Đây là bộ sách tiếng anh về chuyên ngành vật lý gồm các lý thuyết căn bản và lý liên quan đến công nghệ nano ,công nghệ vật liệu ,công nghệ vi điện tử,vật lý bán dẫn. Bộ sách này thích hợp cho những ai đam mê theo đuổi ngành vật lý và muốn tìm hiểu thế giới vũ trụ và hoạt độn ra sao.
Trang 3Clarendon Press Series
aa TREATISE
ON °
ELECTRICITY AND MAGNETISM
BY
JAMES CLERK MAXWELL, M.A LL) EVIN,, P.R.83, LONDON AND EDINBURGH
Trang 5PREFACE
THE fact that certain bodies, after being rubbed,
appear to attract other bodies, was known to the
ancients In modern times, a great variety of other phenomena have been observed, and have been found
to be related to these phenomena of attraction They have been classed under the name of Electric phe-
nomena, amber, #Aextpoy, having been the substance
in which they were first described
Other bodies, particularly the loadstone, and pieces of iron and steel which have been subjected to certain processes, have also been long known to exhibit phe-
nomena of action at a distance These phenomena, with others related to them, were found to differ from
the electric phenomena, and have been classed under
the name of Magnetic phenomena, the loadstone, uưyune,
being found in the Thessalian Magnesia
These two classes of phenomena have since been
found to be related to each other, and the relations between the various phenomena of both classes, so
far as they are known, constitute the science of Elec-
tromagnetism
Trang 6vi PREFACE,
most important of these phenomena, to shew how they may be subjected to measurement, and to trace the mathematical connexions of the quantities measured Having thus obtained the data for a mathematical theory of electromagnetism, and having shewn how this theory may be applied to the calculation of phe- nomena, I shall endeavour to place in as clear a light as I can the relations between the mathematical form of this theory and that of the fundamental science of Dynamics, in order that we may be in some degree prepared to determine the kind of dynamical pheno- mena among which we are to look for illustrations or explanations of the electromagnetic phenomena
In describing the plienomena, I shall select, those
which most clearly illustrate the fundamental ideas of
the theory, omitting others, or reserving them till the
readcr is more advanced
The most important aspect of any phenomenon from
a mathematical point of view is that of a measurable
quantity I shall therefore consider electrical pheno-
mena chiefly with a view to their measurement, de-
scribing the methods of measurement, and defining the standards on which they depend
In the application of mathematics to the calculation of clectrical quantities, I shall endeavour in the first place to deduce the most general conclusions from the data at our disposal, and in the next place to apply the results to the simplest cases that can be chosen
I shall avoid, as much as I can, those questions which, though they have elicited the skill of mathematicians,
Trang 7
PREFACKE., vii
The internal relations of the different branches of the science which we have to study are more numerous and complex than those of any other science hitherto developed Its external relations, on the one hand to
dynamics, and on the other to heat, light, chemical action, and the constitution of bodies, seem to indicate
the special importance of electrical scicnce as an aid to the interpretation of nature
It appears to me, therefore, that the study of elec- tromagnetism in all its extent has now become of the first importance as a means of promoting the progress of science
The mathematical laws of the different classes of
phenomena have been to a great extent satisfactorily
made out
The connexions between the different classes of phe- nomena have also been investigated, and the proba- bility of the rigorous exactness of the experimental laws has been greatly strengthened by a more extended knowledge of their relations to each other
Finally, some progress has been made in the re- duction of electromagnetism to a dynamical science, by shewing that no electromagnetic phenomenon is contradictory to the supposition that it depends on purely dynamical action
What has been hitherto done, however, has by no
means exhausted the field of electrical research It has rather opened up that field, by pointing out sub- jects of enquiry, and furnishing us with means of
investigation
Trang 8Vili PREFACK,
results of magnetic research on navigation, and the importance of a knowledge of the true direction of the compass, and of the effect of the iron in a ship But the labours of those who have endeavoured to render navigation more secure by means of magnetic observations have at the same time greatly advanced the progress of pure science
Gauss, as a member of the German Magnetic Union, brought his powerful intellect to bear on the theory of magnetism, and on the methods of observing it, and he not only added greatly to our knowledge of the theory of attractions, but reconstructed the whole of magnetic science as regards the instruments used, the methods of observation, and the calculation of the
results, so that his memoirs on Terrestrial] Magnetism
may be taken as models of physical research by all those who are engaged in the measurement of any of the forces in nature,
Trang 9PREFACE, ix
There are several treatises in which electrical and magnetic phenomena are described in a popular way
These, however, are not what is wanted by those who
have been brought face to face with quantities to be
measured, and whose minds do not rest satisfied with
lecture-room experiments
There is also a considerable mass of mathematical memoirs which are of great importance in electrical science, but they lie concealed in the bulky Trans- actions of learned societies; they do not form a con- nected system; they are of very unequal merit, and
they are for the most part beyond the comprehension
of any but professed mathematicians
I have therefore thought that a treatise would be useful which should have for its principal object to take up the whole subject in a methodical manner, and which should also indicate how each part of the subject is brought within the reach of methods of verification by actual measurement
The general complexion of the treatise differs con- siderably from that of several excellent clectrical works, published, most of them, in Germany, and it may appear that scant justice is done to the specu- lations of several eminent electricians and mathema- ticians, One reason of this is that before I began the study of electricity I resolved to read no mathe- matics on the subject till I had first read through
Earadays Experimental Researches on Electricity I
was aware that there was supposed to be a difference between Faraday’s way of conceiving phenomena and
Trang 10x PREFACE
they were satisfied with each other’s language I had also the conviction that this discrepancy did not arise from either party being wrong I was first convinced of this by Sir William Thomson *, to whose advice and assistance, as well as to his published papers, I owe most of what I have learned on the subject
As I procecded with the study of Faraday, I per- ceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols, I] also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathcma- ticians,
For instance, Faraday, in his mind’s eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance : Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids,
When I had translated what I considered to be Faraday’s ideas into a mathematical form, I found that in general the results of the two methods coin-
cided, so that the same phenomena were accounted
for, and the same laws of action deduced by both methods, but that Faraday’s methods resembled those
Trang 11
PREFACE, xi
in which we begin with the whole and arrive at the parts by analysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by syn- thesis
I also found that several of the most fertile methods of research discovered by the mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form
The whole theory, for instance, of the potential, con-
sidered as a quantity which satisfies a certain partial differential equation, belongs essentially to the method which I have called that of Faraday According to the other method, the potential, if it is to be considered
at all, must be regarded as the result of a summa-
tion of the electrified particles divided each by its dis- tance from a given point Hence many of the mathe- matical discoveries of Laplace, Poisson, Green and Gauss find their proper place in this treatise, and their appropriate expression in terms of conceptions mainly derived from Haraday
Great progress has been made in electrical science, chiefly in Germany, by cultivators of the theory of action at a distance The valuable electrical measure- ments of W Weber are interpreted by him according to this theory, and the electromagnetic speculation which was originated by Gauss, and carried on by
Weber, Riemann, J and C Neumann, Lorenz, &c is
Trang 12x1 PREFACE
whether potential or force, from the one particle to the other The great success which these eminent
men have attained in the application of mathematics
to electrical phenomena gives, as is natural, addi- tional weight to their theoretical speculations, so that those who, as students of electricity, turn to them as the greatest authorities in mathematical electricity, would probably imbibe, along with their mathematica] methods, their physical hypotheses,
These physical hypotheses, however, are entirely alien from the way of looking at things which I adopt, and one object which I have in view is that some of those who wish to study electricity may, by reading this treatise, come to see that there is another way of treating the subject, which is no less fitted to
explain the phenomena, and which, though in some
parts it may appear less definite, corresponds, as I think, more faithfully with our actual knowledge, both in what it affirms and in what it Icaves undecided
In a philosophical point of view, moreover, it is exceedingly important that two methods should be compared, both of which have suecceded in explaining the principal clectromagnetic phenomena, and both of which have attempted to explain the propagation of light as an electromagnetic phenomenon, and have actually calculated its velocity, while at the same time the fundamental conceptions of what actually takes place, as well as most of the secondary conceptions of the quantities concerned, are radically different
Trang 13
PREFACE, xi
method than attempted to give an impartial description of both I have no doubt that the method which I have called the German one will also find its sup- porters, and will be expounded with a skill worthy of its ingenuity
I have not attempted an exhaustive account of elec- trical phenomena, experiments, and apparatus The student who desires to read all that is known on these subjects will find great assistance from the Traité
d’Electricité of Professor A de la Rive, and from several
German treatises, such as Wiedemann’s Galvanismus, Riess’ Reibungselektricitdét, Beer’s Hinleitung in die Elek-
trostatzk, &c
I have confined myself almost entirely to the ma- thematical treatment of the subject, but I would
recommend the student, after he has learned, expcri-
mentally if possible, what are the phenomena to be observed, to read carefully Faraday’s Laperimental
Researches in Electricity He will there find a strictly
contemporary historical account of some of the greatest electrical discoveries and investigations, carried on in an order and succession which could hardly have been improved if the results had been known from the first, and expressed in the language of a man who devoted much of his attention to the methods of ac- curately describing scientific operations and their re- sults *
It is of great advantage to the student of any subject to read the original memoirs on that subject, for science is always most completely assimilated when
Trang 14XIV PREFACE
it is in the nascent state, and in the case of Faraday’s Researches this is comparatively easy, as they are
published in a separate form, and may be read con-
secutively If by anything I have here written 1 may assist any student in understanding Faraday’s modes of thought and expression, I shall regard it as the accomplishment of one of my principal aims—to communicate to others the same delight which I have found myself in reading Faraday’s Researches
The description of the phenomena, and the elc- mentary parts of the theory of each subject, will be found in the earlier chapters of each of the four Parts into which this treatise is divided The student will find in these chapters enough to give him an elementary acquaintance with the whole science
The remaining chapters of each Part are occupied with the higher parts of the theory, the processes of
numerical calculation, and the instruments and methods
of experimental research
The relations between electromagnetic phenomena and those of radiation, the theory of molecular electric currents, and the results of speculation on the nature
of action at a distance, are treated of in the last four
chapters of the second volume,
Trang 15Art CONTENTS PRELIMINARY
ON THE MEASUREMENT OF QUANTITIES,
1 The expression of a quantity consists of two factors, the nu- " ; 8 9, 10 11 12 13 14 15 16 17 18 19 20 21, 22 33 24 25 26
merical value, and the name of the concrete unit Dimensions of derived units
The three fandamental units—Length, Time and “Mase
"Derived units ¬
Phy sical continuity and discontinuity Discontinuity of a function of more than one variable Periodic and multiple functions vee Relation of physical quantities to directions in space
Meaning of the words Scalar and Vector Division of physical vectors into two classes, Forces ‘and Fluxes Relation between corresponding vectors of the two classes Line-integration appropriate to forces, surface-integration to
fluxes Loo
Longitudinal and rotational vectors
Line-integrals and potentials — sẻ ¬ Hamilton’s expression for the relation between a force and its
potential ees
Cyclic regions and geometry of position
The potential in an acyclic region is single valued System of values of the potential in a cyclic region Surface-integrals
Trang 16xvi Art, 27 28 20 31 32 33 34 35 36 37 38, 39 40 41, 43 44 45 46 47, 48 49 50, 51 CONTENTS, PART I ELECTROSTATICS, CHAPTER I DESCRIFTION OF PHENOMENA
Electrification by friction Electrification is of two kinds, to which the names of Vitreous and Resinous, or Positive and Negative, have been given
Nlectrification by induction eee
Electrification by conduction, Conductors and insulators - In electrification by friction the quantity of the positive clee-
trification is equal to that of the negative electrification To charge a vessel with o quantity of electricity equal and
opposite to that of an excited body seas To discharge a conductor completely into a metallic vessel Test of electrification by gold-leaf clectroscope ‹ Electrification, considered as 1 measurable quantity, may be called Electricity
tees Electricity may be treated ag a physical quantity Theory of Two fluids
Theory of One fluid eee tenes
Measurement of the force between electrified bodies Relation between this force aud the quantities of clectricity Variation of the force with the distance ¬
42 Definition of the electrostatic unit of electricity — Its dimensions tea
Proof of the law of electric force Electric field
Electric potential eee °
Trang 17ue a Toor Se Se te a coo c ie ¬ = GO, GL ie} moa s » CONTENTS, Specitic Iiductive capacity of a dielvetric ‘Absorption’ of electricity
Impossibility of an absolute charge Disruptive discharge —Glow
Brush
Spark “
Electrical phenomena of Tourmaline “ Plan of the treatise, and sketch of its results
Electric polarization and displacement “ The motion of electricity nhalogone to that ofa an incompr ssgible
fluid
32 Peculiarities of the theory of this treatise CHAPTER 11
ELEMENTARY MATHEMATICAL THEORY OF ELECTRICITY Definition of electricity as a mathematical quantity sf Volume-density, surface-density, and line-density
5 Definition of the clectrostatic unit of clectricity tos
Law of foree between clectrified bodies 37 Resultant force between two bodies 38 Resultant force ut a point
Tine-integral of electric force ; eleetromotive force
Eloctric potential
Resultant force in terms of the potential
72 The potential of all points of a conductor is the same 73 Potential due to an clectrified system
74 Proof of the Jaw of the inverse square a 75 Surface-integral of clectric induction ° 76 Introduetion through a closed surface duc to q single centre
of force
77 Poisson's extension of Laplace's equation teas 78 Conditions to be fulfilled at an electrificd surface
Resultant force on an electrified surface The electrification of a conductor is entirely on the wurface _ A distribution of electricity on lincs or points is physically
impossible wk ee ete
Trang 18XVil Art, 81 8ä 86 87 8a, 81), 90, Ol, 92, i CONTENTS, CHAPTER II, SYSTEMS OF CONDUCTORS Page
Qn the superposition of electrified systems te ae BB
Energy of an electrified system agg
General theory of a system of conductors, Coefficients of po- tential Ph HH HH cv á v 89 Coefficients of induction Capacity of a conductor, Dimensions of these coeflivients th HH HH cv vu v00 Reciproenl property of the coeficients ha ve ĐỊ A theorem duetoGren 0 Khó ke cà 92 Relative magnitude of the coeflicients of potential 1 6 gg And of induction ` — °Öò 93
The resultant mechanical force on a conductor expressed in terms of the charges of the different conductors of the system and the variation of the coeflicients of potential 9g 93 The same in terms of the potentials, and the variation of the cocflicients of induction
, eae
44 Comparison of cleetritied 8ÿBEINB ¿2 cố vo vo cu 06 CHAPTER IV,
GENERAL THEOREMS
95 Two opposite methods of treating clectrical questions 98 96 Characteristics of the potential function 19 97, Conditions under which the volume-integral To dF ak eo be ?0 da dy dz JJJ ứ, 3 dy” ede vanishes VU VỤ VU VU ` 100 38 Thomson’s theorem of the unique minimum of Tf 1 Ml (+694 er) dedyd: tg 99, Application of the theorem to the determination of the dis- tribution of electricity 2 u UỐ sa 107
100 Green’s theorem and its physical interpretation = 108
101, Green's functions , „„ Hee ad 208
102, Method of finding limiting values of electrical cocfheicnts 11ã
Trang 19
CONTENTS, XIX
CHAPTER V
MECHANICAL ACTION BETWEEN ELECTRIFIED KODIES,
Art Page
103, Comparison of the force between different electrified systems 119 104 Mechanical action on an clement of an electrified surface 121 105 Comparison between theories of direct action and theories of
stress ww, “sa - 122
106 The kind of stress requir ed to account for the phenomenon 128 107, The hypothesis of stress considered as a step ín clectrical
science ¬ 126
108, The hypothesis of stross sewn to account for the equilibriam of the medium and for the forees acting between electrified
bodics 128
109, Statements of Faraday relative { to the | longitndinal tension and lateral pressure of the lines of foree 6 131 110 Objections to stress ina fluid considered 0.00 6 38) 111 Statement of the theory of electric polarization cà 139 CHAPTER V1 POINTS AND LINES OF EQUILIBRIUM, ị ý Ệ i | ‡
¡ 112 Conditions of a point of equilibrium tae ee BS 113 Number of points of equilibrium 0.0 woe 186 i 114 Ata point or line of equilibrium there is a conical point or a : line of self-intersection of the equipotential surface 4 137
115 Angles at which an equipotential surface intersects itself 138 116 Phe equilibrium of an clectrified body cannot be stable 139
CHAPTER VII
FORMS OF EQUJPOTENTIAL SURFACFS AND LINES OF FLOW, 117 Practical importance of 2 Knowledge of these forms in simple
(uses ` ơơ s s ô142
118 Two clectrified points, ratio 4:1 (Big TQ) 148 119 Two electrified points, ratio 4: 1 (Fig Il) oo «144 120, An electrified point in a uniform field of foree, (Wig TTI) 2 145 121 Three clectrified points Two spherical equipetential sur-
faces (Fig, IV) 2.0 ¬— a 145
122 Faraday’s use of the conecption of lines at foree we, 123 Method employed in drawing the diagrams „147
Trang 20AX Art 124 125 126 197 128 129, 130 13] 132 153 131 135 136, 138 139, 140 141 C CONTENTS, HAPTER VIE,
SEMPLE CASES OF ELECTRIFICATION,
Two parallel planes
Two concentric spherical surfaces Two coaxal cylindric surfaces
Longitudinal foree on weylinder, the ends of which are sup-
rounded by cylinders at different potentials CHAPTER IX, SPHERICAL ILARMONICS,
Singular points at which the potential becomes infinite Singular points of diflerent orders defined hy their axes ,
Expression for the potential due to a sinưular point referred to its axes
This expression is perfectly definite and represents the most general type of the harmonic of 7 degrees
The zonal, tesseral, and soetorial types ¬
Solid harmonics of positive degree ‘Their relation to those of negative degree , Application to the theory of electrified spherical surfaces The external action of an clectrified spherical surface compared
with that of an imaginary singular point at its centre Proof that if P, and Ƒ; are two surface harmonies of different
degrees, the surface-integral II Y, Yd = 0, the intesration being extended over the spherical surface
- Value of | | VY; dS where Y, and ?; are surface harmonies of the same degree but of different types
On conjugate harmonies Fa Cài
If ¥; is the zonal harmonie and ¥; any other type of the same degree ae
I YjdN¥a 22" yp 270 -+ ] dna ¡2!
Whtre 222; is the value oŸ 7; ñE the pole of Y; N
Development of a function in terms of spherical surface har- monies ¬
Trang 21
CONTENTS XM
Art, VPage
142 Different methods of trenting spherical harmonics 171 143 On the diagrams of spherical harmonics, (Figs, V, VỊ, VI,
VILT, IX) ae ae 175
144 If the potential is constant throughout 4 my finite portion of space it is so throughout the whole region contimuious with it within which Laplace’s equation is satisfied « 176 145 To analyse a spherical harmonic into a system of conjugate
harmonics by means of a finite number of measurements at selected points of the sphere wo 177 146 Application to spherical and nearly spherical conduetors 178
CHAPTER X
CONFOCAL SURFACES OF THE SECOND DEGREE 147, The lines of intersection of two Systems and their intereepts
hy the third system 181
148 The churacteristic equation of Vin terms of ellipsoidal co-
ordinates 182
149, Expression of a, 8, y in ten ms of elliptic functions ` 183 150 Particular solutions of electrical distribution on the confocal
surfaces and their limiting forms tà Ö 184 151 Continuous transformation into a figure of rev olution abont
the axis of s T87
152 ‘Transformation into a flere of revolution about the axis of « 188 153 Transformation into a system of cones and spheres 189
154 Confocal paraboloids 189
CHAPTER XI THEORY OF ELECTRIC IMAGES,
155 Thomson's method of electric images 191
156 When two points are oppositely and unequally elects fi, the surface for which the potential is zero is a sphere 192
157 Electric images Lows 193
158 Distribution of leetricity 0 on the sur face of the sphere 195 159 Image of any given distribution of electricity 196 160 Resultant foree between an electrified point and sphere 197 161 Images in an infinite plane pondncling surface 198
162 Electric inversion a 199
Trang 22XXxil CONTENTS, Art Page 165 Finite systems of successive images 203 : TT
166 Case of two spherical surfaces intersecting at an angle 5 204 167, Enumeration of the cases in which the number of images is
finite bees 206
168 Case of two spheres intersecting orthogonally 207 169, Case of three spheres interseeting orthogonally 210 170 Case of four spheres intersecting orthogonally eee BLT 171 Infinite series of images Case of two concentric spheres 212
213
172 Any two spheres not intersecting each other
173 Culculation of the cveflicients of capacity and induction 216 174 Calculation of the charges of the spheres, and of the force
between them Be 217
175, Distribution of electricity on two spheres in contact Proof
sphere ee, 219
176, Thomson's investigation of an clectrified spherical bowl 221 177 Distribution on an ellipsoid, and on a circular disk at po-
tential V a ĐI
178 Induction on an uwninsulated disk or bowl by an electrified point in the continuation of the plane or spherical surface 222 179 The rest of the sphere supposed uniformly electrified 223 180, The bowl maintained at potential Vand uninfluenced 223 181, Induction on the bowl due to nx point placed anywhere 224
CHAPTER XII
CONJUGATE FUNCTIONS IN TWO DIMENSIONS,
182 Cases in which the quantities are functions of and yonly 226
183 Conjugate functions eee 227
184 Conjugate functions may be added or subtracted 0 998 185 Conjugate functions of conjugate functions are themselves
conjugate ¬— 229
186 Transformation of Poisson's "Ôn .aAaAIUỤỪ: 187 Additional theorems on conjugate functions 6 4 239 188 [Inversion iu two dimensions 0.00 232 189, Electric imayes in two dimensions 0 Q 233 190, Newmann’s transformation of thiseuse 0 oe 284 191, Distribution of electricity near the edye of a conductor formed
by two plane surfaces 0, ee 236
192 Ellipses and hyperbolas (ig Xp 237
Trang 23% 3 ae 1& a NT WML ph gee Mt AS 3 PONY T CAN ane CONTENTS, xxHI Art Page 194 Application to two cases of the flow of electricity in a con- ducting sheet , 239
195 Application to two cases of electrical induction 239 196 Capacity of a condenser consisting of a circular disk hetween
two infinite planes 210
197, Case of a serics of equidistant planes cụt of by au plane at right
angles to them 242
198 Case of a furrowed surface 213
199 Caso of a single straight groove vee 243 200 Modifiention of the results when the groove is circular 211 201 Application to Six W, Thomson3 guard-ring , 215 202 Case of two parallel plates cut off by a perpendicular plane
(Pig X{T) been 246
203 Cuse of a grating of par allel wires (lig XTIT) 248 204 Case of a single electrified wire transformed into that of the grating 218 205 The grating used as a shield to pr otect ¢ a ody from electrical influence 249 206 Method of approximation applied to the ¢ case of the, gr ating 251 CHAPTER XIII ELECTROSTATIC |NSTRUMENTS
207 ‘The frictional electrical machine 254
208 The electrophorus of Volta 255
209, Production of electrification by mechanical wor ve: —Nicholson's
Revolving Doubler bees 256
210 Principle of Varley’s and Thomson's clectr ical machines vẻ 256
211 “Thonasons water-droppi machine 259
212 Toltzs cleetrical machine 960
213 Theory of regencrators applied to cleetr ieal machines 260 214 On clectrometcrs und clectroscopes Indicating instruments
and null metnods Difference between registration and mea-
surement “ ¬¬ 262
215, Couloml’s Torsion Balance for measuring charges 268 216 Electrometers for measuring potentials, Snow Harris's and
Thomson's ` 266
217, Principle of the guar a-ring Thomson’ s Absolute Electr ometer 267 218, Heterostatic method 269
Trang 24ợnay§š§ẽ>ằ mm ä<
XXIV CONTENTS
Art
Page
222, Measurement of the potential of a conductor without touching it 276
223 Measurement of the superficial density of electrification The
proof plane 277
224, A hemisphere used asatest 978
225 Acireulardisk tg
226 On electric accumulators, The Leyden jar 281 227 Accumulators of measurable capacity teas oe BBD
228 The guard-ring accumulator Vu vỤ c S 283 229 Comparison of the capacities of accumulators 285
PART IL
ELECTROKINEMATICS,
CHAPTER J, THE ELECTRIC CURRENT,
230 Current produced when conductors are discharged 0, 4, 288 231 Transforenee ofeleetrifieation sò ò 288 232 Description of the voltaic battery 00, 0 4 289 233, Electromotive foree HS SỐ 290 234, Production of a steady current eae ea, 290 235, Properties of the current 6 uc 291
236 Electrolytic nctio 2 su su cu cu 291
237 Explanation of terms connected with electrolysis 999
238 Different modes of passaye of the earrent ` ` BOQ
239 Magnetic action of the current 0 oe 293
240, The Galvanometer ¬ HH HE HH BOY
CHAPTER II CONDUCTION AND RESISTANCE,
241 Ohm’s Law a 995
Trang 25CONTENTS
CHAPTER II]
ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT
Art,
216 Voltrs law af the eontnet foree between different metals at the
same temperature He He Vu cây
247, Effect of electrolytes ”
248 Thomson's voltaic current in which gravity performs the part
of chemical action keene ei
249 Pelticr’s phenomenon Deduction of the thermoelectric clec- tromotive force at a junction ¬
250 Secheck’s discovery of thermoelcctric currents 251 Magnus’s law of « circuit of one metal
252, Cumming’s discovery of thermoelectric inversions
253 Thomson's deductions from these facts, and discovery of the reversible thermal effects of clectric currents in copper and
in Iron oe
254, Tait’s law of the electromotive for co of a ther moeleetrie pair CHAPTER IV
ELECTROLYSIS, Earaday's law of electrochemical equivalents , Clansius’s theory of molecular agitation
Electrolytic polarization «60 ee ‹ Lest of an electrolyte by polarization
Difficulties in the theory of clectrolysis » Molecular charges
Secondary actions observed at the eleetr odes
Conservation of energy in clectrolysis “ Measurement of chemical affinity as an electr omotive foree nm mone b2 b2 Ww b2 © C2 G ƠI CtỢt ti CC NOS SENS so lv Cs w CHAPTER V | ` ELECTROLYTIC POLARIZATION
264 Difficulties of applying Ohm’s law to clectrolytes
265 Ohm’s law nevertheless applicable 266 The effect of polarization distinguished from that of resistance 267 Polarization duc to the presence of the ions at the electrodes
The ions not in a free state
Trang 26CONTENTS, XXVI Art, Page 269 Dissipation of the ions and loss of polarization 321 270 Limit of polarization 32]
271, Litters secondary pile compared with the Leyden ja jar 322
272, Constant voltaic clencnts —Daniell’s cell ¬ 325
CHAPTER VI,
MATHEMATICAL THEORY OF THE DISTRIBUTION OF ELECTRIC CURRENTS,
273 Linear conductors 329
274 Ohm’s Law 3929
275, Linear conductors in series 329
276 Linear conductors in multiple are ,, 330 277 Resistance of conductors of uniform section 331 278, Dimensions of the quantities involved in Ohn’s law 332 279, Specific resistance and conductivity in electromagnetic measure 333 280, Lincar systems of conductors in general , 333 281, Reciprocal property of any two conduetors of the sy stem 335
282 Conjugate conductors 336
283, Ifeat generated in the system 336
281, The heat is a minimum when the current is s distribnted uŒ-
cording to Olun’s law 337
CHAPTER VII
CONDUCTION IN THREE DIMENSIONS,
285 Notation 338
286 Composition and resolution of electri ic currents 338 287 Determination of the quantity which flows through any surface 339
288, Mquation of a surface of flow 310
289 Relation belween any three c systems of surfaces of low 310
290 Tubes of flow 0 oy — „ S40
991, Expression for the components of the flow i in terms of surfitees
of flow te ee eas có oo» 34]
292, Simplification of this expression by a proper choice of para-
meters : ` 341
293 Unit tubes of flow used as a complete method of determining
the current 0 ¿uc 341
29-1, Current-sheety and current- fanetions 342 295 Equation of ‘continuity’ _ 342
Trang 27Art 297 298, 299 300 301 302 303 301 305 300, 307 308 309 310 311 312 313 314 315, 316, 317 318 319 320 321 322 323 324 CONTENTS, XXVU CHAPTER VIII RESISTANCE AND CONDUCTIVITY IN THREE DIMENSIONS Kquations of resistance Equations of conduction Rate of gencration of heat
Conditions of stability eee
Equation of continuity in a homogencous medium
Solution of the equation eee
Theory of the coefficient 7 It probably does not exist Generalized form of Thomson's theorem
Proof without symbols “
Strutt’s method applied to a wire of variable section, —Lower limit of the value of the resistance Higher limit Lower limit for the correction for the ends of the wire Higher limit CHAPTER IX CONDUCTION THROUGH HETEROGENEOUS MEDIA, Surface-conditions Spherical surface Spherical shell ¬
Spherical shell placed in a field of uniform flow
Medium in which small spheres arc uniformly disseminated Images in a plane surface ¬ “ Method of inversion not applicable in three dimensions Case of conduction pưongh a stratum bounded by parallel
planes See ee
Infinite series of images Application to magnetic induction , On stratified conductors Cocflicients of conductivity of a
conductor consisting of alternate strata of two different sub-
stances :
If neither of the substances has the rotator y proper ty denoted by 7 the compound conductor is free from it Tf the substances ure isotropic the direction of greatest resist-
ance is normal to the strata Medium containing parallelepipeds of mother medium
The rotatory property cannot be introduced by means of con-
ducting channels Ko HH man
Trang 28ae XXVHI CONTENTS, CHAPTER X, CONDUCTION IN DIELECTRICS, Art Page
325 Th a strictly homogeneous medium there can be no internal
charge a Lee ou OFA
326 Theory of a condenser in w hieh the dieleeie § is not a perfect
insulator we tee BFS
327 No residual charge due to ‘simple conduction wee 376 428 Theory of a composite nccummlator 0 837 329 Residual charge and electrical absorption Hs ca cà 78
330 Total disehurge nh cv 180
331 Comparison with the condnetion of heat Hà ơn 181 332 Theory of telegraph cables and comparison of the cy uations
with those of the conduction of heat 0.0 8Ị 333 Opinion of Olm on this subject ¬ ¬ BBA 334, Mechanical illustration of the properties of a dieleetrie c “ 8ã
CHAPTER XI,
MEASUREMENT OF TITE ELECTRIC RESISTANCE OF CONDUCTORS 335, Advautage of using material standards of resistance in electrical
Incasurements ., 388
336 Different standards which have been used and different systems
Which have been proposed uc 388
337 The clectromagnetie system of units os .ò — 889 338 Weber's unit, and the British Association unit or Ohm ue B89 339 Professed value of the Ohm 10,000,000 metres per second = 389
340 Reproduction of standards Mee 990
341 Forms of resistanee coils 0 vẻ 391
$42 Coils of great resistance vo uc 392
343, Arrangement of coils in series 2 992
SH, Arrangement in multipleare o.oo 393
345 On the comparison of resistances, (1) Ohm's method 394 $46 (2) Dy the differential galvanometer tae ee BO ‘47 (3) By Whentstune’s Bridge = tee ae B98 348, Estimation of limits of error in the deter mination toe 00 349, Best arrangement of the conductors to be compared 4 400 350 Ou the use of Wheatstone’s Bridge 402 351 Thomson’s method for small YOSISEHNCCR 4 wae AOE
Trang 29
Art
353, Comparison of great resistances by the clectrometer 354 By accumulation in a condenser
355, Direct electrostatic method er
356 Thomson's method for the resistance of a galvanumcter 357, Mance’s method of determining the resistance of a battery 358, Comparison of electromotive forces
CHAPTER XII
Trang 33
ERRATA VOL I
Page 26, 1 3 from bottom, dele ‘As we have made no assumption’, &e down to 1,7 of p 27, ‘the expression may then be written’, and substitute as follows :—
Let us now suppose that the curves for which a is constant
form a series of closed curves, surrounding the point on the surface
for which a has its minimum value, đạ, the last curve of the series, for which a = @,, coinciding with the original closed curve s
Let us also suppose that the curves for which 9 is constant form a serics of lines drawn from the point at which a= a, to the closed curve s, the first, 3,, and the last, Øj, being identical,
Integrating (8) by parts, the first term with respect to a and the second with respect to 8, the double integrals destroy each other The line integral,
A da
zz (
I (A dp,
is zero, because the curve a = a, is reduced to a point at which there is but one value of VY and of a,
The two line integrals, a a -Í Tr) day [ x) da độ da A= By ag da B = Bo destroy each other, because the point (a, 8,) is identical with the point (a, 9) The expression (8) is therefore reduced to de My) ~Ö1 “— ( ag (9) ‹ Bo đủ an
Since tho curve a = a, is identical with the closed curve s, we may write this expression
p 80, in equations (3), (4), (6), (8), (17), (18), (19), (20), (21), (22), for read N
p 82, 1 3, for Al read VI,
» 83, in equations (28), (29), (30), (31 oN read ON
p 83, in equations ; ; » (31), for dae Peed Fy » 1n cquation (29), tnseré — before the second member
p 105, 1.2, for Q read 87Q
p- 108, equation (1), for p read pi ” » (2), for p’ read p, ” ” (3), for o read a”, ” ” (4), for a” read o
p 113, 1.4, for KR read _Rk
» 1.5, for KRR cose read A KR 4T RF’ cose .114, 1.5, for 6; read #,
124, last line, for e,t+e, read ete,
125, lines 3 and 4, transpose within and without; 1 16, for v read V; and 1.18, for V read v,
128, lines 11, 10, 8 from bottom, for da read dz, 149, 1 24, for eyupotential read equipotential
vy
Trang 34Pp Ù- p p p yp p p Pp ) p- p p- 356, equation (12), for 7 real đi p- p P- P ) 397, 1.1, for Be read Ds 159, 163, 1 20, for Ajyestr P98 À¡ vắt
: - Sosy Nin |2» \i— wo
164, equation (34), for (—1 ipo read (~1)-* Pei fz ERRATA, VOL I 1.3, for F read f » 1.2 from bottom, for Af read 3H, 1, 12 20218 179, cquation (76), ƒớ" ¿+1 zzad 9+1 22 ~2 4 # = ee ne mead Fe
185, equation (24), fo eos! read B 2 ce 4 pol
186, 1.5 from bottom, for ‘The surfuce-density on the elliptic plate’ reed ‘The surface-density on cither side of the elliptic plate 186, equation (30), for 2a read 47,
188, equation (38), for m read 27% 196, 1.27, for e e read e, e,
197, equation (10) should be A= Bf 4,
ƯƠ„ ⁄ OF ( fia’)
204, 1.15 from bottom, dele cither
215, 1.4, for J 2k read Jl ok,
; E
234, equation (13), for 22 read on 335, dele last 14 lines
336, 1.1, dele therefore
1, 2, for ‘the potential at © to exceed that at D by BP,’ read a current, C, from V to Y
1.4, for *C to J will cause the potential at A to exceed that ut
4 by the same quantity 2 read X to Y will cause an equal current € from aA to B,
351, 1.3, for RPO+RIAv + Rew? read RK, w+ Rev + Rw,
Pe, av dV dV
» L 5, eaử + 2/// (ur +0 ủy + qs) dudydz 355, last line, for §” read „9
ai diy”
365, in equations (12), (15), (1G), for A read Ar 366, cyuation (3), for =2 peal ~2 ry TQ :
367, 1.5, for 24,8 read 2h,8
368, equution (14), for J read 1 +“
» 404, at the end of Art 350 insert ag follows :—
When y, the resistance to be measured, a, the resistance of the battery, and a, the resistauce of the galvanometer, are given, the best valucs of the other resistances have been shewn by Mr Oliver Heaviside (PAdl Afay., Feb 1873) to be
Trang 35
ELECTRICITY AND MAGNETISM
PRELIMINARY
ON THE MEASUREMENT OF QUANTITIES
1.] Every expression of a Quantity consists of two factors or
components One of these is the name of a certain known quan-
tity of the same kind as the quantity to be expressed, which is taken as a standard of reference The other component is the number of times the standard is to be taken in order to make up the required quantity The standard quantity is technically called the Unit, and the number is called the Numerical Value of the quantity
There must he as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the threc funda- mental units of Length, Time, and Mass Thus the units of area and of volume are defined respectively as the square and the cube whose sides are the unit of length
Sometimes, however, we find several units of the same kind founded on independent considerations, Thus the gallon, or the volume of ten pounds of water, is used as a unit of capacity as well as the cubic foot The gallon may be a convenient measure in some cases, but it is not a systematic one, since its numerical re- lation to the cubic foot is not a round integral number
2.) In framing a mathematical system we suppose the funda- mental] units of length, time, and mass to be given, and deduce all the derivative units from these by the simplest attainable de- finitions
The formulae at which we arrive must be such that a person
Trang 362 PRELIMINARY [3
of any nation, by substituting for the different symbols the nu- merical value of the quantities as measured by his own national
units, would arrive ata trne result,
lence, in all scientific studies it is of the greatest importance to employ units belonging to a properly defined system, and to know the relations of these units to the fundamental units, so that we may be able at once to transform our results from one system to another,
This is most conveniently done by ascertaining the dimensions of every unit in terms of the three fundamental units When a given unit varies as the 2th power of one of these units, it is said to be of 2 dimensions as regards that unit
For instance, the scientifie unit of volume is always the cube whose side is the unit of length If the unit of length varies,
the unit of volume will vary as its third power, and the unit of volume is said to be of three dimensions with respect to the unit of length
A knowledge of the dimensions of units furnishes a test which
ought to be applied to the equations resulting from any lengthened
investigation The dimensions of every term of such an equa- tion, with respect to each of the three fundamental units, must
be the same If not, the equation is absurd, and contains some
error, as its interpretation would be different according to the arbi- trary system of units which we adopt *
The Three Fundamental Units
3.] (1) Length, The standard of length for scientific purposes
in this country is one foot, which is the third part of the standard yard preserved in the Exchequer Chambers
In France, and other countries which have adopted the metric system, it is the métre The métre is theoretically the ten mil-
lionth part of the length of a meridian of the earth measured from the pole to the equator; but practically it is the length of
a standard preserved in Paris, which was constructed by Borda to correspond, when at the temperature of melting ice, with the
value of the preceding length as measured by Delambre The matre has not been altered to correspond with new and more accurate
measurements of the earth, but the are of the meridian is estimated in terms of the original matre,
Trang 37
5.] THE TIREK FUNDAMENTAL UNITS, 3
In astronomy the mean distance of the earth from the sun is sometimes taken as a unit of length
In the present state of science the most universal standard of
length which we could assume would be the wave length in vacuum ofa particular kind of light, emitted hy some widely diffused sub-
stance such as sodium, which has well-defined lines in its spectrum, Such a standard would be independent of any changes in the di-
mensions of the earth, and should be adopted by those who expect their writings to be more permanent than that body
In treating of the dimensions of units we shall call the unit of
length [Z] If Z is the numerical value of a length, it is under- stood to be expressed in terms of the concrete unit [Z], so that
the actual length would be fully expressed by ¢[L}
4.) (2) Time The standard unit of time in all civilized coun-
tries is deduced from the time of rotation of the earth about its axis The sidereal day, or the true period of rotation of the earth ;
can be ascertained with great exactness by the ordinary observa-
tions of astronomers; and the mean solar day can be deduced
from this by our knowledge of the length of the year
The unit of time adopted in all physical researches is one second of mean solar time
In astronomy a year is sometimes used as a unit of time A more universal unit of time might be found by taking the periodic time of vibration of the particular kind of light whose wave length is the unit of length,
We shall call the concrete unit of time [7], and the numerical measure of time 4
5.] (3) dfuss, The standard unit of mass is in this country the
avoirdunols pound preserved in the Exchequer Chambers The
grain, which is often used as a unit, is defincd to be the 7000th part of this pound
In the metrical system it is the gramme, which is theoretically the mass of a cubic centimétre of distilled water at standard tem- perature and pressure, but practically it is the thousandth part of a standard kilogramme preserved in Paris
The accuracy with which the masses of bodies can be com-
pared by weighing is far greater than that hitherto attained in the measurement of lengths, so that all masses ought, if possible, to be compared directly with the standard, and not deduced from
experiments on water
Trang 384 PRELIMINARY, [5- earth is sometimes taken as a unit, but in the dynamical theory
of astronomy the unit of mass is deduced from the units of time and length, combined with the fact of universal gravitation, The astronomical unit of mass is that mass which attracts another
body placed at the unit of distance so as to produce in that hody
the unit of acceleration
In framing a universal system of units we miy either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science ; or, if we expect * soon to be able to determine the mass of a single molecule of a standard substance, we may wait for this determination before fixing a universal standard of mass
We shall denote the conerete unit of mass hy the symbol [4/] in treating of the dimensions of other units The unit of mass will be taken as one of the three fundamental units When, as in the French system, a particular substance, water, is taken ag a standard of’ density, then the unit of mass js no longer inde- pendent, but varies as the unit of volume, or as [2]
If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of [A] are #2 7¬*] For the acecleration due to the attraction of a mass m at a
, ; ne
distance 7 is by the Newtonian Law a Suppose this attraction
to act for a very small time ¢ on a body originally at rest, and to
cause it to describe a space s, then by the formula of Galileo,
" )}
s=1/2= bn a;
2
24
whence m = 2 ra 2 Since 7 and + are both lengths, and é is a time, this equation cannot be true unless the dimensions of jn are [4°7'~*] The same can he shewn from any astronomical equa-
tion in which the mass of a body appears in some but not in all
of the terms f,
* See Prof J Loschmidt, ‘Zur Gréose der Luftinolecule,’ Academy of Vienna
Oct, 12, 1865; GJ Stoney on ‘The Internal Motions of Cases.’ Phil, Mag., Aug
1868; and Sir W Thomson on ‘ The Size of A toms,’ Nudare, March 31, 1870,
Trang 396 ] DERIVED UNITS 5
Derived Units,
6.) The unit of Velocity is that velocity in which unit of length is described in unit of time Its dimensions are [Z7'~"],
Tf we adopt the units of length and time derived from the vibrations of light, then the unit of velocity is the velocity of light
The unit of Acceleration is that acceleration in which the velo-
city increase’ by unity in unit of time Its dimensions are [2 7" *] The unit of Density is the density of a substance which contains
unit of mass in unit of volume Its dimensions are [47Z~*]
The unit of Momentum is the momentum of unit of mass moving with unit of velocity Its dimensions are [J7Z7'~"]
The unit of Force is the foree which produces unit of momentum
in unit of time Its dimensions are [AZ L7'~*]
This is the absolute unit of force, and this definition of it is implied in every equation in Dynamics Nevertheless, in many
text books in which these equations are given, a different unit of
force is adopted, namely, the weight of the national unit of mass ; and then, in order to satisfy the equations, the national unit of mass
is itself abandoned, and an artificial unit is adopted as the dynamical
unit, equal to the national unit divided by the numerical value of the force of wravity at the place, In this way both the unit of force and the unit of mass are made to depend on the value of the force of gravity, which varies from place to place, so that state- ments involving these quantities are not complete without a know-
ledge of the force of gravity in the places where these statements
were found to be true
The abolition, for all scientific purposes, of this method of mea-
suring forces is mainly due to the introduction of a general system
of making observations of magnetic force in countries in which
the force of gravity is different All such forees are now measured according to the strictly dynamical method deduced from our delinitions, and the numerical results are the same in whatever country the experiments are made
The unit of Work is the work done by the unit of force acting through the unit of length measured in its own direction Its
dimensions are [Ä/7*7-*]
The Energy of a system, being its capacity of performing work, is measured by the work which the system is capable of performing
Trang 406 PRELIMINARY, L7
The definitions of other quantities, and of the units to which
they are referred, will be given when we require them,
In transforming the values of physical quantities determined in terms of one unit, so as to express them in terms of any other unit
of the same kind, we have only to remember that every expres-
sion for the quantity consists of two factors, the unit and the nu- merical part which expresses how often the unit is to be taken
Hence the numerical part of the expression varies inversely as the magnitude of the unit, that is, inversely as the various powers of the fundamental units which are indieated by the dimensions of the derived unit,
Ou Physical Continuity and Discontinuity,
7.) A quantity is said to vary continuously when, if it passes from one value to another, it assumes all the intermediate values,
We may obtain the conception of continuity from a consideration of the continnons existence of a particle of matter in time and space, Such a particle cannot pass from one position to another without deseribing a continuous line in space, and the coordinates of its position must be continuous functions of the time
In the so-called ‘ equation of continuity,’ as given in treatises
on Hydrodynamies, the fact expressed is that matter cannot appear in or disappear from an element of volume without passing in or out through the sides of that element
A quantity is said to be a continuous function of its variables
when, if the variables alter continuously, the quantity itself alters continuously,
Thus, if «is a funetion ofa, and if, while a passes continuously from a to 2,, « passes continuously from a, to a, but when ø passes from 2, to x,, % passes from 2,’ to w„, my being different from #,, then « is said to have a discontinuity in its variation with respect to @ for the value 2 = 2,, because it passes abruptly from x, to 2,’ while 2 passes continuousl y through 2
Tf we consider the differential coefficient of œ with respect to x for
the value # = 2, as the limit of the fraction U,,— Uo
#,— 8u
When x, and «, are both made to approach 2, without limit, then, if a and a, are always on opposite sides of 2,, the ultimate value of
the numerator will be w/—m, and that of the denominator will be zero If wisa quantity physically continuous, the discontinuity