and a its radius, the value
of the induction F at the centre of the cylinder is
= 27rl.
v=
When the length of the cylinder is very great as compared with
its diameter, we may take the approximate expression
F = 4 Tl| i--
The induction is then sensibly the same throughout the whole extent
of the median section, and the expression for the total flow of induc-
tion Q which traverses it is
If the cylinder is so long that the quantity in brackets does not
sensibly differ from unity, the flow of induction in the median
section is equal to (2Tra) 2 I. This flow is proportional therefore to
the square of the contour ; it has sensibly the same value in any
given section sufficiently distant from the ends.
MAGNETIC INDUCTION.
CHAPTER IV.
MAGNETIC INDUCTION.
374. GENERAL CHARACTERISTICS OF MAGNETIC INDUCTION.
There is probably no substance which, when placed in a magnetic
field, does not experience the effect of induction that is to say, does
not itself become a magnet, at any rate temporarily.
When the body is isotropic, the axis of induced magnetisation
coincides everywhere with the direction of the magnetic force. In
certain bodies the induced magnetisation is in the same direction
as the force ; these are the bodies which we have called paramagnetic
or simply magnetic. In others the direction of the magnetisation is
opposite that of the force; these bodies are diamagnetic. In the
presence of a pole of a magnet, the nearest part of bodies of the
first class acquires polarity of the opposite kind ; bodies of the
second class acquire a pole of the same kind.
We shall assume that at every point of an isotropic body sub-
mitted to magnetic induction, the magnetisation is proportional to
the resultant of all the magnetic forces which are exerted at this
point. These forces depend not only on the original field, but also
on the magnetism developed by induction on the body itself. If F
is the resultant force, to which the name magnetising force is some-
times given, I the intensity of magnetisation, we may write
(i) I = F.
The factor k, which expresses the ratio of the magnetisation to
the magnetising force, is called the coefficient of induced magnetisation ;
this coefficient is positive or negative, according as the body is mag-
netic (in the ordinary sense of the word) or is diamagnetic.
The hypothesis of the proportionality of the magnetisation to the
magnetising force is verified with close approximation whenever k
INDUCED MAGNETISATION. 359
has a very small value. This is the case with most magnetic sub-
stances, with the exception of iron, nickel, and cobalt. In the case
of iron or nickel, for which the coefficient k reaches very high
values, such as 30 or 40, proportionality exists as long as the force F
does not exceed a certain limit ; when the bodies are magnetised by
the earth, for example. This is also the case with ordinary iron,
twisted iron, cast iron, and steel more or less tempered, the co-
efficient of magnetisation of which is considerably weaker. The
coefficient k is always very small for diamagnetic bodies j it scarcely
amounts to for bismuth, which is the most active body
400,000
of this second class.
If the proportionality between the magnetisation and the mag-
netic force does not exist, we may consider the coefficient k as being
itself a function of magnetisation. We shall first investigate the case
in which this coefficient is constant and the same in all directions
that is to say, in which the body is isotropic and the induced mag-
netisation somewhat feeble.
375. INDUCED MAGNETISATION is PROPORTIONAL TO THE
MAGNETISING FORCE. Consider any given body in the magnetic
field. Let V be the potential of the field and 12 that which is pro-
duced by induced masses, the value of the actual potential U will be
u=v+a
At any given point the components of the magnetising force
parallel to the axis are
X- V- W 7- W
~' "" L ~~-
The expression for the force itself is
dn>
its direction is that of the perpendicular , to the equipotential
surface which passes through the point in question.
360 MAGNETIC INDUCTION.
The intensity of magnetisation I, and its components A, B, and C
parallel to the axes then become
Adding these last equations after having multiplied them by <&?, ^,
</s respectively, we get
If the value of k is constant throughout the body, the first member
of the equation is the exact differential of a function < of the
coordinates, and we have
A- 3 * B- 3 * r- 3 *
** ^ > *-* ">T~ > * T~
It follows from this (332) that the magnetisation is lamellar.
376. THE INDUCED MAGNETISATION is SUPERFICIAL. On the
other hand the general expression for magnetic density
c)B
here reduces to
, /yu ^ 2 u yu\
= -* ++ = ~
Since, from Poisson's equation, we have AU = 4?r/o, we get
= 0, or /> = 0;
that is to say that the magnetic density is zero throughout the whole
extent of the body. The magnetisation is then also solenoidal, and
there is no free magnetism except on the bounding surface of the
body.
SUPERFICIAL INDUCED MAGNETISATION. 361
This conclusion presupposes that the parenthesis i + 4^ is not
zero ; but this latter case never occurs, the absolute value of k for
diamagnetic bodies being far from attaining .
4?r
377. The surface density a- of the induced layer is
cr = l COS0,
6 being the angle of the magnetisation with the perpendicular to the
surface (Fig 84).
Fig. 84.
Let this perpendicular be called n when it is drawn inwards,
and n' when it is drawn outwards, and let a be the perpendicular
to the surface for which the function 3> has a constant value,
we have
o- = I cos 6 = ; = - .
on un
378. The value of the potential due to the induced masses that
is to say, to the surface layer is
-//=?
The function 12 is finite and continuous, and satisfies Laplace's
362 MAGNETIC INDUCTION.
equation, both inside and outside the surface. If we denote by 12' its
value on the outside, for two infinitely near points on opposite sides
of the surface, we shall have the condition
379. EQUATION OF CONTINUITY. COEFFICIENT OF INDUCTION.
The principle of the conservation of the flow of induction (323)
enables us to establish in a very simple manner the conditions
of continuity V, U, and 12, at the surface of the magnetised
body.
Consider two infinitely near points on the perpendicular on each
side of the surface ; let F : be the value of the induction at the point
n the interior, F\ the value of the magnetic force at the external
point. If (F 1 ) n and F n denote the normal components of these two
forces calculated in the same direction, then in virtue of the theorem
of the conservation of flow, we have the condition
The magnetisation being parallel to the magnetising force, the
induction in the present case becomes
it is proportional to the magnetising force.
If we put
we have
and the equation relative to the surface becomes then
() fF.-F.. or M-l*
Thus, for two infinitely near points on either side of the surface, the
ratio of the perpendicular components of the magnetic force is constant.
This is a fundamental deduction from Poisson's theory, which we
have already used (111) in defining dielectrics. The coefficient /*
COEFFICIENT OF INDUCTION. 363
represented the specific inductive capacity of electricity; we shall
here call it the coefficient of magnetic induction. It must not be
confounded with the coefficient of induced magnetism which has
been represented by k.
380. Expressing equation (2) as a function of the potentials,
we get
or
To determine the magnetisation of a body placed in a magnetic
field, and bounded by a surface S, we must find two conditions 12
and 12' which satisfy the following conditions
i st The function is finite and continuous in the interior of the
surface, and satisfies Laplace's equation A12 = 0.
2nd. The function 12' is finite and continuous on the exterior, zero
at an infinite distance, and also satisfies Laplace's equation.
3rd. The functions 12 and 12' are equal to each other on the
surface, and their differentials satisfy the equation of continuity (3).
These functions represent the potential of a magnetic layer
distributed on the surface of the body. The density of this layer at
every point is determined by the variation of the normal components,
which gives
from which is deduced
381. CASE OF Two DIFFERENT MAGNETIC MEDIA. RELATIVE
MAGNETISATION. Let us suppose that the body A, bounded by the
surface S, is situated in a magnetic medium whose coefficient of
induction is //; the theorem of the conservation of the flow of
induction gives still
364 MAGNETIC INDUCTION.
that is to say
or
The functions 12 and ft' which determine the surface layer are
defined by the same conditions as the preceding, with this single
difference, that the equation of continuity (4) contains the co-
efficients of induction of the two media.
The surface density is still given by the perpendicular components
If we put
that is to say
4717*
the expression for the density becomes
47TO- = F n (/x 1 - i), or o- = kf n .
It is this surface layer which determines the motion of the body
A in the medium. It is the same as if the external medium were
suppressed, or more exactly replaced by air, and the coefficient
of induction of the body replaced by another value /*j, or the
coefficient of magnetisation k by a different value k v The apparent
magnetisation I v of the body would thus have for its perpendicular
projection
382. The discussion of this problem gives rise to some con-
clusions analogous to those which are deduced from the principle of
Archimedes for bodies immersed in liquids.
RELATIVE MAGNETISATION. 365
We may, in fact, consider k^ as the relative coefficient of magneti-
sation of a body, in reference to the medium which surrounds it,
k and /', being coefficients of the two media in respect of air. If the
coefficient k of the body is greater than the coefficient K of the
medium, the value of k^ is positive, and the apparent magnetisation
of the body is positive. If, on the contrary, k<k\ the value of k^
is negative and the body will appear diamagnetic. When the co-
efficients k and K are equal, the magnetisation of the body A will
appear to be null, which ought to be the case, for it is in a medium
identical with itself, and the induced magnetism is superficial.
We are thus led to assume that there is no real opposition of
properties between magnetic and diamagnetic bodies, and that the
difference of the effects is due to the greater or less magnetic
character of the external medium. As diamagnetic bodies retain
their characteristic properties in the most perfect vacuum which has
been produced, we must assume, on this view, that a vacuum is a
magnetic medium, and that its coefficient of magnetisation is greater
in absolute value than that of all known diamagnetic substances.
If, on the contrary, we assume the value zero for the coefficient
of magnetisation of vacuum, a negative value must be assigned to
those of all diamagnetic bodies. In this case the coefficient of
induction ft is greater than unity for magnetic bodies, and is less
than unity for diamagnetic bodies. No body is known in which ft is
negative since the coefficient of diamagnetic bodies is never greater
than in absolute value ; we have already said that for bismuth, the
47T r
most diamagnetic of all known substances, k is about = .
400,000
The coefficient of induction of diamagnetic bodies only differs from
unity by an infinitely small quantity. For soft iron and nickel the
coefficient k being comprised between 30 and 40, the value of ft is
nearly 500. The ratio of the two absolute values of k for iron and
bismuth is then almost
40 x 400,000 = 1,6 . io 7 .
We may however remark that the influence of a magnetic
medium could only be exactly compared with that of a fluid, and
the principle of Archimedes be applied, provided that k^ = k - k 1 .
From the preceding remark it appears that this ratio is very nearly
verified for all diamagnetic bodies, and for those also which are
very slightly magnetic; but it would be far from the truth if the
surrounding medium had a coefficient of magnetisation near unity,
MAGNETIC INDUCTION.
and especially if the magnetic properties of the medium were com-
parable with those of soft iron.
383. MAGNETIC SUSCEPTIBILITY AND PERMEABILITY. The
phenomena of magnetic induction may, as we have seen, be
expressed with the aid of two coefficients.
The coefficient of induced magnetisation k expresses the ratio of the
intensity of magnetisation to the magnetic force ; in other words,
the intensity of magnetisation in a field equal to unity. This
coefficient is sometimes known as that of Neumann, who first used
it. Sir W. Thomson calls it the coefficient of magnetic susceptibility.
The second coefficient called /*, is the coefficient of magnetic
induction ; it is the analogue of the specific inductive capacity of a
dielectric in electricity. It is equal to the ratio of the perpendicular
components of the force on the outside and inside of the medium in
question, and is connected with the preceding by the ratio
/A I + 47T&
Sir W. Thomson calls this coefficient /A, the coefficient of magnetic
permeability. The following are his reasons for using this expression
" The analogue corresponding to conducting power of a solid for
heat or, as it is shortly called, 'thermal conductivity' is, in electro-
static induction, the 'specific inductive capacity' of the dielectric; in
magnetism it is not what has hitherto been called magnetic inductive
capacity a quality which is negative in diamagnetics but it is
Faraday's ' conducting power for lines of force,' and in hydrokinetics
it is flux per unit area, per unit intensity of energy. The common
word permeability seems well adapted to express the specific quality
in each of the four analogous subjects. Adopting it we have thermal
permeability, a synonym for thermal conductivity; permeability for
lines of electric force a synonym for the electrostatic inductive
capacity of an insulator; magnetic permeability a synonym for
conducting power for lines of magnetic force; and hydrokinetic
permeability a name for the specific quality of a porous solid
according to which, when placed in a moving frictionless liquid,
it modifies the flow." (Reprint of Papers, 628.)
384. ANISOTROPIC BODIES. We have hitherto only considered
the induction produced in isotropic bodies. The experiments of
Pliicker on bodies with a fibrous texture, and on crystals, have shown
that magnetic action is exerted unequally in different directions.
Poisson had predicted the existence of such bodies, and in order to
explain them on his theory we must substitute for the conducting
UNIFORM MAGNETISATION. 367
spheres, disseminated in a non-conducting medium, equal ellipsoids
turned in the same direction. If the body thus constituted " were a
homogeneous sphere, and it were made to turn without displacing its
centre of gravity, and without any change either in the external
forces or in the function V, the magnitude and directions of the
magnetic forces in this body would nevertheless change. As this
particular case has not yet been met with, we shall for the present ex-
clude it from our researches." (Memoire surla theorie du magnttisme,
"Me'm. de 1'Institut pour 1821-22," Vol. v., p. 278.)
For the reasons which we have already developed in electricity
(215), there must in this case be three principal axes of magneti-
sation. If we place each of these axes successively in the direction
of the field, we shall have between the coefficients , #, k" of sus-
ceptibility, and the coefficients ju, //, p" of permeability, the ratios
When the body is directed in any manner whatever in reference
to the field, we may substitute for the true field three fields whose
directions are rectangular and parallel to the principal axes, and
consider the real magnetisation as the resultant of these three
magnetisations.
This superposition is evidently legitimate in all cases of dia-
magnetic, or of slightly magnetic bodies ; it is only a consequence
of the principle of the proportionality of the magnetisation to the
magnetising force.
385. UNIFORM MAGNETISATION. If the surface of a body is
such that uniform magnetisation in a certain direction produces a
certain constant force, and we place it in a uniform field, the force
of which is parallel to this direction, it will acquire a uniform
magnetisation, since the magnetising force will have the same value
at all points, and that, without its being necessary to make any
restriction on the magnitude of the coefficient k. The magnetising
force F being the sum of the strength of the field <, and of a force
CI due to induced magnetisation, which is evidently proportional to
the intensity of the magnetisation, we shall have
and therefore
368 MAGNETIC INDUCTION.
Since the internal action of a uniformly magnetised body is given
(354) by the partial differentials of the second order of the poten-
tial P of a homogeneous mass, the force can only be constant
provided the function P is represented by a polynomial of the
second degree that is to say, if the body is bounded by a surface
of the second degree. When the coefficient k is very small that
is to say, for all diamagnetic and feebly magnetic bodies we have
sensibly
In this case the magnetisation induced in a uniform field is
under the same laws of induction as are dielectrics, so that all
the results to which we have attained in electrostatics are also
applicable to magnetism, without any other modification than the
substitution of the magnetic potential for the electrical potential,
and of the coefficient of magnetic induction for the specific inductive
capacity.
386. SPHERE. For a uniformly magnetised sphere (355) we
have
The magnetisation produced on a sphere by a uniform field
will be
47T/X+2
M I
The coefficient h or - - is equal to unity for conductors of
p+2
electricity; it is always positive and less than unity for magnetic
bodies, and it differs little from unity when /*, is very great. This
coefficient, on the contrary, is negative and very small for dia-
magnetic bodies.
The value of the magnetic moment of the sphere is
The resultant force within the sphere is
POISSON'S HYPOTHESIS. 369
and the induction
The former of these expressions also represents the total external
force on a point near the equator, and the second on a point near
the pole. The ratio of these two forces is therefore equal to - .
In the case in which /x is very great, the formulae become simpler,
and we have sensibly
387. POISSON'S HYPOTHESIS. If we suppose with Poisson that
a magnetic body is made up of a system of small spheres, which
are absolute magnetic conductors (/x = oo ), disseminated in a non-
magnetic medium, the ratio of the volume occupied by all the
spheres, to the total volume, is expressed (167) by
Taking the value /* = 500 for iron, we get
.-
500 167
But the maximum value which the ratio h can have with equal
7T I
spheres is j= = i - -. We must therefore suppose that
in the present case the volumes of the spheres are not the same,
and that the greater intervals are occupied by spheres of smaller
diameter. It appears, however, difficult to suppose that the adjacent
spheres do not act on each other, and that the magnetisation of
each of them, as is assumed on Poisson's theory, could be solely
dependent on the external field.
B B
37 MAGNETIC INDUCTION.
388. ELLIPSOID. CYLINDER. For an ellipsoid magnetised
uniformly in any given direction, the components of the internal
force parallel to the axes are equal respectively to - AL, - BM, and
- CN (356).
In a uniform field in which the force < makes with the axes
angles whose cosines are A, A', A", the components of the magneti-
sation will be
These equations presuppose that the magnetisation is so weak
that we may admit the effects produced in different directions to
be superposed.
If one of the axes of the ellipsoid is parallel to the direction of
the field the axis of x for instance we have
and however great may be the value of k, the magnetisation will be
uniform.
By the results indicated in (357) we might apply this expression
to several different cases.
For an unlimited cylinder, perpendicular to the direction of the
field (358), we shall have
I-
389. BARLOW'S PROBLEM. The case of a dielectric layer com-
prised between the surfaces of two concentric spheres, and placed in
a uniform field (166) corresponds to that of a magnetic layer of the
same form placed in the same conditions. This question is known
as Barlow's problem.
BARLOW'S PROBLEM.
There are produced then, as we have seen in the case of
electricity (165), two magnetic layers, one M : on the internal
surface S : (Fig. 85), and the other M 2 on the external surface S 2 of
the volume in question.
Fig. 85.
The internal actions F! and F 2 of these two layers are determined
by the equations
-F,
where /5 denotes the ratio ( ) of the cubes of the radii.
\a 2 /
The action of these two layers is constant in the interior of the
small sphere S p and its value is
but it is not constant within the magnetic substances, nor without.
We have then, for the total action <f> - F on the interior,
9 I
B B 2
372 MAGNETIC INDUCTION.
and, if k is very great,
The external potential at P, of the layers M x and M 2 , at a
distance r from the centre, and on a radius which makes the angle to
with the force of the field, is
cos q> (ji- i) (i +p) (i - /3) , cos (o
<?>
putting
from which is deduced, for the action of the induced magnetism in
two points at a distance r on the line OA and on the line OB, that
is at the pole and at the equator,
v a/I-
y\.p ^ I
390. For a solid sphere we have /? = 0, and therefore
The ratio of the actions excited at the same point by a hollow
sphere, and by a solid one of the same external diameter, is then
A
A~o
BARLOW'S PROBLEM. 373
Applying this formula to iron, and taking ^ = 500, we get
A i
112 i-
By giving different values to /?, we find that, as long as the thickness of
the spherical layer is greater than a fifth of the radius, the magnetic
action on the exterior does not differ by o.oi of that which a solid
sphere of the same diameter would produce.
The older experiments of Barlow are in conformity with the
results of this calculation. With spheres of 10 inches external
diameter, Barlow found no appreciable difference between the
actions of two different spheres, one solid and the other hollow,
the thickness of the latter being equal to - of the radius.
2
On the other hand, the action of the hollow sphere was only -
that of the solid one, when the thickness was reduced to about the
of an inch. Taking the value of ft at 500, calculation would
30 j
give about - for the ratio of the two actions.
391. We have seen (386) that the total action in the vicinity of
a solid sphere is
near the pole, and
ear the equator. Still taking //, = 500, as for soft iron, we get
nearly.
374 MAGNETIC INDUCTION.
The force is sensibly zero at the equator, and at the pole its value
is three times that of the force of the field.
If the coefficient p- is near unity, and we put
we get in like manner
3
In the interior of a hollow sphere, the force is
9/x
If the coefficient /* is very great, we may write
i
If the coefficient /* is very near unity, and we again put /A = i + a,
we have
I -- E -- :I _ 2 _ a2(l _ /8) .
* 9
A small magnetised needle introduced into the sphere would
determine there a new induced layer, which would be superposed on
the first, and the distribution of which can be easily calculated, for
the external action of such a needle is equal to that of a uniformly
magnetised sphere ; but the action of this new layer will always be
parallel to the magnetic axis of the needle, and will not influence its
direction. The oscillations of this needle depend then only on the
resultant actions of the external field, and on the layers induced by
the field itself.
ANISOTRpPIC BODIES. 375
392. ANISOTROPIC BODIES. Consider an anisotropic body in a
uniform field. Let , ', k" be the three principal coefficients of
magnetisation, and A, A', A" the cosines of the angles of the strength
of the field with the axes ; the coefficients of magnetisation being
supposed to be very small, the values of the intensities of the three
partial magnetisations will be
and the corresponding magnetic moments
=ul = u<
1 =ul' =u<i>k'\',
From this we deduce, for the resultant magnetic moment,
M 2 =
The resultant magnetic axis of the sphere makes with the axes of
the co-ordinates, angles whose cosines a, a', a" are defined by the
equations
and this axis makes, with the direction of the field, an angle 6 defined
by the equation
COS C/ = aA-faA+aA =
H
Denoting by M' the moment of the couple produced by the action
of the field on the sphere, we have
376 MAGNETIC INDUCTION.
or, replacing M and by their values,
M' 2 = < 2 M 2 (i - cos 2 0) = *A 4 [H 2 - (k\ + k'X" 2 + /T 2 A" 2 ) 2 ]
= i?<? [~(/ 2 A 2 + ' 2 A' 2 + " 2 A" 2 ) ( A 2 + A' 2 + A" 2 ) - (A 2 + k' A' 2 + "A" 2 )
= ty* \ [X'X'(k' - k")J + [ A" A (k" - k)J + [XX' (k - k')
The sphere can only be in equilibrium if the couple produced by
the action of the field is null ; hence the three squares comprised
within the brackets must be separately null. As, by hypothesis, the
co-efficients k, k' k" are different, two of the cosines A, A', X" must be
equal to zero, and therefore one of the principal axes of magneti-
sation must coincide with the direction of the field.
The preceding calculation applies also to the case of a homo-
geneous body of any given shape situated in a uniform field, for the
magnetic moment, in reference to one of the principal axes, is simply
proportional to the volume of the body, and to the component of the
force of the field.
If the field is variable, we may suppose the volume of the body
under consideration to be infinitely small ; the moment of the couple
which tends to turn it about its centre of gravity will still have the
same expression as a function of the strength of the field at the point
occupied by the element of volume.
393. EXPERIMENTAL DETERMINATION OF THE COEFFICIENTS
OF MAGNETISATION. When a cylinder is magnetised in a manner
uniformly parallel with the axis, the action which it exerts on a point
in the interior only depends on the two terminal layers. If the cylin-
der is very long, this action may be neglected for all points whose
distance from one end is very great compared with the diameter ;
the resultant force will then be produced by external masses alone.
If the external field is uniform and parallel to the axis, the magneti-
sation in the greater part of the cylinder will be uniform and
proportional to the strength of the field. In the neighbourhood of
the ends only, the induced magnetisation will be slightly modified ;
the superficial layer, instead of being uniform, and limited to the
terminal surface, will have a more complicated distribution, and will
be partially spread on the lateral surfaces.
According to this the coefficient of magnetisation of an isotropic
substance may be defined as the quotient, by the force of the field,
of the intensity of magnetisation which an infinitely thin cylinder of
the substance acquires when placed in a uniform field; or the
magnetisation which it assumes in a field equal to unity.
DETERMINATION OF THE COEFFICIENTS OF MAGNETISATION. 377
In like manner, the coefficient of magnetisation of an anisotropic
medium in a determinate direction is the longitudinal magnetisation
which an infinitely thin cylinder would acquire, parallel to this
direction, in a field equal to unity.
394. We see also that to determine the coefficient of magneti-
sation of highly magnetic bodies such as iron, we cannot make use of
the external action produced by spheres or by elongated bodies in a
direction perpendicular to the field. For the ratio of the magneti-
sation to the force is respectively equal to
II 21
ATT I ATT 2
1+ r I+ -^
ATTK ATtk
according as the body is a sphere, a disc, or a very elongated
ellipsoid of revolution. These ratios differ too little from the
approximate values obtained by making k infinite, to allow us to
deduce the coefficient k with any degree of precision.
On the contrary, with elongated bodies parallel to the lines of
force, the magnetisation tends to become proportional to /, and
independent of the shape of the body.
395. DISPLACEMENT OF BODIES IN A MAGNETIC FIELD.
ATTRACTIONS AND REPULSIONS. The potential energy of an
infinitely small dielectric sphere (178) in a field is
2 47T 2
This expression represents also the energy of an infinitely small
magnetic sphere, and even of any volume element of a homogeneous,
isotropic substance, whose coefficient of magnetisation, positive or
negative, is very feeble ; we have then = /, and W = -uk >
ATT 2
For a very small displacement, the variation of energy is
"
If the body is magnetic the coefficient k is positive, the energy
diminishes when the body approaches points where the absolute
37^ MAGNETIC INDUCTION.
value of the force is a maximum. A very small magnetic body in a
variable field tends then to move towards points where the force is a
maximum. As there is no absolute maximum of force outside
acting masses (180), it follows that, if the body is left to itself, it
will end by touching the surface of the magnets ; it is therefore
attracted by the magnets.
For diamagnetic substances the coefficient k is negative. A small
diamagnetic body approaches points where the force is a minimum ;
it tends to move more and more away from the centres of force it is
repelled by magnets.
As the field may contain points where the force is null, and which
are then absolute minima for the value of </> 2 , we see that there may
be stable equilibrium for a diamagnetic body in a variable field
outside acting masses.
Faraday had already announced as a result of experiment this law,
that magnetic bodies move towards points where the force is a maximum,
and diamagnetic bodies towards points where the force is a minimum.
It is to Sir W. Thomson that we owe the true interpretation of the
phenomenon.
In a uniform field the energy of a small isotropic, magnetic, or
diamagnetic body is constant, and therefore the force null.
396. For an anisotropic magnetic body the total energy is the
sum of the energies corresponding to the magnetic moments u$>k\,
u<}>k'X', &<>"A", due to the components <A, <A', </>A" of the force
parallel to the three axes ; we have then
W = - u (ktf + k'X'* + k" A" 2 ) .
If the body is compelled to turn about its centre of gravity,
stable equilibrium corresponds to the case in which the energy is a
minimum that is to say, where the expression comprised between
the brackets is a maximum.
As it is necessary, for equilibrium, that two of the cosines A, A'
and A" are zero, this maximum will take place when the quantity in
brackets is reduced to the term corresponding to the greatest of the
coefficients k, k' and k". The axis of greatest magnetisation is then
parallel to the force of the field.
If a body passes from a position in which the force and the
direction of the field are defined by < 1? A 19 A\, A" 3 , to another
DISPLACEMENT OF BODIES IN A MAGNETIC FIELD. 379
position in which the values of the same quantities are </> 2 , A 2 , A' 2 , A",
the change of energy is
W, - W, = - lt-SAj + k'\'\ + k"X'\[ - $*A| + k'X'\ + k"X" J] .
This variation is negative, and the displacement tends to take place
under the influence of magnetic forces alone, when the quantity in
brackets is positive.
If the body is compelled to move parallel to itself, the preceding
expression becomes
W 2 - W, = - - [ktf + 'A' 2 + " A"
We thus see that the body tends in all cases to move towards
points in which the force is a minimum. This tendency will be the
more marked, the greater is the second factor; it is a maximum
when the principal axis of the greatest magnetisation, is parallel to
the field, and a minimum when it is perpendicular to it.
397. If the body is diamagnetic, the values of , k', k" are
negative, and the conclusions are just the opposite of the preceding.
Stable equilibrium in a uniform field takes place when the axis of
feeblest magnetisation is parallel to the direction of the field. In a
variable field the body tends to move in a direction in which the
force decreases, and the action is a maximum when the axis of the
feeblest magnetisation is parallel to the lines of force. These two
causes may act in opposite directions, and produce opposite effects
according as one or the other predominates. In this way may be
explained many experiments which have long appeared contradictory
or paradoxical.
The results would be still more complicated for bodies whose
three principal coefficients of magnetisation are not all of the same
sign.
Such bodies would be magnetic in certain conditions and
diamagnetic in others. None such are known ; but the case might
be realised artificially by placing in a non-uniform field, a crystallized
magnetic sphere surrounded by an equally magnetic liquid, whose
coefficient of magnetisation would be intermediate between the
coefficients of the greatest and the smallest magnetisation of the
sphere. The sphere would be magnetic along the axis of the
380 MAGNETIC INDUCTION.
greatest, and diamagnetic along the axis of the feeblest magneti-
sation ; it would turn in the direction of increasing forces, when the
first of these axes was parallel to the field, and in the opposite
direction when it is the second. These actions, however, would be
so feeble that it would be difficult to make them evident.
398. EQUILIBRIUM OF LONG BODIES IN A UNIFORM FIELD.
We have seen in electrostatics (185) that an elongated conductor
placed in a uniform field is in equilibrium when the axis of the
cylinder is perpendicular or parallel to the force of the field, and
that the equilibrium is unstable in the first case, and stable in the
second.
This must also be the case with a long iron cylinder placed in a
uniform magnetic field, for the magnetisation of a soft iron sphere is
a fraction very near unity, h = - , of the electrification which this
sphere would acquire in a uniform electrical field in which the forces
had the same absolute values.
It has been known in fact, since Gilbert's time, that a soft iron
needle movable about a vertical axis sets in the magnetic meridian,
and that if it were movable about its centre of gravity it would take
up the direction of the dipping needle.
Nevertheless, in order to explain this experiment it is not suffi-
cient to say that the magnet is everywhere magnetised parallel to the
force of the field, for in that case the needle should be in equilibrium
in all positions ; hence the magnetisation of the mass cannot be
uniform.
The couple which acts on the needle when it is oblique to the
forces of the field, is due to the fact that the reactions of the various
particles have modified the magnetisation ; the direction parallel to
the field is that therefore which, in consequence of these reactions,
corresponds to the maximum of magnetisation.
399. Consider, in fact, a series of balls of soft iron B, B', B"
fixed on a non-magnetic axis, and placed in a uniform field ; let a be
the angle of the axis with the direction of the field.
If the balls are so far apart as not to act on each other, the
magnetisation is parallel to the field, and the resultant is null. But
if the distance of the balls is not very great compared with their
dimensions, it is clear that the magnetisation of each of them is
increased by their mutual action, and that it takes place along
directions which make, with the axis, angles o>, c/ . . . smaller than a,
and changing from one sphere to another. Each sphere is no longer
in equilibrium ; it is under the action of a couple, and the whole of
EQUILIBRIUM OF A DIAMAGNETIC BODY. 381
the couples tends to bring the common axis into the direction of the
field. In this position the magnetisation is a maximum.
If, on the contrary, the axis is perpendicular to the direction of
the field, the reciprocal actions tend to diminish the magnetisation of
each of the insulated balls ; the equilibrium is unstable, and the
magnetisation of the system of the spheres is minimum.
Thus the existence of a position of stable equilibrium for a
magnetic needle in a uniform field, implies the existence of inter-
actions between the different magnetic elements which constitute it,
and in contradiction with Poisson's hypothesis on the constitution of
magnetic bodies, which presupposes that such actions do not take
place.
400. The conclusions would be almost the same for the equi-
librium of a diamagnetic body, although the actions are in the
opposite direction.
For the induced magnetisation is then in the opposite direction
to the magnetising force. A series of balls B, B', B", . . ., arranged
on a straight line perpendicular to the field, becomes magnetised in
a direction opposite to that of the field ; the reactions increase then
the magnetising force on each of the balls. This direction corre-
sponds thence to a maximum of magnetisation and to a state
of equilibrium.
Let us now suppose that the line of the balls forms an angle a with
the direction of the field ; since the direction of magnetisation is
inverse, and each pole tends to develop a pole of the same kind in
the nearest part of another ball, the reactions diminish the magnetis-
ing force, and modify its direction; the effect is further the more
marked the smaller the angle a. The couples which act on the
spheres tend to set the axis in a direction parallel to the field. The
magnetisation is then a minimum.
Hence a diamagnetic needle should also take up a direction
parallel to that of the field to be in stable equilibrium.
Nevertheless the coefficient of magnetisation for diamagnetic
bodies is so feeble, that the reactions of the particles may be
neglected and their effect escape all means of observation.
For a diamagnetic needle, provided it is not crystalised, is in
mobile equilibrium in a uniform magnetic field ; in all experiments
in which there seem to be phenomena of direction, the effect is due
to the magneto-crystalline properties (397) of the body in question.
401. EQUILIBRIUM OF BODIES IN A VARIABLE FIELD. In a
variable field the phenomena are more complicated.
Diamagnetic bodies simply follow the law of Faraday that is to
382 MAGNETIC INDUCTION.
say, that each of the volume-elements tends to move towards points
where the force is a minimum, and the movement of the whole of
the system is determined by this tendency of each element.
Consider, for instance, the field produced by the opposite poles
of two identical magnets, or by the two poles of a horse-shoe
magnet, or more simply the field of two equal masses of opposite
signs (Fig. 34).
In the centre of the figure O, at an equal distance from the two
magnets, the value of the force is a minimum in reference to the
diametrical line AA', and a maximum in reference to a direction Oy
perpendicular to the former. A small isotropic magnetic sphere,
which can only move along the right line O^, moves towards the
point O when it is in stable equilibrium; a diamagnetic sphere in
the same conditions would be in unstable equilibrium at the point O,
and would tend to move away to an indefinite distance. Even if this
sphere were absolutely free, and situate on the right line Ojy, at a
small distance from O, it would move away from this point along the
line Oy (that is, perpendicularly to the lines of force) , for that is the
direction in which the force varies most rapidly.
402. A long magnetic needle, movable about the point O, would
set parallel to the line of the poles AA' in stable equilibrium ; each of
the volume-elements would tend towards points where the force is a
maximum.
A diamagnetic needle, on the contrary, would be in stable equi-
librium in a direction perpendicular to the line of the poles.
The needles set then parallel, or transversely to the line joining
two opposite poles, according as the coefficient of magnetisation is
positive or negative. Hence the names paramagnetic, or diamagnetic,
given by Faraday to bodies belonging to the first or second class.
403. We have seen that even in a uniform field, a magnetic
needle places itself parallel to the lines of force, and on the other
hand the different elements tend towards points where the force is
a maximum.
When the two kinds of actions are concordant, as in the pre-
ceding case, the position of equilibrium can be easily determined ; but
it may happen that the tendency of each element to move towards
the maxima of force may have the result of bringing the system into
a direction which is not parallel to the lines of force. The position
of equilibrium depends, in that case, on the conditions of experi-
ment. Suppose, for instance, a series of identical soft iron needles
arranged perpendicularly, and at equal distances from each other,
on a non- magnetic rod, and let this system be placed between the
OSCILLATIONS OF AN INFINITELY SMALL NEEDLE. 383
opposite poles of two magnets. If the needles are at considerable
distances each of them will tend to put itself parallel to the lines of
force, and the entire system will be in equilibrium when perpen-
dicular to the line of the poles. If, on the contrary, the needles are
gradually shortened, or if they are multiplied so that they are almost
in contact, a moment will arrive in which the tendency of each to
move towards points of maximum force will predominate, and the
whole system will now set parallel to the lines of force that is, to
the line of the poles.
It will be seen that all intermediate cases may present themselves,
and even that for a given magnetic system the direction of parallel
or transverse equilibrium depends on the law of variation of the
field in which it is placed.
404, OSCILLATIONS OF AN INFINITELY SMALL ISOTROPIC NEEDLE.
The problem is identical with that which has already been treated
for dielectrics (183, 184).
As a particular case, if the field is symmetrical in reference to the
centre of the needle, the time of the oscillations is given by the
formula
KA + B'
it is independent of the length of the needle. This latter fact had
been found experimentally by Matteucci for non-crystalline bismuth
needles ; the explanation was given by Sir W. Thomson.
In the present case, the coefficient
K*-i-
reduces sensibly to a constant for great 'values of k, and becomes
equal to k for small ones. The method of oscillations could not
then be employed to determine the coefficient of magnetisation of
highly magnetic bodies such as iron ; it serves very well on the
contrary for feebly magnetic or for diamagnetic bodies.
If the field varies in any way, the method of oscillations would
with difficulty give good determinations of the value of k even for
bodies with a very feeble coefficient. The position of equilibrium
of the needle depends then, as we have seen, on the law of the
variation of the field, and on the length of the needle ; this is also
the case with the duration of the oscillations.
384 MAGNETIC INDUCTION.
405. INFLUENCE OF TEMPERATURE. Temperature has a pro-
nounced influence on the value of the coefficient k; there is
nevertheless considerable uncertainty as to the law of the variation.
The fact longest known, and best marked, is that soft iron loses
almost all magnetic properties at a red heat. This is also the case
with nickel at a temperature of 300 degrees ; but with cobalt it only
occurs near the temperature at which copper melts.
If we only take into account temperatures between - 20 and 150
degrees, we find that the inducing power of iron is virtually constant,
although there is reason to think that it increases at first and passes
through a maximum; that that of nickel decreases continuously; and
lastly, that that of cobalt constantly increases. In the case of this
latter metal there is certainly a maximum towards a red heat.
Heat acts also on crystalised magnetic or diamagnetic bodies in
such a way as not only to diminish the coefficients, but also to
diminish the magneto-crystalline properties which are closely con-
nected with the difference of these coefficients. In the case of
bismuth the difference of the coefficients diminishes by one-half
between 30 and 140 degrees ; and for iron carbonate by two-thirds
between the same limits of temperature.
MAGNETISATION. 385
CHAPTER V.
ON MAGNETS.
406. MAGNETISATION. In order to magnetise a body endowed
with coercive force a steel bar, for instance it may be placed in a
constant magnetic field, or its various parts may be successively
submitted to the action of a variable field, like that produced by
rubbing it with a magnet.
This latter method is the oldest, and is that most frequently
employed. Each point takes at every moment a magnetisation
depending on that already obtained, on the actual resultant force,
and, to some extent, on the time during which it acts.
Whatever method may be employed, part of the magnetism
developed is temporary, and disappears with the external forces.
Another part is permanent or residual, and all experiments show that
these two kinds of magnetisation have a maximum limit.
The temporary magnetisation is greater, and the residual magneti-
sation less, with iron than with steel; but both have a maximum
which depends only on the quality of the substance. In the case
of very feeble forces magnetisation seems to be altogether temporary
both for steel and for iron.
The general problem of magnetisation would consist in determining
what would be the temporary magnetisation at each point of a body
of given shape, and nature, subjected to known forces ; and what,
when these forces are suppressed, would be the residual magneti-
sation. This problem has only been solved theoretically in a very
small number of cases.
407. INDUCTION OF A MAGNET ON ITSELF. DEMAGNETISING
FORCE. The total magnetism of a magnet must be considered as
made up of two parts, the one due to magnetic masses kept fixed by
the coercive force, and which may be called rigid magnetism, the
other resulting from the induction of the first on the magnetic body,
and which constitutes induced magnetism.
The internal action of induced magnetism is clearly in the opposite
c c
386 ON MAGNETS.
direction to the force which produces it ; it follows that the induction
of a magnet on itself always tends to diminish the magnetisation, and
acts like a demagnetising force.
The apparent magnetism, or that whose effects we can observe,
arises from the superposition of these two magnetisms. Hence
the determination of the intensity, and of the distribution of the
apparent magnetism, generally presents great difficulties.
The problem is simplified when the demagnetising force is
proportional at each point to the rigid magnetism at this point ; the
law of distribution is then the same as if this secondary effect of
induction did not take place.
In particular, when the rigid magnetism is uniform, the apparent
magnetism will itself be uniform, if the secondary inductive action is
constant in the interior of the magnet.
This condition is realised, as we have seen above, for a uniformly
magnetised sphere ; and also for an ellipsoid with a uniform magneti-
sation parallel to one of the axes, and for a straight unlimited circular
cylinder, magnetised perpendicularly to the axis.
408. Let us first consider a sphere. Let I be the rigid, I' the
induced, and \ the apparent magnetisation ; the demagnetising force
is then (355) equal to - irl v and we have
From which we deduce
i r-i
For an ellipsoid magnetised parallel to one of the axes, the
demagnetising action has the value IjL, I 1 M, or I 1 N, according to
the axis along which it acts (356). It is 471-^, or Try i - e 2 I x for a
disc, according as it is magnetised transversely or parallel to a
diameter (357). For an elongated ellipsoid of revolution, it is 2i?\
b*/2a V
if the magnetisation is transverse, and 4^11 ^ ( * -r i ) if the mag-
netisation is longitudinal (357). ' ^
This latter expression tends towards zero as the ratio - gradually
diminishes. The demagnetising force would be still smaller for a
long cylinder (373).
PARTICULAR CASES OF MAGNETISATION. 387
Hence the shape of thin plates, or of very long cylinders,
is that best fitted for obtaining permanent magnets, for the de-
magnetising force is then the least possible. These are, in fact,
the shapes which have been adopted in practice. Experiment
shows, moreover, that the influence of temper is then far less than
in the case of short and thick magnets. Coulomb had already
observed that tempering has but a very slight influence on the
magnetic rigidity of a steel wire.
409. PARTICULAR CASES OF MAGNETISATION. Sphere. It
follows from the preceding discussion that a solid homogeneous and
isotropic steel sphere, placed in a uniform magnetic field, will acquire
a uniform temporary magnetisation, and will then retain a uniform
residual magnetisation.
The expression for the temporary magnetisation will be of the
form
I- k F
w 4 : '
I+-7JV&
in which the coefficient k must be regarded, not as a constant
quantity, but as a function of the intensity F of the true field ; the
fraction by which the force F must be multiplied to get the magneti-
sation I, tends in fact to become inversely as F that is to say,
equal to -=!, as F increases, since the magnetisation tends towards a
maximum I .
In like manner, the residual magnetisation is a fraction of the
temporary magnetisation ; a variable fraction, and one which tends
towards a limiting value , since the residual magnetisation has a
maximum, and is then , a fraction of the maximum temporary
, , m
magnetisation.
In all cases the law of distribution is the same ; the density at
every point is equal to the perpendicular projection of the magneti-
sation that is to say, proportional to the abscissa of the point
measured from the centre along the diameter, parallel to the
magnetisation. The linear density measured along the same axis is
also proportional to the abscissa. The moment of the sphefre is ul lt
the total mass of each of the layers -, the distance of the poles is
4a .2 4 *
, and each pole is - of the radius from the centre.
3 3
C C 2
388 ON MAGNETS.
410. Ellipsoid. This is also the case with a homogeneous and
isotropic ellipsoid, one of whose axes coincided with the direction of
the uniform field during magnetisation ; it is merely necessary to
replace the factor -TT by a coefficient L which depends on the form
of the ellipsoid (357).
The maximum magnetisation I , and the fraction which deter-
m
mines the maximum residual magnetisation, have values which are
connected with those which correspond to the sphere by ratios
depending on the form of the ellipsoid.
The law of distribution is still known, and the poles are at a
distance from the centre equal to - of the semi-axis parallel to the
magnetisation.
We might in like manner obtain a uniform magnet with a
circular disc magnetised perpendicularly to a plane, or parallel to
a diameter (357).
411. Anchor Ring. A simple case, which can be easily realised
experimentally, is that of a body bounded by a closed tube a
torus or anchor ring, for instance in which the magnetisation
would be everywhere parallel to the axis. The magnet may then
be regarded as formed of simple solenoids closed on themselves
(371). The external action of the system is always exactly null.
412. Cylinder. To the preceding examples, all of which repre-
sent finite volumes, which can be exactly realised, we may add that
of a homogeneous and isotropic unlimited circular cylinder placed in
a uniform field perpendicular to the axis ; the magnetisation is then
represented by the expression
1 = Z F.
I + 2TTK
These cases seem to be the only ones in which the distribution
of magnetism can be theoretically determined, at least when the
coefficient k is not independent of the magnetising force.
413. ANY GIVEN MAGNETS. EXPERIMENTAL METHODS. The
problem of magnetisation for a body of any given form can only be
attacked experimentally by the study of its external actions ; we have
already had occasion to remark that our knowledge of the external
field of a magnet can teach us nothing about the internal distribution
of magnetism ; it only enables us to determine the distribution of the
fictive layer, equivalent to the real magnetisation.
METHOD OF OSCILLATIONS. 389
We shall mention here the principal experimental methods used,
in order to define their theoretical meaning.
414. Oscillations. This method, which was used by Coulomb,
consists in making a very small horizontal needle oscillate in front
of several points of a bar placed vertically in the meridian plane of
the needle. If n and N are the number of oscillations made by the
needle under the sole action of the earth, and under the combined
action of the earth and of the bar, the action of the bar on the
needle (the magnetism of which is supposed unchanged) is propor-
tional to the difference N 2 - 2 of the squares of the two numbers.
We measure thus the perpendicular component of the magnetic
force at the point in question. Coulomb assumed that this perpen-
dicular component was proportional to the density of the superficial
fictive layer at the nearest point of the needle, except quite close to
the end ; in this case he determined the density either by a graphical
method, or by doubling the value obtained for the oscillations of
the needle. It cannot be concealed that this mode of correction
is somewhat arbitrary ; it is, moreover, quite inexact, as we shall
afterwards see (419) that the perpendicular component at a point
is proportional to the density of the corresponding fictive layer, and
that it may directly give the distribution of magnetism.
415. Torsion Balance. A second method, also due to Coulomb,
consists in measuring the repulsion exerted by every point of the
magnet, at a constant and very small distance, on the pole of a long
needle movable in a plane perpendicular to the axis of the bar. If
we regard the pole of the needle as unchanged, the torsion imparted
to the wire by which it is suspended to keep the needle in the desired
position, measures the perpendicular component of the magnetic
force with a certain degree of approximation.
416. Use of Soft Iron. In the two preceding cases it is
assumed that the magnetism of the auxiliary magnet is invariable, so
that the action which it experiences is simply proportional to the
strength of the field. If the oscillating needle is of soft iron, and
the magnetisation of this needle is proportional to the strength of
the field, the action which it undergoes will be proportional to the
square of the perpendicular component.
We may, in like manner, place a piece of soft iron (M. Jamin's
test nail} in different parts of the magnet, and determine the force
necessary to detach it ; this force of tearing away is still within
certain limits proportional to the square of the normal component.
In these two methods, however, we do not take into consideration
either the variation of the coefficient k with the strength of the
390 ON MAGNETS.
magnetic force, nor of the modifications produced by the presence of
soft iron in the magnetic state of the bar exactly in the region we
are exploring. The results furnished by the use of soft iron do not
then seem to be so definite as those obtained with magnets.
417. Measurement of the Flow by Induction Currents.
This method is the only one which gives exact results ; the theory
will be given subsequently. It is sufficient here to remark, that by
means of induction currents we may determine the flow of force, or
the flow of magnetic induction across a closed circuit.
If the bar is anywhere surrounded by a ring formed of one or
many turns and connected with a galvanometer, and if by any
method we suddenly suppress the magnetisation, the momentary
current produced in the ring measures the total flow of induction
which traverses the plane bounded by the ring at the point in
question ; if the ring clasps the bar tightly, the flow of induction
which traverses the ring is that which exists in the section of the
bar itself.
The ring being placed in the same point it is caused to glide
along the axis of the bar to a distance which may be regarded as
infinite ; the current, in this case, measures the total flow of force
emanating from the magnet, measured from the point of departure.
Experiment shows, as indeed is evident from the theorem of the
conservation of the flow of induction, that the current is the same as
in the preceding case.
By measuring in either way the flow corresponding to different
points, we may construct a curve which represents the magnetic
condition of the bar. The curve has a maximum ordinate which
corresponds to the neutral line ; it sinks on each side and becomes
an asymptote to the axis of the bar, which we suppose to be pro-
longed indefinitely. This may be called with Gaugain the curve
of demagnetisation.
If, while the ring is at a point, the abscissa of which is x, it is
displaced by a quantity dx, the current measures the external flow
corresponding to this length dx, or, what is the same thing, the
variation in the internal flow of induction. By successively displac-
ing the ring by equal amounts, we may construct the curve whose
ordinates represent the external flow, and therefore the perpendicular
component at the various points. The ordinates of this curve are
the differentials of the ordinates of the curve of demagnetisation.
This method furnishes then, like the preceding, but in an exact
manner, the values of the perpendicular component at every point
of the bar.
DISTRIBUTION OF THE FICTIVE LAYER. 39 1
418. DISTRIBUTION OF THE FICTIVE LAYER. The fictive layer
is not a layer of equilibrium, but we know that its density at each
point (39) satisfies the ratio
in which F w and F' n denote for two infinitely near points on each
side of the surface, the first outside and the second inside, the per-
pendicular components of the actions exerted by the external masses,
and by the layer. The preceding methods give the component F n ,
but the component F' n is in general unknown ; hence they only
enable us to determine the density of the fictive layer in certain
special cases.
It may happen, in fact, that the fictive layer may replace magnetic
masses which really exist in the magnet not only for external, but
also for internal points ; and this is what takes place in the phe-
nomena of magnetic induction, when the coefficient k is constant.
There is then a constant ratio /* between the external and internal
perpendicular components, and the expression for the density is
47T fJ.
In this case the distribution is completely known when we know
the external perpendicular component at every point. This is not
the case if the coefficient k is variable, and still less if there is rigid
magnetism.
The ordinary methods do not give directly the distribution
of the fictive layer in a magnetised bar; it is incorrect, in par-
ticular, to consider the abscissa of the centre of gravity of the
curve of the perpendicular components as giving the position of the
pole. This is readily seen, if we examine the case of a cylinder
magnetised uniformly in a direction parallel to the axis. We have
seen (373) that its action may be represented by that of two layers,
one negative and the other positive, distributed uniformly on each
of the bases. It is easy to see that the flow of force for the lateral
surface is not zero, although the density is zero. The centre of
gravity of the curve representing the flow across the lateral surface is
in the interior of the magnet, while this pole is exactly situate on the
terminal surface.
39 2 ON MAGNETS.
We may observe that if the perpendicular components do not
give the distribution, they enable us to calculate the total mass of
magnetism by Green's theorem. This mass is obviously null for the
whole magnet; but the total flow of force on either side of the
neutral line is equal to 477 by the mass of the fictive layer corre-
sponding to this side. This total mass is represented by the area of
the curve obtained by taking as ordinates the values found for the
perpendicular component at all points of the axis of the magnet,
imagined to be indefinitely prolonged, or more simply by the
maximum ordinate of the curve of demagnetisation.
419. CYLINDRICAL MAGNETS. Coulomb determined experi-
mentally, and by means of the methods mentioned above, what he
calls the distribution of magnetism in cylindrical needles.
He first found that for short magnets those, that is to say,
whose length is less than fifty times the diameter the perpendicular
force at each point (which he confounded with the density) is pro-
portional to the distance from the middle.
The linear density would then be the same as for a sphere or an
ellipsoid uniformly magnetised.
The curve of distribution is then figured by a right line OB
(Fig. 86), making a certain angle a with the axis OA of the bar.
Fig. 86.
A straight line OB', forming the prolongation of the former, would
represent the negative magnetism on the other half of the bar. The
centre of gravity of the surface is projected, as for a sphere, at a
third of the semi-length of the bar measured from the ends.
This law ought to represent the true distribution of magnetisation
with tolerable approximation, for Coulomb proved that, other things
being equal, the magnetic moment of short bars is proportional to
the cube of the length.
If the bar is long that is to say, if the length is more than fifty
times the diameter d the magnetism is imperceptible for a certain
length on either side of the centre, and may still be represented by a
CYLINDRICAL MAGNETS. 393
triangle CAA' (Fig. 87), the base of which is equal to twenty-five
times the diameter. The angle a of the right line which represents
the densities is constant for bars which only differ in length. The
quantity of magnetism is then constant, and is the same as in a
limited magnet, for which we should have L = 50^; this quantity
may then be represented by a (50^) 2 and the moment by
Coulomb, however, only considered these results as a first
approximation. He observed that if we take equidistant parts from
the end A of a magnet, the successive tangents to corresponding
points of the curve make with each other equal angles. The curve
which satisfies this condition is given by the equation e~^ = cos fix,
which for small values of x merges into an arc of a parabola CB
(Fig. 87) tangential to the axis at a point C, at a distance /, from the
end ; the quantity of magnetism is then proportional to / 3 , that is / 3 ,
and the pole is at a distance from the end equal to -. The magnetic
/ A 4
moment has the value ( L \ bl z .
Fig. 87.
It will be seen that the magnetic moment for a very long cylinder
tends to become proportional to the length, as in the case of induced
magnetisation.
420. EMPIRICAL FORMULAE. These two portions of a parabola
do not represent the distribution of magnetism by a continuous
function. Biot found that Coulomb's experiments are represented
very exactly by the exponential formula
(3)
in which y is the magnetism at a point at a distance x from one end,
a and /* are constants.
394 ON MAGNETS.
Biot arrived at this formula by comparing the magnet to a Volta's
pile, which he considered as being itself a series of plates in which
the electricities of the terminal plates A and B dissimulate quantities
of electricity of opposite signs which vary in geometrical progression
with the number of plates. If N be the total number of plates, the
positive electricity of A dissimulates in the n th plate a quantity of
negative electricity expressed by #a n , and the negative electricity of
B dissimulates, in the same element, a quantity of positive electricity
tfa N - n , so that the quantity of free electricity in this element, sup-
posing the terminal charges to be equal, is
We may get the previous formula from this by putting N = 2/^, and
therefore n = xp, p being the number of pairs for unit length, and
taking p = a*,
It seems difficult to discuss a mode of reasoning which has for
its basis only the vague notion of dissimulated electricity.
It may be observed that if we take the origin of the co-ordinates
in the centre of the bar, instead of at one end, equation (3) may be
written
421. Green, starting from a particular conception of the coercive
force, found that, for a circular cylinder placed in a uniform field
parallel to the axis, the linear density at a distance x from the
middle of a bar whose length is 2/, and radius a, might be expressed
by the formula
(4) \ ~
+e a
or putting - =
EMPIRICAL FORMULAE. 395
in which F represents the strength of the field, and q a constant
given by the equation
Green assumes that the coefficient of magnetisation /, is constant
throughout the whole extent of the body ; in this case the linear
density is proportional to the perpendicular component of the
magnetic force at every point of the surface. Maxwell gives the
following table of the corresponding values of q and of k:
k q k q
oo 0.00 11.80 0.07
336.4 0.01 9.13 0.08
62.02 0.02 7.52 0.09
48.41 0.03 6.32 0.10
29.47 0.04 0.143 1.00
20.18 0.05 0.0002 10.00
14.79 0.06 0.0000 oo
For negative values of k, q becomes imaginary ; the formula does
not seem then to apply to diamagnetic bodies.
Green's formula seems to represent very exactly the distribution
of temporary magnetism in soft iron as well as that of permanent
magnetism in cylindrical bars. Green showed that the value of the
moment which is deduced for a needle of this form,
'\
(5)
agrees remarkably with determinations made by Coulomb with
needles which only differed in length. The agreement ceases, how-
ever, to be very close when the length of the needle is less than
twenty-five times the diameter.
The expression for the area of the curve corresponding to Green's
formula is, for each half of the bar,
it represents the total value of the flow of lateral force. For a very
396 ON MAGNETS.
long cylinder it reduces sensibly to ira^k ; the flow from the ends
may then be neglected.
If we assume that the abscissa of the centre of gravity of this
area determines the position of the pole, we shall obtain the distance
2d of the two poles by dividing the moment m by the mass S ; we
shall thus obtain
(6)
for very long needles, this expression reduces to 2U-J =2U-
that is to say, that the poles are at a distance - from the ends.
q
422. M. Jamin obtained an analogous expression. If y is the
tension at each point, or the density, / and s the perimeter and
section of the bar, and A and c two constants, M. Jamin, in
comparing the phenomenon to the propagation of heat, finds, by
Fourier's laws, the following formula, which agrees with the results
of his experiments:
If the section of the bar is circular and of radius a, we have
* /- = A /-> and putting ^ = 1$, the formula becomes
/=
V"
it becomes identical with that of Green if we put
q 27T#F
^ = cinCi JTi. = - =
M. JAMIN'S FORMULAE. 397
For long bars, this latter condition reduces to
A = 27T0/&F
and shows that the constant A is proportional to the perimeter.
423. Green's formula corresponds to the case of a cylinder
placed in a uniform field parallel to the axis, and for which the
coefficient of magnetisation is constant Professor Rowland has
pointed out .the analogy of this formula with that which expresses
the lateral flow from a pile of the same form placed in a conducting
medium (268). Let us suppose that the flow of magnetic induction
is propagated like the flow of electricity; if we retain the same
meanings for the quantities /o, /a', and R x , and if we replace the
quantity by the force F of the field, and if we call Q the flow
of magnetic induction across a section of the bar at the distance x
from the centre, we have, for the flow in the interior,
(8)
i -
and, for the lateral flow,
These formulae also apply to the case in which the magnet is
solenoidal, bounded by a channel surface closed upon itself, and
the magnetisation of which is everywhere perpendicular to the right
p
section. We have, in that case, Rj0, Q = , and the flow of
P
lateral induction is zero. If F 1 is the magnetic induction, s the
section of the bar, and /* the coefficient of permeability, we have,
further,
and therefore /* = .
ps
424. All experiments go to prove that magnetisation tends
398 ON MAGNETS.
towards a maximum when the magnetising force increases without
limit. If the values of this force be taken as abscissae, and the
values of the magnetisation as ordinates, we obtain a curve like
OBL (Fig. 90), having an asymptote parallel to the axis of the
abscissa, and with a point of inflexion near the origin.
Professor Rowland represents the phenomena in a different way.
Taking the values of induction F 1? as given directly by experiment,
as abscissae, and the values of yu as ordinates, he finds that his
experiments are represented very closely by the formula
(9)
in which #, , c, and d are constants depending on the nature
and quality of the metal.
The curve represented by this equation has the general form
of a parabola with a diameter conjugate with the axis of the
abscissae ; it cuts this axis in two points, and the inclination of
the diameter depends on the constant b. The position of the
points of intersection with the axis depends on the values of- c and
d. The constant a evidently represents the maximum value of /*.
Professor Rowland's experiments give for this maximum at the
ordinary temperatures numbers between 3000 and 5000 in the
case of iron, and 300 in that of nickel. The curve assumes
another form when the temperature changes, and the deformation
appears to be far greater for nickel than for iron.
425. HYPOTHESIS ON THE CONSTITUTION OF MAGNETS.
According to Poisson's theory the magnetisation of a medium is
produced by the separation of the magnetic fluids in the interior
of each particle, and as no limit can be assigned to the quantity of
neutral fluid which can exist in a definite volume, the magnetisation
itself might increase without a limit.
We shall afterwards see how Ampere, starting from the magnetic
properties of electrical currents, was led to assume that each particle
of a magnetic substance is surrounded in the natural state by an
infinitely small electrical current, and constitutes an elementary
magnet. In a magnetic body withdrawn from all external force,
these elementary magnets are only subjected to their mutual actions,
and are turned indifferently in all directions. If the body is sub-
mitted to the action of a magnetic field the axes of the different
magnetised particles tend to take the direction of the field at each
WEBER'S THEORY. 399
point, and the magnetisation which results therefrom is the stronger,
the more these particles are deviated from their original direction.
If the axes of all these particles were parallel the magnetisation
would have reached its maximum value.
But this position can never be attained, owing to the reciprocal
reactions of the molecules. Wilhelm Weber has shown how these
reactions may be allowed for.
426. WEBER'S THEORY. Let us assume with Weber that each
unit of volume contains n magnetic molecules, and that the moment
of each of them is equal to m. If all these molecules were parallel,
the magnetic moment of unit volume would be M = nm> and the
magnetisation of the medium would be at its maximum.
When the medium is in the natural state, the molecules are
turned indifferently in all directions. To express this property let us
draw through the centre of the sphere a radius parallel to each of
the axes of the n molecules ; the extremities of these radii will be
arranged on the sphere in a uniform manner.
The number of molecules the axes of which make, with a
determinate direction, which we take for the axis of x, an angle
smaller than a is - (i - cos a) ; and the number of molecules whose
angles with the axis of x are between a and a + da is equal to
sin ada.
2
Let us now suppose this medium to be in a uniform field whose
intensity X is parallel to the axis of #, and consider the action
which it exerts on a molecule whose magnetic axis makes an angle a
with the direction of the field.
If this molecule were free it would set parallel to the force of the
field, and, all the other molecules undergoing an analogous rotation,
the medium would attain the maximum magnetisation under the
influence of any external force however feeble.
As this is not the case, it must be assumed that each molecule is
impelled to resume its original direction by an antagonistic force,
which arises either from the constitution of the medium itself, or by
the reactions which the magnetised molecules exert upon each other.
The simplest hypothesis is to suppose that this antagonistic force
D is constant, and acts in the original direction of the axis of each
molecule.
The new direction of the axis of a molecule in its position of
equilibrium is then given by that of the resultant of the forces
DandX.
400
ON MAGNETS.
427. To get the direction of the molecule let us draw a sphere
whose radius is equal to the reaction of the medium, and take a
length OS from the centre, equal and opposite to the strength of the
field (Fig. 88).
-
HOO "I '0
Fig. 88.
A molecule, the axis of which was originally directed along the
line OP, is subject to two forces SO and OP, the resultant of which
is SP. If- the resultant S is in the interior of the sphere that is, if
the reaction of the medium is greater than the strength of the field
the axes of the deviated molecules will be still in any direction
whatever, but no longer uniformly.
If the force of the field is greater than the reaction of the
medium, the point S is beyond the sphere (Fig. 89), and the axes of
the deviated molecules are all comprised within the cone TST
tangential to the sphere.
Fig. 89.
Let a be the original inclination of the axis of a molecule to
the axis x,
9 the final inclination,
/3 the deflection a - 6,
R the resultant of the magnetising force X, and of the reaction
D of the field.
WEBER'S THEORY. 401
The condition of equilibrium is
mX sin = mD sin ft = mD sin (a - 6)
from which we deduce
D sin a
(n) tan
X + Dcosa'
428. The structure of the medium being symmetrical in
reference to the axis of x, the strength of magnetisation is given
by the sum of the projections of the magnetic moments of all the
molecules on the axis of x.
The projection of the moment of a molecule is expressed by
mcosd; the number of those which originally made the angle a
with the axis of x, is - sin a da. : the resultant is then
2
cos0sina<a,
C ir n f inn
= m cos0-sinadfa= -
Jo 2 J, 2
or
M
1= --
The triangle SOP gives the equation
from which is deduced
We have further
D 2 = R 2 + X 2 -2RXcos0.
Expressing in this way the angles a and 9 by their values as a
function of R, we get
D D
402
ON MAGNETS.
In the first case, in which we have X< D, the limits of the
integration are R 2 =
and R = D-
In the second case, in which we have X>D, the limits of
the integration are R 2 = X + D and R x = X - D.
All reductions being made, we get then :
When X<D,
X = D,
X>D,
X=oo,
= M.
iD 2 "]
~3^J ;
According to this theory, the magnetisation is at first propor-
tional to the magnetic force until it is equal to the reaction of the
medium, in which case the magnetisation attains two-thirds of its
maximum value. When the magnetic force has become greater,
the magnetisation increases less rapidly, and tends towards a finite
limit.
The curve OL (Fig.. 90), which represents the magnetisation as
a function of magnetic force, consists then of a rectilinear part OA
which is prolonged by the curve AL, the asymptote to a horizontal
straight line CD.
to
Fig. 90.
429. Weber's own experiments agree satisfactorily with this law.
More recent researches, however, have shown that the value
of k cannot be considered constant even for small forces. This
coefficient at first increases regularly up to a maximum, and then
diminishes.
MAXWELL'S THEORY. 403
The magnetisation of iron, as a function of the field, must
therefore be represented by a curve such as OBA (Fig. 90), having
a point of inflexion; this first part of the curve has often been
confounded with the tangent passing through the origin, and
which gives the maximum value for k. Weber's theory does not
account for this variation of the coefficient of magnetisation for
small forces; nor, on the other hand, does it throw any light on
the nature of residual magnetisation.
430. MAXWELL'S THEORY. In order to complete this last
link, while still adhering to the general theory, Maxwell supposed
that the medium had a kind of imperfect elasticity. He assumes
that the axis of the magnetic molecules revert to their original
position, after the suppression of the magnetising force, so long as
the rotation which they experience is below a certain value, but
that their axes retain a permanent deviation ft - /3 Q , when the
rotation /3 has been greater than the limiting value /3 . This
deviation ft - /3 Q characterises the permanent condition of the
molecule.
This hypothesis undoubtedly dbes not represent the exact state
of the phenomena, but it may furnish an approximate idea, and
enable us to submit the problem to calculation.
According to Maxwell, we may deduce the temporary magneti-
sation I and the permanent magnetisation I' by a calculation
analogous to the preceding. Putting
we get thus :
When X<L,
When X = L,
For L < X < D,
D D 2
404
ON MAGNETS.
2 I/ L 2 V]
il' 5*) '
\ / J
For X > D,
Lastly, for X = oo,
Fig. 91 represents the course of the phenomenon for particular
values: M = iooo, L = 3, D = 5. The magnetising forces are taken
as abscissae ; the ordinates of the curve OAB represent the temporary
magnetism, and that of the curve O'A' the residual magnetism, The
former consists at first of a rectilinear portion corresponding to the
values of X comprised between and 3 ; it then rises suddenly, and
rapidly approaches its asymptote. The curve of residual magnetism
JAMIN'S OBSERVATIONS. 405
only commences when X = L; the maximum M' towards which it tend?,
and which is figured by the right line C'D', is equal to 0'8i M.
It may be remarked that the residual magnetism thus calculated,
corresponds to the case in which the magnetisation of the body itself
produces only an inappreciable demagnetising force ; these results
correspond then only to the case of a very long body magnetised
longitudinally.
It is difficult to admit that a discontinuous curve like that which
represents the temporary magnetism can be an exact expression of
the phenomenon. Nevertheless, this theory leads to curious con-
sequences relative to the successive action of magnetising forces of
opposite signs, and which have been verified experimentally.
Let us suppose that a piece of iron after having been submitted
to the action of a force X , has acquired a permanent magnetisation.
A new force - X 2 of the same direction is without effect as long as
it is less than X , and if it is greater than X the residual magnetism
is the same as if the original force had not acted.
If the new force - X 2 is in the opposite direction, it produces a
permanent effect long before it reaches X ; the residual magnetism
seems to be destroyed for a certain value of this force, but the
metal is not in the neutral state, for it is insensible to the action of a
force - X, so long as X is less than X 2 , while a feebler positive force
produces a permanent magnetisation in the original direction.
431. JAMIN'S OBSERVATIONS. Jamin gives a different expla-
nation of these phenomena. He assumes that the action of the
field on a bar extends to a greater or less depth according to its
strength. When the apparent magnetisation has become zero, the
magnetism is not destroyed ; it was merely a case of the super-
position of two opposite magnetisations. An inverse field of less
strength than X 2 has no action on the superficial layer, but a direct
field of less strength forms a new superficial magnetisation, the
action of which is added to that which was previously there.
M. Jamin verified these theoretical ideas by removing the super-
ficial layer of inverse magnetisation, and exposing the subjacent
layer of direct magnetisation. He succeeded in doing this either
mechanically, by grinding or filing away the outer surface of the
magnet, or chemically by dissolving it with acid.
It must, however, be observed that this predominance of the
surface layers is perhaps an accidental phenomenon peculiar to steel,
and simply dependent on the constitution 'of this metal. For, in
the case of highly tempered bars, such as those which are sought
for the construction of magnets, the tempering is necessarily very
406 ON MAGNETS.
unequal; it is more particularly produced near the surface, where
the cooling is very rapid, so that the maximum action of the coercive
force is in the superficial layers. The inductive action and the
demagnetising force, manifest themselves then in conditions quite
different from those met with in homogeneous bodies.
432. INFLUENCE OF TEMPERATURE. The magnetism induced
by the action of a magnet on itself, is perhaps the simplest way of
explaining the influence of temperature.
It is natural to assume that rigid magnetism is not altered by
small changes of temperature, for the magnetisation resumes its
original value when the magnet regains its original temperature ; it
is difficult to suppose that rigid magnetism can repair its losses, for
all the internal actions tend to diminish it. On these considerations,
the temporary enfeeblement of magnetism will be simply due to an
increase in the induced magnetism, and therefore the coefficient of
magnetisation must at first increase with the temperature.
For higher temperatures, above 100 for instance, magnetism
undergoes a distinct diminution ; the rigid magnetism itself has
therefore changed. In these conditions we cannot say whether the
coefficient of magnetism continues to increase with the temperature,
for the enfeeblement is due to a double cause. As iron and steel at
a red heat are no longer attracted by a magnet, we must assume that
the coefficient of magnetisation then becomes null, or at any rate
is extremely small.
It appears then that for iron and steel the coefficient of
magnetisation must at first increase with the temperature and
then diminish to zero, passing through a maximum at a definite
temperature.
If this is the case, a bar magnetised at a lower temperature
than the maximum must lose magnetism when it is heated, and
the converse must take place with a bar magnetised at a lower
temperature than that of the maximum.
Experiment shows that this is the case with cobalt. For iron
and steel the facts hitherto known agree partially with this mode of
view; but there are too few experiments made under well defined
conditions, to enable us to judge how far it agrees with the truth.
Everything seems to indicate that the true phenomena are more
complex.
MAGNETIC PARALLELS. 407
CHAPTER VI.
MAGNETIC CONDITION OF THE GLOBE.
433. GAUSS' METHOD. The representation of terrestrial mag-
netism by the hypothesis of a central magnet, or by equivalent
hypotheses, only constitutes a somewhat rough first approximation ;
the problem is really far less simple. Gauss treated it in a com-
pletely general manner, on the hypothesis that the effects observed
on the surface of the Earth are due solely to the action of magnetic
masses.
Whatever may be the distribution of these masses, whether they
are in the inside or on the surface of the globe, the elementary
actions are exerted inversely as the square of the distance, and the
force at each point is still determined by a potential. The space
surrounding the earth forms the magnetic field of the system, and we
may suppose it divided into layers by equipotential surfaces, corre-
sponding to equidistant values of the potential. The surface, which
corresponds to a given value V, may be formed of one or more
sheets ; but we know that two surfaces of different potentials do not
intersect, and that the force perpendicular at each point is inversely
as the distance of two consecutive surfaces.
434. MAGNETIC PARALLELS. A certain number of these sur-
faces cut the terrestrial globe : magnetic parallels are the lines of
intersection corresponding with the surface of the Earth ; these lines
are level lines. As they belong both to the surface of the Earth,
which we suppose is spherical, and to the equipotential surface, they
are perpendicular at each point to the vertical and to the magnetic
force; they are therefore perpendicular to the magnetic meridian
passing through these two lines, and therefore to the intersection
of this meridian with the surface of the Earth that is, to the mag-
netic meridian. The magnetic parallels form, therefore, on the
surface of the terrestrial sphere, a system orthogonal to the mag-
netic meridians.
408
MAGNETIC CONDITION OF THE GLOBE.
Let us consider the parallels corresponding to two infinitely near
equipotential surfaces V l and V 2 (Fig. 92) ; let ds be the arc of the
Fig. 92.
meridian comprised between them, and dn the perpendicular distance
of the two surfaces at the same point. If F be the magnetic force
and I the inclination, we have evidently
dn ds cos I
from which is deduced
(i)
3V
The horizontal component, perpendicular at every point to the
magnetic parallel, is therefore inversely as the distance of two con-
secutive parallels ; but the total force and the horizontal component
are not necessarily constant along a magnetic parallel, as is the case
on Biot's theory.
435. MAGNETIC EQUATOR. The sum of the magnetic masses
being null for the whole system, and also separately for each of the
magnetised bodies, there is a level surface for which V = ; this
surface cuts the terrestrial globe along its neutral line if it is the
only magnetic body, or in the vicinity if the other magnetic bodies
are sufficiently distant.
The magnetic parallel where the potential is zero is called the
magnetic equator; along this equator the force is not constant, nor
is it necessarily horizontal. On Biot's theory the equator was a line
of which the inclination was null.
TERRESTRIAL MAGNETIC POLES.
409
The magnetic equator separates those points on the earth for
which the potential is positive, from those where it is negative. On
either side of the equator the absolute value of the potential de-
creases continuously.
436. TERRESTRIAL MAGNETIC POLES. The term terrestrial
magnetic poles, is ordinarily applied to those points of the surface
where the potential is a maximum or minimum. A pole is a point
where the level surface becomes a tangent to the surface of the
Earth; the force there is evidently vertical.
The number of poles is at least two, for there are at least two
points at which the level surfaces are tangents to the surface of the
sphere; but there may be a far greater number. Suppose, for
instance, that there are two poles P and P' (Fig. 93) situate in the
Fig. 93
positive region that is to say, on the southern hemisphere. These
poles might belong to the same level surface which had two points
of contact with the surface of the sphere ; but more generally we
shall consider them as belonging to two different surfaces of poten-
tials V m and V' m , V^ being greater than V' m .
Since the points P and P' are points of maximum, the potential
decreases in all directions around each of them, and we may always
choose a value Vj of potential lower than V' m , such that the inter-
section of the surface V 1 with the sphere, gives two closed curves
S and S', insulated from each other, and each of which surrounds
one of the points ; we may then take a value V 2 so small that the
same curve of intersection comprises the two points.
410 MAGNETIC CONDITION OF THE GLOBE.
By making the potential vary continuously from V l to V 2 , we
shall find a value V for which the two curves, previously separated,
come in contact, and merge into a single one S ; the junction may
take place either by a single point of intersection, as in Fig. 93, or
by a greater number of points of intersection or of contact. Let O
be one of these points. It is first of all clear that the horizontal
component there is null, and that therefore the point corresponds to
the ordinary definition of poles ; it is, however, to be observed that
as we move in certain directions we get increasing, and in other
directions decreasing, potentials ; for the former directions the point
O would act like a south pole, and for the second as a north pole.
This is what we may call a false pole.
There cannot thus be two distinct poles in the same hemisphere
without there being at the same time at least one false pole. But
observations give nothing of this kind, and it is only an inexact
interpretation of phenomena which has sometimes led to the con-
clusion, that observations indicate the existence of two poles in the
northern hemisphere.
Near a pole, in fact, the magnetic parallels have an elliptical
shape ; their perpendiculars that is to say, the magnetic meridians
do not coincide in the same point, but the points of convergence, 1
which they show more or less clearly, are the centres of curvature,
and have clearly no relation with the poles.
. Observation leads then to this consequence, that, apart from
purely accidental and local circumstances, there are only two mag-
netic poles on the surface of the Earth a negative pole in the
northern and a positive pole in the southern hemisphere.
It is important to add, also, that terrestrial magnetic poles, such
as we have defined them, have nothing in common with magnetic
poles, properly so called, considered as centres of gravity of positive
and negative magnetic masses. The magnetic axis of the Earth is
the right line along which the sum of the projections of the mag-
netic moments of the various elements is a maximum (297).
437. PROPERTIES OF A CLOSED POLYGON. We know that if we
move a magnetic mass equal to unity from a point Pj where the
potential is V 15 to a point P 2 where it is V 2 , and if we denote by F
the force, by ds the element of the path described by the mass, and
by e the angle of the force with the element, the magnetic work is
expressed by the equation
Yds cos e.
PROPERTIES OF A CLOSED POLYGON.
411
This work is independent of the path traversed, and is zero whenever
we return to the original level surface by making the mass describe
any given closed curve. Suppose that the two points P l and P 2 are
situate on the surface of the Earth, and that the mass is displaced
along this surface; the work of the vertical component is zero at
each instant, the expression for the work only depends on the
horizontal component H, and reduces to
(2)
v.
fp.
-v 2 =
J*l
cose,
the integral of the second member being zero whenever the mass is
made to describe a closed circuit.
That being admitted, let us consider a polygon formed of great
circles passing through the points P , P I} P 2 (Fig. 94). Trace
Fig. 94.
at these various points the geographical meridians P M , P 1 M 1 ,
P 9 M 2 , and the magnetic meridians P D , P^, P 2 D 2 . . .
Let
^o ^i> V-- be the declinations reckoned positively from north
to west;
0.1 the azimuth of the arc P P 1 at the point P , this azimuth
being counted positively from north to east;
1.0 the azimuth of the arc PQ?! at the point P x counted posi-
tively in the same direction ;
c o-u i-o tf 16 values of the angles e at these various points.
412 MAGNETIC CONDITION OF THE GLOBE.
We have
At the point P , e Q ^ = ^ + . 1 ;
*!. 2 = ^ + 1. 2;
' 2 1 = 2 ~^~ * ' 1 '
2>3 = 8 a + 2.3; etc.
On the side PoPj the horizontal component H is not constant either
in magnitude or in direction ; nevertheless, if this side is very small
compared with the dimensions of the terrestrial globe, we may
assume that the value of H is constant, equal to the mean of the
values which it has at the points P and P 1? and put
H cose = -(H cos^.j + Hj cose 1<0 ).
The theorem expressed by equation (2) gives then
We shall have then, for the closed polygon,
cos + O.l + cos
+ 2 [ Hl cos (S, + 1 . 2) + H 2 cos-(5 2 + 2 . 1)]
+ ? [H n cos (8 n + n . 0) + H cos (8 + . )].
Applying this equation to the triangle formed by the stations at
Paris, Gottingen, and Milan, and taking as unknown the value H
at Paris, Gauss found by calculation the value H = 0.5i7, while
observation gave 0.518.
GEOGRAPHICAL CO-ORDINATES.
413
438. INTRODUCTION OF GEOGRAPHICAL CO-ORDINATES. Let
us consider any given point P at a distance r from the centre of the
Earth ; let u (Fig. 95) be the complement P'D of the latitude and /
Fig. 95-
the longitude CQ counted towards the east. We shall decompose
the magnetic force F at the point P into three rectangular forces,
one Z along the vertical and measured positively towards the zenith,
the other X in the meridian and directed towards the north, the
third directed towards the west.
Taking into account the ratios
dx = rdU) dyr sin udl^ dz = dr,
the components of the force become
(4)
r sn u
*.--*.
Tr'
414 MAGNETIC CONDITION OF THE GLOBE.
We have, moreover,
When the point P is at the surface of the Earth at P', we must
take r=a, and equations (4) give
3V
sin u = - .
ol
Since, further, we have
we get
<)X 3 (Y sin?/)
1)7 ~ ~~^u
and, therefore,
For u = that is to say, at the north pole we have Y sin u = 0,
and therefore/ (/) = 0. We get then, finally,
(6)
We are thus led to the remarkable theorem of Gauss :
If for all points of the surface of the earth we know the horizontal
component directed towards the north, that is sufficient to give us the
horizontal component- turned towards the west, and therefore the total
horizontal component.
EXPRESSION OF POTENTIAL. 415
439. EXPRESSION OF POTENTIAL. Whatever may be the mag-
netisation of the Earth, the external potential may be represented, as
we have seen (369), by the expression
which, for a point on the surface, reduces to
We deduce from this, as the components of the Earth's magnetism,
1 3V JA SA
K . . .
a ou uu ou
=-i_^=-!-r '+^2+ i
a sin ull sin u [_ W <>/ " J '
The coefficients A 15 A 2 , A 8 , are functions of the two angles
/ and u. A n is expressed (368) by 2n+i terms in sines and co-
sines. Hence, if we wish to represent the condition of the Earth by
a series of this form, we must determine three numerical coefficients
for A 1? five for A 2 , seven for A 3 , etc.
Gauss found that, in the then existing condition of magnetic
determinations, it was useless to push the development beyond the
fourth term, so that there are still twenty-four numerical coefficients
to calculate.
Every point of the surface gives three equations by the values of
the components X, Y, Z ; hence, if we know these three elements at
any eight places in the earth, we have a complete solution of the
problem. In order to avoid errors arising from neglected terms, and
from inexact observations, Gauss applied the method of least squares
to the data for eighty-four points, taken on twelve equidistant meri-
dians, and seven parallels. The results thus obtained were then
applied to ninety-nine other points.
41 6 MAGNETIC CONDITION OF THE GLOBE.
The formulas calculated by Gauss assign to the two poles the
following positions for the year 1838 :
North Pole latitude 73 35' longitude 95 39' W,
South Pole 72 35' 152 30' E ;
they are, as will be seen, far from corresponding to the ends of the
same diameter.
The true magnetic axis, determined by the condition that the
sum of the projections of the moments is a maximum, is parallel to
the terrestrial diameter which corresponds to that point in the
northern hemisphere the latitude of which is 77 50', and the longi-
tude 63 31' W. Its direction does not coincide exactly with the
line of the poles.
This direction is that for which the coefficient A 1 has its maxi-
mum value (370). The magnetic moment of the Earth is equal
to # 3 K. Comparing this moment with that of a magnetised steel
bar, which weighed about 500 grammes, and had been used in the
absolute determination of the Earth's magnetism, Gauss found that it
was about 8.io 21 times as great. If we suppose the Earth to be
uniformly magnetised, it follows from this number that the magnetic
moment of each cube metre of the terrestrial globe is the same as
that of eight of the magnets used by Gauss. Assuming that the
magnetisation of the bar was also uniform, the intensity of its
magnetisation would be about 2200 times that of the terrestrial
globe.
440. Is THE MAGNETISM OF THE EARTH IN THE INTERIOR
ONLY ? It may be observed that if the acting masses were in part in
the interior and part outside, the potential might be expressed by the
sum of two series
/a\ +i
1 +A t
the former relative to the internal masses, and the second to the
external masses. Denoting by V n the general term of the develop-
ment, we should have then
V =
INFLUENCE OF THE SUN AND MOON. 417
from which is deduced
,
+ ~~
dr r n \rj a r4 \a
For a point on the surface we have simply
(8) dV n+i n
The vertical component
dr
has for its general term
(9) Z^
This equation, combined with the preceding one (8), gives
and we may thus easily separate the effect due to internal masses
from those produced by external masses.
The calculations of Gauss having shewn that the observations
are satisfied by means of the single coefficients A, it follows that
the coefficients B are virtually null ; hence no sensible part of the
terrestrial action is due to external magnetic masses.
441. INFLUENCE OF THE SUN AND MOON. There are, how-
ever, certain periodical variations in the elements of terrestrial
magnetism, which appear to be connected with the apparent motions
of the Sun and Moon, or at any rate to depend on certain accessory
phenomena such as the spots of the Sun. The influence of these
bodies can scarcely be doubted ; everything leads, however, to the
belief that they do not act directly as magnetic bodies, but that their
influence is indirect, and only modifies the magnetic condition of the
terrestrial globe.
E E
41 8 MAGNETIC CONDITION OF THE GLOBE.
A star, whatever may be the distribution of its magnetism, is, in
fact, equivalent for very distant points to an infinitely small magnet,
or to a sphere magnetised uniformly.
Let us denote by :
I the mean intensity of the Earth's magnetisation ;
R its radius ;
I' the mean intensity of the magnetisation of a star ;
R ; its radius ;
7' its magnetic moment ;
D its distance from the Earth.
The value of the action of the Earth at the equator, where it is
a minimum, is (153)
If we suppose that the line of the poles of the star in question is
'directed towards the Earth, which is the most favourable case, the
force F^, which it will exert on the Earth, will be (153)
The ratio of the polar action of the star in question, to the
equatorial action of the Earth, is
D
The ratio is therefore proportional to the magnetisation of the star
and to the cube of its apparent diameter.
The apparant diameter of the Sun, and that of the Moon, are
about 30' that is to say, less than o.oi so that we have
FP 1 1' _ 6
INFLUENCE OF THE SUN AND MOON. 419
If these stars are magnetised like the Earth, the greatest variation
which they could produce at the equator, on the declination, is less
lo- 6 i"
therefore than or , that is to say, absolutely inappreciable.
4 20
To have variations of 10', such as are frequently met with, the
intensity of the magnetisation of the Sun and of the Moon, must be
12,000 times as great as that of the Earth. Now the most powerfully
magnetised steel has not 10,000 times the intensity of the Earth;
hence, to produce a deviation of 10', the Sun and Moon should be
more powerfully magnetised than the best steel magnets.
The same conclusions result from supposing that the Moon, for
instance, is magnetised by the Earth. If the Moon is at the equator,
the action which it experiences from the Earth is
and the value of the intensity of the magnetisation is
From this is deduced
r
t*v m i w (*:
\D) 5 W
Whatever value we assume for the coefficient k if even we
compare the Moon with the very softest iron the ratio of the
magnetisations will be always very small, and the reaction of the
Moon upon the Earth may be completely neglected. Still more
must this be the case with the Sun.
E E 2
420 CURRENTS AND MAGNETIC SHELLS.
PART IV. ELECTROMAGNETISM.
CHAPTER I.
CURRENTS AND MAGNETIC SHELLS.
442. OERSTED'S EXPERIMENT. Older experiments on electrical
discharges had already shewn that the passage of a current in a
conducting wire could modify the magnetism of a steel needle.
These phenomena, to which only small importance was attached,
were a first indication of the relations which existed between elec-
tricity and magnetism. It is only since 1820, in consequence of
(Ersted's experiment, that the existence of these relations has been
made completely evident by the immortal researches of Ampere.
When a straight conductor traversed by a current is brought
near a magnetised needle, the needle is, in general, deflected
from its position. In order to explain in all cases the somewhat
complicated effects which are observed according to the relative
positions of the magnet and the current, Ampere gave a very simple
rule : Suppose an observer placed in the wire in such a manner
that the current enters at his feet and emerges at. his head ; the
observer, turning his face to the needle, always sees the North pole
turn towards his left, which for the future we shall call the left of
the current If the needle were freed from the action of the Earth,
and of any other action than that of the current, it would set at right
angles with the current.
443. MAGNETIC FIELD OF A CURRENT. The fundamental fact
which results from (Ersted's experiment, is that an electrical current
of any given form creates about itself a true magnetic field.
This field possesses all the properties observed in an ordinary
magnetic field, for the actions which it exerts at any point on
equal magnetic masses of opposite signs are equal and directly
MAGNETIC FIELD OF A CURRENT. 421
opposite. The force, moreover, is proportional to the magnetic
mass in question, for if we put near the current a small needle,
which at the same time is under the action of the Earth, the
direction which it takes is independent of its magnetic moment ;
the resultant of the two forces which arise from the terrestrial
field, and from the field created by the current, has thus itself a
fixed direction, and therefore the two forces maintain a constant
ratio.
The action of the current also changes its sign, without
changing its magnitude, when the direction of the current is
simply reversed; thus, when the conducting wire is bent upon
itself, the two portions close to each other, which are traversed
by equal currents in opposite directions, have no action on the
pole of a magnetised needle.
The existence of the field produced by the current may be
rendered evident by the ordinary method of magnetic images.
Thus, if iron filings are scattered on a sheet of paper traversed at
right angles in its centre by a rectilinear current, the filings are
seen to arrange themselves in concentric circles about the path
of the current. We conclude from this that the lines of force are
circumferences whose centre is the axis of the current. The force,
therefore, is perpendicular at each point to the plane passing through
this point and the current ; it is, moreover, turned to the left of the
observer in Ampere's rule.
The successive equipotential surfaces obtained about a rectilinear
current are thus formed by a series of planes passing through the
axis of the wire, and making equal angles with each other. The same
is the case near any given current, so that the equipotential surfaces
are formed about each portion of the wire, making equal angles
with each other.
444, ACTION OF A RECTILINEAR CURRENT ON A POLE.
EXPERIMENTS OF BIOT AND SAVART. Biot and Savart determined
experimentally the magnitude of the force at each point. They
examined the action of a vertical current on a small horizontal
magnetic needle placed at various distances on a right line passing
through the current, and perpendicular to the magnetic meridian.
In these conditions the resultant force is the sum of the horizontal
component H of the Earth's field, and of the force < of the
current.
The needle is first caused to oscillate under the influence of the
Earth alone, then at distances a and a' from the wire under the
combined influence of the Earth and of the current. If #, N, and
422 CURRENTS AND MAGNETIC SHELLS.
N' are the numbers of oscillations of the needle in a given time / in
the three experiments, then if K is a constant depending on the
magnetisation of the needle and on its moment of inertia, we have
From which is deduced
But experiment showed that by employing the method of alter-
nate distances to eliminate the effect of variations in the strength
of the current, the following ratio was always obtained :
It follows from this that <$>a = <$>&', that is to say, that the action
of the current on a point is inversely as the distance.
On the other hand, experiments made on the discharge of
batteries those of Colladon and of Faraday particularly and the
more accurate measurements made with the voltameter, have shown
that 'the magnetic action of a current is proportional to the quantity
of electricity which flows during unit time that is to say, to the
intensity / of the current.
The action exerted by a rectilinear current on a magnetic mass
m at a distance #, may then be represented by the expression
' \
to
in which k is a coefficient to be determined.
The action observed in this experiment, as well as in that of
CErsted, is always the action of a closed current ; but it is easy to see
that if the rectilinear portion is sufficiently great, and the rest of the
current sufficiently distant, the action of this latter part is inappre-
ciable, and the effect observed only depends on the nearest part.
The action of the rectilinear portion may then be considered as
POTENTIAL OF AN UNLIMITED RECTILINEAR CURRENT. 423
equal to that of an unlimited rectilinear current. Hence we arrive
at the following law of Biot and Savart :
The action of an unlimited rectilinear current on a pole is perpen-
dicular to the plane passing through the current and the pole, is directed
towards the left of the current, and is inversely as the distance of the
current from the pole.
A simpler experiment, at any rate in theory, leads to the same
result. Suppose that a portion of the circuit is vertical, and a
magnet placed in any given way, upon an apparatus movable
about an axis which coincides with that of the current. It will be
seen that the movable system is at rest for all positions of the
magnet, whatever be the direction and strength of the current.
It follows hence that the sum of the moments, in reference to
the axis, of the actions exerted on the different masses of the
magnet, is null.
If m is the magnetic mass at a distance a from the axis, we
shall have
If we suppose the magnet reduced to two masses m equal and
of contrary signs, at the distances a and a' from the current, the
equation reduces to
m(<f>a (f> f a') = Q, or <$>a = const.,
that is to say, to Biot and Savart's law.
The experiment carries with it its own verification, for if we cease
to make the axis of rotation coincide with the axis of the current,
the system is displaced, and tends to turn in one or the other
direction to obtain its position of equilibrium.
445. POTENTIAL OF AN UNLIMITED RECTILINEAR CURRENT.
We shall proceed to show that the magnetic field of a current is
defined by a potential that is to say, by a function whose partial
differentials, in reference to the axis of the co-ordinates, represent
the respective components of the force taken with contrary signs.
In the case of a rectilinear current, the equipotential surfaces are
planes passing through the current. Let us take the current for the
2-axis, and a plane perpendicular to the current passing through the
point P (Fig. 96) for the plane xy. If we suppose that the current
goes behind the figure, the force <f> at a point P of the plane, from
Ampere's rule, is perpendicular to PO, and would tend to turn this
424 CURRENTS AND MAGNETIC SHELLS.
point about the current in the direction of the hands of a watch.
Let a denote the angle PO/. For a very small displacement PP'
in the direction of the force, the work on a positive mass equal
to unity will be
dT = < x PP' = <$>ada = kida.
Fig. 96.
As the angle /5, which the right line PO makes with a parallel
P#' to the axis, is complementary to a, we may write
</T = -kid ft.
This work is equal to the corresponding fall of potential - dV ;
we deduce therefrom
and therefore
V = fa'j3 + const.
We may observe that the angle /5 is the rectilinear angle of the
dihedral angle of two planes drawn from the point P, one in the
current, the other parallel to the current and to the jc-axis ; twice this
angle /3 measures the surface o> of the spindle, which the dihedron
cuts through a sphere of unit radius with its centre at P ; we may
accordingly write
ki
V = w + const.
2
The surface o> is evidently the solid angle under which the plane of
xz is seen from the point P, unlimited in one direction, and
bounded in the other by the current that is to say, the apparent
surface of this plane.
POTENTIAL OF AN UNLIMITED CURRENT. 425
We conclude from this that the potential of an unlimited recti-
linear current at a point is, within a constant, proportional to the
product of the strength by the apparent surface of a plane, unlimited
in one direction, and bounded in the other by the current
In order to determine the sign of this apparent surface, we must
remember that, in practice, the unlimited rectilinear current neces-
sarily forms part of a closed circuit, and that if the non-rectilinear
portion is very distant from the point P, the angle under which the
whole circuit is seen, which we may suppose plane, only differs by an
inappreciable quantity from the unlimited plane of which it forms
part. We shall call that face of the current, which is on the left of
the observer placed in the current, and who is looking inwards, the
positive face ; the negative face being that on his right ; and we shall
take the angle w positive or negative, according as the positive or
the negative face is seen from the point P.
446. THE POTENTIAL OF AN UNLIMITED CURRENT is NOT A
SIMPLE FUNCTION OF THE CO-ORDINATES. At a given point, the
angle w only gives the value of the potential of an unlimited current
to within a constant. It is easy to see what is the significance of
this constant. Suppose that a unit positive mass taken at the point
P (Fig. 96) describes a circumference about the point O, in the
direction of the force, and reverts to its original position. The angle
o> has resumed the same value, but the force </> has performed a
ki
work <27T# that is to say, 2irki or 477 , and this mass has traversed
2
the plane of the current by the negative face. For n turns of the
ki
mass, the work would be equal to 471-72 , and the potential would
2 ki
have varied by the same quantity - 471-72 .
ki
On the other hand, the expression w is the work which must
be expended against magnetic forces, in order to bring this mass
from infinity to the point P, without traversing the plane of
the current.
If then, by analogy with the properties of magnetic shells,
we call the potential at a point, the work necessary to bring a
positive magnetic mass equal to unity from an infinite distance,
this potential is expressed by
/ \ TT / N
(2) V = (0-47772 = (to -47772).
426
CURRENTS AND MAGNETIC SHELLS.
The magnetic potential of the current at a point is not, therefore,
a simple function of the co-ordinates, but a function having an
infinity of values, which differ from each other by a multiple of
ki
477 ; that is to say, of the work which would be represented by
the complete rotation about the current of a magnetic mass equal
to unity. This property may be easily generalised.
447. POTENTIAL OF AN ANGULAR CURRENT. Let us consider
two unlimited currents AA' and BB' (Fig. 97) of the same strength,
situated in the same plane, and moving in the directions indicated
by the arrows. Let Q be the projection of the pole P on this plane.
The potential at P of the current AA' is proportional to the
apparent surface of the plane AA'X; that of BB' is proportional
to the apparent surface of the plane BB'X.
With the actual direction of the current, and assuming that their
planes extend indefinitely on the right, these two apparent surfaces
must be taken with contrary signs, and the resultant potential is
equal to their difference. But the part in common, AOBX, dis-
appears; the potential is therefore proportional to the apparent
surface of the angle BOA', diminished by the apparent surface of
the angle AOB'.
On the other hand, the system of two unlimited currents is
identical with that of two angular currents BOA' and AOB', the
first of which turns its positive face to the front, and the second its
negative face.
We may accordingly assert that the potential at a point P of an
angular current, such as BOA', is proportional to its apparent surface,
POTENTIAL OF A TRIANGULAR CURRENT.
427
within a function of the co-ordinates of the apex of the angle ;
a function whose sign depends on the sign of the surface turned
towards the point, and which, moreover, would disappear in
applications.
448. POTENTIAL OF A TRIANGULAR CURRENT. Let us suppose
further that in the same plane there is a third current CC' (Fig. 98),
identical with the former, and forming with it a triangle abc.
Fig. 98.
The potential at P of the two former is proportional to the
apparent surface of the angle BrA', less that of the angle ArB'.
The potential of the current CC' is proportional to the apparent
surface of the plane CC'X taken with the - sign. If we add
together the effects of the three currents, the part in common
BtfM.' disappears, and finally there remain, in the expression of the
potential, the apparent surface of the triangle abc^ and that of the
external angles A^rB', C&A', and B#C', these latter being all taken
negatively.
Let us add to the system three angular currents of the same
strength represented by bent arrows; they will introduce into the
potential the apparent surfaces of these same angles taken positively
this time, so that only the apparent surface of the triangle will
remain. Of all the currents only that circulating round the angle
will remain, for each of the external lines is traversed by equa
currents of opposite signs.
428 CURRENTS AND MAGNETIC SHELLS.
Thus the potential at a point P of a closed triangular current is
proportional, ivithin a constant, to the apparent surface of the triangle
which the current encloses, or to the solid angle under which the
triangle is seen from the point P.
If this angle be called w, we have
ki
V = to + const.
2 ,
The theorem clearly applies to any given quadrilateral; for we
may alwayt divide the quadrilateral into two triangles, and suppose
that along the diagonal are two equal currents in opposite directions.
By this addition nothing is changed in the electrical system, and the
given current is transformed into two triangular currents with their
positive faces on the same side. The potential is the sum of the
two apparent angles of the triangle, or the apparent angle of the
quadrilateral. .
449. POTENTIAL OF ANY CLOSED CIRCUIT. We can draw any
surface through the outline of a closed current, and suppose this
surface divided by two systems of lines, into any number of quadri-
laterals, and of infinitely small triangles with rectilinear sides. If
we suppose the contours of each of these elementary figures to be
traversed by currents of the same strength, and the same direction
as the principal current, we should obtain a system of closed currents
which will be equivalent to the given current, since each of the
interior lines is traversed by two equal currents in contrary directions,
and the only effective portions are those, the whole system of which
forms the given current. As all the elementary currents have their
faces turned in the same direction, the potential of the system is
proportional to the sum of the apparent surfaces of the elementary
currents that is to say, to the apparent surface of the proposed
current.
Hence the potential at any point P of any closed current is given,
to within a constant, by the solid angle under which the contour of the
current is seen from the point P.
450. EQUIVALENCE OF A CLOSED CURRENT AND OF A MAG-
NETIC SHELL. AMPERE'S THEOREM. Let w be the value of the
solid angle under which the contour of the current is seen from
the point P, then from the preceding theorem
(3) V = <j) + const.
AMPERE'S THEOREM. 429
For a magnetic shell of power <, which would be bounded by
the same outline (329), we should have
The two potentials will then be equal, within a constant,
provided that
(4) 7=*- 1.
the symbol I being a new expression for the strength of the current
defined by this condition itself, and which we call the electromagnetic
strength.
The potentials of the current and of the shell for which I = &
are not absolutely identical, but they only differ by a constant, and
their differential coefficients are the same. Hence the actions
exerted by the current and by the shell are the same for each point
of the field. We are thus led to Ampere's celebrated theorem :
The magnetic action of a closed current is equal to that of a
magnetic shell of the same contour.
The positive forces of the current and of the shell correspond,
and are on the left of the observer placed in the current, and who is
looking towards the interior of the circuit.
We have deduced this important theorem from Biot and Savart's
experiment, but we might consider it as an experimental fact verified
in all its consequences, and accept it as a starting point to deduce
from it all the magnetic properties of currents.
451. REMARKS ON THE EQUIVALENCE OF A CLOSED CURRENT
AND A MAGNETIC SHELL. It is important to insist on the con-
ditions of the equivalence of the current and of the shell. We
have seen that with a shell the force is not a continuous function
of the co-ordinates ; it is constant in the interior of the shell, and
changes its sign when one of the surfaces is passed through ; the
lines of force start on each side of the positive face, and are
absorbed by the negative. These sudden changes do not take place
in the case of a closed current ; the force is a continuous function of
co-ordinates, and the lines of force are closed curves which do not
touch the circuit, and do not encounter any acting mass. It will be
seen that this may be the case, without any contradiction ; for the
shell equivalent to the current is only under the condition of being
bounded by the same contour, and we may suppose that when a
430 CURRENTS AND MAGNETIC SHELLS.
magnetic mass is displaced in the vicinity of a current, the equivalent
shell is being constantly deformed, and recedes before it without
ever being met.
The analogy of the two systems becomes closer if instead of
considering the magnetic force of a shell we consider the induction.
We know, in fact, that magnetic induction is a continuous function
of co-ordinates, that the flow of induction is maintained throughout
the entire extent of an orthogonal tube, and that the force and
magnetic induction have the same value for any point outside the
magnetised media. As a particular case, the magnetic induction in
the thickness of a shell is identical with the force which would be
produced there, if the shell, while still retaining the same contour,
and the same magnetic power, were deformed in such a manner as
no longer to include the point in question, and this force is equal to
that of an equivalent current which went along the contour. This
is evidently the same also for any system of currents ; from which is
deduced the general law :
Any system of dosed currents is equivalent to a magnetic system^
and the action of currents at a point is identical with the induction^ at
the same point , of the equivalent magnetic system.
452. RELATIVE ENERGY OF A MAGNETIC SYSTEM AND A
CURRENT. The potential of a current at a point P, is within a
constant equal to -Iw, if we denote by o> the solid angle under
which the negative face of the current is seen. The product - mlu
is the work which would be expended in bringing a magnetic mass
equal to m, from an infinite distance to this point, without traversing
a continuous surface bounded by the current. The potential energy
of the mass m at the point P is then, within a constant, equal
to mliD.
If this mass has passed the surface of the current n times ,by the
positive face to arrive at the point P, the work ml^ir must each time
have been expended; the total work is then
ml (^irn a>).
Conversely, if the mass is left to itself, it tends to turn in-
definitely around the current, and at each turn expends an amount
of energy equal to m^I.
This continuity of motion is not possible with two magnetic
systems, for the potential is then a determinate function of the
co-ordinates ; it would, moreover, be inconsistent with the principle
of the conservation of energy. With currents the movement may
RELATIVE ENERGY OF A MAGNETIC SYSTEM. 431
be continuous, for an extraneous energy necessarily comes into play
in the phenomena, such as that of the chemical actions which take
place in batteries.
If Q be the flow of force of the magnetic system, which traverses
the surface of the current, entering by the negative face, the relative
energy of the two systems is expressed, to within a constant, by
(5) W=-IQ.
When the magnetic system is left to itself, the work ^T of the
magnetic forces, for any infinitely small displacement, is equal and of
opposite sign to the change in the energy, and we have
or
The motion of the system takes place in such a way that the
value of Q tends towards a maximum.
For two successive positions characterized by the indices i and 2,
the work, to within a constant, will be
It is important to remark that in a general way the difference
Q 2 Qj depends not only on the final and initial positions of the
magnetic system, but also on the path pursued by each mass; for
the product ;#47rl should be added to the work of all those which
should have surrounded one of the branches of the current. If a
long and flexible uniform magnet, for instance, were placed near a
current, the positive pole would turn in one direction indefinitely
about the current, and the negative pole in the contrary direction.
Nevertheless, if the contour of the current is rigid, as well as the
magnetic system, all the masses which constitute the magnet will
necessarily make the same number of turns, and in the same time,
and the corresponding work is equal to n^irl^m. As the total mass
of a magnet is always zero, the work of any given displacement only
depends on the initial and final position of the system, and not of
the path traversed ; in this case the work is null when the magnet
reverts to its original position.
It is therefore impossible to obtain the continuous motion of a
magnet by a current which traverses a rigid system ; the reciprocal
432 CURRENTS AND MAGNETIC SHELLS.
action of the two systems is then identical with that of two magnets.
The maximum and minimum values of the flow of force Q correspond
to positions of relative equilibrium, stable in the former case and
unstable in the latter. The motion may be continuous, on the
contrary, if the circuit can be deformed ; if it contains, for instance,
liquid portions, sliding contacts, or if it can be broken in certain
parts while the magnet is being displaced. '
453. RECIPROCAL ACTION OF Two CLOSED CURRENTS. It
may be asked whether a closed current and a shell, which are
equivalent with respect to any magnetic system, are so towards
another current? Thus the current Cj and the shell Sj of the
same contour, are equivalent in their action upon the magnetic
system M 2 ; suppose that this magnetic system is a shell S 2 ; the
reciprocal action which is exerted between S T and S 2 is identical
with that which is exerted between S x and the current C 2 , which
is equivalent to S 2 ; but is this latter action the same as that which
would be exerted between the two currents C} and C 2 ? The
affirmative seems probable ; but this is only an induction, and it
would be easy to find examples, for which the same mode of
reasoning would lead to conclusions which are manifestly erroneous.
Thus, under conditions suitably chosen, it might happen that the
actions exerted upon a magnet by a magnet and by a piece of soft
iron are the same ; we could not conclude from this that the soft
iron and the magnet would be equivalent towards another piece of
soft iron.
It is therefore as an experimental result, and not as a necessary
deduction from theory, that we shall assume the following theorem of
Ampere :
The reciprocal action of two closed currents is identical with that of
two magnetic shells respectively equivalent to each of them.
454. RELATIVE ENERGY OF Two CURRENTS. The value of
the potential energy of two magnetic shells (341) is
W= -**'M.
From Ampere's theorem, that of two closed currents will be ex-
pressed, to within a constant, by the formula
(6) W=-II'M,
in which I and I' are the strength of the two currents, and M the flow
of force which, starting from one of the circuits, traverses the other by
its negative face, the strength in each of them being equal to unity.
ELECTROMAGNETIC ROTATION. 433
The work al of the magnetic force corresponding to an infinitely
small displacement will be given by the equation
(7) </T = -<AV= -IIVM.
455. ELECTROMAGNETIC ROTATION. We have seen that the
reciprocal action of a magnet and of a rigid current cannot produce
a continuous motion. This would also be the case with two rigid
currents, but the impossibility ceases if one of the systems can be
deformed, and the preceding considerations enable us to give a
simple explanation of most of the experiments of this kind.
Consider, for instance, an unlimited rectilinear current, the
outline of which is at O (Fig. 99), and a magnet PP', one of whose
poles P' can slide along a groove AB perpendicular to the current,
while the other pole P may describe a circumference about the
current, by means of a movable contact which opens the passage to
it at each turn. The pole P will rotate for an unlimited time about
Fig. 99.
the current, and apart from friction the velocity will go on increasing,
since the magnetic action of the current performs, at each turn, work
equal to the product of 4?rl by the mass of the pole. A regular
condition is established from the time in which the passive resistances
balance the motive force. We shall see several examples of the
same kind in one of the following chapters.
The action of a current on itself may also give rise to deforma-
tions, or to continuous motions.
Let ACB (Fig. 100) be a portion of a circuit movable about an
axis passing through the two points A and B by which it is attached
to the general circuit. We may suppose that the line A B is traversed
by two currents in contrary directions, of the same intensity as the
current itself, and may thus decompose the system into two distinct
closed circuits s and s'.
F F
434
CURRENTS AND MAGNETIC SHELLS.
If there were no other forces in the field than those arising from
these two circuits, the relative energy of the two currents is
W= -PM.
As the energy tends towards a minimum, the movable part would
displace itself, so that the flow of force M is a maximum.
\
Fig. 100.
If the two contours s and s' are plane, it is clear that the movable
part s would place itself in the plane s 1 so as to form its prolongation :
it is seen by the action which the two equivalent magnetic shells
would exert on each other.
If the general circuit consists of a flexible wire of a given length,
the action of a current on itself would tend to give it the greatest
surface that is to say, to make it take the form of a circumference
of a circle.
If the wire is elastic it will elongate .until the elasticity balances
the electromagnetic forces.
456. FARADAY'S EXPERIMENTS. In certain cases the funda-
mental formula
W= -IQ
appears not to hold, as continuous movements can be produced, even
when the flow of force through the system is zero or is invariable.
Let us consider, for instance, an arc ACB of a plane curve
(Fig. 10 1) movable about a straight line AB, passing through the
axis of a magnet PP', one of the ends being placed between the two
FARADAY'S EXPERIMENTS.
435
poles and the other without, and let us suppose that a current goes
from the point A, to the point B, by the arc ACB. The flow of
magnetic force from the pole P, which traverses the portion ACB of
P U
Fig. 101.
the circuit, seems null, for the pole is in the plane of the circuit ;
and, moreover, the arc ACB seems in all azimuths to have an
identical position in reference to the pole. Yet the arc ACB acquires
a continuous rotatory motion, which, if the pole P is a North pole,
makes it turn in the direction of the hands of a watch for an
observer placed above the point A.
In order to analyze this phenomenon let us replace the current
by the equivalent shell ; we may imagine that the movable part of
this shell is made up of an infinitely elastic plate, which forms a
concave surface behind the pole, and presents its negative face to the
pole. As this surface tends to comprise a great part of the flow, it
will move in the direction indicated, and the motion will be con-
tinuous, as the elastic sheet can fold upon itself indefinitely.
The change in the flow of the force for a rotation of the plane
of the current, will be equal to 2mO ; the work of the electromagnetic
forces relatively to this pole will be 2mOl for the displacement 0, and
for a complete turn, 4ir;//I.
The moment of the couple of rotation in reference to the axis is
then expressed by
477 ml
27T
2ml ;
it is to be observed that this moment is independent of the magni-
tude and shape of the arc.
F F 2
43 6 CURRENTS AND MAGNETIC SHELLS.
The action of the lower pole P' is obviously null ; for any portion
of the flow of force starting from this point, and which meets the
sheet, necessarily cuts it twice, entering first by the positive, and
then by the negative face ; there cannot, therefore, be any variation
of energy on this side, and therefore no cause of motion.
If the two poles were beyond the line AB, which joins the ends
of the movable current, the action of each will be null, and there
will be no rotation. In like manner, if the two poles were in the
interval AB, the total variation of the flow of force relative to any
displacement of the arc will be null, for the two poles will produce
equal and opposite variations. The arc must, therefore, be at rest.
These various experiments are due to Faraday.
457. ANOTHER FORM OF EXPRESSION FOR THE. ELECTRO-
MAGNETIC; WORK. In the preceding example the work 2mW
corresponding to a rotation is equal to the product of the
strength of the current by the flow of force cut by the arc ACB in
the displacement. It is easy to generalise the expression for the
work in this new form.
Let us consider, in fact, a fixed magnetic system, in the field of
which a current experiences any given displacement or deformation.
Let s and s' be the two successive positions of the current (Fig 102),
and Q and Q' the flows of force which traverse the negative face
in the two cases. The corresponding work of the electromagnetic
forces is I(Q' - Q).
Draw two planes P and P' tangential to the two positions of the
circuit ; join the points of contact AA', and BB', and denote by Q x
and Q 2 the flow corresponding to the surfaces A'ACBB'C' and
A'ADBB'D'. We have evidently
Q'-Q=Q 2 -Qr
ELECTROMAGNETIC ACTION ON A CURRENT ELEMENT. 437
But Qo is the flow of force cut by the arc BDA, Q x the flow cut
by the arc ACB in the displacement ; hence we may say that the
work of electromagnetic forces is equal to the excess of the flow cut
by one of the portions of the circuit over the flow cut by the other.
If the forces traverse the plane of the figure from front to back, the
values of the flow are positive for the direction of the current
indicated by the arrow.
For the elements of the curve ACB, the motion is to the right
of an observer who is placed in the current, and who looks in the
direction of the force, and the flow of force cut enters into the
expression of the work with the - sign. For the curve BDA the
motion is towards the left, and the flow of force cut is taken with
the + sign.
If we agree to give the sign + to the flow of force cut by the
circuit when the motion is towards the left of the observer, and the
- sign when it is to the right, we may say that the total work is
equal to the algebraical sum of the flow of force cut by the current.
458. ELECTROMAGNETIC ACTION ON A CURRENT ELEMENT.
We are thus led to consider the action exerted on a current as
resulting from the actions which would be exerted on each of the
elements into which we may suppose it to be decomposed ; it is the
same problem as for a magnetic shell (344). To apply the result
obtained to currents, we must replace the magnetic power of the
shell by the strength of the current, and the force exerted on each
element is expressed by
(8) IF<&sina = L/A,
dA being the area of the parallelogram constructed on Fds. Thus :
The action exerted on a current element placed in a magnetic field,
is equal to the product of the intensity of the current into the area of
the parallelogram, drawn on a right line which represents the intensity
of the field, and on the current element. This force is perpendicular to
the parallelogram, and is directed to the left of the observer placed in
the current, and who is looking in the direction of the force.
If the field is due to a single pole of mass m placed at P
(Fig. 103), at a distance r from the element, we have F = ; it
follows that the reciprocal action of a current element and of a pole
is expressed by
(9) d(f> =
43$ CURRENTS AND MAGNETIC SHELLS.
The action is therefore inversely as the square of the distance
of the pole from the element; it is applied to the element, and
is perpendicular to the plane passing through the element and
the pole.
Y i
> .y
\
A
Fig. 103.
459. RECIPROCAL ACTION OF Two ELEMENTS OF A CURRENT.
We have seen (347) that the action of two shells may be expressed
as a function of the two contours. The action of the two currents
may then be considered as the resultant of the actions exerted
between the elements of the current which constitute it.
This elementary action d 2 (f> is not determinate, but if we assume
that it takes place along the right line which joins the two elements,
it is expressed by
Jr tofc'
If 6 and & are the angles of the two elements with the right line
joining them, and e the angle which the two elements make with
each other, we have
(10) d*ip = + cos - - cos cosO' I dsds'.
The formulae (9) and (10) represent the elementary laws dis-
covered by Ampere.
The method adopted by Ampere to arrive at this result was
entirely different ; it will form the object of the following chapter.
460. ELECTROMAGNETIC INTENSITY OF A CURRENT. We have
hitherto defined the intensity of the current by the quantity of elec-
tricity which passes through a section of the circuit in every unit
of time. The strength thus defined is called the electrostatic intensity;
it may be determined by measurements of capacities and potentials,
or by electrochemical phenomena. The electromagnetic intensity
introduced above (450) is defined by the condition of being ex-
ELECTROMAGNETIC UNITS. 439
pressed by the same number as the magnetic power of the equivalent
shell of the same contour. We deduce from it
k
We shall see later what is the significance of this factor - .
2
With the new expression for the intensity, the action of an un-
limited rectilinear current at the distance a becomes
(12)
Consequently, the electromagnetic intensity equal to unity is that
of the unlimited rectilinear current which at unit distance exerts a
magnetic force equal to 2.
461. ELECTROMAGNETIC UNITS. The change in the expression
for the strength necessarily leads to corresponding modifications in
estimating other electrical quantities. If it is desired that the in-
tensity shall always represent the quantity of electricity which
traverses a section of a conductor in unit time, the equation
will define Q, and therefore the unit of electricity.
If the current produces no other work than that of heating the
circuit, Joule's law will define the expression of the resistance, and
therefore the unit of resistance, by the ratio
W = PR/,
in which W represents the thermal energy produced during the
time /.
The electromotive force, lastly, is given by the equation
W-EI/.
The units thus defined, and which we shall call electromagnetic
units, are those used in the following chapters. We shall subse-
quently establish the relations between them and the electrostatic
units.
44 ELEMENTARY ACTIONS.
CHAPTER II.
ELEMENTARY ACTIONS.
462. AMPERE'S METHOD. The course we have followed is, so
to say, the inverse of that which led Ampere to the law of ele-
mentary actions. The importance of the subject, and the interest
which Ampere's experiments and reasonings present, will justify the
fresh statement of the question which we shall make, based on the
ideas of this illustrious philosopher.
Ampere considered the actions exerted by currents, on magnets
or on currents, as the resultant of the actions of each of the elements
of length, into which the current may be decomposed, and he endea-
voured to deduce from experiment the law of these elementary
actions.
If we inquire to what extent an elementary law thus defined is
directly accessible to experiment, it will be seen that in strictness it
is possible to study the action of a single pole on a current element
by working with a solenoidal magnet so long that the action of the
other pole may be neglected, and with a portion of the current as
small as we like, which is made movable ; this, however, is not the
case when we consider the action of a current element on a pole, or
the interaction of two current elements. Only the entire circuit
of the current, or in all cases a closed current, can be made to act
on a movable current element as on a pole.
The investigation of a mathematical law, in the way in which
Ampere views the problem, corresponds then at least in the second
case to a purely mathematical conception ; but the method is none
the less legitimate so long as we merely propose to determine the
resultant action of the whole circuit, the elementary law being then
restricted only by the condition that the integral relative to a closed
circuit gives a result which agrees with experiment. But it is clear,
also, that the problem thus stated is not completely determinate, and
that there may be several elementary laws which satisfy this funda-
mental condition.
ACTION OF A POLE ON A CURRENT ELEMENT. 441
463. ACTION OF A POLE ON A CURRENT ELEMENT. FUNDA-
MENTAL PRINCIPLES. We will start from the following principles,
some of which may be regarded as evident axioms, and others as
experimental facts :
I. Equality of action and reaction. The action of a magnet on
a current is equal and directly opposite to the action of the current
on the magnet. This general law of Nature is experimentally verified
in the present case ; for if the magnet and the current are connected,
the system if made free does not move.
II. The action changes its sign with the sign of the pole and
with the direction of the current. This fact is a result of experi-
ment. The action remains the same when the sign of the pole and
the direction of the current are simultaneously changed.
III. Principle of sinuous currents. The action of a sinuous
current on a magnet is identical with that of a rectilinear current
which has the same terminals.
In order to verify this principle, Ampere showed that two con-
ducting wires terminating at the same ends, one straight and the
other sinuous, have no action on any magnet when they are tra-
versed by the same current in opposite directions.
Some limitations are here necessary; the sinuous current must
be of the same order of magnitude as the rectilinear current, and be
but little distant from it; nor must it turn about the rectilinear
current This principle, moreover, will only be used to replace an
element by its three projections.
IV. The action of any given magnet, and therefore of a pole, on
a current element, is perpendicular to the element.
Ampere established this principle in the following manner. A
metallic arc of a circle, movable about an axis passing through its
centre, and perpendicular to its plane, can glide on two drops of
mercury by which the current traversing it enters and leaves. Any
given magnet placed in the vicinity leaves the arc at rest. The
action of the magnet is then in the plane which passes through
the axis of rotation, and is therefore perpendicular to the movable
current. The arc, moreover, begins to move when the axis no longer
passes through the centre.
V. The action of a magnet on an element of current is applied
to the element. This results from the following experiment, due to
M. Liouville. Part of the rectilinear current is made movable
about its axis ; with this object, its ends dip in two small mercury
cups by which the current enters. The rectilinear element does not
rotate at all, in whatever manner the magnet is presented to it.
442 ELEMENTARY ACTIONS.
VI. Principle of symmetry. The application of the principle of
symmetry will determine the direction of the force.
We see at first that :
i st. The action of a pole on an element of current perpendicular
to the right line which joins it to the pole, is perpendicular to the
ds
p
Fig. 104.
ds
Fig. 105.
Fig. 106. :
plane passing through the pole and the element. Let us join the
pole P to the centre of the element ds (Fig. 104). We already
know that the action is perpendicular to the element. It is also
perpendicular to the right line PO, for if the figure is turned
through 1 80 about the right line, the force must change its sign
without changing direction (II).
2nd. The action of a pole on a current element, the prolongation
of which passes through the pole, is zero. This action must be
perpendicular to the element ds (Fig. 105); on the other hand, it
should not change in direction when the element is made to turn by
any quantity about the right line PO ; it is therefore null.
Let there now be an element ds (Fig. 106) which makes an
angle a with the right line joining it to the pole; the element of
current ds may be replaced by its two projections ds cos a and
ds sin a, the one along the right line PO, the other in a perpendicular
direction.
The action of the pole on the former is null ; there only then
remains the action of the pole on ds sin a. This latter is propor-
tional, as we have seen, to the mass m of the pole, to the intensity i
of the current ; it is also proportional to the length ds sin a of the
element, and lastly to a certain function of the distance f (r).
Hence, if d<}> is this force, and k a coefficient to be determined by
experiment,
d<j> = mkids sin of (r) .
LAW OF BIOT AND SAVART.
443
The force is moreover applied to the element, and is per-
pendicular to the plane Pds. Its direction is on the right of the
current that is, on the right of an observer placed in the element,
and looking at the pole, for the action of the element on the pole is
in the opposite direction.
VII. Law of Biot and Savart. The experiments of Biot and
Savart (444) demonstrate that the magnetic action of a rectilinear
current on a pole, is inversely as the distance of the current from
the pole.
According to a remark of Laplace, this law is satisfied if we
assume that the action of a pole on an element of current is inversely
as the square of the distance that is to say, if we have /(?) =
We may conversely prove that the law of the square is the only one
which satisfies Biot and Savart's experiments.
Fig. 107.
Consider, in fact, two parallel rectilinear currents, unlimited and
of the same strength AS and A'S', at distances a and a' from the
pole P (Fig. 107). For two elements ds and ds\ comprised between
the same two radius vectors drawn through the point P, and the
distances of which from this point are r and r', we have
ds _ r a
ds' r a '
and therefore
rds' = r'ds
444 ELEMENTARY ACTIONS.
The ratio of the actions d$ and d<$ of the pole on the elements
ds and ds' becomes then
ds sin a
d( r'" r .r ds r a
i r . rds' r a
ds sin a
The actions of the corresponding elements being inversely as the
distances a and ', this will also be the case with the resultants.
This is the law resulting from experiment. The action of a pole on
an element of current is then expressed by
dssina
d<p = mkt - - .
As all the forces are parallel and in the same direction, the action
of the pole on the unlimited rectilinear current is
. (ds sin a (
= mkt mki
J r*
ds cos 8
Measuring the length of the circuit from the point A, we have
dB
i0tan0, ds = a ,
cos 2 #
a? = r^ cos 2 \
from which follows
ds cos 6 i
r 2 = # C(
and therefore
IT
/&' f + 2 2;^/
This force is by symmetry applied at the point A, and the action
of the rectilinear current on the pole is applied at the same point,
but in the opposite direction.
LAW OF BIOT AND SAVART. 445
This result seems at first to disagree with experiment, for the
action of the current on the pole is applied to the pole itself. This
contradiction arises from the fact that in practice the current is
necessarily closed. If, for simplicity, we suppose that the general
circuit is in a plane passing through the point P, the actions d$
and d<$ of two corresponding elements ds and ds' situate at the
angle dQ, are in opposite directions, and inversely as the distances
r and r'. The portion which closes the circuit being supposed to be
very distant, the difference of the two forces is sensibly equal to the
action of the element ds ; but as rd<f> = r'd$, the point of application
of the partial resultant is the pole P. This is also the case for the
general resultant. The action of the whole circuit is sensibly equal
to that of the rectilinear part.
If the intensity is expressed by means of the electromagnetic unit
(460), the action of the unlimited current on the pole m, placed at
the distance a, is expressed by m , and the elementary formula
a
becomes
mlds sin a
(i) d$ = -
or, noting that is the magnetic action F of the mass m at the
point occupied by the element of current,
(2) d<j>
dA denoting the surface of the parallelogram constructed on the
element and on the force F.
The action exerted on the current Ids, situate in a magnetic field,
only depends on the intensity of the field at this point, whatever be
the system from which the force proceeds (458) :
The action exerted on an element of current placed in a magnetic
field is equal to the product of the intensity of the current by the area of
the parallelogram constructed on the element of current, and on the
intensity of the field. This force is perpendicular to the plane of the
parallelogram, and directed to the left of the observer placed in the
current who is looking in the direction of the field.
The plane of the parallelogram to which the magnetic force is
perpendicular, was called by Ampere the directive plane.
446 ELEMENTARY ACTIONS.
Although we have given the name elementary to the force which
we have defined, it cannot so be considered in the strict sense of
the word ; thus, as Ampere observes, " we cannot apply the term
elementary either to a force which is manifested between two
elements which are not of the same kind, or to a force which does
not act along the straight line joining the two points between which
it is exerted."
464. RECIPROCAL ACTION OF A POLE AND OF A CURRENT.
Starting from this elementary law we shall prove as above (346)
that the components of the action of a unit pole, placed at the
origin of the co-ordinates, on a current element ds situated at a
point whose co-ordinates are x, y, and #, are
^ = -^
(3) dri =
It may be observed that the moment dM. z of this force, in
reference to the z axis, is
^M z = xdv) -yd% = \z (xdx +ydy) - (x 2 +/) dz .
The equation
gives
xdx +ydy + zdz = rdr.
It follows that
--\z rdr zdz - r* *' dz\ =- zc
2 ~r*[_ Z r r* ZL
- Id ( - )
r
But - is the cosine of the angle, which the right line r makes
with the z axis; we have then
*/M z = - \d cos y,
RECIPROCAL ACTION OF A POLE AND A CURRENT. 447
so that the moment M 2 of the actions exerted by the pole on any
arc AB has the value
(4) M^ = I(cosy a -cosy & ).
If the circuit is closed, this moment is null, and as the direction
of the z axis has been arbitrarily chosen, we see that the action of a
pole on a closed current passes through the pole. Conversely, the action
of a closed current on a pole also passes through the pole.
465. Instead of following the course taken, and of proving that
Biot and Savart's law is satisfied by an action which is inversely as
the square of the distance, we might have pursued a perhaps more
rigorous course, and have admitted as an experimental fact that the
action of a closed current on a pole passes through the pole.
The moment of the action of a pole on the element ds in
reference to the z axis will then be
and the moment relative to an arc AB
/B fA
M,= -I ry(r)<Jcosy = I ^ 2 /(r
JA JB
Integrating this expression by parts, we get
If the current is closed, the first term of the second member is
null. As the moment must be null, whatever be the shape of the
circuit traversed by the current, the second term must be identical
with zero ; that is to say, the product r 2 f(r) must be a constant, and
therefore the force must be inversely as the square of the distance.
If the arc, without being closed, terminates at two points, A and
B of a right line, about which it can turn, the couple of rotation
will not in general be zero.
Supposing, for instance, that the points A and B are situate on
the same right line which passes through the pole, and on the same
side of the pole, the moment of the forces in reference to this
axis is null, and the current will not acquire any moment of rotation
about the right line.
448 ELEMENTARY ACTIONS.
On the other hand, if -the points A and B are on different sides of
the pole, the angles y b and j a are equal, one to zero and the other
to TT : in this case the couple of rotation will be equal to 2!, and the
axis will turn continuously in the same direction.
We thus again find an explanation of the various special features
in Faraday's experiment (456).
466. The components X, Y, and Z of the action of a current
on a pole, placed at the origin of the co-ordinates, are, from what has
been said,
z = -ydx
If we put
-/
(6) B
ydz zdy
A= - ^- = G cos A,
with the condition
we shall have
X= -IA= -IGcosA,
Y= -IB= -IGcosA,
Z = -1C- -IGcosv,
and therefore
(7)
EQUIVALENCE OF A CURRENT AND A MAGNETIC SHELL. 449
The factor G is the action at the point P of the circuit, when it
is traversed by a current of unit strength. We may represent this
action by a straight line PG proportional to G, and making angles
A, /x, v with the axes.
467. EQUIVALENCE OF A CURRENT AND A MAGNETIC SHELL.
The action of a magnetic field on a current element being identical
with that of the same field on the corresponding element of the edge
of a shell bounded by the current, it follows that the action of the
current on a pole is identical with that of a shell of strength I, the
positive face of which is on the left of the current.
The magnetic potential of a current at a point P is then, to
within a constant, equal to the product of the strength I by the
angle w, under which we see from this point the positive side of a
surface bounded by the circuit that is, the face on the left of an
observer who is going with the current, and looking towards the
interior. The angle to represents also the flow of force which a
mass equal to unity, placed at the point P, would send towards
this surface.
As the components of the force are equal and of opposite signs
to the partial differentials of the potential, we see that the solid angle
corresponding to a surface ^<o, seen from the origin of the co-
ordinates, is given as a function of the contour by the equations
t)o> Cydz zdy
(8)
c)co Czdx xdz
*y~ "J ^
<)a> Cxdy -ydx
^T J ^
468. ACTION OF Two ELEMENTS OF A CURRENT. The action
of two elements of a current may be established by an analogous
method by the aid of some principles and of facts taken from
experiment.
I. Equality of action and reaction. This principle does not
allow of experimental verification in the case of two elements of
currents. It must be regarded as a fundamental hypothesis ; it
carries with it the necessary consequence that the action of two
elements is along the right line which joins them. On the other
hand, the reciprocal action of two elements of current is obviously
G G
45 ELEMENTARY ACTIONS.
proportional to the length of each element, to the intensity of the
current in each of them, and to a function, which remains to be
determined, of the distance of the elements as well as of their
relative distances.
II. The action changes its direction when the direction of one of the
currents is changed ; it remains unaltered when the direction of the
two currents is simultaneously changed. This is a general property of
electrical currents.
III. Principle of symmetry. It follows from this principle of
symmetry that the reciprocal action of two elements a and b (Fig.
1 08), one of which a is in the plane perpendicular to the other in its
middle, is null.
X*
Fig. 108.
For consider a system a'V symmetrical with the first in reference
to a plane P parallel with the element a, and with the right line OC
joining the centres of the elements.
The actions of a on , and of a' on b' are respectively along
OC and O'C, and in the same direction from symmetry. But
the second is none other than the first, in which the direction
of the current has been changed in the element b; the force
should have changed its direction owing to this inversion, hence
it is null.
The force in particular is null, if the element a is perpendicular
to the right line OC, which joins the centres of the two elements, or
directed along this right line. These are two cases which will have
to be made use of.
IV. Principle of sinuous currents. The principle of sinuous
currents may be applied as above (463) and with the same limi-
tations ; we can always replace a current element by its projections
on three rectangular axes.
PRINCIPLE OF SINUOUS CURRENTS.
45 1
Let us consider two elements a and b (Fig. 109) in any position ;
let ds and ds' be their lengths, i and t' the intensities of the two
currents referred to a given unit, 6 and & the angles of their
directions with the right line OO' joining their centres, r the distance
OO', lastly co the angle of the planes drawn through the right line
OO' and the two elements.
Fig. 109.
Let us take the plane which passes through the element ds and
the right line OO' as plane of the figure, and replace each of these
elements by its projections on three rectangular axes ; one of these
axes is the right line OO', the other a right line in the plane of the
figure, and the third a perpendicular to this plane. The element a
has only two projections
a' = ds cos 0,
a" = ds sin B ;
the three projections of the element b are
b" ds' sin & cos w ,
b'" = ds' cos 0' sin w.
The total action consists of the actions of each of these elements
a' and a" on each of the elements b', b"> and b'".
Of these six actions, four are null from the principle of symmetry,
that of a' on b" and b'" t and that of a" on b' and b'".
There only remains to be examined the action of a' on &', and
that of a" on b".
The former is exerted between two elements directed along the
same right line; it might be represented by
ii'dsds cos cos 0'F(r).
G G 2
45 2 ELEMENTARY ACTIONS.
The second is exerted between two elements parallel to each
other, and perpendicular to the right line joining their centres ; we
might represent it by
ii'dsds' sin sin 0' cos wf(r) ,
the two functions of the distance being different since the conditions
are not the same.
The action d^ will then be expressed by the formula
( 9 ) d^ = ii'dsds' [cos cos & F (r) + sin 9 sin & cos (o/(r)] .
If e be the angle of the two elements, we have
cos e = cos cos & + sin sin & cos w,
and we may write
(i o) d*$ = ii'ds ds' [cos 6 cos & [F (r) -f(r)] + cos e/(r)] .
469. DETERMINATION OF THE FUNCTIONS F(r) and /(/). To
determine the functions F(r) and f(r\ we must have recourse to
experiment, and may employ very different methods, according to
the phenomenon to which we apply ourselves. We shall adopt the
following course, which is not perhaps the most rigorous from the
mathematical point of view, but which leads most rapidly to the final
formula.
We start from the two following experiments devised by Ampere.
V. When the homologous dimensions of three similar currents of
the same intensity are in geometrical progression that is to say, as i, m,
;/z 2 , and are moreover similarly placed the actions of the extreme
currents on the intermediate current are equal and of opposite sign. If
this latter is movable along a line passing through its centre of
similitude, and if it is disturbed from its position of equilibrium,
it returns to it of itself that is to say, that the equilibrium is
stable.
Ampere made this experiment with three circular currents situate
in the same plane, the intermediate circuit being movable about an
axis perpendicular to this plane.
DETERMINATION OF THE FUNCTIONS F(r) AND/(r). 453
VI. The action of a closed current on an element of current is
perpendicular to the element. The arrangement of this latter experi-
ment is the same as that for the action of magnets on currents
(463, IV.)
Consider the three similar currents of the first experiment (V.)
For the position of equilibrium, the distances to the centre of
Fig. no.
similitude of the three homologous points, A, B, C (Fig. no) of
the circles, satisfy the ratio
OA OB OC
from which is deduced
and therefore
OB - OC = B C = OA m (i - m),
AB
For three homologous elements of current #, b, and c, the lengths
will be ds t mdS) and m z ds ; the distance of the two former being r,
that of the second to the third will be mr.
If we assume that each intermediate element such as b is in
equilibrium between the two others a and c, which correspond to it,
the entire current S' will be in equilibrium between the two similarly
placed currents S and S". It does not seem that this condition is
always necessary, but it is evidently sufficient, and it enables us to
determine the form of the two functions ~F(r) and/(r).
454 ELEMENTARY ACTIONS.
It follows, in fact, that the action exerted on the element b ought
not to change when a is replaced by c that is to say, ds by m*ds t
and r by mr; we get then, from equation (9), suppressing the
common factor ii'ds ds', and observing that the angles 6 and 6' are
equal, and the angle w is zero,
cos 2 F (r) + sin 2 Of(r) = m* [cos 2 6 F (mr) + sin 2 Of(mr)~\.
This condition should be satisfied, whatever be the particular
values of m, 0, and r; we must have separately
m*f(mr)=f(r).
Making r=i, and m = r, we get
r 2 F(r) = const = /*, ,
r' 88 const M
and, consequently,
Thus the functions F(r) and /(r) are both inversely as the
square of the distance.
The expression for the elementary action then becomes
(u) d^= U * S -\ k cos# cos^' + sin^ sin 0' cos w ,
or
(I2 )
J m >d ^ S> \(k - i) cos cos & + cos el .
470. DETERMINATION OF THE RATIO OF THE Two CON-
STANTS. The last experiment (VI.) enables us to determine the
ratio k of the two constants.
DETERMINATION OF THE RATIO OF THE CONSTANTS. 455
Let us place the origin of the co-ordinates at the centre of the
movable element ds', and the x axis in the direction of the element
itself. The action of an element ds of a closed circuit in which the
intensity is i is expressed, as we have seen, by
= m * S (k-i) cos cos 0' + cos e .
The co-ordinates of the element ds being x, y, z, and its distance
from the origin r, we have
-,
dr
-,
dx
COS = - .
ds
The elementary action may then be written in the form
hii 'dsds ' f x dr dx~\
= - (_!) +
r 2 \_ rds ds\
and the projection of this force on the x axis is
d*$ cos 6' = d^- = hii 'ds 1 \(k - i) + 1
r [_ r 4 r B J
The component parallel to the x axis of the action of the closed
circuit on the element ds' is expressed by
Integration by parts gives
456
ELEMENTARY ACTIONS.
For a closed circuit, the first term of the second member is null ;
we get then
~ hii'ds' ,
Since from experiment this component must be null, we get
0, or k= .
2
With this value of , the elementary action becomes
(i3)
or
ii'ds'ds I 3
cos e - - cos 6 cos | ,
7.--'7,r^ 3'awfrl
^hu 'ds'\ -T---T .
[_r 2 2 r 3 J
471. DETERMINATION OF THE CONSTANT A The components
of the action of the current parallel to the other axes are then
ydx $xydr
r* 2 r*
zdx 3 xzdr
~^~~2~^
di)'= \d^ = hu'ds'\
d= \d^ Z - = hu'ds'{
Integration by parts gives
xydr / i xy\ i Cxdy +ydx
" = V~3^/ 3J ^~
The former term of the second member being null for a closed
circuit, we have lastly
hii'ds' C
- J:
ydx - xdy
hii'ds' Czdx-xdz
DETERMINATION OF THE CONSTANT k. 457
The value of the action F, of the current on unit magnetic mass
placed at the origin, and therefore the intensity of the field which
this current produces at the point where is the element, is expressed
by IG, where I is the electromagnetic intensity of the current (466),
and its components are
z=-ic "'^-^
-I
The three components d%, dtj, d, of the action d$ of the
circuit on the element ds' may then be written
Mi'ds 1
from which is deduced
It follows from this that the two forces F and d$ are perpendicular
to each other.
As the x axis is the only one which is defined, we may choose
the two others in such a way that the magnetic action F of the
current is in the plane xz\ we have then-
Y = 0, B = 0,
X = Fcosa, A = Gcosa,
Z=Fsina, C=Gsina,
a being the angle which the force F on the straight line G makes
with the x axis.
458 ELEMENTARY ACTIONS.
It follows from this that
hii'ds' . hii'ds' .
dn -- G sm a = -- - Jb sin a = a<b .
2 2!
The action of the closed circuit on the element is therefore
perpendicular to the force F and to the element ds' that is to say,
Ampere's directive plane and proportional to the surface of the
parallelogram constructed on the force F, and the element ds'.
If the force of the field F = IG, at the point where is the element
ds\ was produced by a magnetic system, the action in like manner
would be along the y axis, and its value would be I'ds'F sin a, where
I' is the electromagnetic intensity of the current which traverses the
element.
The two actions are in the same direction, and they are pro-
portional; if we assume that they are identical, it will follow
that
A H/
h 2 .
It
As the numerical expression of a magnitude is inversely as the
unit with which it is measured, it will be seen that the constant h is
equal to twice the square of the ratio arbitrarily chosen to measure
the strength of the current in electromagnetic units.
472, If we suppose that the currents have, in the first place,
been determined in electromagnetic units, we have then
we thus arrive at Ampere's formula, which we had already obtained
(459),
(15) d^ = sin sin 6' cos w cos 6 cos 0' ,
(16) d <2 "d/ = -cose cos cos & ,
7-2 [_ 2 J
and which we may write in the more symmetrical form (351)
J r dsds'
ELECTRODYNAMIC UNIT OF INTENSITY. 459
473. ELECTRODYNAMIC UNIT OF INTENSITY. If, with Ampere,
we directly make h=i in formula (13), the strength of the current
will be expressed as a function of a particular unit, which is called
the electrodynamic unit.
This unit will be defined by the formula itself. By making
e=0,
ds = ds' = i ,
i>fVi,
we get
In this case the currents are parallel, of unit length, perpen-
dicular to the line which joins their centres, and at unit distance j
the strength of the current, which is equal for each of them, and
is taken at unity, is such that the reciprocal action is equal to the
unit of force.
Supposing the currents equal, equation (14) will give,
The electrodynamic intensity of a current is equal to its electro-
magnetic intensity multiplied by \/2.
In virtue of the ratio which connects the numerical expression
of a magnitude into the unit which serves to measure it, we see that
the electrodynamic unit of current is equal to the electromagnetic unit
divided by \/2.
474. The identity between the mutual action of currents and
that of the correlated magnetic systems has been confirmed in all
experiments as long, at least, as a steady condition has been estab-
lished in the circuits.
We may cite, for instance, the experiments of Weber on the
reciprocal action of the cylindrical coils with circular bases. This
action is proportional to the strength of the two currents ; it varies
with the relative distance and direction of the coils according to
the same law as that of two magnets whose axes are respectively
parallel to the axes of the coils.
460 ELEMENTARY ACTIONS.
475. FORMULA EQUIVALENT TO THAT OF AMPERE. We have
seen (349) that the action of two elements of the contour of two
magnetic shells, which is equivalent to the elementary electro-
dynamic action, may be expressed in an infinity of different ways,
with this condition that the resultant of the actions of a closed
circuit on an element has a determinate value.
476. (i.) Formula of M. Reynard. The first form which we
have met (348) for the action of ds upon ds' is, by supposing the
element ds' at the origin of co-ordinates and directed along the
x axis, a force whose components are
x r* y
The factor a in these equations represents the product Il'^y', and
x, y, z are the co-ordinates of the element ds.
The force itself is expressed by the formula
ll'ds'ds .
/= sin 6 cos IM,
in which is the angle of the element ds with the right line ?*, and
p! the angle of the element ds' with the plane rds.
If d$ is the angle under which the element ds is seen from the
element ds', an angle which is equal to - - , this formula may still
be written
/=
which is the formula of M. Reynard.
In order to determine the direction of this elementary force, we
observe, in the first place, that it is perpendicular to the element ds'
since/,. = 0.
It is in the plane rds. The equation of this plane, of which
X, Y, and Z are the co-ordinates, is
X (ydz - zdy) + Y (zdx -ydz) + Z (xdy -ydx) = 0.
FORMULAE EQUIVALENT TO THAT OF AMPERE. 461
The intersection with the yz plane is
Y (zdx - xdz) + Z (xdy -ydx) = ;
from which follows
Y_Z
We have further
r 2 ds 2 sin 2 = (ydz - zdyf + (zdx - xdz? + (xdy -ydx)*
which gives
_ a W sin 2 r (ydz -
r L
Now, the expression - : - is the cosine of the angle which
rds sin
the perpendicular to the plane rds makes with the x axis : hence the
quantity in brackets is the square of the sine of this angle, or the
square of the cosine of the angle /*' which the plane makes with
the x axis that is to say, with the element ds', and we have
ads sin cos M' \Vdsds' .
/ = - smtfcos/*.
Thus the action of ds upon ds l is in the plane rds t perpendicular
to the element ds', proportional to the sine of the angle which the
element ds makes with the distance r, and to the cosine of the angle
which the element ds' makes with the plane rds, and lastly inversely
as the square of the distance.
Let us take the plane rds as that of xz, and in this plane the line
OO', which joins the two elements, as the x axis. The force acting
on the element ds' placed at the origin of the co-ordinates is in the
xz plane and is perpendicular to ds'. To obtain its direction, we
must project trie element ds' on the xz plane ; a straight line in this
plane, perpendicular to the projection, will be the direction in ques-
tion ; it is perpendicular to the projecting plane, and therefore to the
element which passes through its foot in this plane.
462 ELEMENTARY ACTIONS.
The components of this force parallel to the axes are
II'dssmB, ll'dzdz'
f x =fcos/3 = ds cos/* cos/3 = 2 ,
A = 0>
> , .' ll'dssmO , . ll'dzdx'
f z =/sm p = ds cos fi! sin p = .
OC 00
If we still denote by & the angle which the right line r makes
with the element ds', and by w the angle of the two planes rds and
rds' , we have
dz' = ds' sin 0' cos w,
<&' = <&' cos 0',
which gives
f x = sin 6 sin & cos o> dsds' ,
f z = ^ sinO cos O'dsds'.
The action of two elements of consecutive currents is evidently null.
In fine, we have not here an equal and opposite action and
reaction, but there is a different action on each of the two elements,
directed perpendicularly to this element, and in the plane determined
by the other element.
The existence of a force perpendicular to the element is incom-
patible with the idea of an action at a distance; but if, on the
contrary, we view electrodynamic forces as resulting from a modifi-
cation in the elastic properties of the medium, we can easily see that
the reaction of this medium on an element of current may be
perpendicular.
477. (n.) General Formula. We may add an exact differential of
the co-ordinates to each of the components f x ,f y , and/ 2 without the
action of the closed circuit on the element ds' being modified. We
may then take as components of the elementary action the following
general expressions in which X, Y, and Z, are any given functions of
the co-ordinates :
GRASSMANN'S FORMULA. 463
478. (in.) Amplre's Formula. If we impose on the elementary
force the condition of being directed along the right line joining the
two elements, we get Ampere's formula ; this formula is the only one
which satisfies the general principle of action and reaction, and
consequently the essential conditions of a true elementary force.
For any other solution, the action of the element ds on the element
ds', will not be equal and directly opposite that of the element ds'
on the element ds.
479. (iv.) Grassmanris Formula. Let us replace the arbitrary
functions X, Y, and Z, respectively by xf t yf, and zf, f denoting a
function of the distance r. The components of the elementary force
will then be
This operation is the same as adding to the force given by
M. Reynard's formula, another force d 2 ^ the components of
which are
The force itself is given by the equation
[XViP = a*[f*ds* + r*(df)* + 2frd(fdr}},
or, taking into account the relation dr = ds cos 0,
~ 2 Wi) 2 = [flr + rdfj
This force makes with the straight line r an angle, the cosine
of which is
xd (xf) +yd (yf) + zd (zf) _ frdr + r*df_ d(rf)
'
464 ELEMENTARY ACTIONS.
Finally, the angle 8 which it makes with the element ds is
or
If we impose on this force the condition of being perpendicular
to the right line which joins the two elements we have then
'(</)- Oj
from which is deduced
/-A, /-;
hence the value of the added force is
i = afds sin = ds sin 9.
It will moreover be seen that this force d 2 ^ makes with the
element ds an angle equal to + -.
2
When the two elements are directed along the same straight line,
as the force d 2 ^ * s nu ll> an< ^ tne force given by M. Reynard's
formula is also null, the action of the two elements is null.
On this hypothesis, which is that of Grassmann, the true force
d^ will be the resultant of a force which is inversely as the square of
the distance, and of a force ^Vi which is inversely as the distance,
is perpendicular to the right line joining the elements, and whose
direction makes with the element ds an angle equal to - + 6.
Several other conditions might be imagined equally compatible
with experiment; but these few examples will suffice to show the
indeterminateness of the problem, and to point out the principal
solutions.
ACTION OF TWO PARALLEL CURRENTS. 465
CHAPTER III.
PARTICULAR CASES.
480. ACTION OF Two PARALLEL CURRENTS. According to
Ampere's formula, two elements of currents parallel to each other,
and perpendicular to the right line which joins their centres, will
attract or repel according as the currents are in the same, or in
opposite directions.
This result is usually verified by bringing a rectilinear current,
which we may suppose unlimited, near a portion of a rectilinear
current movable parallel to itself. The experiment is really more
complicated, for each of the currents in question forms part of a
closed circuit. In whatever manner we may suppose the planes of
the two currents placed in reference to each other, bringing the two
rectilinear portions near each other will increase, for each of them,
the flow of force which it will receive from the other by its negative
surface, and will diminish the relative energy if the currents are in
the same direction ; the converse takes place when they are in
opposite directions.
Let I be the intensity of the unlimited current, I' that of the finite
current, which is parallel to it, and b its length. If we vary the
distance a of the two currents (which we suppose in the same
direction) by da, the variation in the flow of force which enters the
circuit from the movable current is
= -bda = -2lb ;
a a
the force exerted upon the movable part of the circuit is expressed
by I'- = 211'-, it is therefore inversely as the distance a.
da a
481. ANGULAR CURRENTS. Two rectilinear currents placed
near each other tend to set parallel. This result is usually enunciated
H H
466 PARTICULAR CASES.
by saying, that two currents which form an angle with each other
attract if they both approach, or both recede from the apex of the
angle or the common perpendicular, and that in the contrary case
they repel.
The experiment is made by bringing an unlimited rectilinear
current near the bottom of a movable rectangular frame traversed by
the current. The movable frame turns so as to receive, on its
negative face, the maximum flow of force which proceeds from the
rectilinear current. There is no simple expression for the work of
any given displacement ; but the total work corresponding to the
displacement of the frame, from the position in which its plane is
perpendicular to the current, to that in which it becomes parallel, is
proportional to the flow of force which traverses the frame in the
second case. If a Q and 1 are the distances from the unlimited
current of the two sides of the frame parallel to it, and b the length
of one of these sides, we have
and the electromagnetic work is equal to 2 1 !'/. .
O-Q
These movements are easily accounted for by supposing the
currents replaced by equivalent magnetic shells, and considering the
actions of these shells.
We may arrive at the same object on Faraday's plan, by con-
sidering the lines of force and their distribution in the field. The
figured lines of force of the field resulting from the various systems
near each other, are closer in certain regions than in others. If we
represent these lines of force (105) as elastic threads exposed to a
strain in the direction of their length, and to a repulsion in the
direction perpendicular to this, we shall have a very definite idea
of the relative motion which they tend to produce.
482. APPARENT REPULSION OF Two CONSECUTIVE ELEMENTS
OF CURRENT. This important experiment of Ampere consists in
putting the two poles of a battery in connection with two rectangular
troughs containing mercury, and separated by an insulating division.
A copper wire is bent so as to form two horizontal legs floating
on the mercury, and a cross piece (in the form of a bridge) which
connects the two former. When the battery circuit is closed, the wire
is seen to glide along the surface of the mercury, and to recede from
the points by which the current enters.
ELECTROMAGNETIC ROTATION 7 ".
467
Ampere thought this a proof that the two elements of current
directed along the same right line, and in the same direction, repel
each other, as the elementary formula indicates; but it is easy to see
that the interpretation of the phenomena does not entail this conse-
quence.
In this experiment the current traverses a circuit, one of whose
portions is movable, and the surface of which tends to become a
maximum (455). This result may moreover be arrived at directly
by replacing the current by a flexible shell bent upon itself, as
shown in Fig. in. The three shells superposed in the space
ABB'A' do not give rise to any force among them parallel to the
plane of the current ; but their external action is equivalent to that
of a simple shell. The portion aDC& tends to recede, and the shell
extends so as to occupy the greatest surface.
Fig. in.
483. ELECTROMAGNETIC ROTATION. ist. Barlow's Wheel
A toothed metal wheel, movable about a horizontal axis, is arranged
so that one or more teeth plunge with their lower ends in a cup
containing mercury. If the system is traversed by a current
which enters by the axis and leaves by the mercury, the only
action of the current on itself will tend to move the bottom
teeth in a direction which displaces them from the rest of the
circuit so as to increase the total surface; but this action is
generally too weak to overcome friction. A stronger effect is ob-
tained by putting the trough between the limbs of a horseshpe
magnet arranged horizontally. The lines of magnetic force then
traverse the plane of the wheel ; if they are directed from back
to front that is, with the north pole in front the rotation will
be in the opposite direction of the hands of a watch.
In order to get a phenomenon easy to calculate, replace the
magnet by a uniform magnetic field of intensity F, parallel to the
axis of the rotation. Let a be the radius of the wheel, 6 the angle of
H H 2
4 68
PARTICULAR CASES.
two consecutive teeth, and suppose the surface of the mercury
placed so that one of the teeth touches the liquid at the moment the
preceding one quits it. The flow of force through the triangle
formed by the radius, which corresponds to these two teeth, is
F , or sensibly F ; that is to say, the product of the force
2 aW
by the surface of the sector, and the corresponding work is IF .
For an entire turn the work is IFS that is to say, proportional to
the whole surface S of the wheel.
484. 2nd. Ampere's Experiment. This experiment, in which
the rotation of a magnet is produced by a current, is the converse
of that, the theory of which has been given above (456). The
apparatus is arranged so that only one of the poles of the magnet
can traverse the current ; a continuous rotation is obtained in this
way. The magnet (Fig. 112), loaded by a counterpoise of platinum,
floats on the mercury, and can rotate about itself on its own axis ;
the current is brought to the surface of the liquid, traverses the
projecting part of the magnet, and emerges by a fixed conductor
Fig. 112.
which dips in a drop of mercury in the top N. If we suppose that
the current goes rigorously along the axis of the magnet, the work at
each turn for a magnetic mass outside the axis is 47r;;/I, and gives
rise to a couple the moment of which is 2ml. The phenomenon is
really more complicated, because the current traverses the whole
section of the magnet.
Faraday repeated the experiment by placing the magnet outside
the circuit. The magnet is brought to the centre of the vessel by a
metal rod, and the magnet floats in an eccentric position.
In both cases, if the current ascends by the axis, and the top of
ROTATION OF LIQUIDS. 469
the magnet is a north pole, the rotation is opposite to the motion of
the hands of a watch. Faraday's arrangement gives greater friction,
and the rotation is less rapid.
485. ROTATION OF LIQUIDS. When the current traverses a
liquid, the liquid filaments, which coincide with the lines of electrical
flow, may be considered as movable circuits, capable of obeying
electromagnetic actions, and experiment shows that the liquid is
moved along with the current which it carries.
i st. Davy's Experiment. Two platinum electrodes just pro-
ject a very little below the top of the mercury. If the N pole
of a magnet is placed over one of them the negative electrode,
for example a depression of the mercury is observed ; and, at
the same time, a rotation in the same direction as the hands of
a watch.
2nd. M. Jamiris Experiment. The two electrodes of a volta-
meter are placed in the same vertical line, and on the axis
of the poles of a horseshoe magnet. If the liquid molecules in
a filament of the current formed a rigid thread, we should be
in the same condition as in Faraday's experiment, in which
rotation is impossible. The electromagnetic forces really act inde-
pendently, and in the same manner as in Davy's experiment, on
the portions of the filaments which diverge as they start from
each of the electrodes. The liquid divides them into two super-
posed layers which rotate in contrary directions, and the rotation
is made visible by the bubbles of gas which result from the
decomposition of water.
3rd. M. Bertiris Experiment. In M. Bertin's experiments the
movement of the liquid is made visible by small pieces of cork
which float on the surface. The liquid is in an annular dish
containing two rings of metal, one inside the other. If these
circles are electrodes, a series ot either centripetal or centrifrigal
radiating currents is obtained in the liquid. When a magnet is
placed in the axis of the current, the liquid acquires a rotation
in a definite direction in agreement with theory. The direction
of the rotation is not altered, if the central magnet is replaced
by a magnetised tube encircling the dish. For if the north
pole is at the top in the two cases, the flow of magnetic force is
diverted downwards, both inside and outside the hollow magnet.
This is not the case when the magnet is replaced by a coil :
the rotation of the liquid changes its direction according as the
coil is inside or outside the dish, and each of the lines of force
constitutes a closed circuit.
47 PARTICULAR CASES.
486. ELECTRODYNAMIC ROTATION. Consider an unlimited
rectilinear current X'X of strength I (Fig. 113), and a finite recti-
linear current of length a, and strength I, perpendicular to the first
A
a.
Fig. 113-
and in the same plane. If we give the current a a displacement
doc parallel to the current I, then if r Q is the distance BC, the
corresponding work will be
f r o
\.'dx\
J''o
1 a
=
r
The force which acts on the movable current is perpendicular
to its direction, parallel to the unlimited current, and its value is
2117. ( i + ). This current will be carried parallel to itself by a
\ r o/
constant force, and will ascend or descend the unlimited current
according as it is ascending or descending in reference to this latter.
The experiment is ordinarily made by causing a circular current
to act on a portion of a current movable about an axis perpendicular
to its plane and passing through its centre. The movable current
then rotates in the opposite direction to the principal current.
If the movable current is closed, or if its ends are on the axis,
there is evidently no movement, for then each line of force meets
the edge twice.
487. ACTION OF A UNIFORM FIELD. Consider first two un-
limited rectilinear conductors AA', BB' (Fig. 114) parallel to each
other at the distance b, and let us suppose that, while the two ends
are in communication with the poles of a battery, the circuit is
closed by a cross bar CC', movable parallel to itself along the
conductors A A' and BB'.
Let Z be the component of the intensity of the field perpen-
dicular to the plane of the conductors ; for a displacement dx of
ACTION OF A UNIFORM FIELD. 471
the movable bar, the variation in the flow of magnetic force in the
circuit is bUx. In the case of the terrestrial field, and if the rails
are horizontal, the component Z is directed downwards in our hemi-
sphere, and with the direction of the current shown by the arrows,
the movable bridge CC' will recede from AB under the influence of
electromagnetic forces ; it will approach, if the current is in the
opposite direction.
In Ampere's experiment (482) the action of the current on itself
tends to increase the surface, and to repel the movable bridge.
This action and that of the Earth will add themselves or oppose
according to the direction of the current ; the motion of the wire is
more or less easy according to the case.
A"
y i
Fig. 114.
488. Suppose now that the movable conductor forms a closed
circuit ; let S be the surface of this circuit, if it is plane, or the
maximum projection of its surface on a plane, which we will call the
plane of the current. When such a circuit is movable in a uniform
magnetic field, like that of the earth, stable equilibrium corresponds
to the case in which the flow of force across the negative face of the
circuit is a maximum ; the plane of the circuit tends then to set at
right angles to the force. Under the influence of the earth this plane
will be perpendicular to the dipping needle ; the current will move
from east to west in the lower part. If F is the strength of the field,
I that of the current, the potential energy of the current in the
position of equilibrium is
Wj = - ISF ;
when the face of the current is turned upside down, it becomes
W mF
o JLO J7
472 PARTICULAR CASES.
The work done against electromagnetic forces in this operation
is then
If the current is made to turn about a vertical axis we need only
consider the horizontal component H of the field. The work of the
rotation of 180 about the axis from the position of equilibrium
is then
W'-alSH.
If the current turns about a horizontal axis parallel to the
magnetic meridian, the vertical component Z of the field alone
comes into play. For a rotation of 180 from the position of
equilibrium the work is still
W" = 2 ISZ.
The ratio of the works in the two latter cases,
W" Z
is equal to the tangent of the Inclination. Hence, if we could
measure these works, we could determine the elements of terrestrial
magnetism without having recourse to magnets.
If the axis of rotation is in the magnetic meridian, the total
work is null, when the current, which at first was in this plane,
comes back to it after having been turned through 180; the works,
which correspond to the two halves of the rotation, are therefore
equal and of contrary signs.
Lastly, the work would be null for any given rotation if the axis
of rotation were parallel to the direction of the field.
489. ASTATIC CIRCUITS. The work of any given displacement
is still null when the circuit comprises two closed curves, such that
their projections on any plane give two equal surfaces S and S',
surrounded by currents moving' in opposite directions. This is the
condition which movable currents must satisfy, which are arranged
so as not to be under the action of the earth ; what are called astatic
ASTATIC CIRCUITS.
473
currents. Figures 115, 116, and 117 give examples of currents
which realise these conditions.
If the two surfaces S and S' were not equal, the action of the
field would be proportional to their difference S - S'.
H- S'
Fig. 115-
B
Fig. 1 1 6.
Fig. 117.
490. ROTATION OF A CURRENT UNDER THE ACTION OF THE
EARTH. A portion of a current not closed, and movable about
an axis, takes in general a continuous rotatory motion under the
influence of terrestrial magnetism.
We observe, in the first place, that in a uniform magnetic field,
like the terrestrial field, we may always replace a current by its
projection on three rectangular planes ; this amounts, in fact, to
replacing the strength of the field by its three rectangular
components.
Consider any given current movable about an axis and determine
its projections on three planes, one perpendicular to its axis of
rotation, the two others passing through this axis, and such that
one of them is parallel to the direction of the field ; let S, S', and S"
be these three projections. The projection S" perpendicular to the
field will not produce any action. The action on the projection S'
will be purely directive ; the circuit of the current will be carried
along in such a manner that the surface S" is a maximum, and
presents its negative face to the force ; in our hemisphere the current
must be descending in the part turned towards the east. There
remains to be considered the projection S on the plane perpendicular
to the axis. If it is closed, and of a fixed shape, it undergoes no
action ; if part of it is movable, the component of the field parallel
to the axis will have a constant moment relatively to this axis, and
will produce a continuous rotation.
474 PARTICULAR CASES.
491. For instance, let the system be formed of a current OP of
length a (Fig. 118) movable about a vertical axis, one of whose ends
is on the axis of rotation and the other dips in a mercury cup.
The current enters the mercury at A, traverses the two parts ABP
and ACP in opposite directions, and regains the axis by the movable
part PO. Let I be the total strength of the current, x the strength
in the arc B, y in the arc C ; the current will evidently be equal to
1 in the movable portion PO,
Fig. i i 8.
The surface comprised by the horizontal projection S consists
of two parts, one ABPO presenting its negative face to the com-
ponent Z of the terrestrial action; the other, ACPO, its positive face.
The former tends to increase, the second to diminish, and for an
angular displacement of the radius PO, the total work is
-a*(x + y)ZO = -
2 2
This work is independent of the position of the conductor OP;
the force then is constant. The work corresponding to an entire
turn will be
If the current has a vertical projection S', the motion of rotation will
be modified by the directive action corresponding to this projection.
It is easy to see that according to the ratio of the two surfaces S and
S', the initial velocity, and the value of friction, the moment of the
directive action might preponderate over the moment of rotation,
ACTION OF TWO RECTANGULAR CIRCUITS. 475
and keep the apparatus in equilibrium in a position perpendicular to
the magnetic meridian. In the apparatus used for this experiment,
we take a movable current symmetrical in reference to the axis of
rotation. The projection S' is then null, and the couple of rotation,
which would impart to the system a uniformly accelerated rotation if
there were no friction, ultimately makes it rotate uniformly.
492. ACTION OF Two RECTANGULAR CIRCUITS. We may cal-
culate the action of two rectangular frames AC and A'C', the sides
of which are parallel. Suppose, for the sake of simplicity, that
the frames are equal (Fig. 119), and their corresponding summits
Fig. 119.
A and A' on a perpendicular to their plane. The mutual energy
of the two circuits, with currents equal to unity, is expressed on
Neumann's formula (352) by
W= 1 I -dsds'.
flf
The value of cos e is equal to unity for two parallel sides, and null
for two perpendicular sides such as AB and B'C'. The energy thus
becomes
This expression only contains terms relating to parallel wires.
Consider, in the first place, the two sides AB and A'B', of length
a, and at the distance //. Let ds and ds' be two elements, placed
respectively at M and M' and r their distance ; lastly, suppose that
we measure the lengths s and s f from the points A and A'. From
the ratio
47^ PARTICULAR CASES.
we get for the first integration, in which the distance s is taken
constant,
or
/
The second integration relative to ds is easily effected, for we
have in general
l>(-u + ^WT^ 2 )du = ul\-u + ^W^ 2
we get then
p *-*W(^g+g
Jo -*+^ 2 +^ 2
Changing the sign of this expression and replacing /$ by the
distance # of the sides AB and CD', we get in like manner the term
relating to this last side. If the rectangle is a square, h' = ,Ja 2 + h 2 ,
and the two terms of the energy corresponding to the side AB
give
The total energy is then
_ + ,
W J I n-f , o TTi 7
When the distance of the frame is altered by dh, the variation
of the energy ^W, is equal to the work -Mfc of the force F,
PROPERTIES OF CIRCULAR CURRENTS. 477
considered as attractive, which is exerted between the two circuits,
and we have
dh '
We thus obtain, all reductions being made,
If the strengths of the currents in the two frames are respectively
I and I', the expression for the mutual action is
where P is the sum of the terms in the brackets.
493. PROPERTIES OF CIRCULAR CURRENTS. The potential of
a circular current is equal, within a constant, to that of a shell of the
same strength and the same contour. We have given above (368)
the expression of this potential for any given point. If the point is
on an axis at a distance x from the centre, it is sufficient in equation
(16) to make p = 0; replacing < by I, we get
V=27Tl
from which is deduced
a 2 IS
denoting by S the surface of the circle. For a point on the axis the
force is inversely as the cube of the distance to the contour.
This force is a maximum at the centre of the circle ; we
have then
IS I IL
L being the length of the circumference.
47$ PARTICULAR CASES.
This latter result would follow directly from a consideration of
the equivalent shell. Let 2h be the thickness of the shell supposed
to be plane, then denoting by l a the intensity of magnetisation,
The value of the action of the two terminal layers on a point
at the centre is (322)
and the magnetic induction is
We may now reject the shell outside the point in question,
without changing the value of the force (451).
494. ELECTROMAGNETIC SOLENOID. Ampere gave the name
solenoid to a system of equal circular currents, infinitely near, and
infinitely close, equidistant and perpendicular to any given curve
passing through their centre, which is called the directrix.
Let dS be the common surface of the elementary currents, h
their distance, and I the strength of the current. Each elementary
current may be replaced by a magnetic shell of the same magnitude,
of thickness h, and surface density a-, such that we have
As the surfaces in contact of all these shells have equal and
opposite charges, they neutralise each other except at the ends, and
the system is identical with that of a solenoidal filament The
external action reduces then to that of two magnetic masses M
placed at the ends. If / be the length of the solenoid, n the total
number of elementary currents, and n : the number of these currents
in unit length, we have
495. CYLINDRICAL COIL. Let us suppose that a cylinder is
covered with equidistant currents perpendicular to the axis. The
system of these currents forms a kind of cylindrical solenoid, of
CYLINDRICAL COIL. 479
finite transversal dimensions ; it is approximately realised by winding
a wire in the form of a helix on the surface of the cylinder. Each
element of the helix may be replaced by its projections on the
axis and on a plane perpendicular to the axis. If the section of
the cylinder is small, we sensibly destroy the effect of the former by
bending the wire back in a contrary direction parallel to the axis.
Whatever be the diameter, if the individual turns are sufficiently
near, and the coil consists of an equal number of layers in which the
inclination of the windings is alternately in opposite directions, the effect
of the projections on the axis is still sensibly zero, and the external
action differs very little from that of the perpendicular projections.
The system of the currents perpendicular to the axis is equivalent
to a solenoidal magnet of the same form ; we may, in fact, replace
each of them by a shell, and decompose the system into an infinity
of parallel solenoids, each of which is equivalent to a solenoidal
filament.
The action of the system on points outside the cylinder, reduces
then to that of two equal and opposite layers spread uniformly on
the bases, and the density cr of which is n-J..
For internal points the force is equal to the induction of the
equivalent magnetic system. If the cylinder is so long that the
action of the ends may be neglected in part of its extent, the lines of
force are parallel to the axis of the cylinder ; the field is uniform and
its strength is
The flow of induction across the section of the cylinder is
this flow is in the opposite direction to the internal flow from the
bases of the equivalent magnet.
It is, moreover, evident that a coil is not equivalent to a hollow
magnet ; in the hollow magnet all the lines of force, both internal and
external, start from the positive surface, and are absorbed at the
negative surface ; in coils, on the contrary, the internal lines of force
are the continuation of the external lines of force, and form closed
curves which never terminate at magnetic masses.
496. ANNULAR COIL. Suppose a ring to be covered by equal
currents equidistant from each other, and each situate in a plane
passing through the axis ; the system may be decomposed into a
series of solenoids, and it is equivalent to a solenoidal magnet of the
same form (411).
480 PARTICULAR CASES.
All the elementary solenoids comprise then the same number of
currents with the same intensity I, but of different lengths. If n^
is the number of windings comprised between two meridian planes
which make with each other an angle equal to unity, and if x is the
radius of an elementary solenoid, the distance of the successive turns
will be ; the intensity of magnetisation of the equivalent magnetic
filament is then 2L , and the induction, or the magnetic force, ^ HI .
x x
The value of the flow of induction across a surface S, taken in the
/ 7Q
meridional section of the ring, is 473-^1 .
J x
In the case in which the ring is a circular torus (372), we have
--WR
x
The total flow across the section is then
497. CASE OF ANY GIVEN SURFACE. Let us now consider the
general case in which any surface J5f is covered by plane currents of
the same strength, parallel to each other, and at such a distance that
these are n^ in unit length. These currents may be replaced by a
series of parallel solenoids terminating in the surface, and these
solenoids themselves by equivalent magnetic filaments ; in this way
a uniform magnet will be formed, the intensity of magnetisation
of which is n^, and the density at each point of the surface, has
the value n-J. cos 0, where 6 is the angle which the perpendicular to
the surface, at the point in question, makes with a perpendicular to
the plane of the currents.
The internal action of these currents is equal to the induction of
the equivalent magnetic system. In the case of the sphere (355) it
o
is constant and equal to -irn-^L ; the value of the flow of induction
across the great circle perpendicular to the line of- the poles on the
common axis of the currents is
where L is the circumference of the great circle.
AMPERE'S THEORY OF MAGNETISM. 481
The internal field would also be uniform in the case of an
ellipsoid (356).
From this we arrive at a new way of regarding terrestrial
magnetism ; the magnetic action of the earth is equivalent to that
of a series of circular currents situate in equidistant planes perpen-
dicular to the magnetic axis, these currents circulating from east
to west.
498. AMPERE'S THEORY OF MAGNETISM. We see that it is
possible, by means of currents situate in parallel planes, to realise
a system equivalent to a uniform magnet, which has the same
external surface ; the two systems are equivalent for all external
points, and produce the same induction in the interior.
Any given magnet may, in like manner, be replaced by a system
of superficial currents, in so far, at least, as the external action is
concerned.
This action, in fact (315), is equivalent to that of a layer of
total mass null, distributed on the surface. If o- be the density
of the layer at a point, F H and ' n the perpendicular components,
measured from the surface, of the actions which it exerts outwards
as well as inwards, we have (38)
Let us consider the internal potential V of the layer, and the
equipotential surfaces to which the force F' is perpendicular, and
suppose that on each of these surfaces we place equal and opposite
magnetic layers, the density of which, at each point, is determined
by the condition
The external action of this system of surfaces is null. We
observe now that the product cr'dri = -- dV is constant between
4 7T
two equipotential surfaces. If then we connect the negative layer
of the surface Jjf, where the potential is V, with the positive layer of
the following surface ^ 5 at the potential V + dV, we form a shell, the
magnetic power of which, o-'dn', is constant. A current, of the same
strength, which followed on the surface, the curve formed by the in-
tersection of the shell, would have the same action on the outside ;
we could proceed in the same way with all other shells. But in
i i
482 PARTICULAR CASES.
forming this shell a negative corona has been left corresponding to
the difference 2? - J5f' of the two surfaces, and the sum of the external
actions of these corona is equal and of opposite sign to that of the
superficial currents. If d^> and dS are the two corresponding
elements of this corona, and of the surface of the magnet deter-
mined by a tube of force, we have
which gives
<r'd2= dS.
4 7T
The quantity of magnetism contained in the element d^, is
therefore the same as that which the element dS would contain
K
with a density equal to --- - .
4?r
It follows from this that the external action of all the surface
currents defined by the preceding shells, is equivalent to that of a
Fl
layer of total mass zero, the density of which at every point is H -- .
4?r
By continuing the equipotential surfaces, which correspond to the
external potential V, by continuous surfaces, we shall see in like
F n
manner that a layer whose density at each point is + - , may be
47T
replaced by a system of surface currents.
The combination of the two systems of currents is a new system
of surface currents having the same external action as a layer whose
density is (F n + F' n ), that is to say, the density a- of the magnetic
4?r
layer, which is equivalent to the proposed magnet.
The external action of any magnet may accordingly be replaced
by that of a system of surface currents ; but this equivalence does
not hold for points in the interior.
It is possible, however, to obtain an exact representation of all
the magnetic effects, both on the inside and on the outside, if we
replace the magnetic filaments into which any given magnet may be
decomposed (313), by the corresponding electromagnetic solenoids.
Ampere assumes that these elementary solenoids are formed of
real currents circulating in the interior of molecules ; these currents
never pass from one molecule to another, and they exist before
magnetisation, the only effect of which is to direct them.
MAGNETISATION BY CURRENTS. 483
On this hypothesis a magnet is no longer a continuous substance,
but an assemblage of distinct molecules, giving rise to a very com-
plicated distribution of the magnetic force and of the potential. But
a material simplification results, for the force and the magnetic
induction are then defined in the same way, and the force satisfies
Laplace's equation, both inside and outside the magnet.
Ampere's hypothesis raises, however, a difficulty of prime im-
portance, for it cannot be conceived that currents can permanently
exist without the disengagement of heat, and therefore without a
continued expenditure of energy. But, if the currents are supposed
to be localised in the molecules themselves, the constitution of which
is unknown, it is not impossible to assume that the resistance may be
zero, and that the currents exist in a form which cannot be experi-
mentally attacked. The assumption does not therefore necessarily
imply a contradiction.
499. MAGNETISATION BY CURRENTS. Magnetisation by currents
was discovered by Arago in 1820. He had observed that a copper
wire traversed by a current attracts iron filings; every particle of
filing becoming a small magnet, places itself at right angles to the
wire the north pole on the left of the current, as in CErsted's
experiment. Ampere observed that the action of the current on a
bar of soft iron or steel could be greatly increased by coiling the wire
round the bar ; in this way, and particularly with soft iron, temporary
magnets are obtained which are called electromagnets.
Electromagnets may acquire a far greater power than that of even
the best steel bars ; but their principal characteristic is that of almost
instantaneously gaining or losing their magnetic properties. They
have further this curious property, that by suitably coiling the wire
on the soft iron core, the magnetism may be distributed at pleasure,
and any number of poles, or of consequent points^ may be obtained
on the bar.
The calculation of the effects of electromagnets is in general very
difficult, even when the currents which surround the core are equi-
distant and parallel. In this case the system of currents develops an
internal field, the strength of which is defined by the induction of a
uniform magnet bounded by the same surface ; the soft iron core
placed in this field acquires a magnetisation at each point which
depends on the strength of the field, and also on the magnetism de-
veloped by induction on the body itself. The magnetism can only be
uniform then in the particular cases which we have examined (385).
The external action is the resultant of that of the system of currents,
or of the equivalent uniform magnet, and of that of the soft iron core.
I I 2
484 PARTICULAR CASES.
500. EXAMPLES. For a sphere surrounded by parallel and
equidistant currents, the value of the strength of the field < produced
o
by the currents (497) is - irn-J..
The intensity of magnetisation is then (385)
3/*- I ^_^-% I
-?ji,
and the induction
, T
4>= -- *,!,
^ ]
If the sphere is placed in a cylindrical coil which is so long that
the effect of the ends may be neglected, it will also acquire a uniform
magnetisation.
This would also be the case with an ellipsoid, and also with a
sufficient approximation for a cylinder, the axis of which would
coincide with that of the coil.
501. In the case of a cylindrical coil of great length (495), the
strength of the field in the interior is 47^! ; the value of the intensity
of magnetisation of a long cylinder parallel to the field would be
k$ (292) or 47rvfc# 1 I, and the internal induction F l will be 47n<, or
I 6w 2 # 1 I.
If S is the section of the bar, the flow of magnetic induction
across it is
F 1 S = 1 67^18,
and the total flow, including in it the induction 471-^18 of the current,
has the value
Q = 471-;^ (i + 47r/) IS.
This value can be experimentally investigated, and we might deduce
from it the coefficient of magnetisation k.
502. The determination of this coefficient is still more accurate
by means of a piece of soft iron in the form of a torus, which is
surrounded by equidistant currents (496). In this case the
1
strength of the field produced by the currents is - - at each
DETERMINATION OF THE COEFFICIENT OF MAGNETISATION. 485
point. This being the only effective force, the value of the intensity
of magnetisation at the point in question is
The induction is equal to ^-rrkfa so that the total flow of induction
across the section S of the soft iron, comprising still that which arises
from the currents, is
Q =
/d S
*'
If the soft iron does not occupy the whole of the space bounded
by the currents, but merely a portion S' of the section S, the total
flow of induction, across the surface of the currents, is
T f (VS , (VS'1
= 4 7r 1 I J + 4**J
Suppose, for instance, that the section of the iron is a circle of
radius a! concentric with the circular section of a torus ; the total
flow of induction in the torus will be
Q' = 4*^1 [~R ~
-* +
If the section of the coil were a rectangle of height b parallel to
the axis of revolution, and of the thickness 20, the mean radius
being R, we should have
x R-a
IR-a
The iron ring, in like manner, might have a rectangular section
of height b\ of thickness 20', and of mean radius R'; the total flow
of induction would then be
We shall see further on (559) the use which can be made of
these various formulae.
486 PARTICULAR CASES.
503. MEASUREMENT OF CURRENTS. GALVANOMETERS. The
strength of currents is usually measured by the electromagnetic or
electrodynamic actions which they exert, and the instruments which
are used for this purpose are called galvanometers, or electrodyna-
mometers, according as they depend on one or the other of the two
actions.
A galvanometer consists of a magnetised needle, or of any
magnetic system on which a conductor traversed by a current is
made to act; the effect produced is measured by means of an
antagonistic force, such as the torsion of a metal wire, or of a bifilar
suspension, or by the action of an external magnetic field.
Let us consider the simple case of a horizontal magnetic needle
suspended by a wire without appreciable torsion, and placed in the
centre of a frame on which is coiled a wire forming a series of
parallel turns.
If the turns are parallel to the magnetic meridian, and they are
traversed by a current, they produce a magnetic field, the strength of
which is proportional to the strength of the current, and which
may be represented by GI. The horizontal component of the
terrestrial field at this point being H, the horizontal component of
the field is ^/G' 2 ! 2 + H 2 , and its direction makes an angle 8 with the
C*T
magnetic meridian, the tangent of which is equal to .
An infinitely small needle placed at this point, and which at
first was in equilibrium in the plane of the needle, will be deflected
through an angle <5, and from it we may deduce the strength of the
current by the expression
TT
This formula is only exact provided the magnetic field is uniform
throughout the whole space which the needle occupies. When the
needle has a length which is considerable in reference to the
dimensions of the frame, the intensity of the field is not constant,
and the formula for the deflection is less simple. In that case, by
an empirical graduation, we could determine the ratio which exists
between the strength of the current and the deflection produced.
The magnetic moment of the needle has no influence on its
position of equilibrium ; it has no other effect than that of modifying
the strength of the forces, and therefore the duration of the
oscillations of the needle.
TANGENT GALVANOMETER. 487
In order to increase the sensitiveness of the galvanometer that is,
the deflection 8 for a given current we must increase the value of G,
and diminish that of H. The value of G is increased by increasing
the number of turns by Schweigger's method, and by placing them as
near the needle as possible. In order to diminish H, a magnet is
placed at a certain distance, which produces at the centre of the
frame a magnetic field parallel, and in the opposite direction to, that
of the Earth.
Use is sometimes made of a quasi-astatic system of two needles
(299), one of which is inside the frame and the other is outside ;
the action of the Earth on the movable system is then far feebler
without there being any appreciable modification in the action of the
current, which is exerted more particularly on the inner needle. We
may also use two frames, each having one of its needles in the centre,
and pass the current in opposite directions, so that the actions exerted
on the two needles are concordant.
504. TANGENT GALVANOMETER. In order to determine the
absolute value of the strength of a current, besides knowing the
component H of the terrestrial magnetism, we must also know the
constant G of the galvanometer. The name tangent galvanometer \s>
given to a galvanometer, the wire of which has been coiled in such a
manner that this coefficient may be calculated from the dimensions
of the wire and the shape of the frame.
If on the frame a wire L is coiled on a circle of radius a in such
a manner as to make n turns, and if the needle, which is supposed to
be infinitely small, is placed at a point of the axis at a distance u
from the circumference, we shall have (493)
2TTa 2 La A7T 2 ?
G = n = = -
u* fc 3 L
which gives
I = --r- tan 8 = / - } tan 8 .
La
The distance u is equal to a, when the needle is at the centre
of the circle.
If the length is to be taken into account, we must estimate the
strength of the field outside the axis of the currents by the formulas
of (368).
The formula of the tangent galvanometer would be exact and
independent of the length of the needle if the field of the current
were uniform. This would be the case, for instance, with a
488 PARTICULAR CASES.
cylindrical coil (495) or a spherical coil with equidistant currents
(497). If n^ is the number of turns for unit length, we have
o
G = 47r^j in the first case, and in the second G = - Tm v
O
505. ELECTRODYNAMOMETERS. In an electrodynamometer we
measure directly the action exerted between two circuits, one
fixed and the other movable, traversed by the same or by different
currents. Suppose, for instance, that the magnet of a tangent
galvanometer is replaced by a small coil, through which a current
could be passed by a bifilar suspension, and which is in equilibrium
when the axis of the coil is in the magnetic meridian. If a current
I is passed through the wire on the frame of the galvanometer and
a current I' in the coil, the latter is displaced, and by a suitable
torsion a of the suspension, it is restored to its original position.
The magnetic moment of the movable coil is proportional to I',
and may be represented by S* ; the couple produced by the action
of the frame is then GSII'. As the couple of torsion of the bifilar
is proportional to the sine of the angle, if T is the moment of the
couple which corresponds to an angle of torsion equal to ,
GSII' = Tsina.
If the two wires are traversed by the same current I, the expression
becomes
GS
Hence we might determine the strength of the current in
absolute measure if we knew the constants T, S, and G, or we might
leave these constants undetermined, and use the apparatus as an
instrument of comparison. This is the principle of Weber's
experiments.
If we suppose that the current traverses the parallel rectangular
frames (492), as in Cabin's experiments, the intensity might be
deduced from the attractive or repulsive action exerted between the
two circuits.
506. MEASUREMENT OF DISCHARGES. When the duration
of the current is so short that the needle has no time to undergo
an appreciable displacement before the current stops, it has
nevertheleless received an impulse or throw, and acquired a
certain velocity; it is impelled from its position of equilibrium,
and returns to it after a series of oscillations. This is the case,
MEASUREMENT OF DISCHARGES. 489
for instance, of the discharge of a condenser through a conducting
wire in which is a galvanometer; the total quantity of electricity
may be deduced from the angle of throw imparted to the needle.
The strength of a permanent current in the galvanometer in
question* is given by an expression of the form
in which f (B) reduces to the angle 8 when the deflections are very
small. If p, be the magnetic moment of the needle, the action
of the current on the needle produces a couple, the moment of
which is /xGI.
We know, on the other hand, that when a body is movable
about an axis, the product of the moment of inertia K, by the
angular velocity is equal to the moment of the resultant couple
in reference to the axis of rotation. Hence, since the deflection
during the discharge is so small, that the action of the Earth can
be neglected, we have for the needle in question,
v d{ * <-T
K =/xGI.
dt
If dm is the quantity of electricity which flows in unit time
dt, this equation becomes
da) dm
K = MG __
dt dt
From which, if w o is the initial angular velocity, and m the total
discharge, we get
Kw o = /xGw.
The needle, once impelled with this velocity co o , has a vis
K.CO*,
viva equal to - L and it stops at an angle 0, when it has done
work of the same value against the action of the terrestial field.
We have then
TT- a f\
= H/x(i - cos 6) = 2H/x sin 2 - ,
or
n 2 -,
i /HK
m = L \ 2 sm - .
G \ /A 2
49 PARTICULAR CASES.
If the deflections are small enough, we may simply take
i /HK H /~K~
= cV^ eW^'
=
It appears thus that the angle of throw 6 is proportional to
the quantity of electricity which flows during the discharge, and
this law of proportionality will be sufficient for all comparative
experiments.
To determine m in absolute value we must know the constant
G of the galvanometer, and the quantities which come under
the root.
It may be observed that if the needle is left to itself under
the influence of terrestrial magnetism, the time T of infinitely
small oscillations is
from which follows
HT
m = - .
g 7T
As a matter of fact, the true angle of throw is diminished
by the resistance of the medium, and by the induction currents
which the motion of the needle produces in the wire. But if
the oscillations do not diminish very rapidly, this effect is allowed
for by adding to the angle 0, a quarter of the excess of this
deflection over the deflection produced on the same side by the
succeeding oscillation. We shall have, finally,
m =
FARADAY'S DISCOVERY. 491
CHAPTER IV.
INDUCTION.
507. FARADAY'S DISCOVERY. The electromagnetic actions
studied in the preceding chapters are purely mechanical ; they are
exerted on conductors traversed by currents, and correspond to a
permanent condition of currents, and of the magnets near them. In
all cases in which the systems experienced relative displacements, we
have implicitly assumed that those displacements had no influence
on the electric condition of the conductors. Faraday discovered
in 1831 a class of phenomena of a totally different kind which
corresponds to the variable condition of the system ; these
phenomena, which he comprised under the term induction^ are
of an electrical character, and are manifested by the production of
temporary currents in conductors.
The currents which are formed are called induced currents ; the
induced circuit is that submitted to induction ; the term inductor is
applied to the current, the variation in which has been the cause of
the induced current.
508. The phenomena discovered by Faraday may be classed
under several heads :
i st. A closed circuit becomes the seat of a temporary current
whenever a magnet is displaced near it ; or if the magnetisation is
varied ; or still more generally when the magnetic field is modified
in which the circuit is placed. This is magnetoelectrical induction.
2nd. Analogous effects are obtained by substituting a system of
currents for the magnetic system. The circuit in question is
traversed by an induced current whenever the distance, strength, or
form of the external current is altered. The effect is the same as
that which would produce the corresponding modification of the
equivalent magnetic system or the current. This is electro dynamic
or voltaic induction.
3rd. The change of form or of relative position of a closed
49 2 INDUCTION.
circuit, in reference to the magnetic field of a system of magnets or
of currents, is ordinarily sufficient to give rise to an induced current
in this circuit, which comes under one of the preceding heads.
4th. Finally, the mere fact of altering in any way the strength of
the current in a circuit, even when it is withdrawn from any external
action, produces an induction current in this circuit which adds itself
to the principal current, and always tends to counteract the change
of strength which it experiences ; it is a current of self-induction
or an extra-current.
509. Experiment has established the following general facts in
reference to induction currents :
i st. Whatever be the kind of variation which gives rise to an
induction current, two equal variations in opposite directions always
give rise to equal and opposite currents.
2nd. The duration of the induced current is equal to that of the
variation of the inducing system.
3rd. The quantity of electricity set in motion in the induced
current by any operation is independent of the duration of the
variation, and therefore of that of the induced current itself.
4th. Lastly, the nature of the conductor in which the induction
currents are transmitted is only of importance in so far as it affects
the resistance which it brings into the circuit.
510. Examining the various circumstances in which induction
currents are produced, it is easily seen that their common charac-
teristic is that of corresponding to a variation in the flow of magnetic
force which traverses the induced circuit. This is evident for all
the phenomena of the relative displacement of currents or of
magnets; experiment shows, moreover, that any displacement, or
any deformation of the induced circuit which does not modify the
value of the flow which traverses it, never produces induced currents.
This is also the case with the extra current. For a current gives
rise to a magnetic field, and therefore to a flow of force in the
surface of the circuit which it traverses. It is easily seen that
any change of intensity, or of shape, which modifies this flow, may
produce an effect analogous to that which would be produced by the
displacement of an external magnet, giving rise to the same variation.
We are thus led to define the phenomena of induction in the
following manner :
When the flow of magnetic force which traverses a closed circuit is
in any way modified, this circuit becomes the seat of a temporary
current, the duration of which is equal to that of the variation of
the flow.
LENZ'S LAW. 493
This enunciation defines the conditions in which induced
currents are produced. It remains for us to establish the direc-
tion and the magnitude.
511. LENZ'S LAW. A short time after Faraday's discovery,
Lenz enunciated the following law, which establishes a connection
between the induction produced by the displacement of the inducing
system, and the electromagnetic work as defined by Ampere's
formula :
Any displacement of the relative positions of a dosed circuit, and of
a current or magnet, develops an induced current, the direction of which
is such as would tend to oppose the motion.
512. NEUMANN'S THEOREM. Lenz's law, which is of great
practical utility, merely gives the direction of the induced current,
but not the intensity. Assuming, as an experimental fact, that the
induction produced in a very short time is proportional to the velocity
with which the conductor moves, Neumann has given a complete
theory of the induction currents produced in a movable linear
conductor in the presence of any magnetic system. He has thus
demonstrated this theorem, which we shall afterwards meet with
under a more general form :
The electromotive force of induction is equal to the work which
would be done in unit time by the magnetic system, if the intensity of
the current in the induced circuit was equal to unity.
513. THEORY OF HELMHOLTZ AND THOMSON. The existence
of phenomena of induction may be considered as a necessary
consequence of the conservation of energy combined with the
electromagnetic law of Ampere and the law of Joule. This
proposition was first put forth in 1847 by Prof. Helmholtz in his
celebrated memoir on the Conservation of force. Sir W. Thomson
arrived independently at the same conclusions.
Consider an invariable magnetic system, in the vicinity of a fixed
conductor S, in communication with a battery. If the magnet is
stationary, the strength I of the permanent current is determined by
Ohm's law, and if E is the electromotive force of the battery and R
the resistance of the circuit,
(1) E-I.R.
Multiplying both sides by I <#, we get
(2) EI <#=
494 INDUCTION.
This equation expresses that during the time *#, the energy due
to the chemical actions is equal to the thermal energy expended in
the circuit on Joule's law.
Suppose now that instead of being stationary, the magnetic
system moves in accordance with electromagnetic actions. The
external work resulting from this displacement, can only be borrowed
from the sole source of energy in the system (that is, the chemical
action), and the preceding equation must be in default. On the
other hand, there is no reason for supposing that the laws of Faraday
and Joule cease to hold ; in other words, the weights of the bodies
combined in the different couples must still be proportional to the
strength of the current, and the thermal energy disengaged in the
circuit is equal to the product of the resistance into the square of
the strength. Hence, the strength of the current could not retain
its original value.
514. Suppose now that the magnet is displaced in such a way
that the new value of the strength remains constant. So long as
this condition is fulfilled, the excess of chemical work over the
thermal energy expended in the circuit in the time dt, serves to
produce the external work dT corresponding to the electromagnetic
forces. Hence, if I is the strength of the current,
(3) EL#=I 2 R<// + arr.
If Q is the flow of force due to the magnetic system which
traverses the circuit, entering by its negative face, we have
Replacing dT by this value in the equation (3), and dividing
by \dt, we get
(4) E = IR + f-
In order that the strength of the current shall be constant, the
dQ
displacement must take place so that the differential is itself
constant.
Putting I -! = /', equations (i) and (4) give
(5)
THEORY OF HELMHOLTZ AND THOMSON. 495
It will be seen that the differential plays the part of an
at
electromotive force acting in the contrary direction to E, and
capable of producing a current i in the contrary direction to the
principle current, so that the resultant current I still satisfies Ohm's
law, under the form
(6) E-g-IR.
The quantity ~= is called the electromotive force of induction ;
if is equal to the differential, in respect of time, of the flow of magnetic
force which traverses the circuit.
If the value of ^Q is positive that is to say, if the flow of
force increases, the electromotive force of induction diminishes
the strength of the current, and the work of the electromagnetic
forces is positive. If, on the contrary, the value of */Q is negative,
the magnet is displaced in resisting the electromagnetic forces, and
this operation introduces a fresh energy into the system ; the strength
of the current is then greater than in a state of rest.
515. The quantity dm or idt of electricity induced in the wire
is given by the equation
(7) . i^dt=Kdm = dQ.
The total quantity of electricity m corresponding to a finite dis-
placement, for which the flow of force passes from the value Q x to
the value Q 2 , is therefore
516. The establishment of the current in a circuit requires itself
work which we have not taken into account, and this work (to which
we shall subsequently revert) is a function "SP of the strength of the
current During the variable period, the energy of the chemical
action should also furnish the work d"*P which corresponds to an
increase d\ of strength. Equations (3) and (5) then become
(9)
496 INDUCTION.
We get from this, by a reasoning analogous to that which gave
equation (8),
Whatever be the law by which the magnetic system is displaced,
if the strength of the currents is the same at the two limits, the last
term of equation (8') is null. This is the case more particularly
if the limits are chosen before and after the motion, in which case
the two limiting values of the current are equal to I .
With these limitations we may enunciate in a general form the
theorem expressed by equation (8) :
The total quantity of electricity put in motion by any displacement of
a magnetic system, is equal to the quotient of the variation of the flow of
force corresponding to this displacement, by the resistance of the circuit.
517. The preceding results suggest some important remarks.
i st. It is seen, in the first place, that the electromotive force of
induction is of a kind which opposes the motion, for the original
intensity of the current is diminished or increased according as the
magnetic system obeys or resists electromagnetic actions ; this is
Lenz's law.
2nd. If the strength of the current were equal to unity, the
external work dT would correspond to the work in unit time.
The electromotive force of induction is equal to this work ; this
is Neumanris theorem.
3rd. The electromotive force of induction is independent of the
electromotive force E of the battery ; the induction is then the same
however feeble is the strength of the original current. It results
from this, that induction should also take place when the conductor
is neutral, provided it forms a closed circuit. It is in this form that
Faraday discovered induction currents.
It must, however, be observed that if the current was really null
throughout the entire extent of the conductor, the reciprocal action
of the magnet and of the circuit would also be null, and the
preceding considerations would not enable us to foresee the pro-
duction of induced currents. But it may be said that this perfect
neutrality is a state of unstable equilibrium, impossible to realise in
practice, and that an infinitely slight cause, a change of temperature
at any point of the circuit, or the displacement of an external
electrified body, even at a great distance, would be sufficient to
produce a current, however slight, in the conductor, and thus enable
induction to take place.
GENERAL LAW OF INDUCTION. 497
518. GENERAL LAW OF INDUCTION. The preceding reasoning
would apply in the same way, and in almost identical terms, to the
other cases of induction.
It is seen to be evident for electromagnetic induction that is to
say, that which is produced by the displacement of a system of
constant currents, substituted for the magnetic system for we
have demonstrated the complete equivalence of the magnetic fields
produced by currents and by magnets.
In the case of currents induced by a variation in the strength of a
magnet, or of an adjacent current, the result may be considered as
equivalent to that which would be obtained by bringing a magnet or
a current, identical with the variation in question, from an infinite
distance to superpose it on the former.
Experiment shows that extra-currents, produced by deformations
of the circuit itself, or by changes in the strength of the principal
current, are also connected, and in the same manner, with the corres-
ponding variations of the flow of magnetic force.
It may therefore be considered as a general rule thajt any
variation in the flow of force in a circuit, whatever be its origin,
corresponds to a variation of potential energy, and gives rise to the
same electromotive force of induction as if it were produced by the
displacement of an external magnetic system.
This conclusion appears necessary if, abandoning the idea of
actions at a distance, we regard the transmission of electrical and
magnetic forces as due to a modification' of the elastic properties
of the medium ; we can understand then that the only proximate
cause of currents induced in a conductor may be the state of the
medium in which is the conductor, whatever may be the origin of
the forces which are at work in this medium. We may, therefore,
formulate the general law of induction phenomena in the following
terms :
The total electromotive force developed in a circuit at a given time is
equal to the differential \ in regard to time, of the flow of magnetic force
across it.
Or again : The total quantity of electricity induced in a circuit, is
equal to the product of the inverse of its resistance by the total variation
of the flow of force across it.
The flow of force across a circuit at a given time consists of the
flow Q, arising from external bodies, magnets or currents, and of the
flow produced by the current which traverses the circuit itself. Let
L be the value of this latter flow when the intensity of the current
is equal to unity ; it will be equal to LI for strength I, and if E are
K K
498 INDUCTION.
the ordinary electromotive forces at work in the circuit, the general
equation of induction will be
(10) (E
or
519. COEFFICIENTS OF INDUCTION. If the inducing system is
a magnetic shell or a current, the flow Q is equal to the product of a
constant M, by the magnetic power of the shell or the strength of the
current. This constant is a function merely of the form and relative
position of the two circuits ; we know that it has the same value for
the two adjacent conductors (341), and that its value is defined
(353) by the integral
(n) M=- I I Ads'.
-IP
This factor M is called the coefficient of reciprocal induction, or
of mutual induction of the two circuits.
The constant L is an integral of the same form but with this
difference, that the two elements ds and ds' belong to the same
circuit. It is called the coefficient of self-induction.
The value of the coefficient of self-induction L is the limit
towards which M tends when two equal circuits traversed by currents
in the same direction, and of the same strength I, nearly coincide.
For the total flow, which at this instant traverses the system of the
two circuits, is equal to the sum of the flows produced by each of
them that is to say, to 2 LI ; it may also be considered as the sum
2 MI of the equal flows, each one of which starts from one of the
circuits and traverses the positive face of the other.
520. ELECTROMAGNETIC INDUCTION. The general formula,
applied to the case in which the inducing system is a shell of power
3>, gives
(12) (E
Suppose that while the magnetic power <3> is constant, the shell is
brought from an infinite distance to a determinate position in
presence of the induced circuit which we suppose fixed ; we have
then
ELECTROMAGNETIC INDUCTION. 499
If both sides of this equation are multiplied by I, and we
integrate from /=0 to the time t when the shell takes up its final
position, we get
ft ft / , \ t
(13)
The first member of this equation represents the excess of the
chemical energy, furnished by the battery during the time /, over the
energy which appears as heat in the circuit during the same time.
The first term of the second member is the total work of the
electromagnetic actions ; this work depends on the law of the
motion. We may imagine, for instance, that the shell may have
been approached very slowly, so that the induction is very feeble,
and the principal current differs very little from its initial value I ;
in this case, if M is the flow of force corresponding to the final
position of the shell, the electromagnetic work will be equal
to 3>MI .
This latter term represents the change in the potential energy of
the current ; it is zero if the two limiting values of the current are
the same that is, if the magnetic shell is in a state of rest in its
final position.
521. If the power of the magnetic shell, while still at rest,
was variable, the equation would be quite analogous :
(El - RI 2 X/= M I - dt+ ( L
and would lead to the same conclusions.
522. The most general case is that in which the three functions
M, < and L vary simultaneously that is to say, when the magnetic
shell changes its strength, its form, and relative position, and that
the circuit itself is deformed. Equation (12) gives therefore the
electromotive force of induction
LI) d& ,dU dl d"L
(14) e = --^ = M + $--- + L-- + I .
at at at dt dt
If the induced circuit contains no electromotive force inde-
pendent of induction, we need only make E = in the preceding
equations.
K K 2
500 INDUCTION.
523. ELECTRODYNAMIC INDUCTION. If the inducing system
were a constant current, we might replace it by the equivalent
magnetic shell, and thus bring it within the preceding case ; but in
consequence of reactions, the inducing circuit itself will be under
induction, and the strength of the current will no longer be constant.
If R' and L' are the resistance and the coefficient of self-induction
of the inducing circuit, and E' the electromotive force which it
contains, the strength of the current in the two circuits will be
determined at each instant by two simultaneous equations
OiLi/CQRY. (E
+LT).
The complete solution of these equations generally presents
great difficulties, and in the next chapter we shall investigate the
simplest cases in which it can be obtained; but the differential
equations already suggest some important remarks.
If we add these equations, after having multiplied the first by I
and the second by I', we get
(16) (El + ET - RI 2 - RT 2 )<# = L/(MI' + LI) + IV(MI + LT).
The left hand side represents the excess of the energy furnished
by the sources in the two circuits over the thermal energy expended
in the conductor.
The right hand side may be written as follows :
(17) - </(LI 2 + 2MII' + LT 2 ) + - IVL + Htf M + - I'VL';
it represents the total variation of the potential energy of the two
circuits, and the external work.
524, INTRINSIC ENERGY OF THE CURRENT. If the circuits
are fixed both in form and position, the factors L, M, and L' are
constants ; the portion of the energy not converted into heat is
expressed by
[~LI 2 LT 2 ~1
+Mir+
_2
INTRINSIC ENERGY OF THE CURRENT. 501
The term Mil' is the relative energy of the two currents ; it is
the work which would have been necessary to bring the circuits
traversed by the currents I and I' from an infinite distance to their
actual position. Each of the other terms within the parenthesis may
be called the intrinsic energy of the corresponding current; it is
equal to half the product of the coefficient of self-induction by the
square of the strength, and represents the cost of the work of creating
the actual current in each circuit (this not being subject to any foreign
action), or the external work which this current could develop if it
were left to itself, and vanished under the same conditions.
It may be observed that we may write
T T2 T 'T'2 T T
- + Mil' + - - = - (LI + MI') I + - (LT + MI) I'.
2 22 2
Each of the two terms on the right hand side represents the
potential energy of the corresponding circuit ; it is the work which
must be spent in each circuit when the field is brought from zero to
its actual state, and is therefore the work which it would produce if
all the currents were simultaneously annulled. In each of the
circuits this work is equal to half the product of the strength by the
flow of magnetic force which traverses it.
That part of the energy which we have been here considering
is in a form which it is not possible to define in the present state
of science. We cannot say, for instance, whether it is in the state
of ordinary potential energy, like the tension of an elastic body,
or of an actual energy consisting in the motion of a particular
fluid, or again in both of these at once ; nor further, whether it is
localised in the circuit traversed by the current, or diffused through
the whole medium, in accordance with the ideas of Faraday and of
Maxwell.
525. In the general case in which the factors L, M and L' are
variable, the first term of the expression (17) always represents the
variation in the potential energy of the two circuits ; the whole of
the other terms
1 IVL+IIVlI+-IVL f
2 2
represents the work done by the electrodynamic actions of the
conductors, in consequence of their changes of form, or of relative
position.
502 INDUCTION.
Suppose that the two circuits have an invariable shape, and that
they are so displaced that the two strengths I and I' retain constant
values, which are naturally different from the initial or final values ;
as the coefficient M is the only one which changes, the term for
the potential energy and that of the external work are both reduced
to the same value IIWM. We have then, at each moment,
(El + ET - RI 2 - RT V= 2IIVM.
It will thus be seen that the excess of the chemical energy
furnished by the pile over the energy expended in heat, is equal to
twice the external work expended in effecting the displacement;
half this energy is used in producing external work, the other in
increasing the potential energy of the system.
This remark, which is due to Sir W. Thomson, should be com-
pared with the analogous proposition relative to the displacement of
conductors at constant potentials.
ELECTROMAGNETIC RESISTANCE IS A VELOCITY. 503
CHAPTER V.
PARTICULAR CASES OF INDUCTION.
526. ELECTROMAGNETIC RESISTANCE is A VELOCITY. Consider
the case (Fig. 114) of a bar CC' sliding parallel to itself on two
parallel rails A A', BB', at the distance <, situate in a vertical plane
at right angles to the magnetic meridian, the ends of which are
connected by a metal conductor. Suppose that the direction of
the horizontal component of the terrestrial field is from back to
front. When the bridge CC' is moved away from AB, parallel to
itself, it is carried in the direction in which the action of the
Earth would urge it, if the circuit were traversed by a current
going from A to B by the bridge. This motion produces a
current of induction, which traverses the circuit in the opposite
direction that is, which goes from B to A by the bridge.
If we consider that the resistance of the rails may be neglected
in comparison with that of the conductor which joins the points
A and B, and if R is the resistance of the circuit which we suppose
to be unchanged, x l and x 2 the values of the distance AC in the
two successive positions of the bridge, and H the horizontal com-
ponent of the terrestrial magnetic field, the corresponding quantity
M of induced electricity will be given by the equation
RM = Q! - Q 2 = JH ( Xl - x 2 ) .
As the product d(x l -x 2 ) represents the area S described by
the bridge, it follows that
.
M
In this expression the factor H is the intensity of a magnetic
field that is to say, a force which is exerted upon unit mass;
504 PARTICULAR CASES OF INDUCTION.
hence, if r and / are two lengths, and m a magnetic mass,
On the other hand, the electrical mass M is the product by a
time of an intensity of current, or of the magnetic power of a
shell, and we may also write
M = \t = $t = h<rt = h >
m' denoting a magnetic mass of suitable magnitude, and h and /'
lengths. We shall have then
As the three first fractions are abstract numbers, it will be
seen that the resistance expressed in electromagnetic units is equal
to the quotient of a length by a time that is to say, a quantity
of the same order as a velocity.
We may, indeed, easily discover in the experiment itself, a
physical representation of this velocity. Suppose, in fact, that
the cross-bar moved uniformly with a velocity u, and that the
intensity I of the current is measured by the action which it
exerts on a needle placed in the centre of a tangent galvanometer
(504) ; we shall have
On the other hand, MR = HS, or
it follows from this that
tan 8 = -
CLOSED CIRCUIT IN A UNIFORM FIELD. 505
If the velocity u is so great that the action of the current is
equal to that of the Earth that is, that the deflection of the
needle in the galvanometer is 45 we shall have
and if we take fl 2 = L, R = #.
Hence the resistance of the circuit in question is equal to the
velocity with which the bridge must be uniformly moved under
the given conditions, in order that the action of the current
induced in a galvanometer of suitable dimensions may produce a
deflection of 45.
527. The following experiment, suggested by Faraday, may be
considered as an application of the same problem.
Suppose that two electrodes, A and B, are immersed in water
on the opposite edges of a river, of a canal, or of a current in
the sea, and are connected by a metal conductor. If u is the
velocity of the current, and a the distance of the electrodes,
under the influence of the Earth's magnetism, an electromotive
force equal to iSLa will be established between them, which, in a
circuit of resistance R, will develop a current of intensity - .
R
The experiment is not impracticable; but, unless we could work
with very great values of u and a, the polarization of the electrodes
would no doubt make it very difficult to verify the conclusion.
528. CLOSED CIRCUIT IN A UNIFORM FIELD. Consider a
closed circuit, which may be supposed plane (487), and let S be
its surface. Let us suppose it placed in a uniform field of in-
tensity F the terrestrial field, for instance and perpendicular to the
direction of the field. If the frame be made to turn through an
angle a, the variation of the flow of magnetic force is equal to
FS (i-cosa), and the quantity M of electricity put in motion in
the circuit, which we suppose has the resistance R, is given by
the ratio
FS(i -cos a)
-
If the frame turns through 180, face for face, we have
506 PARTICULAR CASES OF INDUCTION.
This quantity of electricity may be measured in absolute value
(506) by the throw of the needle of the galvanometer, which
would enable us to determine the intensity of the magnetic field.
The direction of the current on Lenz's law, is that which ought
to traverse the current in its original position, in order that it may
be in stable equilibrium.
529. DETERMINATION OF THE INCLINATION BY INDUCTION
CURRENTS. We have seen (487) that the tangent of the magnetic
inclination is equal to the ratio of the works which for the same
strength of current correspond to a rotation of 180 of the circuit,
starting from a position at right angles to the magnetic meridian :
ist, about a horizontal axis perpendicular to the meridian; 2nd,
about a vertical axis. This ratio is that of the electromotive
forces of induction for the same displacements, and therefore that
of the corresponding quantities of induced electricity (506) ; a
measurement of this latter ratio will therefore give the inclination.
530. FARADAY'S Disc. A metal disc, movable in a uniform
field about an axis parallel to the direction of the field, forms part
of a circuit which communicates on the one hand with the axis of
rotation, and on the other with a spring which presses on a point
of the circumference. When the disc is put in uniform rotation,
a uniform current is also produced in the circuit. It will be seen
that the arrangement of this experiment is, as it were, the inverse
of that of Barlow (483). -If the plane of the disc is vertical, and
the force of the field F traverses it from front to back, and if
the direction of the rotation is that of the hands of a watch, the
induced current traverses the disc from the centre to the edge.
If a is the radius of the disc, and w the angular velocity, the
electromotive force is
,-^-- 2 F
~ dt 2
An analogous result is obtained, though with a less simple
calculation, by placing the disc between the poles of a horse-shoe
magnet or between the armatures of two electromagnets. In this
latter form, M. Le Roux obtained currents so strong that bright
sparks passed between the disc and the spring.
Moreover, all the experiments, particularly those which were
examined in Chap. III., in which the motion of a conductor is
produced by electromagnetic or electrodynamic actions, would pro-
duce an inverse induction current if the motion was kept up by
an extraneous cause.
TERRESTRIAL CURRENTS. 507
531. TERRESTRIAL CURRENTS. Let us consider, for example,
a sphere magnetised uniformly. Let us suppose that a conducting
arc, resting with one end at the pole and the other on the
equator, turns about the axis with a uniform motion; this arc
will cut the same flow of force as if it were applied on the sur-
face along a meridian. An element ds, the velocity of which is v,
cuts in each unit of time, a flow of force equal to Zvds, Z
being the perpendicular component of the magnetic force on the
corresponding parallel. If V be the velocity at the equator, F^
the magnetic force at the pole, a the radius of the sphere, and A
the latitude of the element ds, we have
z> = VcosA, Z = FpsinA, ds = ad\.
The flow of force cut in each unit of time by the whole arc
is equal to the electromotive force of induction *?, which gives
e VFtf sin A cos \d\ - F v Va.
Jo 2
If the arc is insulated, this value of e represents the difference
of potentials at the two ends.
If the ends of the arc were connected to two bodies of the
same capacity C, these bodies would acquire, after a longer or
shorter time, equal and opposite statical charges, the absolute value
of which would be -Ce.
2
Finally, if the arc were closed by a fixed conductor on the
inside or outside of the sphere, the circuit would be traversed by
a continuous uniform current, from the equator to the pole or
conversely.
As the Earth may be compared to a sphere magnetised uni-
formly, it will be seen that an external arc, which does not share
the rotatory motion, should be traversed by an induction current
from the equator to the pole, for the direction and magnitude of
the induced currents only depend on the relative motion of the arc
and of the magnetic system. It is probable that this induction
plays an important part in certain natural phenomena, such as
the aurora borealis (which seem to be electrical discharges in the
upper regions of the atmosphere), the currents observed on the
surface of the Earth in telegraphic wires, and the perturbations of
the magnetic elements.
508 PARTICULAR CASES OF INDUCTION.
532. VARIABLE STATE OF A CURRENT. The establishment of
a current in a circuit represents a certain amount of work which
is the potential energy of the current; this energy is absorbed at
the starting of the current, and is restored when the electromotive
forces disappear. In all cases, the effects of self-induction, which
are the consequence, determine the law of intensity during the
variable period, whether at the closing or opening of the circuit.
Consider a single circuit. Let R be the total resistance, L its
coefficient of self-induction, and E the electromotive force which
it contains. We have the equation
dt
If we suppose L constant as well as E, the strength at each
moment is given by the formula
(2) l-^ + ^-l^-r,
I being the initial value and I x that which corresponds to the
permanent state.
The total quantity of electricity which passes in time /, is
f' -d g-)
J' l ^
If the time / is sufficiently great, we have simply
C \ L
\o) i \ JR.
Jo
We have also, for a sufficiently long time,
this expression is proportional to the calorific energy expended in
the circuit.
VARIABLE STATE OF A CURRENT. 509
It may be observed that these values of the two integrals are
the same as if there had been a current of strength - during
T
the time - , which had been succeeded by a current of the
R
normal intensity I v during the rest of the time.
Suppose that the electromotive force is constant, and that we
measure the time from the closing of the circuit ; we have then
i.-o,
and therefore
'-
533. The expression ^T represents the strength of the extra
R
current obtained after the moment /. It will be seen that the current
only attains its normal strength after an infinitely long time ; but if
-p
the ratio - is very great, which it is in most cases, the exponential
J_/
tends rapidly towards zero, and after a very short time, the real
strength only differs from the final strength, by a quantity which
may be neglected. In order to calculate after what time this
difference would be below a given quantity, - for instance, we
may put
from which we deduce
The total quantity of electricity which corresponds to the extra
current is
(6)
R 2
it is the same as if the current had had half the intensity of the
11 lE - 2L
normal value - -- m the time .
2 R R
510 PARTICULAR CASES OF INDUCTION.
534. EXTRA CURRENT ON OPENING. Suppose that the per-
manent regime being established, we suddenly introduce a resistance
r in the circuit ; we shall have at the two limits
IO ~R'
E
E
and the strength at any instant is given by the equation
(7) I "
The value of the total quantity of electricity which corresponds
to the extra current is
(8) ' EL
This case has some analogy with that in which the circuit
is broken in air, the resistance r being that of the layer of gas
traversed by the spark on breaking; but in reality this resistance
is far from being constant 'while the phenomenon lasts.
Suppose that instead of breaking the circuit we had separated
it from the battery, by replacing the latter by a wire of the same
resistance, so that the total resistance of the circuit is still repre-
sented by R, which simply amounts to suppressing the electromotive
force. Equation (i) reduces to
(9) L
Determining the constant by the conditions that for /=0 we
p
have I = , it follows that
R
E _R
(10) ! = -* L.
In this case the law of the extra current of opening is the
same as that of the current of closing (533), and the quantities
of electricity put in motion are the same.
VARIABLE ELECTROMOTIVE FORCE. 511
535. VARIABLE ELECTROMOTIVE FORCE. Suppose that the
electromotive force, instead of being constant as with ordinary
batteries, undergoes periodical variations, and is represented by an
expression of the form
(n) E = E sin 27T .
When the steady condition is obtained, the intensity of the
current evidently follows the same period, and may be represented
by the expression
(12)
/ -- (j>\
Substituting this value in equation (i), and determining the
constants A and <f>, by the condition that it shall be satisfied for
any given time, we find
(13)
^2
and
277-L
(14) tan 27Td> = .
TR
It will be seen from this that the effect of the coefficient of
self-induction is to increase the apparent resistance of the circuit.
T
The strength of the current is zero whenever /-<T = # . The
2
value <T expresses the time which elapses from the time at which
the electromotive force is zero, and that when the current itself
passes zero. It is a kind of retardation to the transmission of
the electromotive force, which arises solely from the effects of
induction.
Whatever be the values of L, of R, and T, the maximum
value of 27T(f> is - , that is to say, that the difference of phase <f>
is equal to -. The maximum retardation of the induced current
4
is therefore equal to a quarter of the whole period, or to half
the semi-period.
512 PARTICULAR CASES OF INDUCTION.
During a semi-period, the quantity of electricity which passes
through the circuit is
[~$ f AT
(15) Q = A " sin27r-^= ,
Jo 7T
and the corresponding calorific work
(16) W = RA 2 f 2 sin 2 27r-<//=R .
Jo T 4
From this is deduced, for the mean intensity I' of the current,
disregarding the sign,
(17) r-^,
and for the mean strength I", which would give the same quantity
of heat
^'
536. CURRENT OF DISCHARGE OSCILLATING DISCHARGES.
Let us finally consider, with Sir W. Thomson, the phenomena which
accompany the discharge of a conductor. Let C be the capacity
of the electrified body in electromagnetic measure, Q its charge ;
it is connected with the ground by a wire of resistance R, and
the coefficient of self-induction of which is L. At a given moment
t t the charge of the conductor is Q, and its potential , which
\->
gives the equation
C~ ~dt
Observing that I = , this may be written in the form
dt
(19)
dt* -Ldt CL
OSCILLATING DISCHARGES. 513
The general integral of this equation is
(20) Q = A^ + Ay ;/ ,
p and p' being the roots of the quadratic equation,
which gives
= - + / R2 _L
"~~
According as L^ -- , these roots are real or imaginary.
4
The constants A and A' are determined by the condition that
for /=0, we have Q = Q , and 1 = 0, which gives Q = Ap + A'p.
If the roots of the equation (21) are real, then representing
the radical
Q = Q ^ 2L
(22)
/"R* ~
ical^/- ,
I R\ a, /x R\ -a,"]
-+ r ) e + ( --- r ) e h
2 4 aLy \2 4 aLy
at -at\
'-' )
When the roots are imaginary, we may still take the integral
in the same form, and replace the constants by their imaginary
V~i R2~
--- , we get then
r R i
cosaV + sinaV
(2$
O -
^fLC e '
L L
514 PARTICULAR CASES OF INDUCTION.
We get from these equations, in the two cases,
L//=Q ,
f
J
These results were evident a priori, for the discharge is com-
plete, and the work it produces is reduced to a disengagement of
/e
heat ; the calorific work RIV/ should be equal to the electrical
energy (89) which the conductor had before the discharge, that
iQ
is to say, - .
537. The nature of the discharge is very different according
as the solution is given by equations (22) or equations (23).
In the first case the discharge is continuous. The strength
of the current begins by being zero, passes through a maximum,
and then decreases to zero. The maximum is at the time deter-
dl
mined by the condition -r = ^t r
R \ 2a* R
a ) e = 1-a,
2 L J 2 L
which gives
538. In the second case, the values of Q and of I are given
by the periodic functions; the conductor takes alternately charges
in contrary directions, and the wire is the seat of alternating
currents.
The times of maxima and minima of charge correspond to
1 = 0, that is to say, to
sin a'/=0, or a!t = nir.
OSCILLATING DISCHARGES. 515
The oscillations of the discharge are regular, therefore, and
the value of the complete period T is
27T
The values of the alternate maxima are
+ Qo>
RTT
-Qo* 2La/ >
2R7T
+ Qo* 2La ',
_3Rir
-Qo' 2L *', etc.;
they decrease, therefore, like terms of a geometrical progression,
RTF
the ratio of which is e ^LT'.
The maxima of the intensity of the current in the two directions
d\
correspond to = 0, which gives
2La'
tana/= - ,
R
or
sina7= a'x/CL;
7T T
they are still equidistant, separated by a semi-period = - , and
a 2
succeed the periods at which the current is zero, by the time 0,
defined by the smallest angle which satisfies the condition
The values of the maxima of intensity are successively
2R:r
2a'L, etc.;
they also decrease in geometrical progression.
L L 2
516 PARTICULAR CASES OF INDUCTION.
Disregarding the sign, the total quantity of electricity put in
motion is
_Rir
/ __ R JL _2R!T \ !+<> sL7'
(2 5 ) Q (l + 2* 2La' +2 , 3La' + \ = Q Q _ _.
\ / -**.
i-e 2La'
RTT
This total mass is the greater the nearer the quantity e zLa? is
to unity, that is, the greater the factor - , or the greater is
R
. L
mtl
539. We pass to the case of a continuous current by making
Q
Q o = oo, C = oo, and = 1&. The roots of equation (21) are then
real, and the discharge is continuous ; we arrive thus at the formula
already found (532)
E
i e
If we suppose the coefficient L of self-induction to be very small,
we always fall into the case of real roots ; the coefficient a then
R / 2 L\ R i
becomes equal to ( i-^r^s ) or -z---^ t an d the equations
2Li \ CK. 4 / 2JL CR
(22) become
The current accordingly starts at once with its maximum value,
and then diminishes indefinitely.
It must, however, be remarked that all the preceding discussion
rests on the implied hypothesis that electricity does not possess
inertia, and that the intensity is the same throughout the whole
extent of the wire. These hypotheses appear justified in all cases
of permanent currents, or of slow variations, but we can no longer
assume them in the case of sudden discharges ; the results at
which we have arrived can only then be considered as a first
approximation.
CASE OF TWO CIRCUITS. 517
540. CASE OF Two CIRCUITS. Let us consider two adjacent
circuits C and C', whose relative position is fixed, and which contain
electromotive forces ; we shall represent by E, R, and L the electro-
motive force, the resistance, and the coefficient of self-induction of
the first circuit ; by the same quantities accentuated, the analogous
quantities of the second ; and by M the coefficient of mutual
induction. All the coefficients being supposed constant, we shall
have the two simultaneous equations
M + L + RI-E = 0,
dt dt
(27) dl dl ,
M + L' + RT-E' = 0,
dt dt
the solution of which is of the form
RI -E=
'
The coefficients A, B, A', B' are constants to be determined from the
conditions in respect to the limits.
Expressing the condition that these values of the strength satisfy
the differential equations (27), whatever be the time, we find that the
exponents p and p are roots of the equation of the second degree.
/ M 2 \ /R R'\ RR'
(29) ( i -- U2 + / _ + _ \p+ -- = o.
\ LL'/ \L L'/^ LL'
These roots are always real ; again they must be negative, seeing
that, under no circumstances, can the strength increase indefinitely
with the time. Hence we must have LL/>M 2 , which moreover is
evident (519); we could only have LL' = M 2 if the two circuits
coincided.
It follows also from this condition that if the coefficient of self-
induction L of a wire is very small, the coefficient of mutual
induction M of this wire with any other wire is also very small.
541. Consider the case in which E' = 0, that is to say in which
the second circuit contains no electromotive force.
If we only wish to determine the quantities of electricity m and
m' which traverse the two circuits in time /, we may calculate the
518 PARTICULAR CASES OF INDUCTION.
integrals \dt and I'dt by equations (27). If I and I' are the
intensities of the two currents at the beginning of the time in
question, we shall have
Suppose that we close the circuit, and that we consider the
phenomenon after a sufficiently long time, we shall have
i =o, i;=o,
i-f, r-oi
and therefore,
(-D-K'-i)-
<"> M EM V
m : = 1= .
R' RR'
If, on the contrary, we open the circuit C, after the steady con-
dition has been attained, we shall have at the two limits,
i =0, r = o.
The quantities of electricity induced in the two circuits are then
equal, and of opposite signs to those of the preceding case which
was evident, for the variation of the flow of force was the same in
both cases.
It is to be observed that the extra current of C is independent
of C', and, on the other hand, that the induction on the circuit C'
only depends on its resistance R', on the coefficient of mutual
induction of the two currents, and on the strength I of the permanent
current in the circuit C. The direct consideration of the flow
of force still enables us to foresee these results.
If we wish to know the strength of the currents at each instant,
the solution of the problem must be completed by determining
the constants.
CURRENT ON OPENING. 519
542. CURRENT ON OPENING. Let us first consider the case
in which, having closed the current in the principal circuit C, con-
nection with the battery is broken by leaving the circuit open;
an induced current is formed in the secondary circuit C', but the
principal current is entirely broken. The first of the equations (27)
no longer holds, and the second reduces to
L'
dt
it is identical with equation (9), and therefore the law of the extra
current will be the same in both cases.
In like manner, on the hypothesis that the opening of the
circuit C was instantaneous, and that the current is not prolonged
with a variable resistance, as in the case in which a spark is
produced, we may determine the initial value of the current pro-
duced in C'.
Let us integrate the second of these equations (27), taking E' = 0,
from /=0, to a time T which is infinitely small in comparison with
the duration of the induction current in C' ; denoting by l\ the
intensity of the current induced at the time r, we shall have the
equation
_M-+L'I;+R' f T r<#=o.
R Jo
If we make T diminish towards zero, the last term tends itself to
zero, seeing that T has an infinitely small value, and that the intensity
I' of the current retains a finate value ; we have then, in the limit,
ME M
(33) ' 1 "
Thus the initial intensity of the induced current is to the
intensity of the inducing current I in the ratio of the coefficients
M and L'.
Hence, at any given instant, we shall have, by equation (10),
ME _RL<
(34) 1' = --'.
52O PARTICULAR CASES OF INDUCTION.
From this is deduced, for the quantity of electricity put in
motion,
L'RR' RR''
and, for the calorific work expended in the circuit,
543. CURRENT ON CLOSING. The moment the inducing
current is closed, the two circuits react on each other, and we must
take into account the two simultaneous equations (27).
If we put
2 R'
(35)
the roots of the equation are
/ + R / L)-2RMa
2(LL'-M 2 )
2(LL'-M 2 )
The coefficients will be determined by the aid of equations (27)
and (28) by the condition that, for /=0, we have 1 = and I' = 0,
which gives
M L
Jx JK
M L'
- (A P + Ep) + ( A> + B >') = ,
K. K.
We get then
_ E( if/ R'L-RL'\ ft / R'L-RIA P 't~]
! = <!-- ( i+ \e -(i )e
R( 2\_\ 2 RMa / \ 2 MRa /
(3 6 )
E / ** P
I = ( e -e
2 Ra\
CURRENT ON CLOSING. 521
The differential of the latter equation
dV E / ft , P'/\
= ( pe - pe )
dt 2 Ra V /
shows that, in the secondary circuit C', the induced current, which
is always negative, since we have p>p in absolute value, starts
from zero, without its initial differential being zero, passes through
a maximum, and then decreases to zero.
It will be seen that the inducing current commencing at zero,
-p
increases progressively to its maximum value relative to the
R
permanent state.
If we suppose the two circuits C and C' identical, the formulae
reduce to
_ Ef i/Jk --*_
I = i - - ( e L+M - e L-M
(37)
E r _^_ _RL_-I
I'= e L+M-<? L-M .
2 R|_
If, further, the two circuits are in contact, the coefficients L
and M are very slightly different, and we have sensibly
E r i _Rfi
i --e SL
4 J
I' = -- e SL .
2R
In this latter case the strength of the direct current produced
on opening the circuit (533), is
(39) r, = -*:'.
The two currents attain their maximum, for / = 0, and this
maximum is twice as great for the direct as for the inverse current.
544. Two CIRCUITS WITH VARIABLE ELECTROMOTIVE FORCE.
Suppose we have
522 PARTICULAR CASES OF INDUCTION.
and that the second circuit is closed without containing any electro-
motive force.
When the permanent state is attained, the two currents are still
periodic, and as in (355), we may write
I = A sin 27T ( 6 )
(40) V T /
Putting
M2R '
T 2
/=L _4, l! _M ! L_
T 2 47T 2
R'2 + 1_ L '2
T 2
the condition that the differential equations are satisfied for any
given time by their values of E, I, and I', gives
A2 _
(41)
27T/
tan 27rd> = -- .
Tr
These expressions have the same form as above (535). It will
be seen that the effect of the presence of the second circuit is to
increase the apparent resistance of the first, and to diminish the
apparent coefficient of self-induction.
We shall find, in like manner, for the second circuit,
(41)
tan 27T</>' =
27T LV + R7
TELEPHONE AND MICROPHONE. 523
The amplitude A' is at first inversely as the period, so long
as the oscillations are not rapid ; for a very short period we shall
have
A'_M
A~L'"
This problem corresponds to the case of an induction coil,
the inducing current of which was sinusoidal.
545. TELEPHONE AND MICROPHONE. For a circuit placed in
a variable magnetic field, equation (10) of (518) becomes
(43) -dt dt dt
Let us suppose that the circuit contains a constant electromotive
force, that the factor L is constant, and that the coefficient Q
varies periodically. This is the case, for instance, of an electro-
magnet, in front of which a magnet is made to oscillate. We
may write, in that case,
. 27T/
1 ~ 7 F'
The result is the same as if the electromotive force of the
battery experienced a periodical variation equal to Qj cos 21? .
When the permanent state is attained, the induced current
oscillates according to the same period, and may still be represented
by the expression
I = L + A sin 27r (
\ m
Substituting this value in equation (43), we deduce from it
27I-L
tan
TR
4^! QL
^2
T 2
If the electromotive force E is zero, the current transmitted
is periodical ; this is the case of the telephone.
524 PARTICULAR CASES OF INDUCTION.
All the other quantities being constant, it may happen that
the resistance R varies periodically ; the current will still be
periodical, and the amplitude of the variations will be proportional
to the strength of the current which is produced with a constant
resistance that is to say proportional to the electromotive force.
Such a current will produce periodical induced currents in an
adjacent circuit. This is what takes place in the microphone when
the variations in resistance are produced by the vibrations of two
bodies in contact, such as pieces of carbon; it is also the case
with the selenium photophone, in which the variations in resistance
produced by the intermittent action of light are utilised.
The same result would be attained if the coefficient of self-
induction L was variable, which might be attained either by
periodically altering the configuration of the circuit, or by oscillating
the iron armature of an electromagnet.
546. INDUCTION IN AN OPEN CIRCUIT. We have hitherto only
considered closed circuits. If the wire submitted to induction is
not closed, we may assume that the effect of the electromotive forces
of induction is to tend to move the electrical masses towards the
end of the wire, and to set up for a moment a definite difference
of potential between these ends. On Maxwell's theory of displace-
ment this problem does not present any fresh difficulties, for the
circuits are always closed by the dielectric. It seems natural,
moreover, to extend the theory to the case of open circuits,
especially if the circuit contains a great number of turns (as in
the case of coils), and if the conditions are such that the current
could at each instant be regarded as identical throughout the whole
length of the wire.
Let us consider, for instance, the induction of two adjacent
circuits (540), and suppose that the ends of the induced wire
communicate separately with the armatures of a condenser of
capacity C. If we open the inducing circuit, and call V the
difference of potential of the armatures of the condenser at the
period / after this break, the intensity I' of the induced current
will be defined by the equations
L' + RT + V = 0,
'<**
We may further suppose that the insulation of the condenser
INDUCTION IN AN OPEN CIRCUIT. 525
is not absolute, and that the resistance of the dielectric, instead
of being infinite, is represented by R r The latter equation should
then be replaced by
from which follows
R'WV i /R' \
^ + L 7 / ^l/OVR^ V
This equation is the same as equation (19), and has an integral
of the form V = Ke^ + AW. If the roots of the quadratic equation
analogous to equation (21), which determines the coefficients p and
/o', are imaginary, as, for the most part, will be the case unless the
resistance R x of the dielectric is very small, the potential V is
represented by a periodical function of decreasing amplitude. The
value of the period is (538)
27T
~7 = rr~/R'
The constants of the general integral will be defined by the
condition that for /=0 we have V = and I' =11. Putting
i R'
a = + , we thus get
t^-Kj JL
I', f?
V = -e * sin a/.
C
On the other hand, the initial intensity IJ of the induced
current is given as a function of the inducing current I (542)
by an equation (33) in which the resistance of the circuit does
not enter. We get then, by substitution,
M _?
V = I CLV^
M T _
We may control this result by connecting the ends of the wire
at a given period, and for a very short time, with the binding screws
526 PARTICULAR CASES OF INDUCTION.
of an electrometer, which determines the difference of potential.
Experiment has shown that the phenomena are in complete agree-
ment with this theory.
If we detach the condenser at the ends of the wire, the
electricity set in motion by the electromotive force of induction
charges the wire throughout its entire extent, especially near the
ends, and the intensity of the current is not at each moment the
same throughout the entire length of the wire. It may also happen
that the electricity thus accumulated on the surface of the wire
escapes across the surrounding medium by a series of branch
circuits. These two circumstances complicate the problem, and
the preceding equations no longer present an exact solution, but
the difference of potentials at the end of the wire is in most cases
still represented by a periodical oscillation of decreasing amplitude.
547. LAWS OF BRANCH CURRENTS IN THE VARIABLE STATE.
The law which in a closed polygon connects the strengths of the
currents with the electromotive forces (211), also applies in the
variable state, provided that to the ordinary electromotive forces
we add those arising from the effects of induction.
We shall first examine the case in which the current bifurcates
between two points A and B, along conductors r and /, which
contain no electromotive forces. To define our ideas, we shall
assume that these conductors are wound as coils, and we shall
call L, L', and M their coefficients of induction. If, at a given
time, we denote by i and /' the strengths of the currents in the
two branches, we have
(L-M)*WY' + -(L'-M)*'.
dt dt
Let us first consider the case in which the second wire is
uncoiled ; the coefficient L' is very small, as is also the coefficient
of mutual induction M (540). Equation (44) reduces to
(47) -r7' + Ly = 0.
at
As long as the general current is increasing, we see that the
strength { in the rectilinear branch is greater than it would be
in the permanent state. The branch which contains a coil has
therefore an apparent resistance which is greater than its real
resistance. The difference is greater the greater is the coefficient
LAWS OF BRANCH CURRENTS. 527
L, and the more rapid is the variation in strength ; when the
strength is decreasing the reverse is the case. The effect is the
same in the general case, in which none of the coefficients is zero,
and equation (46) shows that each conductor behaves for an
increasing current, as if it had a greater resistance than for the
permanent state.
Let m and m' be the quantities of electricity which traverse
the two conductors in the time /, we have
rm
- r'm' + [U - LV + M (i ' - 1)] J = .
In the case of a discharge, the initial and the final strengths
are zero ; if the coefficients L, L', and M were constant, the term
within the parenthesis is null, and we have
rm r'm' = Q.
It follows that the total quantities of electricity were divided
between the two branches according to the ordinary law, although
at each instant the division was different; there is therefore ulti-
mately compensation.
This conclusion is only valid when the two circuits have
produced no external work ; and particularly that during the
discharge no magnet shall have been moved near one of them.
Hence, in measuring discharges, it is difficult to use a galvanometer
with a shunt.
548. If the principal current is sinusoidal with the amplitude
I , the two branch currents will ultimately have the same periodic
character with an unequal difference of phase. By calculations
which are analogous to the foregoing, we shall find the following
relations between the amplitudes A and A', and the difference of
phases < and </>',
A 2 A' 2 i;
Lr'-LV+M(r-r')
27T
tan 27rc =
+ (L + L'-2M)(L-M)
528 PARTICULAR CASES OF INDUCTION.
There is evidently a difference of phase between the principal
current and the sinusoidal electromotive force which produces it.
549, Let us further consider as an application the experiment
by which Faraday proved the existence of the direct extra current
which is produced when the circuit is broken. A battery is closed
by a circuit with two branches, like that which we have been con-
sidering; one of the wires, of the resistance r', is rectilinear, the
other, of the resistance r, consists of a coil formed of a great
number of turns.
If E is the electromotive force of the battery, R the resistance,
and I the principal current, after closing the circuit we shall have
the equations
(48) E-RI
L
dt
Putting
and observing that the currents are zero for /=0, we deduce for
the current which traverses the coil at the period /",
r'E / _PlL_\
1 = ( I-e L(K+rO ) 9
P* \ )
and for that which traverses the rectilinear wire,
If, when once the steady condition is established, we break
the principal circuit, so that the spark on breaking has no ap-
preciable duration, making R = oo, equations (47) give
and therefore
L-
EXTRA CURRENT. 529
The integral of this equation determines the induced current
which traverses the two branches. Observing that the current i is
defined by the permanent state, we shall have for the time t' after
opening the circuit,
- e
P 2
This current in the rectilinear branch is in the opposite direction
to the primitive current.
The total quantity of induced electricity is
EL r'
m -
If the needle of a galvanometer placed on a rectilinear wire abuts
against an obstacle which prevents it from obeying the permanent
current, this needle will be driven in the opposite direction by
the induction current, when the principal circuit is opened. This
method was used by Faraday.
550. If we regard the problem in all its most general forms,
we may propose to determine the division of the currents in the
cases in which the conductor is made up of a network of wires
forming polygons of any given form.
Let us suppose that a portion of the network is made up of a
polygon whose sides are A 1? A 2 , A 3 ..... Let z\, / 2 , z 3 ---- be the
intensities at a given moment of the currents which traverse the
various branches; r lt r 2 , /* 3 .... their resistances; Ej, E 2 , E 3 the
electromotive forces contained in each of them; L 15 L 2 , L 3 ----
their coefficients of self-induction; M 1>2 , M 1-3 , M 23 ... the co-
efficients of mutual induction of each branch on another part
of the circuit, in which the intensity of the current is /'. For
each side of the polygon we have an equation, such as the
following, relative to the side A l :
from which is deduced for the contour of this polygon,
M M
530 PARTICULAR CASES OF INDUCTION.
The expression ^Mi represents the sum of the products of
each coefficient of mutual induction by the sum of the intensities
of the current in the two corresponding wires. If there is a
magnetic or electrical system external to the first, the corresponding
variation of the flow of magnetic force Q in the polygon will also
produce an electromotive force. We shall thus have for the
contour of the triangle,
(48)
Integrating this expression between and /, we get
' I
If all the coefficients of induction are constant, and the initial
and final intensities are the same in each of the wires, the second
term of the second member is null. Finally, if the external work
is also null, we get simply
irdt,
or if m is the quantity of electricity which passes in a wire of
resistance r, and supposing the resistances constant,
= ^ mr.
Comparing this equation with that given by KirchhofFs law
(211), we see that they are of the same form, and we may
deduce from it the following theorems :
i st. The distribution of the discharges in any given circuit is
inversely as the resistances, when the form of the circuit does not
change, and the initial and final intensities are the same in each
wire, and there is no external work.
2nd. Even when the former conditions are not fulfilled for all
the wires, the equations of distribution do not contain the co-
efficients of self-induction of the branches in which the final and
initial intensities are the same.
551. PHENOMENA OF INDUCTION IN TELEGRAPH CABLES. In
studying the propagation of electricity in cylindrical conductors
INDUCTION IN TELEGRAPH CABLES. 531
during the variable state, we have disregarded the effects of in-
duction, which are due to changes in the strength of the current.
From this there arises a fresh cause of delay in establishing and
in suppressing the principal current, and this retardation can no
longer be calculated as we have done before (535), since the
duration of the propagation is very considerable as compared with
the duration of the phenomena of induction ; it is not possible to
assume then that the strength of the current has the same value
at every instant in the whole extent of the circuit.
If there are a series of alternate charges and discharges in
the wire, as is the case with telegraphic signals, we may consider
the phenomenon as being due to a variable electromotive force,
and effects will be produced analogous to those which have been
mentioned above.
The cable may, moreover, be considered as near other ones,
which react on the first either by their mere presence, or by the
variations in their own currents which traverse them. The resul-
tant effects are very manifest in air wires resting on insulators.
Finally, when several conductors are enclosed in the same
dielectric sheath, as is the case with subterranean, or submarine
cables, the potential at each point of one of the wires depends on
the charge of the adjacent wires. From this follows a new kind
of influence or induction, purely electrostatic in character, and
which Sir W. Thomson called peristaltic, to distinguish it from
that of Faraday. This phenomenon is completely analogous with
the reciprocal influence of elastic tubes connected lengthwise, which
are filled and surrounded by the same liquid, when the liquid is
made to circulate in one or more of the tubes, while the ends of
the others are opened or closed, or in any other special condition ;
a closed tube would correspond to an insulated conducting wire,
and an open tube to an uninsulated wire.
From these suggestions it will be seen how complex is the pro-
blem of the propagation of electricity, if we take into account all the
circumstances which may have an influence on the phenomenon.
552. CALCULATION OF THE COEFFICIENTS OF INDUCTION.
SOLENOIDS. The preceding examples show the importance of the
part which the coefficients M and L play in the calculation of the
phenomena of induction. We shall examine a few simple cases,
in which these coefficients may be easily estimated.
Let us, in the first case, consider a cylindrical solenoid so long
that for a considerable portion of its length we may disregard the
action of the ends.
M M 2
532 MAGNETIC INDUCTION.
If S is the section of the cylinder, which we will assume to
be circular, I the strength of the current, and n^ the number of
turns for unit length, the flow of force or of magnetic induction
is equal to 47r 1 IS (495) ; and, when the current is equal to unity,
the value of this flow is
= 471-^8.
Suppose that ri additional turns of any given diameter are
wound on the cylinder, the flow of force of the first circuit
traverses ri times the surface of the second ; the coefficient of
mutual induction is then
(49) M = '^=4irXS.
The same flow traverses the surface of the first circuit n times
for unit length, so that the coefficient of self-induction of the
solenoid for unit length, is
(50) L 1 = 1 ^=4ir!S.
The values found for the coefficients M and Lj are a maximum,
for the true flow of force is less than 47r 1 IS, and this flow
diminishes as we come near the ends of the solenoid, where it
becomes even less than 27r 1 IS. For the magnetic induction in
the uniformly magnetised cylinder, which is the equivalent of a
solenoid (373), is equal to the resultant of the force qn-nj. parallel
to the axis, and of the action of the two terminal layers of uniform
densities <r = 1 I, which is in trie contrary direction. But in a
section near the positive surface, this surface sends towards the
interior a flow of force equal to 2ir 1 IS, to which must be added,
in order to obtain the true flow, the portion of the flow from the
negative surface which traverses the same edge.
Suppose that the ri turns in question form a solenoid, concentric
with the first and of the same length /. Let us denote by r^ and r^
the two radii, and by n^ and n 2 the number of turns in unit length
of each of them. Disregarding the action of the ends, we may write
M = n% Ig N" =
putting
CONCENTRIC COILS. 533
The factor N represents the total number of turns of the outer
solenoid, and g the flow of force produced in the inner solenoid
by a current equal to unity.
553. CONCENTRIC COILS. Let us in like manner consider
two concentric coils composed of several layers of wire closely
pressed together, so that in unit surface of a meridian plane, the
number of wires is respectively n\ and n y For layers of the
thicknesses dr^ and dr^ and of lengths equal to unity, the number
of wires will in like manner be n\dr v and n\dr y We shall have then
for each coil, the integral being extended to the entire thickness.
Denoting the extreme radii by y l and x- for the first coil, and by
y 2 and x 2 for the second, we get
If we disregard the effect of the ends, the value of the co-
efficient of mutual induction will be
(51) M = N-= ***&& - *J (y\ -
When the layers of the two coils are in contact, we may put
# 2 = y l =y. If we wish to arrange the intermediate radius so that
the coefficient M is a maximum, we must satisfy the condition
= - i+- + -i
y r
Since the latter member of this equation is less than unity, the
outer coil must not be so thick as the inner one.
534 PARTICULAR CASES OF INDUCTION.
In order to obtain the coefficient of self-induction of a coil,
we may assume, for instance, that two identical coils are super-
posed, and we may determine what is then their coefficient of
mutual induction (512). If in this way we make in the preceding
equations
we get
(52) L =
O
554. COILS WITH A SOFT IRON CORE. Let us assume that
the inner coil contains a cylindrical core of soft iron of radius a,
the coefficient of magnetisation of which is k. The value of N
does not change ; but so long at least as we remain within the
limits within which magnetisation is proportional to the magne-
tising force, the magnetic induction in the space occupied by
the soft iron is equal (379) to the original value of the force
multiplied by i + 4wvfc. In the case of a solenoid the value of
g then becomes
g
If the coil consists of several layers, we have
it follows that
(53) M
;
- *,) X - *J + l w(y, - *,)
When the iron core entirely fills the inner cavity, and the
two coils are in contact, we may put
we get then
(54) M = ^ **n\f%(z -y) (y - x) / + xy + ^ + i zM .
COIL WITH A SOFT IRON CORE. 535
If the radius of the core is the only variable, and we wish to
arrange it so that the coefficient of mutual induction is a maximum,
the partial differential of M in respect of x should be null ; it
follows that
I27T/&
or sensibly 3^=27, since the coefficient of magnetisation k is
very great.
If the external radius z is given, as well as the way in which
the wires are coiled, and we wish to arrange x and y so that
DM DM
M is a maximum, making the partial differentials - and -^~
<te Dj/
equal to zero, we shall find in like manner
34
Lastly, the coefficient of self-induction of a coil which has the
external radius z, and the internal radius y, and which includes
a soft iron core of radius x, will be expressed by
(55) L = ** n \l(z -y? z* + zy +/ + 1 2****
The problem we have here discussed is that which occurs in
the construction of induction coils.
555. ANNULAR COILS. In a coil formed by regularly winding
a wire on a circular ring, the coefficient g (496) becomes
Suppose that a second wire is coiled n' times round the first in
any way whatever; the coefficient of mutual induction will be
(56)
PARTICULAR CASES OF INDUCTION.
If the first coil contains a soft iron core of revolution about
the same axis, and of section S', the value of g is then
When the core fills the whole cavity of the solenoid this value
reduces to
)/?
In this case the coefficient of mutual induction on an external
wire which makes ri turns, is
(57)
(Vs
J- 1
and the coefficient of self-induction of the coil itself, which contains
2irn turns, is
(58) L=87T 2 ^(l+ 4 7r) ^
We have seen (502) what are the values of the integral I -
in certain special cases. J x
556. ELECTRICAL MOTORS. Electrical motors are machines
containing conducting wires, electromagnets, or permanent magnets,
and so arranged that when a current produced by an extraneous
source is introduced, the electromagnetic or electrodynamic actions,
exerted between the different parts, are used to produce a relative
displacement of these parts. The continuity of the motion is
obtained either by sliding contacts, or by commutators which
alter the direction of the current in the machine.
These machines may be so made as to work uniformly when
the strength of the current is constant, as with Faraday's disc
(530); but more frequently the actions are periodic, and the
relative motion of the parts, whether oscillating or rotating, produces
an electromotive force of induction E of a periodic character ; in
ELECTRICAL MOTORS. 537
order that the motion may be kept up, the current in one part
of the circuit must be reversed by a commutator twice during each
period, so that the electromotive force of induction be at each
instant in the opposite direction to E that of the exciting current.
When a regular state is attained, the work of the reciprocal
actions is entirely consumed in overcoming external resistances,
for the velocity resumes the same value at the end of each
period.
The energy expended by the source, during each period $, is
equal to the external work increased by the energy which corre-
sponds to the heating of the conductors. If R is the resistance
of the circuit, we have then,
PEL#= fpR^+
Jo Jo
(59) E \dt= FR<#+ EI<#.
The reversals of the current by the commutator produce sparks
which absorb part of the work, and which affect the equation if we
take count of the corresponding variations in the resistance R of
the circuit.
In the general case the values of the integrals depend on the
law according to which the current varies during a period ; but
if the strength is sensibly uniform, equation (59) reduces to
(60) E -E = IR.
The efficiency p of such a motor is the quotient of the external
work El in unit time, by the total energy expended E I, or the
p
ratio of the electromotive forces. If I is the strength of the
E o
current which the electromotive force E would produce in the
circuit at rest, we have
E IR I
(61) p = =i =i .
EO E I
The expression for the external work itself is
(62) EI
53$ PARTICULAR CASES OF INDUCTION.
As the external electromotive force is constant, this work is a
maximum when the motion reduces the strength of the current
by one-half, and the efficiency is then equal to 0.50.
If the external work is very small, the velocity of the motor
increases very rapidly with or without limit according to the mode
of construction; the electromotive force of induction tends to
become equal to the external electromotive force, and the efficiency
tends towards unity.
557. ELECTROMOTORS. When an electrical motor is set in
motion by an extraneous machine, it becomes the seat of an
electromotive force, opposite in sign to that which would produce
the motion, and the circuit which constitutes it is, in general,
traversed by an electrical current. The apparatus is then a
producer of electricity, or an electromotor.
Suppose that owing to any temporary cause there is in the
circuit a current of the strength / ; if the work E/, absorbed by
the electromotive force of induction, is greater than the thermal
energy disengaged in the conductors, the current will go on
increasing until
(63) E I = PR.
If the condition E>/R holds for an infinitely small current,
the machine once in motion will prime itself, and for a steady con-
dition will produce a current denned by the preceding equation
(61). When, on the contrary, E</R for an infinitely small current,
the external work cannot create and maintain an electrical current,
unless there is artificially introduced into the circuit a current of
such strength that the condition E>/R is realised, after which
the extraneous electromotive force could be suppressed without
the current ceasing.
A machine used as an electromotor is therefore capable, or not,
of creating an electric current, or of keeping up a current already
established, according to the value of the total resistance, or, what
produces an analogous effect, according to the kind of external
work which the current is to produce. Since the electromotive
force of induction, other things being equal, is proportional to the
velocity of the machine, we see that for a given total resistance,
and given external work, the electromotor will be able to produce
and maintain a current the more rapidly, the greater is its velocity.
Let us consider two extreme cases :
ist If a machine consists of permanent magnets which produce
an invariable field in which the conducting wires move (take
ELECTROMOTORS. 539
Faraday's disc, for instance), the electromotive force is simply
proportional to the number n of turns, or of oscillations of the
machine in unit time, and may be represented by E r When the
velocity is constant, such a machine acts exactly like an ordinary
battery. It can always produce a current in a metallic circuit,
since the condition E>/R always holds when the current is very
feeble.
2nd. If the machine is composed of fixed wires and of movable
wires, or of two systems of electromagnets (provided that we remain
within the limits in which the magnetisation is proportional to the
magnetising force), the electromotive force is proportional to the
strength of the current, and may be represented by nAI. A
machine of this kind can only produce and keep up a current
when the velocity is so great that #A>R. For any value of n
-Tt
greater than , the strength of the current increases until the
A
resistance of the circuit, owing to the heat disengaged, has reached
the value nA.
Suppose that the current has to overcome an electromotive
force E' external to itself.
The value of the efficiency of the apparatus, as in the case of
the battery, is ,
and the work utilised in unit time is
E'(E-E')
R
In the former case the conditions of efficiency, and of maximum
work, are the same as for a battery.
In the second case, in which E = AI, we have
The efficiency is constant, and the work utilised is proportional to
the square of the current.
If the external work is nothing more than that of setting in
motion a second machine identical with the first, the electromotive
forces E and E' are proportional to the number of turns n and n'
of the two machines ; on the other hand, these electromotive forces
54 PARTICULAR CASES OF INDUCTION.
are proportional to the same function <j>(I) of the intensity. The
efficiency
_
P ~
is about equal to the ratio of the velocities of the two machines,
and the useful work in unit time is expressed by
E'l
As the strength I is a function of n - ri, the maximum useful
effect can only be determined if we know the form of the function <.
558. The preceding considerations only apply rigorously to the
case of a uniform current like that obtained with a machine such
as Faraday's disc. In a periodical machine, for instance, a frame
turning with a uniform motion about a vertical axis under the
influence of the terrestrial field, the phenomena are more com-
plicated.
If the machine is used as an electromotor, the ends of the
wire being joined by sliding contacts with an external conductor
in such a way that the coefficient of mutual induction of the two
parts of the circuit may be neglected, the current should satisfy
equation (518)
in which L is the coefficient of self-induction of the whole circuit,
R the resistance, and Q the flow of terrestrial force across the
frame. If S is the surface of the frame, H the horizontal com-
ponent of the terrestrial field, and T the period of rotation, we
may write
Q = HS sin27r-.
/
The electromotive force is sinusoidal, like that which we investi-
gated in (535). When the permanent regime has been established,
the current is itself periodic, and the work necessary to keep up
the motion corresponds altogether to the heat disengaged in the
circuit, which for unit time gives
* H2S2
\
APPLICATION TO THE STUDY OF MAGNETISM. 541
Instead of sending the alternating current into the external
circuit, a commutator may be placed on the axis of rotation,
consisting of two half discs communicating separately with the
ends of the frame, and of two springs connected with the external
circuit. If the springs are so adjusted as to change the connections
the moment the current is null that is to say, at a period <T
after the passage of the frame through a plane perpendicular to
the magnetic meridian the direction of the current in the external
circuit is always the same, and there is no change in the law
of the phenomena, as the commutator does not cause any loss
of energy.
If we use the frame as a motor by connecting it with an external
electromotive force E , it will be necessary to use a commutator
which changes the direction of the current twice in each period,
in order to maintain the motion. There is then a loss of energy
by the sparks at the movable contacts which must be allowed for,
and the calculations are far less simple.
It is impossible to treat here in greater detail the question
of induction-electromotors. We shall return to the subject in the
second part of this work.
559. APPLICATION TO THE STUDY OF MAGNETISM. We are
now in a position to justify the method mentioned in (417) for
studying the distribution of magnetism, a method employed in
the experiments of Van Rees and of Gaugain.
If a magnetised bar is surrounded at a point by a coil
formed of ri turns, connected with a galvanometer, and if the
coil is suddenly made to slide through a certain distance, parallel
to the bar, coming to rest in its fresh position, a certain quantity
of electricity m passes through the coil ; if R is the total resistance
of the circuit, the expression r is equal to the flow of magnetic
n
force, which proceeds from the magnet between the two positions
of the coil. Working in this way by successive displacements,
we can determine the law of variation of the flow of the lateral
force.
If the coil is at first in the middle of the magnet, or more
exactly at the neutral point, and it is suddenly removed to a great
distance, we shall get the total flow of force from the magnet,
and therefore the total mass of free magnetism contained in the
corresponding portion of the bar.
If the auxiliary coil thus surrounds either the centre of a long
cylindrical coil, containing a bar of soft iron, or any point of an
542 PARTICULAR CASES OF INDUCTION.
annular coil, we can at a given moment suddenly establish or
suppress a current of known strength I in the magnetising coil.
The expression , which in the induced current corresponds
n
to the make or break of the principal current, represents the total
flow LI of magnetic induction which traverses one of the turns.
We could then determine experimentally the value of L, and
from it deduce the coefficient of magnetisation k, particularly by
formulae (54) and (57). This is the principle of the method
recently employed by Prof. Rowland.
560. WEBER'S HYPOTHESIS ON MAGNETISM AND DIAMAGNETISM.
We have seen above how Ampere explains magnetism by
molecular currents; we may now examine the physical properties
of these currents.
Consider one of the currents defined by the values L, I, and R,
and let Q be the flow of external force in its contour; as the
electromotive force is null, equation (10) of (518) becomes
-
dt
We must also assume that the resistance is zero ; it follows that
(64) LI + Q = const = LI .
The strength I is that of the current which would traverse
the circuit in question, if the external flow of force were zero. If
we suppose I = (that is, the molecular current originally null),
which would correspond to the case of a magnetic medium in
the neutral state, we have finally
LI--Q.
The current induced in the molecule by an external field
produces therefore a flow of force contrary in sign to Q. In
other words, the magnetisation equivalent to the current is of
opposite sign to the magnetising force; the magnetisation of
magnetic bodies cannot therefore be explained solely by currents
induced in the molecules of the medium.
561. Such currents can, on the contrary, account for diamagnetic
phenomena. Weber's hypothesis assumes that in each molecule of
a diamagnetic medium there are channels along which currents may
circulate without resistance. If these channels were in all directions,
the molecule would be a perfect conductor. With a linear current
WEBER'S HYPOTHESIS ON MAGNETISM AND DIAMAGNETISM. 543
which was originally zero, the strength is given by equation (64).
If 6 is the angle which the magnetising force X makes with the
perpendicular to the plane of the circuit, we have
Q = XAcos(9.
The magnetic moment of the current is I A, and its projection
in the direction of the magnetising force is
V A2
IAcos0=- -cos 2 <9.
L
Suppose that there are n molecules in unit volume, and that
the axes of the circuits are distributed indifferently in all directions.
The zone corresponding to the angle dQ about the direction of the
magnetising force is 27rsin#</0, so that the mean value of cos 2 is
n A 2
The magnetic moment is therefore --- X ; the magnetisation
3 L
is directly opposed to the magnetising force, which is in conformity
with the phenomena of diamagnetism, and the value of the coefficient
of magnetisation is
If the distribution of the axes of the molecular channels is
A 2
not uniform, the sum V cos 2 # extended to the whole of the
* L
molecules will have different values according to the direction of
the magnetising force, and we thus come upon the known properties
of anisotropic diamagnetic substances.
562. Suppose that each molecule is a perfect conductor, or
(what amounts to the same thing) that it is surrounded by a layer,
the conductivity of which is perfect.
The total flow of magnetic induction LI + Q which traverses any
circuit traced on the surface is constant.
It follows that the perpendicular component of the force at each
point of the surface is constant. If the flow of induction which
proceeds from the molecule is originally null, any external magnetic
system will produce induced currents, such that the resultant
544 PARTICULAR CASES OF INDUCTION.
magnetic induction will be null on the whole surface, and in the
interior of the layer which surrounds the molecule.
If we assume that the molecule has the form of a sphere of
radius r, the currents on the surface produce in the interior a force
- X, equal and of opposite sign to the external magnetic force ;
the sphere is magnetised uniformly with an intensity (355) equal to
, and the value of its magnetic moment is - - ^X.
If we assume that in the unit volume there are n equal
spheres, so small and so far apart that they do not act on one
another, the mean strength of the magnetisation of the medium
will be - - ^X = - - , h being the ratio of the sum of the volumes
2 07T
of the small spheres to the total volume of the space which contains
them.
563. ABSOLUTE CONDUCTING SCREENS. It is clear that these
conclusions from formula (63) would apply also to a surface of
finite extent which possessed absolute conductivity ; the induced
currents which any variation of the field would produce in this
surface, would always be such that the flow relative to each portion
of the surface would be constant in other words, that the per-
pendicular component of the magnetic induction at each point would
retain a fixed value. If then this component part was null at a
given moment, it would remain null whatever were the variations
of the field. It follows from this that a closed or unlimited surface,
of resistance zero, is a complete screen for all points in the interior
against the effects of the variation of the field on the other side
of the surface ; these effects reduce to the production of surface
currents, which keep the field in the interior constant at zero.
The distribution of magnetic forces about a perfect conductor
is quite comparable to the distribution of velocities in an incom-
pressible fluid which surrounded the same bodies.
564. In order to explain magnetic phenomena on these con-
siderations, it must be assumed that the primitive current in a
molecule, or about a conducting channel, is not null, and we take
the general equation (64). Considering always the case of a circuit,
the perpendicular of which makes an angle with the direction
of the magnetising force, we shall have
LI + XAcos(9
L o>
or _ _ XA
I = I cos
J_J
ABSOLUTE CONDUCTING SCREENS. 545
If the primitive current I was null, the moment of the action
of the field on the molecule will be
X 2 A 2 X 2 A 2
IAX sin 6 = sin cos = sin 29.
L 2 L
7T
The equilibrium will be stable, for = - , that is when the plane of
the current is perpendicular to the direction of the field. This is
the case, for instance, of a ring which is suddenly brought into
a very powerful field.
In the general case, the moment of the couple which acts on
the molecule may be written
X 2 A 2
- cos 6 sin - I AX sin 6 = mX sin [BX cos - i J,
putting w = I A and B= -.
LI
If we assume with Weber (427), that the reaction D of the
medium is constant in magnitude and in direction, we shall
have the equation of equilibrium,
X sin 0(i - BX cos 6) = D sin(a - 0).
The component, parallel to the magnetising force X, of the
magnetic moment of the molecule is
(XA
I o cos ) A cos 6 = m cos 6>(i - BX cos (9).
If the coefficient B is very small that is, if the primitive
molecular current I is very powerful we arrive again at Weber's
formula for the magnetisation of magnetic substances. If this co-
efiicient B is very great we obtain the phenomena of diamagnetism.
In intermediate cases, magnetisation will first of all be pro-
portional to the magnetising force for small forces, it will then
pass through a maximum, and afterwards decrease. Experiment
does not seem to show any phenomenon of this kind, so that the
occurrence of currents of molecular induction cannot be con-
sidered proved.
NN
546 PROPERTIES OF THE ELECTROMAGNETIC FIELD.
CHAPTER VI.
PROPERTIES OF THE ELECTROMAGNETIC
FIELD.
565, MAXWELL'S THEORY. The preceding considerations are
sufficient, as we have seen, to account for all the phenomena of
induction in linear conductors ; but it is useful to regard the problem
from another point of view, in which the influence of the medium
is brought out, as in electrostatics. We shall explain here the
principles of Maxwell's theory.
566. EQUATIONS OF THE MAGNETIC FIELD. Suppose, for
greater generality, that the conductors are situate in a magnetic
medium whose coefficient of permeability (383) is equal to /*.
When the total flow of magnetic induction Q through a closed
circuit is annulled, it becomes the seat of a total electromotive
force which puts in motion a quantity of electricity equal to .
K.
This total electromotive force may be regarded as the resul-
tant of the elementary electromotive forces acting on each of
the elements of the circuit, and arising from the condition of the
medium. At each point the elastic reaction of the medium, due
to the suppression of the forces, has a determinate direction; it
would produce on an element of the conductor ds, situate in the
same direction, an electromotive force proportional to the length of
this element, and which may be represented by ]ds ; the electro-
motive force on an element which made an angle with the
direction, would be equal to ]ds cos c.
If the medium is homogeneous and isotropic, the electromotive
force J for unit length is a function of the co-ordinates ; it may be
replaced by its components F, G, and H, parallel to the axes, and
which would produce the electromotive forces ~Fdx, Gdy, and Hdz
EQUATIONS OF THE MAGNETIC FIELD. 547
on the projections dx, dy> and dz, of the element ds. We shall
have then the equation
the integral being extended to the whole circuit.
On the other hand, if we consider any given surface S bounded
by a circuit, and we denote by X, Y, and Z the components of the
magnetic force at a point of the surface, and by a, ft, and y the
cosines of the angles of this force with the perpendicular, the total
flow of magnetic induction across the circuit is also expressed by
the integral /* (Xa + Y/3 + Zy) dS, extended to the whole surface.
567. In order to determine the relations between the electro-
motive force J and the magnetic induction, we shall calculate the
value of the integral Jds cos e for an infinitely small rectangular
circuit with its centre at the origin of the co-ordinates, perpen-
dicular to the x axis, and the sides of which are equal to dy
and dz.
Let F , G , H be the values of the functions F, G, H at the
origin; we shall have, for the top side of the rectangle,
and for the bottom side
If we follow the contour in the direction in which the current
would be produced by the suppression of the field, the sum of
the two terms of the electromotive force corresponding to the
sides parallel to the y axis, is G 3 dy G 2 dy ; we get thus
c)G
G-^dy - G 2 dy = dzdy.
02
We shall have, in like manner, for the two other sides,
<)H
- Hj dz + H 2 dz = -- dydz.
N N 2
PROPERTIES OF THE ELECTROMAGNETIC FIELD.
The electromotive force corresponding to the contour is then
Ufrcos=( - \dydz.
On the other hand, this expression should also represent the
flow of magnetic induction through the circuit dydz from the
negative face ; as the expression for this flow is i&dydz, the com-
ao -m
ponent /*X is equal to .
The other components of magnetic induction parallel to the
y axis and the x axis, will satisfy analogous conditions, which is
given by the equation
Let us now consider any given closed circuit ; we may divide
the surface into elements by two series of arbitrary curves. If we
make the summation Jds cos e of the electromotive forces along
the contour of all the elements, we shall obtain the total flow of
magnetic induction across the circuit, and this sum will reduce to
the single terms furnished by the primitive contour, for the sum
of the portions of the curve common to two contiguous elements
is null. The total electromotive force of a circuit, only depends
then on the values of the components F, G, and H at the
different points of the contour, and these components are con-
nected with the magnetic induction by the equations (2).
568. We may observe that the component F represents at a
given point, and for unit length parallel to the x axis, the total
electromotive force which corresponds to the suppression of the
field. This electromotive force may also be regarded as the in-
tegral, in respect of time, of the elementary electromotive forces
which correspond to the gradual suppression of the field. The
EQUATIONS OF THE MAGNETIC FIELD. 549
product of the electromotive force P, which acts at a given time
on unit length parallel to the axis, by the time dt, is equal to the
corresponding diminution of the value of F, which gives
or
P- ^
5?
In like manner, if Q and R are the analogous components
along the other axes, we shall have, for the determination of the
electromotive force at each instant, the three equations
p- 3F
Tt'
.
dt
The function - P expresses the difference of potential, which is
produced at the point in question between the two ends of a
length equal to unity, parallel to the x axis, in consequence of
the variations which at the same time take place in the value of
the flow of magnetic induction.
If there were also electrical masses producing the potential ^
at the same point, the total difference of potential dx will be equal
D D\l/
to -dx - ^-dx : but this latter part is an exact differential which
Dt Dx
itself disappears when the formula is applied to a closed circuit
569. EQUATIONS OF CURRENTS. Let z/, v, w be the com-
ponents of the current at a point that is to say, the quantities of
electricity which in unit time traverse unit surface, perpendicularly
to each of the axes.
We know that the work of a current I on a unit pole, movable
on a closed curve which traverses the plane of the current (452),
is equal to 4?rl for each complete turn of the pole about the
current.
If we apply this property to the case of a pole which traverses
a rectangular circuit dydz, the centre of which is at the origin of
550 PROPERTIES OF THE ELECTROMAGNETIC FIELD.
the co-ordinates, that is to say, which enclose a surface across which
the strength of the current is u dy dz^ and if we determine the same
work by the magnetic forces, we find by a calculation analogous to
that applied to (567), the three new equations
ax
- .
oy f&
570. POTENTIAL ENERGY OF CURRENTS. Let us consider
various currents C^ C 2 , C 3 , . . . in which the strengths are Ij,
I 2) I 3 . . . ; let Lj, L 2 , L 3 be their coefficients of self-induction and
M 1-2 , M 1-3 , M 2<8 , . . . their coefficients of mutual induction.
The flows of force which traverse each circuit are
and the potential energy of the whole of the currents (524) is
(5) W = -(Q 1 I 1 + Q 2 I 2 + ) =
From equation (i), we have therefore
Taking u, z>, and w in the same signification as above (569),
the components of the current across the section S of one of
the conductors are
!$- 8.
ds
EQUATIONS OF CURRENTS. 551
If, further, we observe that the product Sds represents an
element of volume, we shall have
W = - (Tu + Gv + llw) dxdydz.
Replacing finally the components u, u, w by their values taken
from equations (4), we get
f f f F /c)Y dZ\ /c)Z <)X\ /<)X 7)Y\"1
W= F( 1 + G( r ) + H( r--r- ) \dxdydz.
&TT I \ oz oy I \ ox oz I \vy vX J I
J J J L. \ / \ / \ /J
We may integrate by parts each of the terms, which give, for
instance,
r C>Y r *F
F dxdydz = Wdxdy - Y dxdydz.
Repeating the same operation for all the other terms, we get
~i
J
^ , , , , X I \dxdydz.
If we extend this expression to the whole space contained in a
very distant surface, which comprises the whole system of the
magnets and of the currents, the first integral relative to the surface
itself is null, for the components of the magnetic force are inversely
as the cube of the distance. On the other hand, replacing in the
second integral the terms comprised within the brackets by their
values drawn from equations (2), we get
W= (X 2 + Y 2 + Z 2 )^X*>
or, if <f> is the magnetic induction at a point that is to say, the
product of the force by /*,
552 PROPERTIES OF THE ELECTROMAGNETIC FIELD.
The potential energy of currents may therefore be expressed by
an integral (5) which contains the currents themselves, or by an
integral in respect of all parts of the space in which are magnetic
forces. In the former case, the reciprocal action of the currents is
considered as being directly exerted at a distance ; in the second
case, this action results from the elasticity of the intervening medium.
If we adopt this point of view, we see that the potential energy of the
medium at each point, for unit volume, is expressed by
571. RELATIVE DISPLACEMENT OF CIRCUITS. The formulae (3)
correspond to the case in which, the circuit being fixed, induction is
due to the mean variation in the field ; the quantities F, G, and H
are at each point mere functions of the time. But if, while the field
is variable, we displace or alter the configuration of the circuit,
the co-ordinates x, y, z should themselves be considered as functions
of the time. Equation (i) should then be written
The electromotive force utilised in the time dt is - </Q, so that
the electromotive force of induction e at a given moment is expressed
by
*=-^=- ^T-+^^+T-^- 77*
,
" ds ~b ^s ^s J
+
s s s
RELATIVE DISPLACEMENT OF CIRCUITS. 553
Consider the first term of the integral, and replace the partial
"v*
erentials
term becomes
differentials and by their values taken from equations (2); this
or
Analogous results will be obtained by treating the second and
third terms in the same way. We may, moreover, observe that we
have
^rr+F <fr = F-,
and that the groups of this form disappear when the integral is
extended to a closed circuit.
The value of the electromotive force reduces then to
-
Each of the three groups contained in the parenthesis represents
the electromotive force which, at a given instant, acts on unit length
parallel to one of the axes. If, further, we suppose that the field
contains electrical masses giving a potential \//, the most general values
of the components P\ Q', J? of the electromotive force will be
<>
554 PROPERTIES OF THE ELECTROMAGNETIC FIELD.
572. EQUATIONS OF THE ELECTRICAL FIELD. Let us consider,
lastly, an isotropical field in which continuous currents may
coexist with currents corresponding to a variation in the electrical
displacement (126).
If /> & & are the components of the electrical displacement, the
true quantities of electricity u', v', w\ which in unit time traverse unit
surface perpendicularly to the axis, consist of continuous currents
#, v, w (569), and of variations of electrical displacement. We
have then
(II) *-" +
The equations of the currents (4) become
ax
When the medium is a dielectric, it establishes equilibrium
between the electromotive force, whose components are given by
equation (10), and the elastic reactions developed by the displace-
ment. If is the value in electromagnetic units of the coefficient
K.
of electrical elasticity that is to say, the ratio of the electromotive
force to the displacement we shall have
EQUATIONS OF THE ELECTRICAL FIELD. 555
In the case of a conductor, on the contrary, the displacement is
null, and if c is the specific conductivity of the medium, Ohm's law
becomes
Pc=u t
(14) Qc=v,
Finally, if p is the density of the free electricity at the point in
question, we shall have, with a dielectric, the condition
and, in the case of a conductor,
TNn ~t\i/
(16)
The twenty values F, G, H ; P, Q, R; X, Y, Z ; u, v, w; f, g, h;
u', v', w' ; p and ^ can be determined from the twenty equations in
numbers (2), (10), (n), (12), (13), (14), (15), and (i 6), when the
conditions of the problem in each special case are known.
556 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
CHAPTER VII.
PHENOMENA OF INDUCTION IN NON-LINEAR
CONDUCTORS.
573. MAGNETISM OF ROTATION. Following up an observation
by Gambey on the deadening of the oscillations of a compass
needle, Arago showed, in 1824, that a magnetised needle placed
above a disc which is in rapid rotation, is carried along by the
disc, and tends to acquire a rotation in the same direction.
The action exerted on a pole in these three conditions has
three components; the one tangential which impels the pole in
the direction of the rotation, another perpendicular which tends
to remove it from the disc, and finally a third directed along the
radius. This latter is null when the pole is at a distance from
the axis equal to about two-thirds of the radius of the disc ; nearer
the axis the pole is attracted to the centre ; when nearer the edges,
it seems repelled towards the edges.
The motion of the needle is more marked with a good conductor
like copper, than with a metal which does not conduct so well,
such as brass, and particularly antimony. When there are breaks
in the continuity of the disc (as for instance with a radial saw cut)
the effect produced is enfeebled.
These phenomena were at first ascribed to a special form of
magnetisation, and were known as magnetism of rotation. They
are really produced by induction currents developed in the metal;
but it was only after Faraday's great discovery that they were
ascribed to the true cause.
574. CONDUCTING SHELLS. The problem raised by Arago's
experiment is that of induction in a conductor of two dimensions.
Maxwell solved this problem in a very elegant manner, by the use of
a method analogous to that of electrical images.
CONDUCTING SHELLS. 557
Let us consider an infinitely thin homogeneous conductor, which
may be supposed reduced to a surface, and in which exist, for any
reason, electric currents which are not brought from without by elec-
trodes these currents are necessarily closed, and the stream-lines
cannot intersect each other. The annular space comprised between
two infinitely near currents, may be regarded as a linear circuit
traversed by a current of strength d&. This current may be replaced
by a magnetic shell of the same strength and the same contour.
If the surface of the conductor is thus cut into infinitely thin
bands by the stream-lines, it is seen that for an external point, the
sum of the currents will be equivalent to a complex shell (333), the
magnetic strength < of which is equal at each point to the sum of
that of the superposed shells. On the positive face of the shell
the currents are in the opposite direction to the hands of a watch,
about the spaces where the value of 4> is a maximum. This value
is null at the edge if the plate is bounded.
Along a current line the value of < is constant ; the current lines
are equipotential lines of the function <. An element dn of a
line , at right angles to the current line, is cut perpendicularly by
rffc
a current of strength dn, in the direction of the right of an
dn
observer, who, along this line, would move towards points where
the function < increases. Finally an element ds of any given curve
t3>
is cut by a current of strength ds' t but which is no longer per.
pendicular.
The expression for the magnetic potential on the outside is
= Utfco
This function is discontinuous on traversing the surface ; the
two values V 2 and V 1 which it assumes on either side of the shell,
on the positive and on the negative face (330) are connected by
the equation,
The component of the magnetic force perpendicular to the
surface is continuous, for it represents, on either side, the flow of
induction for unit surface. This is the case also with the tangential
d<l>
component along a stream-line, for then we have - = .
Of
558 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
On the contrary, the tangential component along a perpendicular
to the stream-lines is discontinuous, and on both sides of the
surface we have
on on an
The value of V may also be expressed (360) as a function of
the potential Q of a layer which covered the same surface, and
the density of which is equal at each point to the strength < of
the shell.
575. CASE OF A PLANE SHELL. Let us consider, as a special
case, a plane conducting plate in the plane xy, and suppose that
the positive face of the currents is at the top ; the potential of
the corresponding magnetic shell at a point whose co-ordinates are
x, y y and z, is expressed by (362)
The function Q which represents the potential of a layer whose
density at each point is equal to the magnetic strength of the shell,
is symmetrical in respect of the xy plane, and does not change
when z is replaced by -2. The function V, on the other hand,
changes its sign with z, and its absolute value is the same at two
points which are symmetrical in reference to the shell. We have
therefore, for corresponding points of the positive and of the
negative face,
The components X and Y of the magnetic force parallel to
the axes on the positive face, and the values of these components
X' and Y' on the negative face, are given by the equations
Y= _^__ 27r ^
JL. ^ / _
oy oy
PLANE SHELL. 559
576. Let us now investigate the currents in the plate. The
component u of the current parallel to the x axis, which intercepts
unit length parallel to the y axis, is
and the component v of the current parallel to the y axis
If a- be the resistance of the plate for unit surface, the fall of
electrical potential for unit length will be <ru parallel to the x axis,
and (TV parallel to the y axis ; this fall of potential is but the electro-
motive force in the same directions. Hence, from the expressions
found above (568), we have for a field variable with the time,
The equations (2) give, for the positive face,
27T
from this follows
- <)G
The general equations (567) which connect the components of
the total electromotive force of induction with the magnetic
potential give, in the present case,
3G 3H _ DV_
^z *by "bx
560 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
These equations are satisfied by putting
F JQ r JQ
~ G= '
577. If, at the same time, there is an external system of magnets
or of currents, we know (498) that its action on a surface
surrounding it is equivalent to that of a suitable system of surface
currents. .The magnetic potential of this system, in the positive
surface of the shell under consideration, might then be expressed
by a function Q', analogous to the function Q ; we shall have then
which gives
Putting = R, equations (4) become
27*
Integrating the first with respect to y, or the second with respect
to #, we get
For a complete integration we should add an arbitrary function of
/, but it must be observed that this function will disappear whenever
we take a partial differential in respect of x or y to calculate the
components of the current ; it need not then be taken into account.
578. Let us suppose, at first, that there is no external magnetic
body that is to say that Q' is null. This would be the case of a
system of currents set up in the shell and left to themselves ; these
currents would act upon each other by their mutual induction, and
MAGNETIC IMAGES. 561
would rapidly lose their energy owing to the resistance of the
conductor. Equation
gives then
(7)
Hence the value of the function Q, at the time /, in a point at a
distance z from the plane on the side of the positive face, and the
co-ordinates of which are x, y, z, is the same as for the time /= 0, at
the point x, y, and z + R/.
It follows that if a system of currents has been established in an
unlimited and uniform plane, and then left to itself, the magnetic
effect of these currents on a point on the side of the positive face, is
the same as if the plane moved parallel to itself along the normal and
on the negative side with a constant velocity R. The diminution of
electromotive force in consequence of the enfeeblement of the
currents is exactly represented by the diminution of the magnetic
field which results at each point from this imaginary motion.
579. MAGNETIC IMAGES. The integral of equation (6), in
respect of /, gives for points on the surface of the shell
(8)
If we suppose that, Q and Q' being at first null, the external
system is suddenly created on the positive side, in such a manner
that the corresponding potential Q' passes suddenly from to Q',
we shall have at the outset and for the surface of the shell, since
the integral is null,
Q=-Q'.
Hence, for all points of the plate, and therefore for all points of
the negative face, the initial system of currents produces an effect
equal, and of opposite sign, to that of the real system placed on the
positive side. Their effect is then the same as that of a magnetic
system identical and of contrary sign to the real system, and which
would coincide with it.
o o
562 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
For points on the positive side, the effect of the currents is the
same as that of a system of the same sign as the real systems, and
which would be symmetrical with it in respect of the conducting
plane ; we shall call it the positive image of the system.
The action of the currents of each side of the shell may then be
considered as produced by an image of the magnetic system, positive
or negative that is to say, of the same sign as the system, or of the
contrary sign, according as the point in question is on the positive or
negative side of the shell.
If the conductivity of the shell were infinite, we should have R = ;
the second member of the equation (8) will always be zero, and the
condition Q = - Q' will be always satisfied. The plate will be an
absolute screen (563) for all points on the negative side. The
currents will be permanent, and their effect will be represented at
each instant, for all points of space, by that of one or the other of
the two fixed images.
In a real sheet the resistance R has a finite value. The currents
produced by the sudden introduction of a magnetic system begin at
once to decrease, and their effect on each side is at each instant
represented by that of two images of the system, which would recede
perpendicularly from the sheet on each side with the velocity R.
580. INDUCTION OF A MOVABLE MAGNETIC SYSTEM. The
principle of images enables us to determine induced currents by the
variations of any given magnetic system on the positive side of the
shell.
The function Q', which determines the magnetic action, will
-\r\' ^)]y
vary by dt, while the system itself will vary by -^rdt. We may
consider this latter system as being itself a magnetic system, and
suppose that at the instant / there is suddenly formed on the
dM
negative side of the sheet a positive image of ^T^ which then
moves perpendicularly away with a constant velocity R. If the
system varies continuously, we may suppose that the different images
of the variations, relative to the different intervals of time, move
according to the same law as soon as they are formed, and thus form
continuous trails of images.
581. Suppose, for instance, that a positive pole +m moves in a
right line with a constant velocity #, parallel to the shell, and let us
assume that this pole has been suddenly created at the point A
(Fig. 120), which gives rise to an image +m at the symmetrical
INDUCTION OF A MOVABLE MAGNETIC SYSTEM.
563
point B. After an infinitely small time /, the pole comes to A'
(Fig. 121), at the distance ut ; it is as if we suddenly and simul-
taneously brought a pole - m to A, and a pole + m to A', producing
images of the same sign at B and B' ; but at this moment the first
image + m, which was at B, has come to C at the distance R8/. At
the period 28?, the mass +m is at A" (Fig. 122). There are then
two images equal to -m at C and B', and three positive images at
B", C' and D ; and so on.
LA'
X
X X'
X X'
! I i
I
I
7
B'
?..
1
y~
Fig. 120.
Fig. 121.
Fig. 122.
If the motion is continuous, it will be seen that the action which
the movable pole undergoes is that of two magnetic lines, one
positive and the other negative. If U is the resultant Ju 2 + R 2 of
the two velocities, the density of these two lines will be - and
Uo/
their distance yr&/ t ^ ie P r duct of the density by the distance is
Ru
then m . These two lines form an unlimited magnetised ribbon,
which starts from the point symmetrical with the actual position of
the pole, is situate in a plane passing through the trajectory of the
pole, and makes with the plane of the sheet an angle whose tangent
R
is equal to .
u
582. If the pole + m, instead of describing a right line, rotates
uniformly about an axis perpendicular to the sheet, we may suppose
that the band which precedes, forms a helix on the right cylinder
which has the circumference described by this pole for its base.
002
564 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
For a magnet reduced to its two poles, and turning about its
centre, the induced currents are also equivalent to the system of two
magnetic ribbons coiled on the same cylinder. Lastly, the currents
produced by the displacement of any magnetic system, rotating
uniformly, are equivalent to a system of magnetic ribbons which
correspond, point to point, to the different masses of the system.
583. CALCULATION OF THE ACTION OF INDUCED CURRENTS.
To calculate the effect of these images, let us denote by Q T
the value of the potential Q, determined by the currents of the
shell at the point whose co-ordinates are x, y, z + RT, and at the
period / - T ; by Q' T , the value of the potential Q', determined by the
magnetic system at the point x, y, (z + RT), and at the same
period / - T. The potential Q T , being a function of the co-ordinates
x t y t 2 + Rr, and of t - T, we have
,, JQ, R 3Q, JQ,
~ =:K ~~
Equation (6) applied to this function becomes then
Integrating this equation in reference to T between the limits
T = 0, and T = co , we shall have the value of the function Q for the
period /, which gives
()
r/03
<V T ~~(V
Jo
The function Q on which the solution of the problem depends,
since it enables us to calculate the action of induced currents at
each point, is thus determined by the function Q' T defined at each
instant by the condition and the motion of the external magnetic
system.
584, CASE OF A SINGLE POLE. We may apply this method
of calculation to the case, considered above, of a single pole of
mass m, which moves uniformly in a rectilinear path in the
presence of an unlimited conducting plane ; but it is simpler
to treat the problem directly by the consideration of magnetic
images.
CASE OF A SINGLE POLE.
565
Let us first examine the action which is exerted on a point
A (Fig. 123), by a homogeneous magnetic line of density A, situate
along the right line XX'. The action of an element MM' or ds, at the
distance r from the point A, is equal to . Let us denote by da
the angle between the two lines AM and AM', and let fall the two
x'
M: M
Fig. 123.
perpendiculars MQ upon AM' and AP, or h on XX'. Then from
the similar triangles APM and MQM' we have the ratio
MQ AP
MM'~AM'
or
rda. h
from this follows
Xds_ da_ Ma_ NN'
7* it" "FT &
The action of the element ds on the point A is therefore equal
to that of the element NN' of the circumference of radius ^, which
has the same density A, or to that of the corresponding element
of the circumference of radius i, where the density is -. If
the line in question was bounded at the points B and B 15 its
action is equal to that of the arc of circle bb v We at once see
that this latter action is proportional to the length of the chord,
which gives
A. , . a 2\ . a
/ = 2ti sm - = sm - .
J h 2 2 h 2
566 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
585. Let us now suppose that a movable pole, of mass m,
moving uniformly in a straight line with the velocity #, and having
been in motion from an infinite time before the period in question,
is at the point A, at a distance c from the conducting shell X'X
(Fig. 124), and comes from a direction AjA such that it has not
yet traversed the shell. Let be the angle of this direction with
the perpendicular N to the shell.
N
Fig. 124.
The angle a, which the magnetic trails B'jB' and BjB, relative
to the two successive positions A' and A of the pole make with N,
is defined by the triangle BB'C, which gives
sma =
u sin
TT~
CASE OF A SINGLE POLE. 567
The distance of the two lines along B'B is uSt, and, along the
common perpendicular, RS/ sin a. Lastly, the density of each of
them is
m
Let us draw the right line AQ parallel to the magnetic trails;
let a and a' be the angles of the arcs bq and b'q' which correspond
to them, h and h' the perpendiculars AP and AT'. The actions
f and f of these two lines are respectively directed along the
bisections of the angles a and a', and in opposite directions ; they
<x
make with the perpendicular, angles respectively equal to - and
a -- , and we have
2m a
2m
The components Z and X of the resultant force, measured
perpendicularly, and parallel to the sheet, the first upwards, and
the second in the direction of the motion of the movable pole, are
a
Z =/ cos
F 2 sin cos (a )~|
a / a\ WSina 2 \ 2/
"/COS (a --)=__!_ __ _J
a .a' / a'\ _
sin 2 - sin sin I a
(a\ 2m 2 2 \ 2/
-/"uiL-F 1 -a J-
X=/sin--/' sin
2
We have further
h = 2C sin a,
h' h
or, disregarding the infinitely small quantities of the second order,
568 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
The triangle ABB', gives also
a- a sin
or
us'mO Usina
a =a
2C
Substituting these values of h' and a! in the expressions of the
components, and neglecting quantities of the second order, we get
~4' 2 U
m R a
X= - s^r tan .
4^ 2 U 2
The force itself makes with the surface an angle /?, determined
by the conditions
Z U-R a
The induced currents act thus in opposition to the motion of
the pole, as might be foreseen ; but it is not directly opposed.
If the motion of the pole is perpendicular to that of the
shell, we have = 0, a = 0, and U = R + ; we get then
m
f-i = -
The action is the same as that of a single mass equal to
m - -- , situate at each instant at the point B symmetrical with
the position of the pole, or of a mass - placed at the base
of the perpendicular let fall from the pole upon the plane.
If the pole is displaced parallel to the plane, we have
JS.
CASE OF A SINGLE POLE. 569
which gives
Z = --^- F= mU
m
~
The force is perpendicular therefore to the direction of the
magnetic trail; it is the same as if there were a mass equal to
m U 2
- ._ TT . in the plane m its direction.
4 (R + U)^
We may suppose again that the force F is produced by an in-
finitely small magnet situate in the plane. Applying Gauss' formula
(154), for instance, we find that this magnet is situate behind the
projection of the pole, at a distance x, defined by the equation
the magnetisation being parallel to the direction of the motion.
The moment of this magnet would moreover be easy to calculate.
The component X is a maximum, for a given value of R, when
u=i.2 f jR; it is zero for R = 0and R = co.
The component Z tends to move the pole from the plate; it
increases with the velocity, and tends towards the value , when
the velocity tends towards infinity.
586. In the case of a uniform rectilinear motion parallel to
the plane, we may consider the phenomena in still another
manner.
The magnetic ribbon which starts from the point B (Fig. 125),
consists of two magnetic lines, the density of which is , and
the horizontal distance uSf.
Let us denote by s the distance M'B from any point M', and
by x' the distance M'K. For an element ds of the ribbon the
magnetic moment or' is
u
?3' = m ds = mds sin a .
As x' s sin a, we have &' = mdx.
570 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
The action on A of this infinitely small magnet placed at M',
at the distance /, is equivalent to that of an infinitely small magnet
of moment 57 placed at M, at the distance r, and such that
*/.
Fig. 125.
If x and c are the distances MO and AO, we have
c c x cot a
r x
r' x 2C + x cot a
2C
From which we get
m c - x cot a
4 *
It is seen that the action of the induced currents on the point
A is equivalent to that of a complex solenoid (327) situate on the
plane starting from the point O, in a direction contrary to that
of the motion, and the magnetic strength of which at each point
will be
CT m c - x cot a
dx 4 c
We find accordingly, for the components X and Z of the force,
the same values as before.
ARAGO'S EXPERIMENT.
571
587. ARAGO'S EXPERIMENT. This method may be generalised.
Suppose that the pole describes a circle of radius , in the opposite
direction of the hands of a watch when looked at from above.
The action of the induced currents is that of two homogeneous
helices of contrary signs, or of a helicoidal ribbon BM' (Fig. 126)
Fig. 126.
coiled on the cylinder, with the axis of rotation for its axis, and
passing through the pole. Each element of this helix is magnetised
along a tangent to the cylinder, drawn perpendicularly to the axis,
and its action on the pole is equivalent to that of an infinitely
small magnet situate at a point M in the conducting plane. The
locus of the point M is the perspective curve of the helix seen
from the point A. If r is the radius vector MO, the angle
which it makes with the tangent at the point O, and observing
that the angle 6 is half the angle </> of the two planes passing
through the axis, and through the points B and M', we have, by
the triangles AMO and AM'K,
__, T;r 20 sin .
p M K 2 a sin
c AK 2C + a<j> cota c+a6cota'
This curve consists of a series of closed rings which have a common
tangent at the point O.
572 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS.
The arc KM' being equal to a$ or 2aO, the magnetic moment
T3 of the element of the helix at M' is zmadO. The magnetic
moment GT of the corresponding magnet at M is given by the ratio
CT 7 2ma
Observing that we have
r MO
r' M'K 20sin<9'
it follows from this that
m
4 a 2 sin 3
The magnet TS at the point M is parallel to the magnetisation
of the helix at M' ; it makes therefore the angle 6 with the radius
vector of the perspective curve.
The calculation of the force at A would be very complicated, but
it is evident that the portions of the curve corresponding to the first
part BM' of the helix are predominant. From the direction of the
elementary magnets on the perspective curve, we see that the action
on the point A will have a vertical component, another directly
opposite to the velocity of the pole, and a third directed towards the
centre of the circumference which it describes. The entire system
is then equivalent to a small magnet placed behind the point O,
perpendicular to a radius of the disc, which makes a certain angle
from the opposite side of the motion with the radius corresponding
to the pole, the magnetisation being in the direction of the motion if
the pole in question is a north pole.
If the plane was unlimited, this magnet would be at a distance
from the axis greater than that of the pole ; but the action of the
edges is to bring it more and more towards the centre in proportion
as the radius diminishes. We thus find all the peculiar features of
Arago's experiments, among others the fact that the radial component
is centripetal so long as the pole is away from the edges, and that it
becomes centrifugal as it approaches them.
588. If the pole describes any given curve parallel to the plane,
we should obtain in the same way, by the trail of corresponding
magnetic images, the magnetisation at each point of the perspective
INDUCTION ON ANY GIVEN CONDUCTOR. 573
curve. The action on the point A in the case of an unlimited plane,
will still have a vertical component, another directly opposed to the
motion, and a third perpendicular to the trajectory of the pole, and
directed towards the concavity of the curve.
589. DAMPING OF COMPASS NEEDLES. The action of a con-
ducting disc on a magnetic system in motion is used in compasses,
and galvanometers, to deaden the oscillations of needles, in the form
in which the phenomenon was first observed by Gambey.
This reciprocal action is equivalent to a kind of friction which
hinders the relative movement of the two systems ; from this follows
an absorption of energy which exactly corresponds to the heating of
the conductor by induced currents.
590. INDUCTION ON ANY GIVEN CONDUCTOR. More generally,
whenever a conductor of any given form is displaced in a magnetic
field, induced currents result, which oppose the motion j but the
calculation of the effects of induction presents in that case the
greatest difficulties, for the three dimensions of the conductor come
into play. Faraday observed in this way that if a copper cube
suspended by a thread is placed between the poles of an electro-
magnet and is made to rotate rapidly, the cube will stop when
a current passes through the coils, and a considerable resistance
is experienced to its being made to rotate again.
Foucault conceived the idea of utilising this experiment to
render evident the heating of a conductor. By means of a system
of toothed wheels worked by a handle, he maintains a conducting
disc in rotation between the poles of a very powerful electromagnet ;
the work expended is considerable, and the temperature of the disc
rises very rapidly. The measurement of the work expended, and
of the corresponding heat even furnishes a means of determining
the mechanical equivalent of heat; this is the principle of the
method used by M. Violle.
574 OPTICAL PHENOMENA.
CHAPTER VIII.
OPTICAL PHENOMENA.
591. FARADAY'S DISCOVERY. After prolonged researches, which
for a long time were unfruitful, Faraday discovered in 1845 that
a transparent body, though itself destitute of rotatory power, becomes
capable, under the influence of magnetism, of rotating the plane of
polarization of a luminous ray. The effect is at its maximum when
the polarized ray traverses the body parallel to the lines of force ;
it is zero when the two directions are at right angles.
This phenomenon, which was first observed in the case of heavy
flint glass, is produced in all single refracting liquids and solids ;
the action of magnetism is less perceptible in double refracting
bodies ; it is extremely feeble in gases and vapours, and it is only by
quite recent experiments that it has been ascertained to exist.
Bodies naturally endowed with rotatory power give rise to the
same phenomenon ; the two rotations become added or substracted
according to their respective directions.
592. POSITIVE AND NEGATIVE BODIES. All the bodies examined
by Faraday rotate the plane of polarization in the same direction
under the influence of magnetism ; it is that direction of the current
which, revolving around the ray, would give to the field its actual
direction. All these substances are diamagnetic. Verdet found that
most magnetic substances (for instance, solutions of ferric chloride
in alcohol or ether) cause the plane of polarization to rotate in the
opposite direction. If we consider the former rotation as positive,
we may in general, though not with absolute strictness, assert that
diamagnetic substances turn the plane of polarization in the positive
direction, and magnetic substances in the negative direction.
593. There is an important difference in the way in which the
rotation of the plane of polarization takes place, according as we
consider the natural rotation or the magnetic rotation. In both
VERDET'S LAWS. 575
cases the angle of rotation is proportional, other things being equal,
to the thickness of the medium traversed ; but in quartz, in solution
of sugar, in essence of turpentine, the rotation is connected with the
propagation of light in such a manner that it is always in the same
direction for the observer who receives the rays. It follows from
this, that if the ray, after having traversed the transparent substance,
returns on its own path after having been perpendicularly reflected,
it undergoes a rotation which is equal and opposite to the first,
and the plane of polarization reverts to its primitive position at
its starting-point.
The magnetic rotation, on the contrary, is independent of the
direction of the propagation, and only depends on the direction
of the magnetic force. The ray, which returns on its own path
after a normal reflection, undergoes a rotation in the same absolute
direction, which adds on to the first ; in this way, by causing the
ray to be perpendicularly reflected an unequal number of times,
272 + 1, we may observe the same rotation as if it had traversed
a layer of the substance zn + 1 times the thickness.
594. VERDET'S LAWS. Verdet proved, experimentally, that for
a homogeneously polarized ray, the rotation of the plane of polari-
zation is proportional :
ist To the thickness traversed;
2nd. To the component of the magnetic force in the direction
of the ray ;
3rd. To a coefficient depending on the nature of the body,
and which is positive or negative, according as the body is dia-
magnetic or magnetic.
These laws may be summarised in the following statement :
The rotation of the plane of polarization between two points is
proportional to the difference of magnetic potential between these
points.
Let V and V be the values of the potential at two points A
and A' in the path of the ray ; the angle 9 by which the plane of
polarization is turned between these two points will be expressed
by the ratio
= <o(V-V),
w being the rotation which for a given substance corresponds to a
difference of potential equal to unity. This quantity is known as
Verde 'fs constant ; it defines the magnetic rotatory power of the
body.
576 OPTICAL PHENOMENA.
Verdet demonstrated that when a salt is dissolved in water the
water and the salt each bring into the solution their special rotatory
power; the rotation produced by the solution is the algebraical
sum of the rotation due to each of the bodies composing it. Thus,
water having a positive rotatory power, and ferric chloride a negative
rotatory power, a solution of ferric chloride rotates the plane of
polarization in one direction or the other, according to its concen-
tration. The law may therefore be considered to be general.
595. MAGNETIC ROTATORY DISPERSION. For the same body
the value of the constant <o varies with the wave length. The
direction of the variation is the same as for the natural rotation ;
in both cases, in fact, the rotation is approximately in the inverse
ratio of the square of the wave length. In fact, the products of
the rotation by the square of the wave length increases as the
wave length diminishes. Both in the case of natural and of
magnetic rotation, the substances for which the increase is most
marked are just those with the greatest dispersive power.
Some months after the publication of Faraday's discoveries, Sir
George Airy remarked that the phenomena could be accounted
for, by adding to the known equations for the vibratory motion
of isotropic substances, certain terms proportional to the differentials
of the odd orders of the displacements in respect of the time.
Among the various formulae at which we arrive, by making special
hypotheses as to the nature of the terms to be added to the
equation, the following
, dn\
= m ( n- A ) ,
d\)
in which m is a constant, and n the refractive index of the substance
for a ray of wave length A, gives an almost complete agreement
with experiment. The constant m, as we shall afterwards see,
will be inversely as the magnetic permeability.
M. H. Becquerel observed that the quotient of the rotatory
power by the product n 2 (n 2 -i) varies very little with different
substances, and that this quotient is constant for bodies of the
same chemical family.
596. MR. KERR'S EXPERIMENT. When a ray of polarized light
is reflected from the pole of a magnet, its plane of polarization,
from Kerr's experiments, experiences a manifest rotation; it is
advantageous to make the reflection perpendicular in order to avoid
the effects of elliptical polarization. On a positive or north pole,
EXPLANATION OF ROTATORY POLARIZATION. 577
the rotation is towards an observer's right that is to say, inversely
as the current which would produce the magnetisation. It is
difficult to assert whether this is a simple magnetic rotation due
to the gas, or, as Kerr believes, a new phenomenon.
597. EXPLANATION OF ROTATORY POLARIZATION. The
theoretical principles put forth by Fresnel to explain the rotatory
polarization of quartz and of active liquids, may be applied to
magnetic rotatory polarization.
It is known that a ray of light polarized in a plane is equivalent
to two rays polarized circularly in opposite directions of the same
period, moving with the same velocity, and the amplitude of whose
vibration is half that of the resultant rectilinear vibration.
In order to get a conception of each of these circular rays,
we shall assume that, all the molecules in the same straight line
being disturbed from their position of equilibrium, and arranged
along a helix having this right line as axis, a uniform rotation
about the axis is imparted to the system. Each point of the
system, which represents a vibrating molecule, describes a circum-
ference about the axis, and at the same time the helix acquires
an apparent longitudinal motion which represents the propagation
of the undulation. The wave length, which is the distance traversed
during a period, is represented by the thread of the screw. If the
screw is right-handed, like an ordinary one, the vibration is in the
direction of the hands of a watch for an observer towards whom
the propagation takes place. The ray is said to be circularly
polarized to the right, or more simply, that it is a right circular
ray. The circular ray is left when the vibration is in the contrary
direction to the hands of a watch for an observer towards whom
it is moving.
If we superpose in this way two helices in opposite directions,
starting from the same point A, and if each molecule of the
medium shares this double motion, the successive positions which
it will occupy in consequence of the two circular vibrations, will
always be symmetrical in reference to the plane passing through
the ray, and the point A ; the resultant vibratory motion is always
in this plane, and therefore the ray remains rectilinearly polarized
in its original plane.
It may be assumed that matters take place in this way when
a ray polarized rectilinearly traverses a transparent isotropic sub-
stance in the natural state, like Faraday's flint glass. But if this
glass is placed in the magnetic field for instance, inside a cylindrical
coil and if the light is propagated parallel to the lines of force,
p P
578 OPTICAL PHENOMENA.
and in that direction, the ray is still polarized on emergence, but
the plane of polarization has turned through a certain angle in
the direction of the external currents of the coil that is, towards
the left of an observer who receives the ray. The rotation is
towards the right, on the contrary, if the light travels in the
opposite direction.
Hence, on the hypothesis of circular vibrations, it is necessary
that during its passage through an active medium, one of the rays
should have got an advance of phase over the other equal to
twice the angle of rotation of the plane of polarization. For flint
glass and diamagnetic substances, the left circular rays gets in
advance when the propagation is in the direction of the lines of
force ; the reverse is the case for magnetic media.
In any case the conception of two inverse circular rays which
travel through a medium with different velocities is not a mere
hypothesis; experiment shows, in fact, .that the refractive index
of a given circular ray differs according as it traverses the sub-
stance submitted to the action of magnetism in the direction of
the lines of force, or in the contrary direction.
M. Cornu has shown, moreover, that in both cases the variations
in the indices are equal and of opposite sign to that which the ray
would have in the same medium when not under the action of
magnetism.
598. This difference of phase may be variously explained ; it
may be assumed that the period is the same in both rays, but
the velocity of propagation is different ; or, that, while the velocity
of propagation is the same, the period is no longer equal for the
two rays, and is different for each of them from what it is in the
external medium ; or, lastly and what is, perhaps, most probable,
having regard to the ordinary laws of dispersion that the period
is modified as well as the velocity of propagation.
It is in general impossible to conceive a permanent vibratory
state with a period different from that of the cause which produces
it; but, in the present case, the difficulty does not seem to exist,
if we assume that the medium which transmits light itself possesses
a rotatory motion in a determinate direction ; the period of the
relative motion would be the same for both rays, and the same as
in the external medium ; the advance of phase would be due solely
to th'e difference of the absolute motion, and precisely equal to the
half of this difference.
In all cases when the two circular rays emerge from the medium,
they assume the same period and the same velocity of propagation,
EXPLANATION OF ROTATORY POLARIZATION. 579
and they reconstitute a rectilinearly polarized ray. The difference
of time that these two rays of velocities V and V" take to traverse
a thickness e of the medium is
[ I
-"v 7
If T is the period of the ray in air, interference occurs on
emergence between two rays whose difference of phase on entering
was
As these rays have the periods T' and T" in the medium, the
numbers m and m" of oscillations which they have made are
=- and m"T"
which gives
The difference of phase on emergence is then
/ , A f T / i i\ i /i i\-|
(2) 27r( m" - m +- ) = 27rH ( + - ) -- ( + - )
\ T/ |_ V '\T" T/ V'\T' T/J'
and the rotation of the plane of polarization
(3) e=
If V and X denote the velocity, and the wave length of the
ray in air, we may write
(4) ^ r
P P 2
580 OPTICAL PHENOMENA.
599. Whatever be the cause which modifies the circular vibra-
tions, it is to be foreseen that the effects, being very small, become
equal and of opposite signs on the two inverse rays, and that they
are proportional to the magnetic force X.
Hence, if U is the velocity, and T the period of the ray in
question, if the magnetic field were suppressed, and if n is the
refractive index corresponding to the velocity U,
(5)
and
The rotation of the plane of polarization becomes, in that case,
The field being supposed uniform, the product ^X represents the
difference of potential at the entrance and emergence of the ray ;
we shall have then
For ordinary rotation, Fresnel assumes that the period does not
change. If this be admitted, the coefficient ft is null ; if n' and n"
are the refractive indices of the two circular rays, we have simply
_.. V V\ 27T72 -H 47772
(8) o> = - / =- = a
600. This would be the rotation if the velocity of propagation
were independent of the wave length ; but it must still be observed
that the rotatory polarization brings about a dispersion between the
two circular rays.
EXPLANATION OF ROTATORY POLARIZATION.
581
The velocities V' and V correspond to two different lengths
A/ and A", and the values of U and of #, which are introduced into
the second term of the equation (4), refer to the propagation of the
vibrations of these two lengths of wave in the medium in the natural
state. Hence, restricting ourselves to terms of the first order, we
may write
(5)'
n' =
dn lX K
^A A^
dn
dX
it follows from this that
AT
(9) -i
A
We have, moreover,
7^'-^
^"-A);
i dn
_(r-V) .
aX^A J
A" V
I+aX + - (A"- A)
^ /2 A
which gives, to the same degree of approximation,
A" -A' / \dn\
= -A( i- - ) = -A.
7
The magnetic rotation then becomes
(10)
=a ^,_x-).
d\
We see that the result, independent of the dispersion, must be
multiplied by the factor (i --- ) .
n d\
The coefficient a is itself a function of the wave length. To
agree with the formula of (595) which best satisfies the experiments,
the coefficient must be proportional to . We shall see from what
A
considerations this formula may be theoretically deduced.
OPTICAL PHENOMENA.
601. OBSERVATIONS OF SIR W, THOMSON. According to Sir
W. Thomson, the phenomena of magnetic rotatory polarization
would appear to confirm Ampere's ideas on the ultimate nature of
magnetism.
"The magnetic influence on light, discovered by Faraday, de-
pends on the direction of motion of moving particles. For instance,
in a medium possessing it, particles in a straight line perpendicular
to the lines of magnetic force displaced to a helix round this line
as axis, and then projected tangentially with such velocities as to
describe circles, will have different velocities according as their
motions are round in one direction (the same as the nominal
direction of the galvanic current in the magnetising coil), or in the
contrary direction. But the elastic reaction of the medium must
be the same for the same displacements, whatever be the velocities
and directions of the particles; that is to say, the forces which
are balanced by the centrifugal force of the circular motions are
equal, while the luminiferous motions are unequal. The absolute
circular motions being therefore either equal, or such as to transmit
equal centrifugal forces to the particles initially considered, it follows
that the luminiferous motions are only components of the whole
motion ; and that a less luminiferous component in one direction,
compounded with a motion existing in the medium when trans-
mitting no light, gives an equal resultant to that of a greater
luminiferous motion in the contrary direction compounded with
the same non-luminous motion.
"I think it is not only impossible to conceive any other than
this dynamical explanation of the fact, that circularly polarized
light transmitted through magnetised glass parallel to the lines of
magnetising force with the same quality, right-handed always, or
left-handed always, is propagated at different rates according as its
course is in the direction, or is contrary to the direction in which
a north magnetic pole is drawn; but I believe it can be demon-
strated that no other explanation of that fact is possible. Hence
it appears that Faraday's optical discovery affords a demonstration
of the reality of Ampere's explanation of the ultimate nature of
magnetism ; and gives a definition of magnetisation in the dynamical
theory of heat.
"The introduction of the principle of moments of momenta
('the conservation of areas') into the mechanical treatment of
Mr. Rankine's hypothesis of ' molecular vortices,' appears to in-
dicate a line perpendicular to the plane of resultant rotatory
momentum (' the invariable plane ') of the thermal motions, as the
ELECTRICAL DOUBLE REFRACTION. 583
magnetic axis of a magnetised body, and suggests the resultant
moment of momenta as the definite measure of ' magnetic
moment.' The explanation of all phenomena of electromagnetic
attraction or repulsion, and of electromagnetic induction, is to be
looked for simply in the inertia and pressure of the matter of
which the motions constitute heat. Whether this matter is or is
not electricity whether it is a continuous fluid interpermeating the
spaces between molecular nuclei, or is itself molecularly grouped
or whether all matter is continuous, and molecular heterogeneous-
ness consists in finite vortical or other relative motions of contiguous
parts of a body, it is impossible to decide, and perhaps in vain to
speculate, in the present state of science." (Reprint of Papers^
p. 419.)
ELECTRICAL DOUBLE REFRACTION. It is known that whenever
a singly refracting transparent body is subjected to a mechanical
action, this body acquires for the moment double refracting pro-
perties, the axis of double refraction being directed along the line
of pressure or of traction.
Kerr has shown that any singly refracting solid or liquid placed
in an electrical field acquires a transient double refraction ; the
axis of double refraction coincides with the line of force, and
according to the nature of the body, the velocity of the extra-
ordinary ray is greater or less than that of the ordinary ray.
If 8 is the intensity of double refraction that is the difference
of path between the ordinary and extraordinary ray for unit thickness
of the dielectric and if V is the difference of electrostatic potential
between two points at a distance d, the electrical force F in the
V
region in question is equal to . Kerr deduces from his experi-
ments the ratio
V 2
k being a constant characteristic of the body, positive or negative
according to circumstances. It follows from this that the intensity
of electrical double refraction is proportional to the square of the
electrical force.
We have seen (107) that the dielectric may be considered as
subjected to a strain in the direction of the lines of force propor-
tional to the square of the force. It appears then that the phe-
nomenon observed by Mr. Kerr may be considered as an accidental
double refraction, due to the electrostatic tension of the medium.
584 ELECTRICAL UNITS.
CHAPTER IX.
ELECTRICAL UNITS.
603. FUNDAMENTAL UNITS. DERIVED UNITS. In the phe-
nomena of electricity and magnetism experimenters have for a
long time evaluated the various quantities only as functions
of arbitrary units, the choice of which was determined in each
case by the convenience of experiments. This method, even when
the units employed were suitably denned, has the inconvenience
not only of making it very difficult to compare the results obtained
by different observers, but particularly of masking the relations
which may exist between various orders of phenomena. It is then
of the greatest importance, for the progress of science, to arrive
at a common understanding as to the choice of the units, and at
the same time that the units adopted shall have that character of
co-ordination amongst themselves which constitutes the superiority
of the metrical system.
The units corresponding to the different kinds of magnitudes
may, in fact, be chosen arbitrarily, and independently of each
other ; but there is an obvious advantage in making them depend
on as small a number of simple units as possible. Thus, in
geometry, the unit of surface and the unit of volume may be
derived from the unit of length. In Kinematics we introduce
with the velocity a new idea and a new unit that of time. The
study of dynamics leads to a third unit, independent of the two
former the unit of force or the unit of mass. All mechanical
magnitudes may thus be evaluated as functions of the three units
of length, of time, and of mass, or of length, of time, and of force.
In a co-ordinated system, the irreducible units are called fundamental
units ; the others are called derived units,
All magnitudes which we deal with in electricity and magnetism
have been denned by their mechanical properties ; they may
therefore be measured, like the mechanical quantities themselves,
DIMENSIONS OF A DERIVED UNIT. 585
as a function of the three fundamental units of length, of time,
and of mass.
A system of measurements based on these principles is called
an absolute system, the word absolute being employed in contra-
distinction to the term relative, which would characterize a system
of measurements independent of each other.
604. DIMENSIONS OF A DERIVED UNIT. Let n be the numerical
expresssion of a quantity, that is the number of units which it
contains, and [N] the magnitude of the unit of comparison ; if this
number be taken equal to [N'], the magnitude to be measured
will be expressed by another number ri , and we shall have the
ratio
which gives
jjfJNj
n [NT
It follows from this that the ratio of the numerical values of a
given quantity is equal to the inverse ratio of -the magnitudes which
have served to measure it.
When the unit is a derived unit, and it varies in consequence
of a change in the magnitude of the fundamental unit, in order to
learn this latter ratio we must know in what manner the derived
unit depends on the fundamental units. The relation of a derived
unit to the fundamental units determines the dimensions of this unit.
We shall represent the fundamental units of length, mass, and
time, by the symbols [L], [M], and [T], and the magnitude of any
unit by a letter enclosed within a square bracket [x]. The dimensions
of the unit of surface will be represented by the symbol [L 2 ], and
those of volume by the symbol [L 3 ] ; that is to say that the
unit of surface varies as the square, and the unit o'f volume as
the cube of the unit of length.
More generally if the dimensions of a derived unit are expressed
by the symbol [L*MT r ], and if we take the values L, M, T, and
L', M', T' successively, as fundamental units, the ratio of the derived
units in the two systems will be
[N] \L \M T
586 ELECTRICAL UNITS.
605. DERIVED MECHANICAL UNITS. The principal derived
units in mechanics are velocity, acceleration, force, work or energy.
Velocity \v\. Velocity v is the path traversed by a moving body
in unit time, or the quotient of a length by a time. Hence, the
dimensions of velocity will be expressed by the formula
Acceleration [y]. Acceleration y, is the ratio of the increase
of velocity to the increase of time ; it is therefore the quotient
of a velocity by a time, and we have for the dimensions of the
unit,
Force [/]. Force / is the product of a mass by an accele-
ration, which gives
[/] = [LMT-].
Work, Energy [W]. Work or energy is the product ot a
force by a length ; vis viva, which is a quantity of the same kind,
is the product of a mass by the square of a velocity. In both
cases we have
[W] = [L 2 MT- 2 ].
The unit of force is that which acting on unit of mass for
unit time imparts to it unit acceleration.
The unit of work is the work produced by unit of force, when
its point of application is displaced in its own direction by unit
length.
These two latter units are not those in ordinary use ; the
weight is commonly taken as unit of force : for instance a
gramme or a kilogramme, and the kilogrammetre as unit of
work. This amounts to choosing the unit of force, instead of
the unit of mass, as the third fundamental unit. The choice of
a weight like that of the kilogramme in the Archives at Paris,
as the unit of force, has this inconvenience, that if this body, or
any other equivalent, is moved to another part of the globe, its
true weight will no longer represent the unit of force, in consequence
ELECTRICAL AND MAGNETIC DERIVED UNITS. 587
of the change of the intensity of gravity ; the mass of a body,
on the contrary, is an invariable quantity wherever it may be
placed.
It is easy to see what is the ratio between these two units ;
the formula p = mg, in which m represents the mass of a body,
and / its weight in a place where the acceleration of gravity is g,
shows that if the mass of a body is unity, its weight will impart
to it an acceleration equal to g, and is equivalent to g times the
unit of force as defined above, the acceleration being expressed as
a function of the length taken as fundamental unit.
Thus, if we take the metre and the mass of the kilogramme as
units, the unit of force is - - kilogramme, or about 100 grammes.
9'oi
With the kilogramme as unit of force, the unit of mass is that ol
a body weighing 9*81 kilogrammes.
606. ELECTRICAL AND MAGNETIC DERIVED UNITS. The most
important electrical magnitudes are the quantity of electricity, the
strength of the electrical field, the potential or the electromotive force,
the capacity, the strength of the current, the resistance, etc. We have,
in like manner, for the magnetic units the quantity of magnetism,
the intensity of the magnetic field, the magnetic strength of a shell, etc.
All these magnitudes are connected by the ratios which define them,
and if one is given the others follow from it.
In order to have an absolute system, the quantity which serves
as the starting-point must be measured directly in mechanical units.
Thus, the quantity of electricity might be defined by Coulomb's
law (7), or the quantity of magnetism by the corresponding law
(293), or again the strength of the current, by Ampere's electro-
dynamical law (473). Out of this arise three systems of absolute
measure, which are independent, and incompatible, in which the
various units are differently connected with the fundamental units,
and to which the names electrostatic system, electromagnetic system,
and electro dynamic system have been given.
There is no theoretical reason for preferring one system to the
others ; two of them, however, possess a greater practical importance :
these are the electrostatic and electromagnetic systems. The units
of the electrodynamic system only differ, moreover, by a numerical
factor from the corresponding electromagnetic units, and their
applications are less simple. We shall restrict ourselves to the
first two ; and shall represent the quantities measured in electrostatic
units by small letters, and expressions of the same magnitude in
the electromagnetic system by capitals.
588 ELECTRICAL UNITS.
607. ELECTROSTATIC SYSTEM. Quantity of Electricity [q\.
Coulomb's law gives, for the repulsion f exerted between two equal
masses q at the distance */,
or
-djf.
From which is deduced, for the dimensions of the unit of
electricity,
Surface density. Electrical displacement [a-]. The density is
the quantity of electricity for unit surface (18) ; the displacement
is the quantity which has traversed unit surface (126); we shall
have then,
Electrical force. Strength of the field \H\. This is the force
which acts on unit mass at the point in question, which gives
for the dimensions of the unit,
they are the same as that -of density, as could be foreseen from
Coulomb's theorem (35).
The j##w of electrical force is the product of electrical force into a
3 1
surface; its dimensions are [L 2 M 2 T~ 1 ].
Specific inductive capacity []. The specific inductive capacity
is a number in the electrostatic system.
Electromotive force or electrostatic potential \e\ The potential
of an electrical mass at a distance d, is the quotient of the mass
by this distance ; we have then
Electrostatic capacity \c\. The capacity of a condenser is the
quotient of its charge by the difference of potential of the armatures,
and we have
c- 9 -, or , = L.
ELECTROSTATIC SYSTEM. 589
Electrostatic capacity is therefore a length, as we have already
seen (73).
Strength of current [/']. The strength of a current is the quantity
of electricity which traverses the section of a wire in unit time,
or the quotient of a quantity ^, by the time which it takes to
traverse this section, We have accordingly
or
Resistance \r\. The resistance of a conductor is defined by
Ohm's law (204). It is the quotient of the electromotive force
between two points, by the strength of the current, which gives
-., and
Electrostatic resistance is therefore the inverse of a velocity,
as has already been proved (206).
Quantity of magnetism [/]. In the electrostatic system, the
quantity of magnetism is defined by the condition that the action
of a magnetic pole of mass q', on a portion of current whose strength
is / and whose length / is very small, at a distance d from the pole,
and perpendicular to the right line which joins its centre to the
pole, shall be defined by the elementary law (458)
or
from which is deduced
Magnetic density [o-'J. The surface density being the quantity of
magnetism for unit surface, we have
59 ELECTRICAL UNITS.
Magnetic force. Magnetic field \h'~\. This is the force which
acts on unit of magnetic mass ; hence,
[>'] = [/] [/-'
Magnetic potential [<?']. The magnetic potential is the work
of the magnetic force that is to say, the product of this force by
a length; we shall have then
Strength of magnetic shell [<]. The magnetic strength of a shell
is the product of the surface density by the thickness of this shell,
which gives
[</,] = [L~*M*].
608. ELECTROMAGNETIC SYSTEM. Quantity of magnetism [Q].
In the electromagnetic system, the starting-point of the measure-
ments is the definition of the quantity of magnetism by Coulomb's
law,
= ; where [Q']
They are, of course, the same dimensions as those of the unit
of electricity in the electrostatic system.
Surface magnetic density [.2T']. Magnetic force ; strength of the
field [H'J. Magnetic potential [E']. It follows from the preceding
remark that the dimensions of these various units will be the same
as those of the corresponding electrical quantities in the electrostatic
system ; that is to say,
ELECTROMAGNETIC SYSTEM. 59 1
Strength of magnetic shell [<]. Strength of the current [I]. As
the magnetic strength of a shell is the product of a surface density
by the thickness, we have
These dimensions are the same as those of potential, as could be
foreseen (329).
The dimensions and the value of current strength, are the
same as the magnetic power of a shell.
Quantity of electricity [Q]. The quantity of electricity being
the product of a current strength by a time, we have
These dimensions are the same as those of the quantity of
magnetism in the electrostatic system ; hence, in the electromagnetic
system, the surface density, the force, and the electrical potential will
have the same dimensions as the corresponding magnetic quantities
in the electrostatic system.
Specific inductive capacity [K]. The specific inductive capacity
(126) is inversely as the coefficient of electrical elasticity of the
medium that is to say, proportional to the ratio of the displacement
to the corresponding force; which gives
it is therefore equal to the inverse of the square of a velocity.
Resistance [R]. The resistance of a conductor may be defined
by Joule's law (244), which gives
from which is deduced
The electromagnetic resistance is therefore a velocity ; we have
obtained this result directly (407). Suppose that the two rails, and
the bar in the experiment assumed in this paragraph, are without
ELECTRICAL UNITS.
appreciable resistance, and that there is no other resistance in the
circuit than that of the wire which connects the two points A and
B. The resistance of this wire will be equal to the absolute unit, if
the bar, being equal to unit length, and moving in unit field with unit
velocity, perpendicular to the lines of force, gives rise to a current
capable of producing in the wire unit of energy per second in the
form of heat.
Electromotive force [E]. The electromotive force is deduced
from Ohm's law
and its dimensions are
Capacity [C]. Capacity being the ratio of the quantity of
electricity which charges a condenser, to the difference of potential
of the two armatures, we have again
or
609. DIMENSIONS OF THE PRINCIPAL UNITS. We might in the
same way determine the dimensions of the other quantities which we
have not examined. We shall give in the following tables the
dimensions of the most important quantities.
FUNDAMENTAL UNITS.
Length ......... [L],
Mass ...... . . . [M],
Time ......... [T].
DERIVED MECHANICAL UNITS.
Velocity . . ........
Acceleration ........ [LT~ 2 ],
Force ........... [LMT" 2 ],
Energy ......... [L 2 MT~ 2 ].
DIMENSIONS OF THE PRINCIPAL UNITS. 593
DERIVED ELECTRICAL UNITS.
Electrostatic Electromagnetic
System. System.
31 11
Quantity of electricity
Electrical surface density . . . . ^ r ,_i,, -, rT !
Electrical displacement . . .
Electrical force
Electrical field
31 51
Flow of electrical force . . .
Specific inductive capacity .... I [L 2 T 2 ]
Electrostatic potential ^ rT i,,ir^_n r T f i,
Electromotive force.
Electrostatic capacity
Strength of current.
Resistance [L^T]
DERIVED MAGNETIC UNITS.
Electrostatic Electromagnetic
System. System.
1 i .11
Quantity of magnetism [L 2 M 2 ] [L 2 M 2 T 1 ]
3. ! 11
Surface density [L~ 2 M 2 ]
Magnetic force ") JL i,
Magnetic field ) [L 2 M 2 T~
Flow of magnetic force [L^M^T" 2 ]
3 1
Magnetic potential
Magnetic power [I
Magnetic moment [L^M 2 "] [L 2 M Y T~ 1 ]
li 11
Strength of magnetisation .... [L~ 2 M 2 ] [L~" 2 "M^"T~ ]
Coefficient of magnetisation . . . ") r T _ 2 r r o-i
Magnetic permeability )
3 i 1 i
Verdet's constant [L~ 2 M~ 2 T 2 ] [L~ 2 M~
Coefficients of mutual induction and^ ^ -1T 2 1 FT 1
of self-induction )
QQ
594 ELECTRICAL UNITS.
610. RELATIONS BETWEEN THE Two SYSTEMS OF UNITS. In
order to establish a relation between the corresponding units, we
may compare their dimensions, or equalise the numerical expression
of the same quantity as a function of each of them. Consider, for
instance, the different expressions for the same quantity of energy W ;
we shall have the equalities .
Vf = eq =EQ,
W = *V =E 2 C.
From this we deduce the constant value a,
These being the ratios between the numerical values, we shall
have for the ratios of the corresponding units (604)
__
RTF!
The constant a denotes then the number of electrostatic units
[q\ of electricity which there are in an electromagnetic unit [Q].
Since the electromagnetic resistance R is a velocity, and the
electrostatic resistance r is the inverse of a velocity, the ratio
- or p=-^ is the square of a velocity. As this ratio is equal to
r [KJ
# 2 , it follows that the constant a is itself a velocity.
A great many experiments have been made in order to deter-
mine the value of this constant. There are clearly as many methods
as there are quantities which can be measured in electrostatic as
well as in electromagnetic units. All the results obtained range
about the number which expresses the velocity of light in air. It
is probable that this is not an accidental coincidence, and that
the equality of the two numbers arises from a correlation in the
PRACTICAL SYSTEM OF UNITS. 595
nature of the phenomena. The velocity of light is very approxi-
mately 300,000 kilometres per second. This is the number which we
shall take for the ratio a, expressing it as a function of the length
which has been chosen as fundamental unit.
611. CHOICE OF FUNDAMENTAL UNITS. The choice of units
adopted as units of time, of length, and of mass, is obviously arbitrary.
As unit of time the second, the sixtieth of the minute of the mean
time adopted by all civilised peoples, naturally suggests itself; as unit
of length we may take either the metre or a decimal of a metre. As
unit of mass it is advantageous to take the mass of unit volume
of water at its greatest density ; we thus retain the advantage that
the specific gravity of water is equal to unity, and that the weight
of a body is equal to the product of its volume by its density.
The absolute system which would be least foreign to the
habitudes established in the use of the metrical system, would
consist in taking the decimetre and the mass of a kilogramme
as fundamental units.
Gauss and Weber, who introduced into science the first absolute
system, had chosen the millimetre and the mass of a milligramme.
The British Association adopted the centimetre and the gramme on
the proposition of Sir W. Thomson. These latter units were defi-
nitely adopted for electrical and magnetic measurements by the
International Congress of Electricians, which met at Paris in 1881.
612. ABSOLUTE C.G.S. SYSTEM. It has been agreed that the
units derived from the centimetre, from the mass of a gramme, and
from the second of mean time, shall form the absolute system,
properly so-called, and which will be denoted by the symbol C.G.S.
These units have not received any special names. Thus the
unit of force C.G.S. is the force which, acting on the mass of a
gramme, imparts to it in a second the acceleration of a centimetre.
It follows from this that a gramme is g units of force C.G.S., and a
kilogramme g.io 3 C.G.S. units, g being expressed in centimetres.
In like manner, a kilogrammetre is g. io 5 , that is to say, 981. io 5 ,
or about io 8 C.G.S. units of work. On this system the value of a
is 3.io 10 .
613. PRACTICAL SYSTEM. The values of the absolute units of
the C.G.S. unfortunately do not stand in any convenient relation
to the magnitudes we have to measure in practice. Thus, the
absolute unit of the C.G.S. electromagnetic resistance is scarcely
the resistance of a twenty-millionth of a millimetre of copper
wire a millimetre in diameter, and the unit of electromotive force
would be the one-hundred-millionth part of that of a Daniell.
Q Q 2
59^ ELECTRICAL UNITS.
Accordingly the Committee of the British Association, founded in
1 86 1 for the purpose of establishing a rational system of electrical
measurements, was led to choose units more suited for practical
needs, and to give special names to these units so as to facilitate
their adoption. This system has been ratified in the following
form by the Congress at Paris :
The practical unit of resistance is equal to io 9 C.G.S. absolute
units, and acquires the name of Ohm*
The Volt is the practical unit of electromotive force ; it is equal
to io 8 C.G.S. units. f
The Ampere is the current produced by the electromotive force
I of a volt in a circuit having the resistance of an ohm ; it is equal
to io- 1 C.G.S.
The Coulomb is the quantity of electricity which, in a second,
traverses the section of a conductor which is conveying the current
of an ampere; the coulomb is equal to ro" 1 C.G.S.
The Farad is the capacity of a condenser whose armatures
acquire a difference of potential of one volt when the charge is a
coulomb; the farad is io~ 9 C.G.S. units.
In certain applications it is useful to express different magni-
tudes by means of units which are a million times as small or as
great as the corresponding practical unit. These new units are
called by the same name with the prefix mega or micro, according
as they are multiplied or divided by a million,
Thus the megohm is equal to io 6 ohms ; the submultiple called
the microhm is io~ 6 ohms. In like manner, for capacity, the
millionth of the farad, or the microfarad, is equal to io~ 6 farads
or io- 15 C.G.S. units.
* The Committee of the British Association made numerous experiments in
order to determine the value of the Ohm and to construct material standards
presenting the same resistance. The first investigations appeared to show that
the Ohm is represented with a close degree of approximation by the resistance
of a column of mercury at 0, a square millimetre in cross section, and 104 centi-
metres in length ; but it appears that certain errors were made in the calculation,
and that this length should be increased by about one per cent. that is, raised
to 105 centimetres. Recent investigations lead to the same result, but the
question does not seem to have been definitely solved. As it is not certain, on
the other hand, that solid metals do retain their electrical properties without
change, the Congress decided that the Ohm should be represented by a column of
mercury at zero, having a cross section of one square millimetre, and that an
international commission should settle by fresh experiments the exact length
of this unit.
f The electromotive force of a Daniell with sulphuric acid is about 1.08 volts.
COMPARATIVE VALUES' OF THE PRINCIPAL UNITS. 597
The microfarad is really the practical unit of capacity, for the
value of the farad is far too great. For instance, the electrostatic
capacity of the Earth is equal to its radius R ; its electromagnetic
capacity C is equal to the quotient of the radius by 2 , which gives
708. lo-is CG.S. units,
3.io
that is to say, 708 microfarads.
It is important to observe that the practical units themselves
constitute an absolute system, in which the fundamental units are
[L] = io 7 metres, or the quarter of the terrestrial meridian,
[M] = io~ n of the mass of a gramme,
[T] = a second.
Another remark, which is of some practical utility, is, that if
we divide the expression for electrical work by g expressed in
metres that is to say, practically by io we get its value in
kilogrammetres.
The practical unit of work is obtained, for instance, if we multiply
a volt by an ampere, which gives io 7 C.G.S. units. But we
have seen (612) that a kilogrammetre is equivalent to io 8 C.G.S.
units that is to say, to ten times as much. The same quantity
of work in kilogrammetres will then be expressed by one-tenth
of that value.
614. COMPARATIVE VALUES OF THE PRINCIPAL UNITS. We
shall collate in the following table the values of the practical units
in C.G.S. absolute units, and also as functions of the units of
Gauss and Weber, which have been employed in a certain number
of papers on electricity.
FUNDAMENTAL PRACTICAL C.G.S. UNITS OF
UNITS. UNITS. UNITS. GAUSS AND WEBER.
Length, io 7 metres, Centimetre, Millimetre,
Mass, IO~ U gramme, Gramme, Milligramme,
Time. Second. Second. Second.
Resistance Ohm io 9 io 10
Electromotive force Volt io 8 io 11
Current Ampere io 1 io
Quantity Coulomb lo" 1 io
Capacity Farad io~ 9 io~ 10
59 8 ELECTRICAL UNITS.
615. PHYSICAL CONCEPTION OF THE VELOCITY a. We may
give a physical representation of the velocity a which expresses
the ratio of the electrical units in the two systems.
Suppose, for instance, that a sphere of radius R, or a con-
denser of the same capacity, were charged in such a manner that
its electrical potential is equal to unity, and is discharged n
times in the time / through a conductor ; the mean strength of the
current . in electrostatic units will be - . If we determine n in
such a manner that the strength of this current is equal to the
electromagnetic unit, the expression will represent the number
of electrostatic units of electricity which are contained in an
electromagnetic unit ; that is to say, the value of #, and this
expression is a velocity.
616. Maxwell points out another mode of representing this,
based on the hypothesis that the external action of an electrical
mass in motion is equivalent to that of a current.
Consider a plane covered with a uniform charge of electricity
of density <r, and moving in its own plane with a velocity u.
Each band of unit breadth, and parallel to the direction of the
motion, is the equivalent of a current whose intensity is <ru in
electrostatic measure, and - - in electromagnetic measure. Sup-
pose, now, that a second plane parallel to the first, at a distance
<5, moves in the same manner, and in the same direction, with a
velocity u', and let a-' be its density. Two kinds of actions are
produced between these planes ; an electrostatic repulsion in virtue
of charges of the same kind, and an electrodynamic attraction due
to parallel currents in the same direction.
Let us now take in the second plane a band of length /, and
of infinitely small breadth b, and in the former plane an unlimited
band of breadth dx, at a distance x, from the projection of the
band bl. The electromagnetic action exerted by this unlimited
band on the first situated at the distance \/<5 2 + x 2 , is expressed
by (480)
vu ar'u' T / <r<r'uu' T dx
2 dx b = = 2 bl-===,
and its component df along the perpendicular to the plane is
cra-'uu' , , ^" !V
df= 2 bl
a 2
PHYSICAL CONCEPTION OF THE VELOCITY a. 599
In order to have the total action of the first plane on the
movable portion in question bl of the second, we must integrate
this expression from x= - oo to #=+co, which gives
+co
=2 ~~
or
On the other hand, the electrostatic charge of this surface is
bl<r'. As the action of the former unlimited plane on the unit of
mass is equal to 27ro-, the repulsion /' exerted on this surface is
perpendicular, and its value is
/' = 2ir<r<r'bl.
If these two actions are equal, the same would be the case
for all other portions of the second plane, and there would be
equilibrium between them. For this uu' must be equal to # 2 , or,
if the velocities u and u' are equal, u = a.
The constant a is therefore such that if two unlimited parallel
planes, uniformly electrified, moved in the same direction with
this velocity, their electrodynamic action would counterbalance
their electrostatic repulsion. As the velocity a is that of light, the
experiment cannot be realised in this form.
617. In order to evaluate the order of magnitude of the effects
which may be obtained, we may observe that an unlimited band
of breadth ft, and density o-, movable in its own direction with a
velocity u, is equivalent to a current whose electromagnetic in-
tensity is b. If we suppose it placed at a distance e from a
similar band, and if the condenser thus formed be charged to an
electrostatic potential V, we shall have (74)
Now, by means of electrical machines, we can get potentials
equal to 100,000 Daniell's cells that is to say about io 5 volts, or
600 ELECTRICAL LIMITS.
io 5 .io 8 C.G.S. units. With such machines, we shall have, in
electrostatic units,
10"'. 10 10
13
3
which gives virtually o- = .
If we assume d=io and e=i, the electromagnetic intensity
of the current will be
u.-zoo u
As a volt in a circuit of n ohms, gives a current of strength
r= C.G.S. ,
ion
it will be seen that in order to have the same current the band
must move with a velocity,
io 7 100000
metres.
It must be observed that the current I, produced by the
motion of an electrical body, is far more difficult to observe than
that of an ordinary electrical cell, since it must act directly on
the needle, and without any effect of multiplication.
Rowland has proved experimentally that the rotation of an
electrified disc produces a sensible effect on a magnetised needle,
and that the action is of the same order as that which would be
given by the preceding considerations.
AMPERE'S HYPOTHESES. 60 1
CHAPTER X.
GENERAL THEORIES.
618. AMPERE'S HYPOTHESES. In order to establish the ele-
mentary formula of electrodynamic actions, Ampere relied solely on
the hypothesis of central forces, and on certain experimental facts,
without any particular view as to the nature of electrical currents
themselves. Nevertheless, from the year 1822, he "endeavoured to
account for the force which is exerted between two elements of
conducting wires, by the action of the fluid, which is distributed in
space, and whose vibrations produce the phenomena of light."
Ampere pointed out another mode of conceiving the phenomena :
" If we suppose that molecules of electricity put in motion in
conducting wires by the action of the battery, are continually
changing their places, uniting every moment to form neutral fluids,
separating, and then quickly uniting with other molecules of the
fluid of the opposite kind, it is not contradictory to assume that
from the actions which are inversely as the squares of the distances
separating the molecules, a force might be produced between
two elements of conducting wires, which depends not only on
their distance, but also on the directions of the two elements
along which the electrical molecules are moving, uniting with
molecules of the opposite kind, and then separating the moment
after to unite with others." Mkmoires de VInstitut for 1823,
pp. 294 and 299. Ampere did not follow the development of these
ideas any further ; he did not think the time had come to do it
with utility.
The hypotheses of Ampere have been taken up from time to time
by various physicists, particularly by Weber and by Maxwell : we
shall give a summary of the theories proposed.
619. FORMULAE OF GAUSS AND OF WEBER. If we bring into
play the reciprocal actions of the electrical masses which circulate in
conductors, the action of the two electrical masses m and ;;/' must be
602 GENERAL THEORIES.
a function not only of the distance r, but also of their relative
motion, and the problem thus stated appears indeterminate. One
hypothesis is to assume that this action, while directed along the
right line joining them, and proportional to the product of the
masses, and inversely as the square of the distance, comprises a
term proportional to a power of the relative velocity u of the two
masses, and another term proportional to a power of their relative
velocity parallel to their distance. These powers must be even,
if the action is not to be modified when both the directions of the
two motions and of the currents themselves are changed, and
the problem is satisfied by taking them equal to 2, which gives
the elementary law
Weber examined first the simple cases of two elements in the
prolongation of each other, or perpendicular to the same right line.
In the former case, it must be assumed that the action of the two
masses contains a term which depends on their relative velocity,
and he supposed that this term was proportional to the square of
the velocity. The second case led Weber to bring in the acceleration
along the same right line, and the simplest hypothesis was to assume
that this fresh term was proportional to the acceleration. From
this follows another elementary law
It remains to determine the coefficients a, ft, a! and /?', which
enter into these two expressions (i) and (2).
620. Consider, in the first place, two bodies moving respectively
on two fixed curves s and /, with the constant velocities v and v'.
When the two moving bodies traverse the elements ds and dsf, which
are at the distance r t and make with each other an angle e, we
have
ds ds'
v = and v= -
FORMULAE OF GAUSS AND WEBER. 603
From this is deduced
dr ^rds ^r ds' ^r <V-
,
~7^ = v ^2 +2VV ^^~> + v T^-
dt 2 ^>s 2 2lw to 2
If we suppose that the two curves s and / convey currents of
strengths I and I', then from Ampere's hypothesis (351), and
considering the force to be repulsive, the action of two elements
ds and ds' is
2ll'dsds f 3 ar3r-] _ zlTdsds' |
C + ~ ~~~
621. Suppose now that the element ds contains electrical masses
m and m v moving respectively with the velocities v and v v and
that in like manner the element ds' contains masses m' and m\, with
the velocities if and if v If we evaluate the actions of the masses
m and ;;/ x on the masses m' and m' v from the formulae (i) and (2),
the resultant should reproduce Ampere's laws in one or other of
the two forms, and therefore only contains terms in which the
product vv' of the velocities is a factor.
The terms containing the squares of the velocities are, to within
a factor,
(tntf + a*ifj) (m f + m\) t and (wV 2 + m\v'\) (m + m^.
In order that these terms may be null, we must have
(4) wz> 2 + #*!#! = , or
These two conditions are realised simultaneously, if we assume
with Weber that an electrical current of strength I is formed by
two currents of contrary electricities, moving with the same velocity
v in opposite directions, and each having one-half the intensity. It
is even necessary to assume that the algebraical sum of the electrical
masses which exist in each element of current in the permanent
state is null, if the condition is to be satisfied (203) that the density
in the conductor is null; but, without stating anything definite
on the ratio of electrical masses of opposite signs, it would be
sufficient that the sum mv 2 + m^v\ were null in each element that
is to say that there were electrical masses of contrary signs with
the same vis viva.
604 GENERAL THEORIES.
The quantity of electricity which traverses the section of the
first conductor in unit time is equal to mv + m^ ; if a denotes the
number of electrostatic units in the electromagnetic unit, we
have
mv + m l v l =
m'v' + m\v\ = aVds' .
Assuming the existence of equal and opposite currents, these
equations become
(6)
\ / / , ft i ,
2m v = alas .
When we evaluate the action of the masses m and m l on the
masses m' and m' v the terms which depend on the product of the
velocities are, on the first hypothesis,
(mv + mM) (m'v' + m'^) (mv + m^) (m'v' 4- 0/X) cos e,
that is to say,
and with the hypothesis of Weber,
(mv + m-p-d (m 'v' + m\v'^) |~ , dV l)r dr~\
j (X -^- 13 i
r* [_ ^sts ^ ds'\
2ll'dsds' f 4 Vr ^r^r~\
- a*fi' '- - + a 2 a'-
r* [_ DsW ^^'J
The term, which is independent of the relative motion, is equal in
both cases to MI ' , ^. This term should be null if
r 2
there are two equal currents in opposite directions in each wire ;
if not it will represent an electrostatic action between the conduc-
tors, a phenomenon which up to the present time experiment has
not ascertained to exist.
FORMULAE OF GAUSS AND WEBER. 605
In order to satisfy Ampere's law, it is necessary that, in the first
case,
and, in the second case,
a*p = r, 0*0.'=--.
2
The expressions (i) and (2), which give the elementary action of
the two electrical masses, become then
dt
i/ d*r i /dr
I r
\~\ Ti 2 d**Jr~\
I =mm\ I- -
/ J L r2 Vr dt<L
622. The former expression (i)', which occurs in Gauss's
manuscripts, is incompatible with the principle of the conservation of
energy, for it would lead to the conclusion that a limited physical
system can produce an indefinitely increasing quantity of energy.
Weber's formula, on the contrary, is compatible with this prin-
ciple ; for the expression of the force (2)' may be considered as
the differential coefficient in respect of r, taken with the contrary
sign, of the function
(7) </>= -
The work done by the repulsion of a fixed mass on a movable
mass is equal to the difference \l/ - ^ of the values of the function ^
relative to these two masses, for the initial and the final position.
The function ^ may be considered as representing the potential
energy of the system of the two masses ; it only depends on their
distance, and on their relative velocity along the right line r; it
resumes the same value when one of the masses describes a closed
path in reference to the other, and possesses the same velocity at the
same points.
Since induction is a consequence of the law of Ampere, and of
the principle of the conservation of energy, Weber's formula, which
equally well satisfies both conditions, must give the induction.
606 GENERAL THEORIES.
623. This formula also enables us to obtain directly the relative
potential energy of two closed circuits. For, if we replace by its
at
value as a function of the velocities of the electrical masses, we get,
from Weber's hypothesis, that the potential of one of the elements
on the other is expressed by
, ds') = - ITdsds - .
r
The potential energy of the two circuits is then
COS
dsds .
This is the formula of Neumann, as found above (353).
When the currents are constant, and traverse circuits of constant
form, the resultant of the actions exerted by one of the currents on
any mass m' whatever of the other is perpendicular to its trajectory.
624. PHENOMENA OF INDUCTION. Let us now suppose that
the currents move, and that the strengths change. The distance r^
instead of being, as above, simply a function of two independent
variables s and s', is further a function of the time, and we have
r=<f>(s,s',t). For a given value of /, the function <j> represents the
distance of the two elements of the circuit ; if / is variable, and we
consider s and s' as functions of /, the value of <j> represents the
distance of two electrical masses in motion on movable conductors.
If the conductor has a constant section, the velocities do not at all
depend on s and s' as independent variables, for at each instant the
strength is the same in all parts of the circuit. Hence, for the
relative velocity of the two masses, we shall have
dr ^r ^>r ^r
and, considering the differential coefficients and as functions
^s V
of s and s' alone, the relative acceleration will be
d*r JPr ,TPr , dV ^v^r V> Tnlr Wbr *&r
_ = , 2 _ +2Z;z; _ + z;2 __ 2 + __ + ___ + ,__ + ^___ + __.
PHENOMENA OF INDUCTION. 607
The velocity and the acceleration refer to the electrical
Dr ^r
masses, while the terms and - of the second member refer
of ot*
to the distances of the two elements of the two circuits.
The mechanical action of ds on ds' will be obtained as above
by taking the sum of the actions which the masses of the elements
ds exert on those of the elements ds'. With Weber's formula and
hypothesis, it is easily seen that in this sum there only remain, as
before, terms in z/z/, and with coefficients already found. It follows
from this, that in the variable state, the mechanical action is at each
instant conformable with that which Ampere's formula would give.
625. The electromotive force which acts on the element ds' is
the force which tends to separate the equal masses of opposite signs
contained in this element, and to carry them in opposite directions.
We shall obtain the value by taking the difference of the actions
exerted in the direction of the element ds', on each of the masses
which it contains, by the two masses of the elements ds. But when
we add the actions of the two masses + m and - m of the element ds
on one of the masses m of the element ds t the terms which remain
dv
are terms in v, vv' and , which change sign at the same time as m.
dt
Among these the only ones which remain in the final difference are
those which change sign with the velocity v, whatever may be the
sign of v'. These terms reduce to two : one arising from ( J ,
. d^r *bu Dr
and which is 2V = -^-, the other arising from , which is -r-r-
osot dt* ot Ds
The difference thus calculated is equal to
4mm' \~ DvDr Dr T)r~\ i F DIDr "tor Dr~\
r --- v -- = r --- I dsds'
a*r* L ^ fc fc *J r* |_ T>t Ds Ds *J
taking into account equations (6), and supposing the intensity equal
to unity in the element ds'.
We must take the component of this action along ds', and
therefore multiply the preceding expression by ; observing that
we have
lit <tf lit r
608 GENERAL THEORIES.
the elementary electromotive force becomes
The total electromotive force produced in the circuit s' by the
circuit s, is obtained if we integrate this expression in reference to
s and to s'. As the intensity I is merely a function of the time, and
as the limits of the integral are themselves independent of the time,
(9)
We shall have then (353)
(10) E
_d C Ci^r^r
~~d* J J ^^
rr-M
JJ r
A 1 J r dt dt
/ /
which gives the general expression (518) of the electromotive force
produced in a circuit by an external current. We shall find in like
manner the other cases of induction.
626. VARIOUS ATTEMPTS AT A THEORY. Numerous attempts
have been made, after the example of Weber, to bring under one
and the same theory the phenomena of statical electricity, of per-
manent currents, and the effects of induction, and to establish a
connection between electricity, magnetism, and light.
Gauss expressed the opinion that electrical actions cannot take
place instantaneously, and that we should have a key to electro-
dynamical phenomena, if we could discover the law of the propa-
gation of electrical forces.
Guided by these considerations, several mathematicians have
treated the problem. For instance, the phenomena of induction
may be explained by assuming that the electrical potential is
propagated in a medium with a certain velocity which would be the
same as that of light, according to B. Riemann, or of a totally
different order, according to the theory of C. Neumann.
M. Betti compares the action of currents to that of a system of
elementary magnets tangential at every point to the contour of the
circuit, and periodically polarized in opposite directions, and he
considers the magnetic force as transmitted in the medium with a
certain velocity.
ELECTROMAGNETIC THEORY OF LIGHT. 609
t _,
M. Lorenz has shown that by adding, to the equations given by
Kirchhoff for electrical currents, suitably chosen terms, which do
not at all affect any experimental conclusion, we get a new series
of equations which indicate an action from layer to layer in the
medium, and a phenomenon of undulation travelling with the
velocity of light. He thus arrives at results similar to those which
Maxwell had deduced from an entirely different theory.
M. Edlund has attempted to show that electrical phenomena,
both statical and dynamical, may be explained by the aid of a single
fluid, which in all probability is nothing but the ether.
M. Edlund assumes that all bodies in the neutral state contain a
normal quantity of ether, and that a positive or negative electrifi-
cation corresponds to a share of ether, greater or less than that
of the normal charge. It is easy to deduce from this that the
action of two bodies is proportional to the excess of their
respective charges over the normal charges.
An electrical current is then only a transport of ether in a given
direction ; if we assume that the action of the two masses only
depends on their velocity and on their relative acceleration along
the right line joining them, then by reasoning analogous to that of
Weber, and determining certain coefficients by the identification of
the formulae with the results of experiment, we arrive at an expla-
nation of Ampere's laws and the phenomena of induction.
All the preceding theories imply the existence of an intermediate
medium ; for if any mechanical effect, force, or potential, is trans-
mitted with a finite velocity from one particle to another, it follows
that a medium of a suitable structure must have been the seat of this
action while this effect had quitted the first particle and had not
yet reached the second. Maxwell has taken the properties of this
medium into account, and has thus established remarkable numerical
relations between the phenomena of electricity and of light, which
are supported by experiment.
627. ELECTROMAGNETIC THEORY OF LIGHT. We have seen
on several occasions how favourable the various phenomena of
electricity and magnetism are to Faraday's conception, which consists
in giving up the idea of actions at a distance, and considering forces
as transmitted by the elastic reaction of an intermediate medium.
This is a hypothesis which at the present day forms the basis of the
physical theory of light, but it would be contrary to the spirit of
science to assume that there are as many different media as there are
phenomena to explain, as was formerly done by the distinct hypotheses
of calorific fluid, of electrical fluids, and of magnetic fluids.
R R
6 10 GENERAL THEORIES.
The great problem which the philosophy of science raises is to
know the constitution of the single medium by which all physical
phenomena may be explained. If calculation shows that electro-
magnetic phenomena are propagated not only in air, but in all
bodies, with the velocity of the propagation of light, the question
would have made a great step; for it would be shown that this
medium exists, and that in all probability electrical and luminous
phenomena are only different manifestations of the properties with
which it is endowed. Such is the conclusion from Maxwell's theory.
Faraday's discovery of the action of a magnetic field on the polari-
zation of the light which traverses it, would be a natural consequence
of the connection which the common medium establishes between
the two orders of phenomena.
628. GENERAL EQUATIONS. In order to determine the con-
ditions of the propagation of an electromagnetic disturbance in a
medium, we shall suppose this medium at rest that is to say, not
subject to any other motion than that resulting from the dis-
turbance itself.
Equations (n), (13), and (14) of (572), give
From which is deduced
4?r cV
an equation which may be written in the symbolical form
The medium being fixed, the differentials of the co-ordinates
in respect of time are null. Equations (10) of (571) give then
from this follows
(i 2 )
GENERAL EQUATIONS. 6ll
On the other hand, we have by equations (12) of (572),
and the equations (2) of (567) give
fii- - J=i + -- + J -AF = AF,
in which
AF- ^ F
~ + + '
Substituting this value in equation (13) we get
(14)
Eliminating u' between equations (12) and (14) and repeating
the analogous operations for the other co-ordinates, we get finally
Taking the partial differentials of these equations in reference
to x, y, and z respectively, and adding them, we get
(16)
\ 4
R R 2
6l2 GENERAL THEORIES.
When the medium is not a conductor, ^=0, and the value of A^,
which is proportional to the density of the free electricity, is
c) 2
independent of the time. There remains then = ; that is to
say that is a linear function of the time, or a constant. These
two functions 6 and ^ play no part therefore in the phenomena
due to periodical disturbances.
629. PROPAGATION OF UNDULATIONS IN A DIELECTRIC. In
the case of a dielectric, equations (15) may be reduced to
_- .
c) 2 G
(17) K/i -AG = 0,
<) 2 H
K, -AH = 0.
These equations define the manner in which the functions F, G,
and H vary with the time, and therefore the propagation of electro-
magnetic disturbances ; they are of the same form as that of vibratory
motion in a solid elastic body.
The velocity V of the propagation of a disturbance is given by
the expression
(18) V
630, PLANE WAVES. Suppose, in fact, that at an instant the
electromagnetic disturbances form a plane wave perpendicular to
the axis of z. The medium will be traversed by plane waves parallel
to the first, and all the quantities, whose variations determine these
waves, are simply functions of z and of /, independent of x and
y. Equations (2) of (567) become then
PLA^E WAVES. 613
In like manner, the equations analogous to (13) give,
We see already that the electrical perturbation is also in the
plane of the wave, and perpendicular to the electromagnetic dis-
turbance, for if we have Y = (that is if the electromagnetic
disturbance is parallel to the x axis), we shall have u' = 0, and the
electrical disturbance will be parallel to the y axis.
For a non-conducting medium, equation (12) and the analogous
relations to the other co-ordinates give
- ,
< 2 o)
from this follows
1J7 2 ""
= _ _
a/*~K/*&*' a/ 2 ~ ~w'
7) 2 H
The integral of these latter equations is
(22) H
an expression in which A and B are functions of z. This quantity
H is therefore constant, or varies proportionally with the time.
In any case it does not intervene in the propagation of periodical
phenomena,
6 14 GENERAL THEORIES.
The integrals of the two first equations are expressions of the
form
F-/i(*
The values of F and of G consist of two distinct parts. The
former does not change when we make successively = 0, /=0,
or z = V, and / = i ; it represents a plane wave which moves
parallel to the z axis with a velocity equal to V. The second
also represents a plane wave which moves in the opposite direction
with the same velocity.
631. A magnetic disturbance in the form of a plane wave
produces then two plane waves moving on each side with the
same velocity.
When G = 0, the magnetic force is parallel to the y axis and
equal to - ; the electromotive force is parallel to the x axis
and its value is - . If we assume that the phenomena are iden-
tical with those of light, the present case corresponds to a ray of
polarized light. The plane of polarization would coincide, either
with the plane of magnetic disturbance, or with the plane of
electrical disturbance which is perpendicular to it.
If the original disturbance is periodic, and forms a simple
vibration proportional to sin 271- , the same character will be met
with in each of the planes parallel to the original wave, and the
wave-length of the phenomenon is the distance VT traversed by the
propagation of the motion during a single period.
If the disturbance is circular, that is, if it may be figured by
a movable body which describes a circumference in a uniform
motion, the same character will be reproduced in the waves
propagated, and the planes passing through the radius and the
magnetic force, or the electrical displacement, are always perpen-
dicular to each other. This would be the case with a circularly
polarized ray of light.
632. DISTRIBUTION OF THE ENERGIES. We have seen (120)
that in an electrified system, the energy of the medium for unit
volume is equal to half the product of the displacement by the
electrical force. In like manner, in the field of a system of
currents (570) the energy for unit volume is equal to the quotient
of the square of the induction by
VELOCITY OF THE-, PROPAGATION OF LIGHT. 615
Suppose that the plane wave in question is polarized, and that
the electrical disturbance is directed along the x axis. We shall
then have X = Z = 0; z/ = a/ = 0; ^=^=0; Q = R = 0; G = H = 0.
The electrical energy for unit volume is expressed by
2 STT 8a-\V
and the electromagnetic energy
These two expressions are equal, for if we multiply the two
members of the first of the equations (21) by the equal factors
<>F , 3F is
and - , and integrate with respect to /, we get
The total energy of the medium in which the waves are propa-
gated is therefore half in the form of electrostatic energy, and
half in the form of electromagnetic energy.
Let / denote each of these energies for unit volume. In virtue
of its electrical state (104) the medium is subject to a tension -
parallel to the x axis, and a pressure of the same value parallel
to the axis of x and z. In virtue of its electromagnetic condition,
the medium is subject to the same actions, except that the x axis
must be replaced by that of y t and conversely. These actions
destroy themselves in the plane of the wave, and there remains a
pressure p parallel to this plane equal to half the total energy
for unit volume.
A ray of light produces therefore in the medium a pressure
parallel to the direction of the motion, and would exert a repulsion
on a plate of metal which it encountered. It is possible that this
effect may have some part in the motion of the radiometer.
633. VELOCITY OF THE PROPAGATION OF LIGHT. The true
control of this theory is, then, that in all media the velocity of
6l6 GENERAL THEORIES.
the propagation of the magnetic disturbances is the same as the
velocity of light.
Let us suppose, in the first case, that the medium in question is
air. The coefficient K would be equal to unity if the electrostatic
units had been adopted. In the electromagnetic system, the value
of this coefficient (608) is . It follows from this that
a?
.
N/K
Hence the velocity of the propagation of an electromagnetic
disturbance in air is equal to the ratio of the units ; this ratio
ought then to be equal to the velocity of the propagation of light.
Now experiment gives values for these two velocities which differ
extremely little from 300,000 kilometres per second, and the most
recent researches agree in giving numbers which are the nearer
each other, the more exact the measurements have been. Such a
coincidence cannot be due to accident, and Maxwell's theory finds
thus a most striking experimental confirmation.
634. SPECIFIC INDUCTIVE CAPACITY. Let us now consider a
dielectrical medium, the refractive index of which is n, and its
specific inductive capacity greater than that of air. If V is the
velocity of the propagation of light in air, and V its velocity in
the medium in question, we have
72V' = V, or
On the other hand the velocity V" of electromagnetic disturbance
will be obtained if we replace K by K', which gives
K'V" 2 = KV 2 =i.
TC'
In order that V" and V shall be equal, we must have n 2 = .
K
It follows then from this theory that the specific inductive
capacity of a dielectric with respect to air is equal to the square
of its refractive index.
A difficulty here presents itself in the experimental verification
of this conclusion, which arises from the dispersion of refracting
media. As the refractive index varies with the wave-length the
most natural idea would be to take the limiting value of the index :
ANISOTROPIC MEDIA. 617
that is to say, that which corresponds to the greatest wave length.
For paraffine, for instance, the refractive indices of the extreme
luminous rays vary from 1.43 to 1.45, and the best experiments
show that the specific inductive power is equal to 2.29, the square
root of which 1.51 is not greatly different from the refractive index.
The agreement is much less satisfactory with most of the trans-
parent solid dielectrics, such as the different kinds of glass, Iceland
spar, fluor spar, and quartz ; their specific inductive capacity is always
higher, and is sometimes twice the square of the refractive index.
This is also the case with the animal and vegetable oils, according
to the recent experiments of Dr. Hopkinson.
For gases, in which the refraction is less, and the dispersion
may be neglected, the refractive power n 2 - i is proportional to the
specific gravity, or to the pressure, if the temperature is constant ;
it ought to follow from this that the specific inductive capacity
also increases in proportion to the pressure, and by the same co-
efficient as the refractive power. This conclusion seems to have
been verified by the researches of M. Boltzmann.
The experimental control cannot therefore be considered suffi-
cient to confirm the theory; but too much importance must not
be attached to this apparent disagreement, if we take into account
the fact that the specific inductive capacity diminishes with the
duration of the electrification. But the period of electrical oscil-
lations, which must be assumed in order to explain luminous
phenomena, is out of all proportion with the shortest interval
of time that can be realised in electrical experiments.
In any case, this correlation between the electrical and optical
properties of a medium may be considered as, at all events, a first
approximation to a theory which remains to be more minutely
developed.
635. ANISOTROPIC MEDIA. In order to extend the theory to
anisotropic media, we should, in strictness, know the relation between
the molecular constitution of a medium and its electrical properties ;
but without making any hypothetic supposition, it is sufficient if
we assume that the specific inductive capacity of the medium is
not the same in different directions ; in other words, that the
electromotive force, instead of being proportional to the displace-
ment, and in the same direction, is connected with the displacement
by a system of linear equations, as for the phenomena of thermal
expansion.
In this case, there are three rectangular directions, along which
the electromotive force is in the direction of the displacement ;
6l8 GENERAL THEORIES.
if we take these directions as axes of co-ordinates, and call K x ,
K 2 , and K 3 , the three principal values of the specific inductive
capacity, or a, &, and f, the three principal velocities of propagation,
we may write
(24)
In a non-conducting medium, where the electrical density is
constant at each point, the general equations of propagation become
then
IT _AF--
1 a/2 3*~<j2 3/2 '
, ae
(25 ) K = AG -
ae
If /, m, and are the cosines of the angles which the perpen-
dicular to a plane wave, moving with the velocity V, makes with
the axis, we may write
Ix + my + nz- V/= o>.
If we represent by F", G", H" the second differentials in respect
of a/ of the different functions F, G, and H ; equations (25) become
/ V 2 - a 2 \
( + / 2 )F"
V * 2 /
/V 2 - tf \
(26) ^-
(V 2 -^ 2
-
^ 2
BAD CONDUCTORS. 619
Eliminating the functions F", G", and H" between these three
equations, we get
/ 2 m* n*
(27) - + - + -- = 0.
This is an equation of the same form as that which determines
the velocity of propagation of light in double refracting media.
For a given direction it gives two velocities corresponding to two
distinct waves which move in the same direction. If the wave,
for instance, is perpendicular to the x axis, we have ;/z = 0, = 0,
and the values of the velocity are b and c.
636. If the medium is symmetrical in reference to one axis
(for instance, the x axis), the two velocities b and c are equal, and
equation (27) reduces to
/2( V 2 _ P)* + (L - /2)(V 2 - 2 )(V 2 - 2 ) = 0.
For a wave perpendicular to the axis, /= i ; there is then only
one velocity of propagation, V = , whatever be the direction of
the electrical and magnetic disturbances in the plane of the wave.
If the wave is parallel to the axis, /=0, and we have two velocities
of propagation, V = a, and V = . The wave which is propagated
with the velocity ^, has the character of the ordinary wave in
optical phenomena, and corresponds to an electrical disturbance
perpendicular to the axis. As this wave is polarized in the plane
of the axis and of the ray, we see that the plane of polarization
of the light is perpendicular to the plane of the electrical dis-
turbance.
637. BAD CONDUCTORS. Let us suppose that the medium is
an imperfect insulator, and is isotropic, and that the effects of
displacement and of conductivity are of the same order. The
energy is then partially transformed into heat, and the wave which
is propagated is gradually weaker.
Let us consider a plane wave perpendicular to the z axis, the
disturbance being parallel to the x axis. Equations (15) give then
BF **F 2> 2 F
(28) 47ir + K = .
It W W
The integral is a function of the form
F = e~ pz cos (nt qz) ,
620 GENERAL THEORIES.
and the coefficients ought to satisfy the conditions
This expression represents a wave which is propagated parallel to
the z axis, with a velocity V equal to -, and the amplitude of which
?
rapidly diminishes.
The value of the coefficient of absorption p is 2wrV. The
absorption of light ought therefore to increase with the electrical
conductivity. Experiment does, in fact, show that most transparent
substances are dielectrics, and that all good conductors are very
opaque.
This ratio, however, is not absolute, for certain metateare trans-
parent in very slight thicknesses, and many dielectrics are opaque.
We ought to exclude electrolytes, which are almost all transparent,
for the decomposition which accompanies the passage of electricity
completely changes the nature of the phenomenon, and we are no
longer dealing with a mere effect of conductivity.
638. CONDUCTING BODIES. Let us consider, finally, an isotropic
conducting medium, or at any rate a medium in which the phe-
nomena of conduction predominate over those of electrical displace-
ment. If we neglect then the terms containing the factor K,
in equations (15) as well as the functions and ^, we get
^F
W = AF,
<)G
(29) 4,1* = AG.
Each of these equations has the same form as that which gives
the diffusion of heat in Fourier's theory.
For, if k is the coefficient of thermal conductivity of an isotropic
medium (70), and if the function F be considered as giving the
temperature at each point, the expression AF represents the flow
<)F
of heat which in unit time penetrates unit volume ; is the
corresponding rise of temperature, so that the coefficient qirck is
the calorific capacity for unit volume.
ROTATORY MAQNETIC POLARIZATION. 62!
Electromagnetic properties, once established in a medium, ex-
perience therefore a diffusion analogous to that of heat ; but we
must remark that the coefficient of conductivity of the medium k,
which would produce the same calorific diffusion, is inversely as c.
The diffusion of the electromagnetic effects is then inversely as the
electrical conductivity, so that a medium which had a perfect
conductivity would offer an absolute obstacle to this diffusion.
Consider, for instance, the case of a linear conductor surrounded
by a conducting medium. The moment the principal current is
established, the induced current in the surrounding medium has the
same strength, and their action on a distant point is null ; the per-
manent state is only established after the induced currents have been
nullified by the resistance of the medium. But in the degree in
which the induced current is enfeebled, a current in the same
direction is produced round it, so that the space occupied in the
medium by the induced current increases in proportion as the
intensity diminishes.
If the principal current is kept constant, the induced currents
diffuse gradually ; when the permanent regime is set up, the values
of AF, AG, and AH are null throughout the medium, and only
retain finite values in the portion occupied by the circuit of the
current.
639. ROTATORY MAGNETIC POLARIZATION. The ordinary theory
of undulations assumes that luminous phenomena are produced by
the vibrations of ether ; but it must be admitted that a formal
explanation of this kind is outside the range of experiments. We do
not know, in fact, what is the true nature of light. The only thing
which may be considered to be proved, is that in a ray of light there
is a mechanical effect of the nature of a vector in geometry, that is
characterized by a magnitude and a direction; this direction is
perpendicular to the ray, and it varies periodically in the same plane
when the ray is polarized. This is a conclusion from the phenomena
of interference.
In the case of a circularly polarized ray, the magnitude of this
mechanical effect, of this vector, is constant, but its direction turns
about the ray, and produces a complete revolution in each period.
When such a ray traverses a medium under the action of a magnetic
force, its velocity of propagation is modified ; it must be concluded
thence that there is, in the medium, some rotatory motion, the
axis of which is parallel to the direction of the magnetic forces.
This rotation does not apply to any finite portion of the medium
taken as a whole, and it must be assumed that it is confined to the
622
GENERAL THEORIES.
smallest particles of bodies, each of which turns about its own axis.
This is the hypothesis (601) of molecular vortices.
Maxwell explained in this way the phenomena of rotatory polari-
zation by the conception of molecular vortices; but, without entering
upon an explanation of this theory, we may arrive at the same result,
as Professor Rowland showed, by bringing in a new action dis-
covered by Mr. Hall.
640. HALL'S PHENOMENON. Let ABCD (Fig. 127) be a cross
cut in a very thin metal sheet a gold leaf, for instance ; the two
ends A and B of the principal branch are connected with the poles
of a battery, the ends C and D of the cross piece are connected with
a galvanometer. The apparatus may easily be arranged so that
none of the current traverses the galvanometer.
Fig. 127.
When this conductor is placed in a very strong magnetic field, so
that the lines of force are perpendicular to its plane, a permanent
deflection of the galvanometer shows that a constant current traverses
the galvanometer. If the current goes from A to B in the principal
branch, and the lines of force traverse the plane of the figure from
front to back, the branch current goes through the galvanometer
from D to C, when the conductor consists of gold, silver, platinum,
or tinfoil, and in the opposite direction when the metal is iron. The
action ceases to be perceived when the thickness of the conductor is
increased.
In the first case, the current is drawn in the direction of the
electromagnetic force which would be exerted on a wire parallel to
AB, and traversed by a current from A to B ; this may also e said
to be the same in the second case, since in the interior of an iron
plate, owing to the magnetisation, the direction of the lines of force,
and of the electromagnetic force, have changed their sign.
HALL'S PHENOMENON, 623
Explained in this manner, Hall's phenomenon would seem to be
in contradiction with the opinion generally adopted, that in electro-
magnetic phenomena the action is exerted on the supports of the
currents, and not on the current itself. But, however we may
explain the experiment, it follows that a magnetic field in the
stationary state develops an electromotive force which tends to move
electricity in the direction of the electromagnetic action that is, to
the left of an observer placed in the current, and who is Booking in
the direction of the magnetic force.
As the effect in question is very small, the most natural hypo-
thesis, which moreover is approximately verified by Hall's experi-
ments, is to assume that it is proportional to the electromagnetic
force.
641. GENERAL EQUATIONS. Let A, B, C be the components of
this new electromotive force, and suppose' that we are dealing with a
magnetic medium. The component A, which acts along the x axis,
is the algebraical sum of the two actions exerted on the components
v' and w' of the flow of electricity along the y axis, and along the
z axis ; the former is proportional to - Zz/ and the second to Yw'.
If y is the coefficient of proportionality we shall have
(30) B=y(Z'-X/),
C = y(Xz/-Y<).
The components of the total electromotive force of the field
become then
<3<> (> i
^ = ~1J7
Introducing the new electromotive forces, equations (15) give
0, etc.
624 GENERAL THEORIES.
As the functions ^ and 6 play no part in the periodical phenomena
if the medium is not a conductor, we shall have
(33)
a/ 2 a/
642. PROPAGATION OF A PLANE WAVE. Let us consider a
plane wave perpendicular to the z axis. We need only take into
account the component Z of the magnetic force parallel to this axis ;
in this case, if the field is constant, equations (33) reduce to
/a 2 F V\ a 2 F
Kji ( + yZ )= ,
V a/ 2 a/
(34)
Equations (2) and (12) of Arts. 567 and 572 give
Y i
= + -= --- r-,
ax i a 2 G
From this follows
PROPAGATION OF A PLANE WAVE. 625
Substituting these* values in equations (34), we get
/Vr yz yo\ _yp
^ " ~ " '
(35)
KM -^+
These equations have a solution of the form
Y =r cos (// - qz] cos #*,
(36)
G = r cos (/^ - qz] sin ws.
The coefficients will be determined by the condition that the
differential equations are satisfied for all values of z and of /, which
gives
(37)
Z? 2
7 - = 0.
The values of F and G are projections on the axes of an
electromotive force r cos (// qz\ which makes with the axis of x
an angle mz proportional to the thickness.
643. ROTATION OF THE PLANE OF POLARIZATION. The
phenomenon represents then a ray of light, which is propagated
along the z axis with a velocity V = -, the period of whose
vibration is T = ; the wave length that is, the space traversed
*. p 2TT 27T
during a period is A = -- = .
q p q ^
This ray is rectilinearly polarized ; but the plane of polarization
rotates as in an active medium, and the complete rotation is effected
27T
in a time T = .
m
s s
626 GENERAL THEORIES.
The equations (37) give
y Z^ 2 ?ryZ
m
2/A 47T 2/xA 2 '
p i / mZK i /'
~~jn / V' +y -9?"my>
VK?'\ 8,r VK/x'V 4,r*
As the term in y is very small, we may take the square root as
an approximation, and we get finally :
87T 2
We draw from this the following conclusions :
i st. When a polarized ray travels along the direction of a
magnetic line of force, the plane of polarization is rotated in a
direction which depends on the sign of y.
This is the phenomenon discovered by Faraday, with the
inversion for magnetic bodies observed by Verdet.
2nd. The velocity of propagation is increased by the electro-
magnetic action ; but this effect is no doubt too feeble to be
made evident.
If A and V are the wave length and the velocity of the
ray in vacuum, when it is withdrawn from the magnetic action,
and n the refractive index on passing into the medium in question,
we have A = A and V = V . The time which this ray takes in
traversing a thickness e of the medium is = ; the rotation 6
v v o
which the plane of polarization experiences is
ROTATION OF THE YLANE OF POLARIZATION. 627
and the rotation w, for unit difference of potential, becomes
If the refractive index were independent of the dispersion of
the medium, the rotation of the plane of polarization will be
inversely as the square of the wave length, which is approximately
the law of magnetic rotation.
We have seen (597) that, to allow for the dispersion of the
medium, this result must be multiplied by the factor (i ) ;
\ n A/
we get then
irn
This, apart from the factor /*, is the formula at which Maxwell
had arrived by the theory of molecular vortices, and that which
best accords with experiment. We see that, other things being
equal, the rotatory power is inversely as the coefficient of per-
meability /A.
s s 2
628 SUPPLEMENTARY.
CHAPTER XI.
SUPPLEMENTARY.
644. CONCLUSIONS FROM CARNOT'S PRINCIPLE. Sir W.
Thomson showed that the principles which serve as the basis
of the theory of heat that is to say, the principle of the
conservation of energy and Carnot's principle render it possible
to establish some important properties relating to electrical and
magnetic phenomena.
Whenever a body loses heat or changes 'its dimensions, in
opposition to external forces which tend to deform' it in the
contrary directions, it does work. Whatever be the cycle of
transformation, the external mechanical work only depends on the
initial and final state of the body. In all cases, this mechanical
or calorific work corresponds to a loss of energy of the body in
question.
The intrinsic or potential energy of a body is the total work
which it could do if it were indefinitely cooled, or if it were
expanded or contracted to an unlimited^ extent, according as the
molecular forces are attractive or repulsive. There is no means
of measuring this energy, nor even of knowing whether it has a
finite value for a limited mass ;. but we may measure the changes
which it undergoes, starting from a determinate condition, which
is taken as the normal state.
The mechanical state of a homogeneous body which has
undergone any homogeneous deformation that is to say, a
deformation which is reproduced in the same manner in each
element of volume may be expressed by six independent
variables : for instance, the lengths of the sides and the value
of the angles of a parallelopipedon, or the six elements of an
ellipsoid, which would always contain the same portion of a solid.
645. The potential energy E of a body for unit weight, starting
from the normal state, is a function of its mechanical condition
CONCLUSIONS FROM CARNOl'S PRINCIPLE. 629
(form and dimensions) and of its temperature. When the body
undergoes any transformation, it absorbs a certain quantity H of
energy which depends on the deformation it has experienced, and
on the variation of temperature. If only one of the variables x,
varies by dx, and the temperature by dT, we may write
(i) dH = adx + MT.
The functions a and / have an obvious physical meaning. If
we divide them by the mechanical equivalent of heat, the former
represents the latent heat relative to the variable x, and the second
/ the specific heat for a constant mechanical state.
The work dW done by external forces only depends on the
change of form, and we have
The increase of potential energy is the sum of these two expressions,
which gives
(2) dE =
For any given closed cycle, the total variation of potential
energy is null ; the elementary variation must then be an exact
differential of the independent variables, which gives the condition
(3)
DT
This equation may be considered as expressing the principle
of the conservation of energy, or the mechanical equivalence of
heat.
646. In order to apply Carnot'si principle, the cycle of trans-
formations must be reversible, and the final state of the body
must be identical with the initial state.
The sum of the quotients of the calorific energy absorbed by
the corresponding absolute temperature is then null, and we have
(<*'*,( f*i* : ":Lft\i't
j T~J VT ;+ T y~'
630 SUPPLEMENTARY.
The expression in the parenthesis should also be an exact
differential, from which it follows that
~oa a <)/
(4) ^~T = ^'
OL T ox
This equation (4) may also be considered as the translation
of Carnot's principle.
Comparing equations (3) and (4), we deduce
An analogous equation would be obtained for any other
independent variable. If A is the whole differential coefficient
of the calorific energy absorbed for any transformation of the
body at a constant temperature, which corresponds to the latent
heat relative to this transformation, and if B is the corresponding
differential coefficient of the external work, we shall have
(6) A--T*.
^T
647. We may deduce the following conclusions from this
equation (6) :
Whenever the external work takes place in a direction such
that the rate of change is negative, the value of A is positive. In
other words, whenever the deformation produced by the external
work is such that a deformation of the same kind could be
produced by a cooling of the body, this work is accompanied by
a disengagement of heat, and therefore by a rise of temperature.
The reverse is the case if the rate of change were positive. Hence
it is that a gas becomes heated when it is compressed, and cooled
when it expands.
As solids expand as a general rule when the temperature rises,
we see that the uniform compression of a solid will also produce
a rise of temperature.
It is otherwise with bodies whose expansion is abnormal, such
as water at a temperature below 4, and iodide of silver at ordinary
temperatures. The compression of these bodies would produce a
lowering of temperature.
CHANGES OF TEMPERATURE DURING MAGNETISATION. 631
In like manner also a stretched metal wire should become
cooled when it is twisted to a greater extent, if we assume as
certain that the coefficient of torsion diminishes as the tem-
perature rises. A twisted wire ought also to become more
heated, independently of the external work, when it is allowed
to untwist.
In any case, the amount of energy H, absorbed or disengaged,
may be deduced from the external work and the properties of a
body. Without dwelling further on the purely calorific phenomena
which could thus be deduced from Carnot's principle, we shall
examine some deductions from equation (6) relative to electrical
or magnetic phenomena.
648. CHANGES OF TEMPERATURE DURING MAGNETISATION.
The relations we have already pointed out between the changes of
temperature and the coefficients of magnetisation enable us to
predict the following results :
i st. If we work at a temperature below redness, but so high that
the coefficient of magnetisation is decreasing, a piece of soft iron
should become heated when it is slowly brought near a magnet, and
cooled when it is removed. We assume that the motion is slow, so
as to avoid the influence of induction currents.
The reverse would be the case at ordinary temperatures, if the
coefficient of magnetisation, as seems probable, increases with the
temperature.
2nd. Cobalt should behave like iron that is, become cooled
when it is brought near a magnet at the ordinary temperature ; and
become heated, on the contrary, if we work at a higher temperature
than that of the maximum of magnetisation.
3rd. For nickel, there is no maximum of magnetisation ; at all
temperatures this metal ought to become heated when it is brought
near, and cooled when it is moved away from a magnet.
More generally, nickel and cobalt at ordinary temperatures
ought to become cooled when the motion requires an external
work opposed to that of the magnetic forces. For nickel at any
given temperature, and for the two former metals, at temperatures
higher than those of the maximum of magnetisation, any displace-
ment which requires a work opposed to magnetic actions produces,
on the contrary, a heating of the body.
4th. In a magnetic field, a crystal becomes cooled when its axis
of greatest magnetic induction, or of least diamagnetic induction,
passes from a direction parallel, to a direction perpendicular to that
of the field.
632 SUPPLEMENTARY.
649. ELECTRICAL HEATING OF TOURMALINE. Pyroelectrical
phenomena give rise to analogous considerations.
The pyroelectricity of crystals is explained, on Faraday's theory
(118), by assuming that the crystal is in a state of electrical
polarization, the external effect of which is equivalent to that of two
layers, of equal masses and contrary signs, distributed on the surface.
When the crystal is at a constant temperature, the surrounding
medium, either by its own conductivity, or by the surface of
the crystal itself, soon acquires a superficial electrification, which
neutralises the former, and annuls its action on any external point.
When the crystal is broken perpendicularly to the electrical axis,
the whole of each of the two fragments then are electrified in
opposite directions, not only by the new layers which the polarization
on the broken surfaces produces, but also in consequence of the
electrification induced on the old surfaces, the equilibrium of which
is broken.
When the temperature changes, the electrification changes
also ; but the equilibrium produced by the electrification of the
surrounding medium is only set up by degrees, and more or less
slowly, according to the conductivity of the medium or of the
surface of the crystal.
If this explanation of pyroelectricity is correct, it follows that a
pyroelectrical crystal should be heated or cooled when it is moved
in an electrical field, like magnetised iron in a magnetic field.
Tourmaline becomes heated if it is displaced in such a manner
that the influence of the field tends to increase its polarization, and
is cooled in the contrary case.
The effect produced on tourmaline does not depend on the
electrification of the surface, and we arrive at this remarkable re-
sult; a pyroelectrical crystal which appears in the natural state, its
properties being neutralised by the electrification of the medium,
undergoes, when it is displaced in a field, the same variations of
temperature as if its electrical properties were apparent that is to
say, as if it had been raised to a high temperature, then dried, and
rapidly cooled.
650. PRINCIPLE OF THE CONSERVATION OF ELECTRICITY.
Whenever a system of bodies, disconnected from any external bodies,
is the seat of any electrical phenomenon (8), the total quantity of
electricity which it contains remains unchanged. This principle is
verified in all experiments, and it is a consequence of Maxwell's
views as to the constitution of the medium which serves for
conveying electrical force. Although we are not able to affirm that
PRINCIPLE OF THE CONSERVATION OF ELECTRICITY. 633
the total quantity of electricity in nature is strictly null, we may
assume at least that actual physical phenomena produce no change,
and that it remains constant in the same sense as the total quantity
of energy or of matter. In other words, a quantity of electricity
may be considered indestructible by any other cause than a quantity
of electricity of the opposite sign. M. Lippmann has showed that
this principle leads to conclusions analogous to those of Carnot's
theorem ; when it is associated with the principle of the conservation
of energy, we may deduce the explanation of a certain number of
phenomena, and moreover predict other phenomena which have not
yet been observed.
Suppose that a body A traverses a closed cycle in a system
that is to say, that, after having undergone a series of transformations,
it returns to its primitive state the sum of the quantities dm of
electricity which it has received during the cycle is zero, so that
we have
(7)
This condition necessitates that the elementary increment dm
can be integrated that is to say, is the exact differential of a
function of the independent variables. If the phenomenon only
depends on two variables x and y, which is the most general case,
we may write
(8) <fcw
and the condition of integrability is
Equation (9) may be considered as in some sense the expression
of the principle of the conservation of electricity; following M.
Lippmann, we shall give some of the applications.
651. ELECTROCAPILLARY PHENOMENA. Lippmann has observed
that the capillary effects manifested between mercury and acidulated
water depend on the difference of potential of the two liquids, and
conversely that the difference of potential of the liquids is modified
when the magnitude of the surface of contact is changed by external
forces. This reciprocity of phenomena is a consequence of the
preceding principle.
634 SUPPLEMENTARY.
Let S be the surface of mercury in contact with acidulated water,
A the capillary tension of the liquid, and x the excess of potential
of the water over that of the mercury ; when, for any cause whatever,
the surface is increased by */S, the work of the capillary tension is
Let us now suppose that a quantity of electricity dm, furnished
by an extraneous source, reaches the surface of the water ; there
will be an increase dx of the difference of potential, at the same
time as a dilatation dS of the surface. The quantities x and S are
then independent variables of the phenomenon. Since, other things
being equal, the mass dm is proportional to the surface, we may
write
(10) dm = YSdx + XdS.
The factor X represents the capacity of unit surface at constant
potential, and Y the electrical capacity of unit surface, the surface
being constant and the potential variable. The principle of the
conservation of electricity gives the condition
On the other hand, the electrical work xdm, introduced into
the system, produces an increase of potential energy of the surface,
and an external work - AdS. If this operation be repeated several
times in opposite directions, so as to return to the initial state, and
that there has been neither gain nor loss of heat, the variation of
energy of the system will be null, which gives
J
that is to say, replacing dm by its value,
(12)
I
As this expression must be null for any closed circuit, we have
also
GAS CONDENSERS. 635
From these two equations (n) and (13) we deduce
dA c) 2 A
;, Q
= - S dx- -d= -d S
The capacities X and Y are therefore functions of the capillary
tension. It follows from this that if this tension is a function of the
difference of potential, as experiment shows, this capacity cannot be
null. If the surface is deformed, while the difference of potentials is
kept constant, there should from equation (10) be a production or
absorption of electricity ; and, if we work at a constant charge, we
modify the difference of potentials. These two orders of phenomena
are therefore correlated, which is what experiment confirms.
652, GAS CONDENSERS. Boltzmann has proved, in conformity
with Maxwell's theory (634), that the capacity of a condenser, whose
two coatings are separated by a layer of gas, varies proportionally
with the pressure. The converse must follow, that the pressure of
a definite mass of gas, placed between the coatings of a condenser,
is a function of the difference of potential.
The two independent variables on which the phenomenon here
depends, are the difference of potential x, and the pressure p of
the gas. When the positive armature receives a quantity dm of
electricity, we have
(14) dm = Cdx + hdp.
The factor C represents the capacity of the condenser at constant
pressure, *h a coefficient which experiment shows is positive, for
the capacity increases with the pressure, and which is determined
by Maxwell's theory. The principle of the conservation of elec-
tricity gives
ac M
Let us now consider a closed cycle, without change of tempera-
ture. The work required to increase by dm the charge of the
positive coating is equal to xdm ; on the other hand, a mass of
636 SUPPLEMENTARY.
gas in contact with the condenser produces an external work pdv,
when its volume increases by dv. If the gas and the condenser
return to their original condition, the change of energy of the system
is null, and we have
(16)
which requires that the expression within the parenthesis shall be
an exact differential.
The volume v of the gas in question is a function of the pressure
/, and perhaps also of the difference of potential x ; we shall
therefore write
(17) dv
and the course of the reasoning will show whether the coefficient
a differs from zero. As this expression is also an exact differential,
we deduce from it
Substituting in the expression (16) the values of dv and of dm,
we get
(
[(O - op) dx + (hx -bp)dp~\ = Q.
The condition of integrability is then
(19)
(JJJ UA
Taking into account equations (15) and (18), this condition
reduces to
a= -h.
The coefficient a is thus different from zero, and negative. It
follows then, from equation (17), that, at a constant pressure, the
volume of a mass of gas surrounding a condenser should diminish
proportionally to the difference of potential of the armatures. This
result seems to have been verified by Quincke at any rate for
carbonic acid.
ELECTRICAL DILATATION OF GLASS. 637
653. ELECTRICAL DILATATION OF GLASS. A Leyden jar dilates
when it is electrified, and contracts as soon as it is discharged.
This phenomenon, foreseen by Volta, was demonstrated by M. Govi,
and M. Duter has shown that the expansion of gas is proportional
to the square of the difference of potential of the armatures.
Let us consider the phenomenon in the form given to it by M.
Righi, that is to say, a condenser formed of a glass tube whose
two faces are covered with tinfoil; let / be its length, and let us
suppose that at the same time the tube is exposed to a strain in
the direction of its length, which is represented by the weight /.
As the length is a function of the difference of potential x,
and of the stretching weight /, we have
(20) dl=adx + bdp.
The coefficient a is positive, and measures the electrical elon-
gation, and b is the coefficient of elasticity of the tube.
If we assume that the tube undergoes no permanent defor-
mation, it follows that
(21) S* = S~-
op ox
On the other hand, it is to be presumed that the charge of
the condenser is also a function of the stretching weight, so that
we may put
(22) dm
C being the capacity of the jar, and h a coefficient which we do
not know a priori to be different from zero.
The principle of the conservation of electricity gives
DC M
(23) =
The variation of energy of the jar, for an increase dm of the
charge, and an elongation dl, is
xdm +pdl= (Cx + ap} dx + (hx + bp)dp.
638 SUPPLEMENTARY.
As this expression must be an exact differential, we deduce
from it
(24)
or, taking equations (21) and (23) into account,
It follows from equation (22), that at a constant potential, the
charge of electricity increases with the stretching weight, and that
with a constant charge, the potential diminishes when the stretch-
ing weight increases that is to say, when the tube is elongated.
Experiment further shows that the elongation is proportional
to the square of the difference of potential, and that, k being a
constant,
A/=/b; 2 .
It follows from this, that
a/
a = ~
and, from equation (23),
The capacity of the jar increases, therefore, proportionally to
the stretching weight. .
It may be observed that the electrical attraction of the two
armatures of a condenser would also crush the intermediate layer,
and give analogous effects ; but it does not seem that this cause is
sufficient to explain the phenomena.
654. COMPRESSION OF TOURMALINE. We shall analyze in the
same way a phenomenon recently discovered by MM. P. and
J. Curie. When a tourmaline is compressed along its axis, the
crystal becomes electrically polarized in the same direction as
that which would be produced by a rise in temperature. This
polarization is proportional to the compression, and disappears
with it. Other hemihedral crystals, such as quartz and topaz,
behave like tourmaline when they are compressed along a hemi-
hedral axis.
ELECTRICAL COMPRESSION OF TOURMALINE. 639
Let us suppose that the bases of a prism of tourmaline are
covered with plates of metal A and B, one of which B is
connected with the earth, -while the other can be connected with
a source of constant potential.
The pressure p and the potential x of the armature A, may
thus be varied by making the system pass through a closed cycle,
and we may take these two quantities as independent variables.
The quantity dm of electricity which the plate A receives is
expressed by
dm = Cdx + hdp.
The coefficient C is the capacity of the armature A at constant
pressure, and h is a negative coefficient if the end A of the crystal
is positively electrified by compression. The principle of the
conservation of electricity gives then
(25)
If / is the length of the crystal, we may also form the
equation
dl = adx + bdp ,
in which b is the coefficient of elasticity of the crystal.
If we apply the principle of the conservation of energy as
we have done above for M. Righi's tube, we deduce from it
a= -h.
As the value of h is negative, equation (26) shows that a
tourmaline becomes longer, when it is electrified in the same way
as it would be by a rise of temperature, and this elongation is
proportional to the potential. A change of structure results, and,
no doubt, also an alteration of optical properties analogous to
that observed by Dr. Kerr in transparent bodies.
The factor h is constant, since, according to MM. Curie, the
electrification is proportional to the pressure ; hence the capacity
of a condenser with a plate of tourmaline is independent of the
compression to which the crystal is subjected.
655. GENERALISATION OF LENZ'S LAW. In all the preceding
cases the converse of the phenomenon, the existence of which is
640 SUPPLEMENTARY.
demonstrated by the principle of the conservation of electricity,
is of such a nature as to oppose the production of the original
phenomenon. We thus rediscover, in a more general way, Lenz's
law (511) relative to the phenomena of induction.
These few examples will be sufficient to show how the general
principles of the science enable us to connect the most varied
phenomena, and even to determine their numerical relations,
without its being necessary to know the intimate nature of the
forces which come into play.
END OF VOL.
INDEX" OF SUBJECTS. 641
INDEX OF SUBJECTS.
PART L STATICAL ELECTRICITY.
CHAPTER L INTRODUCTORY.
PAGE
Electrification I
Conductors. Insulators I
Distinction of Two Electricities 2
Electrical Actions. Electrical Masses , 2
Positive and Negative Electricity 3
Electrical Force 3
Distribution of Electricity 4
Electricity of Contact 4
Electrification by Influence. Induction 4
Electrical Equilibrium 5
Dielectrics 5
Localisation of Electricity on the Surface 5
Induction on a Closed Conductor 5
Addition of Charges 6
Hypothesis as to the Nature of Electricity 6
Electrical Density. Electrical Thickness 7
CHAPTER IL ON POTENTIAL.
Reciprocal Action of Two Electrified Bodies 9
Electrical Field 9
Lines of Force i 9
Definition of Potential 9
Equipotential Surfaces. Electromotive Force - 10
Expression of Force as a Function of Potential 1 1
Equilibrium of Conductors 12
Numerical Value of Potential 13
Potential in the Case of the Law of the Square of the Distances .... 13
On the Flow of Force 15
Green's Theorem 16
Equations of Laplace and Poisson 20
Distribution of Electricity on the Surface of Conductors . . . ., . . . 21
Green's Formula 22
TT
642 INDEX OF SUBJECTS.
PAGE
Tubes of Force 24
Coulomb's Theorem 25
Corresponding Elements 25
Uniform Field 26
Electrified Surface separating Two Dielectrics 26
Electrostatic Pressure 28
Consequences of the Distribution of Electricity on the Surface of Conductors 30
Actions of Spherical Layers 32
Action of a Sphere consisting of Homogeneous Layers 34
CHAPTER III GENERAL THEOREMS.
Emission and Absorption of Force by Electrical Masses 36
The Potential can neither have a Maximum nor a Minimum outside the
Acting Masses 36
Singular Points of Equipotential Surfaces 37
Points and Lines of Equilibrium 38
There is only One State of Equilibrium 40
Theorems relating to Closed Surfaces 41
Faraday's Law 44
Gauss' Theorem 46
M. Bertrand's Corollaries to Gauss' Theorem 46
Earnshaw's Theorem 47
CHAPTER IV. ELECTRICAL EQUILIBRIUM.
Conditions of the Equilibrium of Conductors 51
General Observations 52
Relation of Charges to Potentials 53
Analogies of the Problem of Electrical Equilibrium 55
Electrical Capacity 59
Sphere 59
Ellipsoid 60
Power of Points 62
Circular Plate 63
Concentric Spheres 63
Condensers 65
Leyden Jar 65
Concentric Cylinders 67
Capacity of Telegraph Cables 69
Plane Condensers 69
Capacity of a System of Conductors 7 1
Batteries 71
Charge by Cascade 7 2
Reciprocal Influence of Two Insulated Conductors. Murphy's Method . . 74
Reciprocal Action of Two Electrified Conductors 75
INDEX OF SUBJECTS. 643
CHAPTER V. WORK OF ELECTRICAL FORCES.
PAGE
Electrical Energy 77
Energy of a Single Conductor 77
Energy of a System of Conductors 78
Discharge of Batteries. Quantity Battery 80
Discharge of a Battery in Cascade 81
Electrical Work in the Displacement of Insulated Conductors .... 82
Conductors at Constant Charge 83
Application to the Theory of Symmetrical Electrometers 85
CHAPTER VI. ON DIELECTRICS.
Function of the Dielectrical Medium 87
Expression of Force as a Pressure 87
Tension and Repulsion of Lines of Force 92
Energy of the Dielectric Medium 92
Specific Inductive Capacity 94
Electrical Absorption 95
Polarization of the Dielectric 95
Definition of Dielectric 96
Refraction of the Flow of Force 98
Tubes and Flow of Induction 99
Characteristic Equations of Induction IOO
Observations on the Fictive Layer IOI
Charges of Two Corresponding Elements IOI
Energy of a System in the Case of any Given Dielectrics IO2
Comparison with Thermal Phenomena 103
Change of Potential Produced by Interposing a Dielectric 104
Maxwell's Theory of Displacement 109
CHAPTER VII. PARTICULAR CASES OF EQUILIBRIUM.
Representation of the Electrical Field. Lines of Force Ill
Uniform Field 112
Field Symmetrical in reference to a Plane 113
Cylindrical Systems 114
Two Parallel Lines .' 115
Several Parallel Lines 116
Two Lines of Opposite Signs 117
Two Equal Lines of Opposite Signs 119
Systems of Revolution 120
Case of a Single Mass 121
Any Two Given Masses 122
Two Equal Masses of the Same Sign 123
Two Unequal Masses of the Same Sign 124
XT 2
644 INDEX OF SUBJECTS.
PAGE
Two Equal Masses of Opposite Signs 125
Electrical Moment ' 127
Principle of Images 128
Induction in a Medium consisting of Two Dielectrics separated by a Plane . 130
Three Dielectrics separated by Parallel Planes 131
Two Equal Masses of Opposite Signs infinitely near each other . . . .134
Induction on an Infinitely Small Body 139
.Polarized Sphere. Layers of Gliding 139
Conducting Sphere in a Uniform Field 142
Uninsulated Conducting Sphere in a Uniform Field 145
Dielectric Sphere in a Uniform Field 146
Concentric Spherical Layers in a Uniform Field 148
Poisson's Hypothesis of the Constitution of Dielectrics 151
Two Unequal Masses of Opposite Signs 154
Electrification of a Sphere under the Influence of a Point 159
Image of any Given System 162
Reciprocal Action of Two Spheres 163
Motion of Small Bodies in the Electrical Field 166
Direction of a Dielectric Needle in a Variable Field 169
Action of a Field on a Conducting Needle 173
CHAPTER VIIL SOURCES OF ELECTRICITY.
Volta's Discovery 175
Electromotive Force of Contact 175
Volta's Laws. Law of Contact 176
Law of Successive Contacts 177
Exception to the Law of Successive Contacts. Electrical Batteries . . .178
Conclusions relative to the Distance of Atoms 179
Contact of Dielectrics 181
Electrification by Friction 182
Electrical Machines 182
Essential Parts of Electrical Machines 183
Limit of the Charge 183
Yield of Machines 185
INDEX OF SUBJECTS. 645
PART II. ELECTRICAL CURRENTS.
CHAPTER L PROPAGATION OF ELECTRICITY IN
THE PERMANENT STATE.
PAGE
Permanent Condition 186
Analogy with Thermal Phenomena 186
Ohm's Theory 187
Kirchhoff's Hypothesis 188
Superposition of Permanent States 188
In the Permanent State the Density in the Interior of a Conductor is Null 189
Linear Conductors. Ohm's Law 190
The Resistance of a Conductor is the Inverse of a Velocity 191
Physical Meaning of this Velocity 192
Any Given Linear Conductors 193
Kirchhoff's Laws 194
Resistance of a Multiple Conductor 195
Heterogeneous Linear Conductors 196
Case in which the Circuit contains Electromotive Forces ../... 198
Conductors of any Given Form. Electrodes 199
Heterogeneous Conductors 2OI
Anisotropic Conductors 202
Conductors in Two Dimensions 204
Resistance of a Conductor of any Given Form 206
Distribution of Electricity on Linear Conductors 207
Propagation in a Wire when there is a Loss on the Surface 208
Resistance of a Conductor where there is a Loss by the Sides .... 210
CHAPTER II. VARIABLE STATE.
Application of Fourier's Formulas 213
Variable State in a Cylindrical Conductor 214
Duration of the Relative Propagation 216
Unlimited Wire 216
Momentary Contacts 220
Electrical Wave 221
Wire of Finite Length 227
Momentary Contacts 230
Use of Condensers 232
Propagation in Dielectrics 232
Residual Charge of Condensers 233
646 INDEX OF SUBJECTS.
CHAPTER III ENERGY OF CURRENTS.
PAGE
Disengagement of Heat 236
Joule's Law 237
Connection between Ohm's and Joule's Laws 238
Peltier's Phenomenon 239
Chemical Decomposition 241
Faraday's First Law 242
Polarization of the Electrodes. Capacity of Polarization 243
Secondary Currents 244
Successive Chemical Actions of the Current. Faraday's Second Law . . 245
Electrochemical Equivalents 247
E. Becquerel's Law 247
Electrical Couples 248
Depolarization by Diffusion 248
Volta's Couple 249
Unpolarizable Cells 250
Cells with Two Liquids 251
Electrostatic Phenomena in Piles or Batteries 252
Uninsulated Battery 252
Insulated Battery 252
Representation of Potentials in the Interior of the Battery 254
Battery Placed in a Conducting Medium 255
Electrocapillary Phenomena 258
CHAPTER IV. THERMOELECTRICAL CURRENTS.
Seebeck's Discovery . ' 259
Laws of Thermoelectrical Currents 260
Law of Volta 260
Law of Magnus 260
Law of Successive Temperatures (Becquerel) 260
Law of Intermediate Metals (Becquerel) 261
Phenomena of Inversion 261
Graphical Representation of the Phenomena 262
Conclusions from Volta's Law 263
Consequences of Inversion 265
Sir W. Thomson's Theory 266
Thomson Effect 267
Thermoelectrical Powers 269
Neutral Point 271
Specific Heat of Electricity 272
Electromotive Force of a Thermoelectrical Couple 273
Tait's Hypothesis 275
Electrical Convection of Heat 277
Nature of the Peltier Phenomenon 278
INDEX OF SUBJECTS. 647
PART III. MAGNETISM.
CHAPTER I. PRELIMINARY.
PAGE
On Magnets 280
Magnets Natural and Artificial, Permanent and Temporary 280
Magnetic and Diamagnetic Substances 281
Distribution of Magnetism in Magnets. Poles 281
The Two Kinds of Magnetism 282
Law of Magnetic Actions 282
Magnetic Masses 283
Magnetic Field 284
Definition of Poles. Magnetic Axis of a Magnet 284
The Magnetic Mass of a Magnet is Zero 285
Magnetic Moments 285
Action of a Uniform Field on a Magnet 287
Astatic Systems 287
Magnetic Polarity. Rupture of a Magnet 288
Induced Magnetisation 288
Soft Iron. Coercive Force 289
Influence of Temperature 290
On Magnetic Fluids 290
Definition of Terrestrial Magnetic Elements 291
Distribution of Terrestrial Magnetism 294
Hypothesis of a Terrestrial Magnet 294
Variations of Terrestrial Magnetism 296
CHAPTER II. CONSTITUTION OF MAGNETS.
Magnetic Filaments 298
Free Magnetism 298
Uniform Magnet 299
Any Given Magnet 299
Potential of a Magnet 299
A Magnet is Equivalent to a Magnetic Surface 300
Poisson's Theory , 301
Sir W. Thomson's Theory 302
Intensity of Magnetisation 302
Expression for Potential 303
Uniform Magnets 305
648 INDEX OF SUBJECTS.
PAGE
Force in the Interior of a Magnet 307
Magnetic Force 309
Magnetic Induction 309
Line, Tube, Flow of Induction 310
Different Kinds of Magnets 310
Magnetic Solenoids. Magnetic Power of a Solenoid 310
Solenoidal Magnets . . . 312
Magnetic Shells 312
Lamellar Magnets 316
Potential of Magnetisation 316
Potential of a Solenoidal Magnet 318
Potential of a Lamellar Magnet 318
Potential of Induction 322
Potential Energy of Magnets 323
Energy of a Magnetic Shell 324
Action of a Field on a Shell 326
Reciprocal Action of Two Shells 330
Relative Energy of Two Shells 335
Neumann's Formula 337
CHAPTER III. PARTICULAR CASES.
Potential of a Uniform Magnet 338
Sphere 339
Ellipsoid 340
Cylinder Magnetised Transversely 344
Potential of Magnetic Shells 345
Potential of a Circular Layer 348
Potential of a Uniform Circular Shell 352
Potential of a Spherical Layer 353
Solenoidal Magnets 355
Cylinder 356
CHAPTER IV. MAGNETIC INDUCTION.
General Characteristics of Magnetic Induction 358
Induced Magnetisation is Proportional to the Magnetising Force . . . 359
The Induced Magnetisation is Superficial 360
Equation of Continuity. Coefficient of Induction 362
Case of Two Different Magnetic Media. Relative Magnetisation .... 363
Magnetic Susceptibility and Permeability 366
Anisotropic Bodies 366
Uniform Magnetisation 367
Sphere , 368
Poisson's Hypothesis 369
INDEX OF SUBJECTS. 649
Ellipsoid. Cylinder 370
Barlow's Problem 370
Anisotropic Bodies 375
Experimental Determination of the Coefficients of Magnetisation . . . 376
Displacement of Bodies in a Magnetic Field. Attractions and Repulsions . 377
Equilibrium of Long Bodies in a Uniform Field 380
Equilibrium of Bodies in a Variable Field 381
Oscillations of an Infinitely Small Isotropic Needle 383
Influence of Temperature , . 384
CHAPTER V. ON MAGNETS.
Magnetisation 385
Induction of a Magnet on Itself. Demagnetising Force 385
Particular Cases of Magnetisation. Sphere 387
Ellipsoid 388
Anchor Ring. Tore 388
Cylinder 388
Any Given Magnets. Experimental Methods 388
Oscillations 389
Torsion Balance 389
Use of Soft Iron 389
Measurement of the Flow by Induction Currents 390
Distribution of the Fictive Layer 391
Cylindrical Magnets 392
Empirical Formulae 393
Hypothesis on the Constitution of Magnets 398
Weber's Theory 399
Maxwell's Theory 403
Jamin's Observations 405
Influence of Temperature 406
CHAPTER VI. MAGNETIC CONDITION OF THE
GLOBE.
Gauss' Method 407
Magnetic Parallels 407
Magnetic Equator 408
Terrestrial Magnetic Poles 409
False Pole. Properties of a Closed Polygon 410
Introduction of Geographical Co-ordinates 413
Expression of Potential . 415
Is the Magnetism of the Earth in the Interior only? 416
Influence of the Sun and Moon 417
650 INDEX OF SUBJECTS.
PART IV. ELECTROMAGNETISM.
CHAPTER I. CURRENTS AND MAGNETIC SHELLS.
PAGE
CErsted's Experiment 420
Magnetic Field of a Current 420
Action of a Rectilinear Current on a Pole. Experiments of Biot and Savart 421
Potential of an Unlimited Rectilinear Current 423
The Potential of an Unlimited Current is not a Simple Function of the
Co-ordinates 425
Potential of an Angular Current 426
Potential of a Triangular Current 427
Potential of any Closed Circuit 428
Equivalence of a Closed Current and of a Magnetic Shell. Ampere's Theorem 428
Remarks on the Equivalence of a Closed Current and a Magnetic Shell . . 429
Relative Energy of a Magnetic System and a Current 430
Reciprocal Action of Two Closed Currents 432
Relative Energy of Two Currents 432
Electromagnetic Rotation 433
Faraday's Experiments 434
Another Form of Expression for the Electromagnetic Work 436
Electromagnetic Action on a Current-Element 437
Reciprocal Action of Two Elements of a Current 438
Electromagnetic Intensity of a Current 438
Electromagnetic Units 439
CHAPTER II. ELEMENTARY ACTIONS.
Ampere's Method 440
Action of a Pole on a Current Element. Fundamental Principles . . . 441
Reciprocal Action of a Pole and of a Current 446
Equivalence of a Current and a Magnetic Shell 449
Action of Two Elements of a Current 449
Determination of the Functions Y(r) and f(r) 452
Determination of the Ratio of the Two Constants 454
Determination of the Constant h 456
Electrodynamic Unit of Intensity 459
Formulae Equivalent to that of Ampere 460
Formula of M. Reynard 460
General Formula 462
Ampere's Formula 463
Grassmann's Formula 463
INDEX" OF SUBJECTS. 65 1
CHAPTER III. PARTICULAR CASES.
PAGE
Action of Two Parallel Currents 465
Angular Currents 465
Apparent Repulsion of Two Consecutive Elements of Current 466
Electromagnetic Rotation. Barlow's Wheel 467
Ampere's Experiment 468
Rotation of Liquids 469
Davy's Experiment 469
M. Jamin's Experiment 469
M. Berlin's Experiment 469
Electrodynamic Rotation 470
Action of a Uniform Field 470
Astatic Circuits 472
Rotation of a Current under the Action of the Earth 473
Action of Two Rectangular Circuits 475
Properties of Circular Currents 477
Electromagnetic Solenoid 478
Cylindrical Coil 478
Annular Coil 479
Case of any Given Surface 480
Ampere's Theory of Magnetism 481
Magnetisation by Currents. Consequent Points 483
Examples of Magnetisation 484
Measurement of Currents. Galvanometers 486
Tangent Galvanometer 487
Electrodynamometers 488
Measurement of Discharges 488
CHAPTER IV. INDUCTION.
Faraday's Discovery 491
Lenz's Law 493
Neumann's Theorem 493
Theory of Helmholtz and Thomson 493
General Law of Induction 497
Coefficients of Induction 498
Electromagnetic Induction 498
Electrodynamic Induction 500
Intrinsic Energy of the Current 500
CHAPTER V. PARTICULAR CASES OF INDUCTION.
Electromagnetic Resistance is a Velocity 503
Closed Circuit in a Uniform Field 505
Determination of the Inclination by Induction Currents 506
652 INDEX OF SUBJECTS.
PAGE
Faraday's Disc 506
Terrestrial Currents 507
Variable State of a Current 508
Extra Current on Opening 510
Variable Electromotive Force 511
Current of Discharge. Oscillating Discharges 512
Case of Two Circuits 517
Current on Opening 519
Current on Closing 520
Two Circuits with Variable Electromotive Force 521
Telephone and Microphone 523
Induction in an Open Circuit 524
Laws of Branch Currents in the Variable State 526
Phenomena of Induction in Telegraph Cables 530
Calculation of the Coefficients of Induction. Solenoids ...... 531
Concentric Coils 533
Coils with a Soft Iron Core , . 534
Annular Coils 535
Electrical Motors 536
Electromotors 538
Application to the Study of Magnetism 541
Weber's Hypothesis on Magnetism and Diamagnetism 542
Absolute Conducting Screens 544
CHAPTER VI. PROPERTIES OF THE ELECTRO-
MAGNETIC FIELD.
Maxwell's Theory 546
Equations of the Magnetic Field 546
Equations of Currents 549
Potential Energy of Currents 550
Relative Displacement of Circuits 552
Equations of the Electrical Field 554
CHAPTER VIL PHENOMENA OF INDUCTION IN
NON-LINEAR CONDUCTORS.
Magnetism of Rotation , 556
Conducting Shells 556
Case of a Plane Shell 558
Magnetic Images 561
Induction of a Movable Magnetic System 562
Calculation of the Action of Induced Currents 564
Case of a Single Pole 564
Arago's Experiment 571
INDEX "OF SUBJECTS. 653
Damping of Compass Needles 573
Induction of any Given Conductor 573
CHAPTER VIIL OPTICAL PHENOMENA.
Faraday's Discovery 574
Positive and Negative Bodies 574
Verdet's Laws. Verdet's Constant 575
Magnetic Rotatory Dispersion 576
Mr. Kerr's Experiment 576
Explanation of Rotatory Polarization 577
Observations of Sir W. Thomson 582
Electrical Double Refraction 583
CHAPTER IX. ELECTRICAL UNITS.
Fundamental Units. Derived Units 584
Dimensions of a Derived Unit 585
Derived Mechanical Units 586
Electrical and Magnetic Derived Units 587
Electrostatic System 588
Electromagnetic System 590
Dimensions of the Principal Units 592
Relations between the Two Systems of Units 594
Choice of Fundamental Units 595
Absolute C.G.S. System 595
Practical System 595
Comparative Values of the Principal Units . , 597
Physical Conception of the Velocity a 598
CHAPTER X. GENERAL THEORIES.
Ampere's Hypotheses 601
Formulae of Gauss and Weber 601
Phenomena of Induction 606
Various Attempts at a Theory 608
Electromagnetic Theory of Light 609
General Equations 610
Propagation of Undulations in a Dielectric 612
Plane Waves 612
Distribution of the Energies 614
Velocity of the Propagation of Light 615
Specific Inductive Capacity 616
654 INDEX OF SUBJECTS.
PAGE
Anisotropic Media 617
Bad Conductors 619
Conducting Bodies 620
Rotatory Magnetic Polarization 621
Hall's Phenomenon 622
General Equations 623
Propagation of a Plane Wave 624
Rotation of the Plane of Polarization 625
CHAPTER XL SUPPLEMENTARY.
Conclusions from Carnot's Principle 628
Changes of Temperature during Magnetisation 631
Electrical Heating of Tourmaline 632
Principle of the Conservation of Electricity 632
Electrocapillary Phenomena 633
Gas Condensers 635
Electrical Dilatation of Glass 637
Compression of Tourmaline 638
Generalisation of Lenz's Law 639
THOS. DE LA RUE AND CO., PRINTERS, BUNHILL ROW, LONDON.
THOS. DE LA RUE & GO'S LIST.
Thos. De La Rue & Co.'s List.
In Preparation. Demy 8vo.
WELLS 01 DISEASES OF THE ETE:
A TREATISE ON CLINICAL OPHTHALMOLOGY.
EDITED AND REVISED THROUGHOUT BY
M. MACDONALD McHA'RDY,
PROFESSOR OF OPHTHALMOLOGY IN KING'S COLLEGE, LONDON J OPHTHALMIC SURGEON
TO KING'S COLLEGE HOSPITAL;
AND SURGEON TO THE ROYAL SOUTH LONDON OPHTHALMIC HOSPITAL.
FOURTH EDITION,
With, a new and original series of Ophthalmoscopic Drawings, reproduced in fac-simile by
ehromo-lithography ; and numerous additional Engravings.
NEW AND CHEAPER EDITION.
Third Edition. Crown 8vo. Cloth. Price 10s. 6d.
BILLIARDS,
By JOSEPH BENNETT, Ex-Champion. Edited by "CAVENDISH."
WITH UPWARDS OF 2OO ILLUSTRATIONS.
" Everyone who plays billiards and who does not
now-a-days? ought to possess a copy of 'Billiards,'
by Joseph Bennett, Champion, edited by 'Cavendish.'
To the third edition, just issued, has been added a
new chapter on winning-hazards, and some amend-
ments have been made in the body of the work.
* Though it is impossible to learn to play
billiards from a book, the instructions on every pos-
sible detail are so minute and so copiously illustrated
by diagrams, that they must be of great assistance
even to those who are considerably experienced in
the history and mysteries of winning and losing
hazards. " Society.
"'Billiards,' a carefully compiled and accurately
illustrated work, which explains the game both from a
scientific and a practical point of view." Sporting
Times.
CLAY ON WHIST.
A new and improved Edition, Cap. 8vo. Price 3s. Gd.
LAWS OP SHORT WHIST,
TREATISE ON THE GAME.
BY JAMES CLAY.
" It is well known that James Clay was the greatest
master of whist of his day. He was also a most able
exponent, in writing, of the principles which he prac-
tised. Since the last issue, whist has not
stood still. Certain alterations in the method of play
have been made, which have secured the favour of
many of the best whist-players; and to omit all
mention of these would be to leave the book behind
the age. A difficulty, according to the
Duke of Wellington, was something to be overcome.
And overcome this difficulty has been, by the simple
expedient of recording in a preface such modifications
of Clay's views as came within the personal knowledge
of his sons. * The publishers should be
awarded a fair share of praise for the admirable finish
of the volume." Field.
" The new edjtion of the treatise on whist by the
greatest of modern whist-players will be welcome to
all admirers of the noble game. The author's sons,
with commendable judgment, have guarded them-
selves against making any alteration in the body of
their late father's work. But in a modestly written
preface they tell us what Mr. Clay thought of some
recent innovations in the game as taught and prac-
tised by other professors. Among the
authorities on the new game Mr. Clay will always hold
a conspicuous place ; and we are glad to find that
there remain no longer any divergences between him
and other writers as to certain new-fashioned yet very
legitimate refinements in the game. In this new
edition several obvious errors in the text have been
corrected, while the print and paper are excellent."
St. James's Gazette.
Thos. De La Rue & Go's List.
Demy 8vo. Cloth. Price 25s. With Map, and Illustrations on Wood.
JUI&LE LIFE H lOIA,
JOURNEYS AND JOURNALS OF AN INDIAN GEOLOGIST,
BY V. BALL, M.A., F.G.S.,
Fellow of the Calcutta, University, and Assistant, Geological Survey of India.
" We can only allude to the many telling pictures of
nature in her more unfamiliar aspects ; curious facts
and discussions on vexed points in natural history;
notes on the economical products of the country ;
traits of native character and manners, with other
interesting matter, which, though scattered broadcast
through the volume, is traceable by means of the in-
dex." Athenaeum.
" Mr. Ball appears to have laboured both in and out
of season, and as the result we have a work which
forms an important contribution to the history of the
fairest possession of the British Crown. Times.
" This is a record of work of a peculiar and scientific
kind, carried out in the teeth of great obstacles, mani-
fold privations, and trials of climate. It introduces us
to tracts little known except to political officers de-
puted to put down wicked customs, or to soldiers who
have had to penetrate the fastnesses of some rebellious
and wayward chief. And the story is told in a simple,
straightforward, and unaffected style." Saturday Re-
ijieTU.
"This work is the day-to-day record of the expe-
riences of a scientific man in parts of India which are
little known even in these times of general travel and
research." Pali Mall Gazette.
Crown 8vo, 2 vols. Cloth. Price 15s.
STRAME STORIES PROM A CHIIESE STUDIO.
TRANSLATED AND ANNOTATED
BY HERBERT A. GILES, OF H.M's CONSULAR SERVICE.
"This collection of Chinese stories is exceedingly
curious as well as entertaining, and Mr. Giles appears
to be excellently qualified for the task he has under-
taken." Times.
"We must refer our readers to Mr. Giles's volumes,
where, if they themselves bring to their perusal a spirit
still capable of enjoying the marvellous, they will find
a great deal that is fuD both of interest and instruc-
tion." Pall Mall Gazette.
" Any one who reads this book with care will not only
be delighted with the stories that are told in it, as works
of art, out will get a much better knowledge of the true
character of Chinese life than could well be got in any
other manner." Scotstnan.
" Under this title Mr. Herbert A. Giles has translated
and annotated a series of Chinese Stories, which are to
the Chinese what the 'Arabian Nights' are to the Ara-
bians." Triibner's Literary Record.
Crown 8vo. Cfoth. Price 7s. 6d.
TOTAGES OP THE ELIZABETHAN SEAMEN.
A SELECTION FEOM THE OEIQINAL NARKATIVES IN
HAKLUYT'S COLLECTION.
EDITED, WITH HISTORICAL INTRODUCTION, BY
E. J. PAYNE, M.A.
FELLOW OF UNIVERSITY COLLEGE, OXFORD.
"Mr. Payne has skimmed for them the very cream
of Hakluyt's collection, and has successfully guarded
them against the chance of stumbling on a single dull
piece of writing. * Unless English school-
boys are greatly changed for the worse, there ought
to be many a one among them whom they will cheat
of his hour of play as much as ever did Scott's
' Marmion' cheat their grandfathers. * * What
a happy use might be made of such a collection of
voyages as this if out of it geography were taught !
* * We must thank the author for thus placing
within the reach of the general reader a book which is,
in so high a degree, both interesting and instructive."
Saturday Review.
" We wish we had space to do more than heartily
recommend this book. It is an excellent
one for boys; and the introduction traces clearly the
causes of the vast change which put England in the
place of Spain in the New World." Graphic.
"In a country like our own, where every one is
interested in maritime affairs, this volume, which
throws so much light on the discoveries and conquests
made by various nations in America, will prove very
generally acceptable."^. James's Gazette.
" Those who would like to be well grounded in the
history of our sailors when they were the grandest
men in the world would do well to peruse this excellent
volume." Daily Telegraph.
" Mr. E. J. Payne has produced a book which is of
great value historically, and which will have besides
much interest for all young people, and others who
delight in stories of nautical adventure. * * *
The book is admirably got up in all respects, and will
certainly meet with general approval." Scotsman.
Thos. De ba Rue & Go's List.
WORKS BY "OATEITBISH."
THE LAWS AID PRINCIPLES OP WHIST.
THE STANDARD WORK ON WHIST. 8vo., Cloth, Gilt Extra. Greatly en-
larged, and revised throughout. I3th Edition. Price 5-y.
" We are happy to present him as a contemporary."
Times.
" 'Cavendish' appears to possess a rare combination
of attributes." Bell's Life.
" We know of no book on whist undertaken on so
satisfactory a plan." Field.
" I urge all those who desire to become whist
players of the highest order to give a very careful study
to ' Cavendish." y. C.'s Treatise on Short Whist.
" An exquisite new edition of this master authority
upon the King 1 of Games has just been issued. *
It has been produced in a manner worthy of
a book which is a very bijou among handbooks
'Cavendish' being indisputably the authority among
Whist players." Sun.
" Those who can learn Whist from books may learn
it from this book with pleasure, the maxims being
explained with great clearness." Pall Mall Gazette.
THE LAWS OP PIQUET,
As ADOPTED BY TH"E PORTLAND CLUB. Edited by " CAVENDISH ;" with a
TREATISE ON THE GAME, by "CAVENDISH." 2nd Edition. 8vo., Cloth,
Gilt Extra. Price 3^. 6d.
THE LAWS OP PIQUET,
As ADOPTED BY THE PORTLAND CLUB, printed in sheet, enamelled and var-
nished, suitable for framing and hanging up in Card Rooms. Price 2s. 6d.
THE SAME,
In neat Frame. Price los. 6d.
THE LAWS OP E'CARTE,
ADOPTED BY THE TURF CLUB, with a TREATISE ON THE GAME, by
"CAVENDISH." 8vo., Cloth, Gilt. Price 2s. 6d.
ROUND &AMES AT CARDS.
By " CAVENDISH." 8vo., Cloth, Gilt. Price is. 6d.
THE GAME OP BEZIQUE.
8vo., Cloth, Gilt. Price is.
THE &AMES OP LAWN TENNIS
(WITH THE AUTHORIZED LAWS)
AND BADMINTON.
Third Edition. 8vo., Cloth. Price is.
CARD ESSAYS, CLAY'S DECISIONS,
AND
CARD-TABLE TALK.
By " CAVENDISH." Cr. 8vo., Cloth. With Portrait of the Author. Price 7-y. 6d.
5
Thos. De La Rue & Co.'s List.
THE POCKET SEEDS By " CAVENDISH."
PRICE SIXPENCE EACH.
POCKET GUILE TO POLISH BEZIQUE,
A pleasing variety of ordinary Bezique.
POCKET GUIDE TO BEZIQUE,
" With a single exception, every publisher has thought proper to print rules which are absolutely unintelligible.
* * * Intending players should buy Bezique only when ' Cavendish's ' rules are sold with it." Pall Mall Gazette.
POCKET GUIDE TO WHIST.
" It is 'Cavendish ' condensed. * * * A pretty little drawing-room table ornament." Field.
POCKET RULES FOE, LEADING AT WHIST.
" The prettiest of De La Rue's Pocket Series." Field.
POCKET LAWS OF WHIST.
" The Pocket Laws are the only ones ' by permission ' from the authorized Club Code." Field.
POCKET GUIDE TO ECARTE,
1 ' Very carefully done, and gives information as to playing the game that we suspect is not generally known. " Queen.
POCKET GUIDE TO EUCHRE.
The national game of the United States.
POCKET GUIDE TO SPOIL-FIYE.
" No living writer assuredly is better qualified than ' Cavendish ' to deal with these and kindred topics, seeing
that in addition to a complete mastery of his subject matter, and a thoroughly logical method of treatment, he
possesses in an eminent degree the rare gift of expressing his ideas in the plainest and simplest language." Bell's Life.
POCKET GUIDE TO CALABRASELLA,
POCKET GUIDE TO CRIBBAGE.
POCKET GUIDE TO SIXTY-SIX. POCKET GUIDE TO CHESS.
POCKET GUIDE TO DRAUGHTS AND POLISH DRAUGHTS.
POCKET GUIDE TO GO-BANG.
POCKET GUIDE TO BACKGAMMON AND RUSSIAN BACKGAMMON.
POCKET GUIDE TO IMPERIAL.
POCKET GUIDE TO FIFTEEN AND THIRTY-FOUR PUZZLES.
Price Is.
THE SPOT-STROKE.
BY JOSEPH BENNETT, EX-CHAMPION. EDITED BY "CAVENDISH."
"We should recommend those lovers of the game whose manipulation is not equal to their desire, to diligently
practise the stroke with Bennett's book for their guide. * * It is very neatly got up, and the diagrams are
beautifully clear and accurately drawn." Sporting Life.
THE LAWS OF CEOQUET.
ADOPTED AT THE GENERAL CONFERENCE OF CROQUET CLUBS. 8vo.
Paper Covers. Price 6d.
6
T/ios. De La Rue & Go's List.
DE LA RUE'S IOELIBLE DIARIES
AND
RED LETTER CALENDARS.
POCKET DIARIES.
DE LA RUE'S IMPROVED INDELIBLE DIARIES AND MEMORANDUM BOOKS,
in three sizes, fitted in Velvet, Russia, Calf, Turkey Morocco, Persian, or French
Morocco cases ; plain or richly gilt, with gilt clasps or elastic bands, in a great
variety of styles. All these Diaries are fitted with electro-gilt indelible pencils.
Also supplied in enamelled paper covers.
A size 3^ x ij4 inches.
B ,, 3^X2^ ,, also same size, F F (oblong).
G 4^x2|<
CONDENSED DIARIES AND EMA&EMENT BOOKS,
In three sizes (A, B, & C, as above), and in a great variety of Plain and
Ornamental leather cases ; they are also published in enamelled paper covers,
suitable for the Card Case or Purse.
COMPANION MEMORANDUM BOOKS,
For use with the Condensed Diaries ; A, B, & G sizes, as above.
N.B. All Condensed Diary and Calendar Cases (except the Tuck) are fitted with an extra elastic band for
the reception of these books.
PORTABLE DIARIES.
Thin, light, and flexible, in a variety of leather cases, adapted for the Pocket.
A, B, & C sizes.
DESK DIARY.
DE LA RUE'S IMPROVED DIARY AND MEMORANDUM BOOK ; for Library or
Counting-house use. E size, J$4 x 4%" inches.
POCKET CALENDARS.
DE LA RUE'S RED LETTER CALENDARS AND ALMANACS, in three sizes
(A, B, & C, as above), in enamelled paper covers, suitable for the Card Case
or Pocket Book. Also interleaved ; and in Russia, Persian, and French
Morocco cases.
'TIMER " CONDENSED DIARY,
In elegant sliding cases, extra gilt. Adapted for the Pocket or Reticule.
ORNAMENTAL WALL CALENDARS
In great variety. Printed in gold and colours, from original designs. Royal 8vo.
MONTHLY TABLET OR EASEL CALENDARS.
Printed in Colours, in a variety of shapes and sizes. In Gilt and Nickeled Metal,
Leather, and Leatherette Cases.
Thos. De La Rue & Go's List.
THE AITI-STYLOGEAPH
(HEARSON'S PATENT)
THOS. DE LA RUE & CO., LONDON,
Sole Licensees.
A self-feeding reservoir pen, nibbed in the ordinary way, and changeable at pleasure.
Is ready for instant use WITHOUT ADJUSTMENT, and will write with any ink-black,
red, or copying. Preserves the ordinary characteristics of the hand-writing, and may
therefore be used for signatures. Only requires to be re-filled after several days' use,
and may be carried in any position, or in the pocket, without risk of leakage.
PRICES OF THE ANTI-STYLOGRAPH.
DESCRIPTION.
POCKET
SIZE,
4-% in. long,
DESK
SIZE,
6 in. long.
Fitted with Non-Corrodible Pen, in neat cardboard box,")
with filler complete . \
j. d.
2 6
s. d.
3 6
Fitted with Palladium Pen, iridium-pointed, in cloth-covered")
box, with filler, complete )
5 6
6 6
N.B. These Pens are strongly recommended, being as flexible as
steel and as durable as gold.
Fitted with Gold Pen, iridium-pointed, in Morocco box,")
with filler, complete )
10 6
ii 6
PATENT NON-CORRODIBLE PENS,
With fine, medium, or broad points, for refitting the Anti-Stylograph.
ONE SHILLING PER BOX.
THE
AITI-STYLOGKRAPH WALLET
(REGISTERED).
THE MOST COMPACT WRITING CASE.
Size only 5% x 3% x % inches.
Containing Large 8vo. Note Paper and Envelopes to match ; Small 8vo. Note Paper and En-
velopes to match ; Spaces for Post Cards, Visiting Cards, and Stamps; Blotting Pad, Memorandum
Tablet, Telegram Forms, and
POCKET SIZE ANTI-STYLOGRAPH,
Fitted with Non-Corrodible Pen-Leather, No. 774, 10/6 ; Russia Leather, No. 778, 14/6
Pitted with Palladium Pen, Iridium-jwinted Leather, No. 780, 13/6 ; Eussia, No. 784, 17/6
8