UNIVERSITY OF 6AUFQW
MAR 1 2 1923
DIVISIOS
PRACTICAL ELECTRICITY
Practical Electricity
A Laboratory and Lecture Course
For First Year Students of Electrical
Engineering, based on the Practical
Definitions of the Electrical Units
BY
W. E. AYRTON
If
F.R.S., ASSOC. M. INST. G.E.
Past President of the Institution of Electrical Engineers, Late Professor
of Electrical Engineering, Central Technical College
REVISED AND LARGELY REWRITTEN BY
T. MATHER
Wh. Sch., F.R.S., M.I.E.E.
Professor of Electrical Engineering, Central Technical College,
Imperial College of Science and Technology.
South Kensington
WITH OVER 300 ILLUSTRATIONS
CASSELL AND COMPANY, LTD
London, New York, Toronto and Melbourne
JS ../,
f i
First published February 1887.
'Reprinted November 1887, November 1888, January 1890.
January and October 1891, October 1892, March and August 1893,
October 1894, January 1896.
New Edition November 1896. Reprinted December 1897,
/w/y 1900, March 1902, /7*6» 1903, October 1906
New Edition February 1911. Revised Edition March 1914.
Rtprinted December 1916, ^/oy 1919, March 1920.
New Edition August 1921.
AM RIGHTS RBSERVRD
PREFACE
TO THE FOURTH EDITION
A NEW edition of the book being required, advantage has been
taken of the occasion to bring the work up to date. Also to
modify, where convenient, the symbols used, in accordance
with the list adopted by the International Electrotechnical
Commission in 1913. A copy of this list is given in Appendix
VII.
The sections on Dry Cells have been rewritten, and that dealing
with Storage Cells amplified. The addendum to Appendix I,
relating to the practical electrical units, has been revised and
extended to include more recent work in this subject.
I am again indebted to Mr. Maurice Solomon, of the General
Electric Company, for valuable information ; to Mr. R. W.
Cooper, M.A., Messrs. Benn Brothers, and Edison Accumulators,
Limited, for the use of blocks ; and to the India Rubber and
Gutta Percha Company. My best thanks are -also due to Dr.
Chas. Chree, M.A., F.R.S., for magnetic data, and to Mr. F. E.
Smith, F.R.S., O.B.E., for help in connection with absolute
measurements of the primary electrical units. The whole
world is deeply indebted to Mr. Smith for the masterly way
in which he has originated and carried out the researches on
electrical standards at the National Physical Laboratory for
many years past. His work has placed Britain well ahead of
other nations in this branch of precision measurements.
T. MATHER.
PREFACE
TO THE THIRD EDITION
MAINLY owing to the numerous public calls on the time of the
late Professor Ayrton, and to some extent because of the impaired
health resulting from his many and strenuous labours for the
advancement of technical education, the second of the two
volumes in which it was intended to issue the second edition ol
" PRACTICAL ELECTRICITY," was not completed. Shortly before
the Professor's much lamented death, it was decided that the book
should be re-written and published in a single volume of somewhat
larger size, in the joint names of Professor Ayrton and myself.
The result is the present work, which deals with the matters
treated of in Volume I. of the 1896 edition, as well as those which
the second volume was intended to contain.
The arrangement of the book follows, in the main, that of
the original work, but an attempt has been made to illustrate
in greater detail the intimate relations that exist between electri-
cal and mechanical quantities, and to show how the practical
system of electrical units is founded on the C.G.S. system of
mechanical units. It is hoped that this treatment will enable
beginners to realise that definite ratios must exist between the
electrical and mechanical units of power and energy, instead of
regarding these relations as somewhat mysterious. Students
should make themselves acquainted with the C.G.S. system of
mechanical units before commencing the study of electricity.
The experimental proof of Ohm's Law, -which formed a pro-
minent feature in previous editions, has been improved by
using a zero electro-dynamometer to measure current strengths
instead of a calibrated galvanometer.
Many new figures and new examples have been added, and
new chapters on the potentiometer, the induction of electric
currents, and on the magnetisation of iron, are now included.
As the measurement of electric energy has become, of late
years, a subject of much technical and commercial importance,
more space is devoted to electric meters than in former editions.
viii PREFACE
Some of the meters described are intended for continuous currents,
whilst others may be used both for continuous and alternating
currents ; but as this work deals with continuous currents only,
no reference to the use of meters for alternating currents is made
in the text. No attempt has been made to treat the subject of
meters completely, for their number and diversity are now so
great that such treatment would alone require a whole volume.
A knowledge of sizes of wires, and the resistances of copper
wires is of considerable utility, so a table of the Legal Standard
Wire Gauge is printed as an appendix, giving, in addition to
diameters and areas of cross -section, the relations between Length
and Resistance, Resistance and Weight, and Weight and Length,
both in British and Metric measure. Other tables, very useful
in making calculation of windings of instruments and machines,
give the number of wires per lineal inch or centimetre, the
number of turns of wire per square inch or per square centi-
metre of windings, as well as the resistances per cubic inch
and per cubic centimetre for the various sizes of copper wire
insulated in several different ways. Tables of this kind, based on
the thicknesses of insulated coverings adopted by the London
Electric Wire Company, have been in use at the Central Technical
College since 1890. For the calculation of the present tables I
am indebted to my son, W. H. Mather, who has also re-worked
many of the examples. The tables have been checked by Mr.
S. S. Watkins, A.C.G.I., B.Sc.
Although the book is primarily intended for students following
a first year's lecture and laboratory course in Electrical Engineer-
ing, it deals with most of the subjects required for the intermediate
examinations in Electricity and Magnetism in the universities
of London and the provinces. I therefore hope it will be useful
in assisting students to acquire the knowledge necessary for these
examinations, not as a " cram book," but as one that, with the
help of laboratory instruments and apparatus such as are
described and illustrated, will give them a sound quantitative
knowledge of the several phenomena which form the basis of
electrical science and industry.
The " Short History of the Absolute Unit of Resistance, etc."
written by the late Professor Ayrton, has been reprinted and
extended by a statement of the work done since 1896, and of the
resolutions adopted by the " International Conference on
Electrical Units and Standards," held in London in October,
1908.
Acknowledgment is due and is hereby tendered to several
correspondents for pointing out a few errors and misprints in
PREFACE ix
previous editions. Notice of similar imperfections in the present
work will be greatly appreciated.
My thanks are also due to Mrs. Ayrton, who has kindly read
the proofs and made several valuable suggestions, to my colleague
Prof. G. W. O. Howe, M.Sc., and my assistants, Mr. F. E. Meade,
Mr. S. S. Watkins, A.C.G.I., B.Sc., for help in collecting data,
etc., to Mr. F. E. Smith, A.R.C.S., of the National Physical
Laboratory ; Mr. Maurice Solomon, A. C.G.I., of the Birmingham
Carbon Works ; The Electrician Publishing Company ; and to
the several firms who have furnished blocks and information
about their manufactures, which have been useful in bringing
the book up to date.
T. MATHER.
PREFACE
TO THE SECOND EDITION
EXACTLY ten years have elapsed since the preface to the first
edition of this book was written — a decade which has seen a vast
development in the applications of electricity to industrial
purposes, and the springing up in all parts of the kingdom
of Technical Schools and Colleges where much attention is devoted
to the study of electrotechnics. Hence, to-day it is far more
easy for a student to connect his experimental apparatus with
the electric light mains and use a comparatively large current at
a pressure of 100 volts, than it was in 1886 to obtain a small
current at a much lower pressure from the battery which he
had to set up for the purpose. This possibility of carrying out
the experiments on a larger scale has led to considerable simpli-
fication in certain cases ; for example, in experimentally deter-
mining the heat equivalent of electric energy, it is no longer
necessary to distract the beginner's attention with a variety of
corrections for the loss of heat, &c.
After many issues of the book had appeared in its original
form, it seemed desirable to bring it up to date ; and since the
practice, not unfrequently resorted to by writers, of inserting
a number of new patches in an antiquated ground work, would be
out of place in a book which had been written to aid electro-
technical teaching and not for purposes of profit, a proposition
was made to entirely rewrite it. This the publishers accepted ;
and, guided by the success which the book had achieved, they
generously, and I anticipate wisely, modified the arrangements
so as to justify my devoting a large amount of time to the pre-
paration of what in reality is an entirely new book, although
called by its old title " PRACTICAL ELECTRICITY."
The reception of the first edition took me by surprise. I
anticipated that the book would be regarded as " faddy,"
and that the critics, while admitting that perhaps it would do
well enough for my own classes, would not recommend its use
PREFACE xi
for students in general. It did not occur to me that the world
was ready for using such a text-book and prepared to adopt the
methods of teaching advocated in its pages. To-day, however,
the following reasons suggested in the original preface for even
elementary students of electricity spending much time in the
laboratory would be advanced by many teachers : —
" One of the great difficulties experienced by people in master-
ing the quantitative science of electricity, arises from the fact that
we do not number an electrical sense among our other senses,
and hence we have no intuitive perception of electrical phenomena.
During childhood we did not have years of unconscious experi-
menting with electrical forces as we had with the forces connected
with the sensations of heaviness and lightness, loudness and
softness, heat and cold. Beyond a shock or two taken perhaps
from some medical galvanic apparatus, or from a Leyden jar,
our senses have never been affected by electrical action, and hence
we ought to begin the study of electricity as a child begins its
early education. Quite an infant has distinct ideas about hot
and cold, although it may not be able to put its ideas into words,
and yet many a student of electricity of mature years has but
the haziest notions of the exact meaning of high and low poten-
tial, the electrical analogues of hot and cold. That it is desirable
that students should learn physics, as they learn to ride the
bicycle, by experimenting themselves, is now generally admitted,
and this is especially true in the case of electricity, since it is by
experimenting, and only by experimenting, that a student
can obtain such a real grasp of electricity that its laws become,
so to say, a part of his nature."
" Hence, in the courses of electricity which I arranged at the
City and Guilds of London Technical College, Finsbury, and at
their Central Technical College, Exhibition Road, for every
hour that a student spends at lecture, he spends several in the
laboratory.''
When Dr. Hopkinson this year, 1896, in his Inaugural Address
as President of the Institution of Electrical Engineers, advocated
commencing the study of electricity with the electric current,
more than one teacher testified to the value of the method by
claiming it as his own, apparently forgetful that when his order
of treating the subject was introduced by the author in 1879 there
was no precedent for such an innovation. Indeed, when even
seven> years later there appeared the first edition of " PRACTICAL
ELECTRICITY " it was thought advisable to introduce the method
by inserting the following explanatory paragraphs : —
" Readers who have been accustomed only to the ordinary
Xll
PREFACE
books, commencing with certain chapters on statical electricity,
continuing with one or more on magnetism, and ending with
some on current electricity, will be surprised at the arrangement
of the subjects in this book, and will probably be astonished at
what they will condemn, at the first reading, as a total want of
order. But so far from the various subjects having been thrown
together haphazard, the order in which they have been arranged
has been a matter of the most careful consideration, and has been
arrived at by following what appears to me to be the natural
as distinguished from the scholastic method of studying electricity.
I have endeavoured to treat the subject analytically rather than
synthetically, because that race of successful experimental
philosophers — children — adopt this method.
" For example, it is not by studying geometrical optics, much
less physical optics, that an infant gradually learns to appreciate
the distance of objects ; and later on it is not by studying a
treatise on struts, nor by listening to a course of lectures on
structures, that the child finds out that the table has legs, hard
legs, round legs. Feeling, looking, trying, in fact a simple course
of experimental investigation, gives a child its knowledge ;
and this, therefore, I venture to think, is the method we should
adopt when commencing the study of electricity."
" The subject of current is treated first, because in almost
all the industries in which electricity is practically made use of,
it is the electric current that is employed ; also, because currents
can be compared with one another, and the unit of current (the
ampere) denned, without any knowledge of potential difference
or resistance. Potential difference is next considered, and resist-
ance the last of the three, because the very idea of resistance implies
a previous acquaintance with the ideas of current and potential
difference, since the resistance of a conductor is the name given
to the ratio of the potential difference (measured electrostatically)
between its terminals to the current passing through it. And
it is Ohm's experimental proof that this ratio was constant for a
given conductor under given conditions, together with the num-
berless experimental verifications that this conclusion has received,
that has led to resistance gradually coming to be considered as
a fixed definite property of a given conductor like its weight or
length."
The international, or Board of Trade, unit of P.D. — the volt —
cannot, however, be defined until the definition of the unit of
resistance — the ohm — has been fixed, because for legal purposes
the units of current and resistance have been taken as the primary
ones, and Ohm's law has been employed to fix the third or derived
PREFACE
xin
unit — viz. that of P.D. Hence, the actual sequence adopted in
the present volume is (i) current and the ampere ; (2) the relative
measurement of P.Ds. with a zero electrostatic voltmeter ; (3)
Ohm's law ; (4) resistance, and the ohm ; (5) the volt, and
current-voltmeters. Electric energy and power, with their
units — the joule and the watt — are next treated ; and, lastly,
the conception of the E.M.F. in a circuit, and the necessity for
the E.M.F. of a good cell being constant, are derived from the
laws of energy.
If should be obvious that any method of trying to experimen-
tally prove Ohm's law with a current-voltmeter, such as may be
found in certain text-books, begs the question. If a voltmeter
be used it must be of the electrostatic type, and to simplify the
definition of one P.D. being twice another this electrostatic
voltmeter should be a zero instrument, which, without the need
of any independent electrification, would be suitable for measuring
P.Ds. no larger than those commonly employed in laboratories
for sending currents. Such an instrument I have long felt the
need of, and now — thanks to the ingenuity of Mr. Mather — it is
available for use, and will be found described for the first time in
pages 163-166* of the present volume.
It will be observed that the apparatus required for each experi-
ment is mounted complete on a board. This is to enable it to be
easily carried backwards and forwards between the laboratory
and the lecture-room without disarranging it. At first sight
it might appear that the student, finding each set of apparatus
joined up quite complete, with current laid on all ready
for carrying out the experiment, would be deprived of all incentive
to exercise his own ingenuity in overcoming experimental difficul-
ties, and, therefore, would fail to acquire habits of self-reliance.
For first year students, however, I have found it a good plan to
have each set of apparatus complete in position ; firstly, because
it is only with some such arrangement that fifty or more students
can commence work almost simultaneously, and in the course of
two or three hours have all performed some quantitative experi-
ment ; secondly, because when the apparatus is so arranged that
even beginners can perform several experiments successfully, they
acquire faith in the possibility of success, and are less discouraged
with the difficulties they subsequently meet with when selecting
and arranging the apparatus for conducting some investigation.
The practical side of electricity has grown so rapidly that
the original single volume has expanded into two. The present
volume of the rewritten book is intended to assist students in
* Pages 134-137 of third edition.
XIV
PREFACE
acquiring experimentally an exact working knowledge of current,
difference of potentials, resistance, energy, and power, with their
electric transmission, cells and their cost of working. This
subject of the cost of converting chemical energy into electric
energy is not, as far as I am aware, to be found in any text-book.
Hence, in view of the " booms " in primary batteries, which
appear to be periodic, the question of cost has been entered into in
considerable detail.
The past four years have seen the legalisation in several
countries of an international system of electrical units, so that,
while the units of length, volume, mass, and money vary from
country to country, there is now but one ampere, one ohm
and one volt throughout the whole world ; a fact of which electri-
cal engineers may feel justly proud. Some thirty pages at the
end of the book are, therefore, devoted to " A Short History of
the Absolute Unit of Resistance, and of the Electrical Standards
of the Board of Trade."
In spite of the fact that the present volume contains some
140 pages more than the original book, the subjects of secondary
cells, electric quantity, coulombmeters, capacity, &c., have had
to be left for a second volume. This has arisen not merely from
primary cells, including dry cells and the Clark's standard cell,
having been treated somewhat fully, but from the subjects of
electric energy and power, the various meters used for measuring
these quantities, the efficiency of electric transmission, the ratio
of the power received to the maximum power receivable in various
cases of transmission, &c., having been entered into at length in
consequence of the commercial importance that the electric
transmission of energy now possesses. And it may be mentioned
that generally where problems of maxima or minima have been
considered, attention has been directed to the kind of change that
is produced in the value of the quantity under consideration, when
the value of the variable is altered from that required to make
the quantity a maximum or a minimum.
In fact, the aim has been to treat a few subjects fairly
thoroughly in a simple manner, and not to prepare a list of
short instructions for carrying out a large number of experiments,
nor to write a treatise, mainly of value as an electrical dictionary,
which should give a little information about everything that can
be comprised under the head of electricity, whether it be electric
eels, the history of the invention of the telegraph, the aurora,
or the earliest forms of fractional machines.
In the letterpress, small capitals have been used to represent
instruments, parts of apparatus, &c., while large capitals
PREFACE xv
systematically stand for electric quantities other than resistances,
these being throughout designated by small letters in italics.
Thus, A, A, a stand respectively for an ammeter, the current in
amperes flowing through it, and its resistance in ohms. *
In the preface written in 1886 it was mentioned that, with the
exception of two or three blocks that had been lent, the 180
figures had been specially drawn for the book, and were not
time-honoured representations of historical apparatus. Of these
180 figures only 64, however, have been employed in the present
volume, partly because the fresh matter required many new
figures to illustrate it, and partly because several of the blocks
specially executed for the original book have lost their freshness
from the appreciative use of them by other writers. Hence, 183
of the 247 figures contained in the present volume will not be
found in the former book, and 163 of these fresh illustrations have
been specially drawn for this new edition.
A large number of new examples have been added, and any
that have been reproduced from the original book have been
reworked, either to check the accuracy of the results, or because
the so-called legal units referred to have been replaced by those
that have now been adopted internationally.
My thanks are due to my past and present assistants — Dr.
Sumpner, Mr. Haycraft, and Mr. Severs — for much assistance in
the preparation of this book ; to my daughter for compiling a
very comprehensive and judiciously arranged index ; and to
Messrs. Spiers, Twyman, and other students for carefully exam-
ining the proofs. In conclusion, I desire to express to-day even
more warmly than in October, 1886, my indebtedness to Mr.
Mather for the very earnest, thoughtful, and painstaking way
in which for many years he has assisted me in developing the
course of instruction for students of electrical technology, of
which the present volume represents part of the elementary
portion.
W. E. AYRTON.
October, 1896.
* As "International Symbols" have been largely used in the present
edition, this rule no longer holds. — T. M., 1921.
CONTENTS
CHAPTER I
THE ELECTRIC CURRENT AND ITS MEASUREMENT
SECTION FACE
1 . What is meant by an Electric Current, and by its Direc-
tion of Flow i
2. Production of an Electric Current : Electric Circuit . 2
3. Cells and Batteries 3
4. Conductors and Insulators 3
5. Properties of an Electric Current 4
6. Current Strength . 7
7. The Strength of an Electric Current ; by which of its
Properties shall it be directly measured ? . . 12
8. Definition of the Unit Current ; Ampere . . . 18
9. Electrolysis ; Electrochemical Equivalent . . . 21
10. Definition of Unit Quantity of Electricity : Coulomb . 24
n. Definition of the Direction of the Current : Ions . . 26
12. Objection to the Usual Mode of Constructing Voltameters 30
13. Description of Practical Forms of Sulphuric Acid Volta-
meters 31
14. Relative Advantages of Voltameters and Galvanometers 33
15. Measurement of Current by Galvanometers : Tangent
Galvanometers . . . . . . . 36
1 6. Meaning of the Relative and the Absolute Calibration of a
Galvanometer ....... 39
17. Comparison of Tangent Galvanometer with a Voltameter 40
1 8. Absolute Calibration of Tangent Galvanometer . . 42
19. Calibrating any Galvanometer by Direct Comparison
with a Tangent Galvanometer . . . . 43
20. Graphically Recording the Results of an Experiment . 44
21. Practical Value of Drawing Curves to Record Graphically
the Results of Experiments .... 47
22. To Construct a Galvanometer Scale from which the
Relative Strengths of Currents can be at once
ascertained . . . . . « 5°
CHAPTER II
MAGNETIC FIELDS
23. Magnetic Fields . ».' 53
24. Lines of Magnetic Force . . . . . .56
25. Strength of Magnetic Poles » 59
xviii CONTENTS
SECTION J»AGH
25^. Hibbert's Magnetic Balance 61
256. Balance for Finding Strength of Pole .... 62
26. Magnetic Moment ....... 63
27. Absolute Measurement of Strength of Magnetic Field
and of Magnetic Moment ..... 66
28. Mapping Magnetic Fields . . .68
29. Comparing the Relative Strengths of different Parts of a
Magnetic Field by the Vibration Method . . 72
30. Comparing the Relative Strengths of different parts of
a Magnetic Field by the Magnetometer Method . 74
300. Difference of Magnetic Potential : Equipotential Sur-
face 79
Addendum to Chapter II. : Electric Lines of Force and Elec-
trostatics 8 1
CHAPTER III
GALVANOMETERS, ELECTRODYNAMOMETERS AND AMMETERS
31. The Tangent Galvanometer 83
32. Adjusting the Coil of a Tangent Galvanometer . . 84
33. Scale for a Tangent Galvanometer . . . 86
34. Tangent Law 88
35. Variation of the Sensibility of a Tangent Galvanometer
with the Number of Windings, and with the
Diameter of the Coil 90
36. Values in Amperes of the Deflections of a Tangent Gal-
vanometer controlled only by the Earth's
Magnetism 96
37. Pivot and Fibre Suspensions 99
38. Sine Galvanometer 100
39. Electrodynamometers ....... 102
40. Construction of Galvanometers in which the Angular
Deflection is directly proportional to the Current 105
41. Galvanometers of Invariable Sensibility . . . 107
42. Permanent Magnet Ammeters 109
43. Moving Coil Ammeters . . . . . .112
44. Soft Iron Ammeters : Spring and Gravity Control . . 118
45. Hot- Wire Ammeter 124
CHAPTER IV
DIFFERENCE OF POTENTIAL AND RESISTANCE
46. Difference of Potentials 126
47. Potential of the Earth arbitrarily called Nought ; Posi-
tive and Negative Potentials . . . . 132
48. Measurement of Potential Difference
49. Electrometer ....
50. Ohm's Law ...»
51. Resistance
52. Ohm : Unit of Resistance
133
134
138
142
143
CONTENTS xix
53. Resistance Coils and Resistance Boxes . . .145
54- volt 148
55. Ohm's Law applicable to Complete Circuits : E.M.F. 149
550. Electromagnetic Definition of E.M.F 151
56. Current Method of Comparing P.Ds. and Resistances . 153
57. Reason for using High Resistance Galvanometers for P.D.
Measurements, and Low Resistance Galvanometers
for Current Measurements . . . . . 154
58. Voltmeter 155
59. Resistances of Ammeters and Current Voltmeters . . 158
60. Ammeters used as Voltmeters 158
61. Moving Coil Voltmeter 160
62. Calibrating a Deflectional Voltmeter . . . .160
63. Voltmeters used as Ammeters 163
64. Gold-Leaf Electroscope . . . . . .166
65. Sensibility of Gold -Leaf Electroscopes . . . .168
CHAPTER V
GALVANIC CELLS
66. Chemical Action in a Simple Voltaic Element . . 170
67. Daniell's Use of a Depolariser : Two-Fluid Cell . . 173
68. Local or Prejudicial Action J75
69. Gravity Daniell's Cells . .... 178
70. Minotto's Cell
71. Resistance of Daniell's Cells
72. Grove's and Bunsen's Cells
73. Potassium Bichromate Cell
74. Storage or Secondary Cell .
75. Leclanche Cells
76. Dry Cells ....
77. Hellesen and Dania Dry Cells
78. G.E.C. and Obach Cells
79. Blue Bell and Columbia Cells
80. Extra-Sec and Inert Cells .
81. Edison-Lalande Cell
179
1 80
183
185
187
189
193
196
197
198
198
199
82. Standard Cells, Clark's and Weston's Cells. . . 200
83. Calculation of the E.M.F. of a Cell from the Energy
Liberated by the Chemical Action . . . 206
CHAPTER VI
RESISTANCE ! ITS LAWS AND MEASUREMENT
84. Comparing Resistances : Voltmeter and Ammeter
Method 210
85. Ohmmeter : Megger ....... 211
86. Simple Substitution Method of Comparing Resistances 214
87. Differential Galvanometer, A Null Method . . . 216
88. Wheatstone's Bridge : its Principle . . . .218
89. Wheatstone's Bridge : its Use and Simple Method of
Constructing 221
xx CONTENTS
SECTION PAGE
90. Bridge Key . . . . . . . . 225
91. Use of a Shunt with the Bridge 227
92. Meaning of the Deflection on a Bridge Galvanometer 227
93. Conditions Affecting the Resistance of a Conductor . 228
94. Variation of Resistance with Length s . . . 229
95. Variation of Resistance with Cross-Section . . . 230
96. Variation of Resistance with Material . . .231
97. Resistance of Metals and Alloys per Centimetre Cube
and per Inch Cube . . . . . .231
98. Resistance of Metals and Alloys for a Given Length and
Weight * . .234
99. Variation of Resistance with Temperature . . 236
100. Conductors of Large Specific Resistance have Small
Temperature Coefficients ; : .. . . 239
101. Conductivity and Conductance ..... 242
102. Comparison of Electric and Heat Conductivities . 243
103. Resistance and Conductance of Several Conductors in
Series or in Parallel . . . . . . 244
104. Currents in Parallel Conductors 248
105. Kirchhoff's Rules 248
106. Shunts . . . . . . . . 251
107. Multiplying Power of a Shunt . ••-•'. . 252
1 08. Usual Method of Constructing a Shunt Box • . . 253
109. Increase of the Main Current produced by Applying
a Shunt . . ... ... . . 255
no. Principle of Universal Shunts ..... 259
in. Method of Constructing a Universal Shunt Box, and its
Advantages 260
112. Standard Resistance Coils . . . . . . 264
113. Ordinary Forms of Wheatstone Bridge . . .265
114. Portable Forms of Wheatstone Bridge . . . 269
115. Dial and Bar Patterns of Bridge . . . .271
CHAPTER VII
ELECTRIC ENERGY AND POWER
1 1 6. Work done by a Current . , . . . .273
117. Electric Unit of Energy : Joule 277
ti8. Heat Produced by a Current . . . . .277
119. Measuring the Heat Equivalent of Electric Energy . 278
120. Power 282
121. Electric Unit of Power : Watt 283
122. Joule's Law . ....... 285
123. Instruments for Measuring Power : Wattmeters . 286
124. Commercial Forms of Wattmeters . . . .288
125. Joule — or Energy Meter : Clock Form . . . 290
126. Board of Trade Unit of Energy . . . . 294
127. Energy Meter : Motor Form 296
128. Quantity or Ampere-hour Meters .... 302
129. Electric Transmission ol Energy . . . . . 308
130. Power Developed by a Current Generator . . . 312
CONTENTS xxi
131. Connection between the E.M.F. of a Battery, the P.D.
between its Terminals, the Resistance and the
Current 313
132. Electromotive Force of any Current Generator . . 314
133. Power Absorbed in the Circuit Exterior to the Genera-
tor : Back E.M.F 315
134. Distribution of Power in an Electric Circuit . . 318
135. External Circuit that Receives Maximum Power from
a Given Current Generator . . . . . 319
136. Arrangement of n Cells -to give Maximum Power to an
External Circuit of Fixed Resistance . . 325
137. Minimum Number of Cells required to give a fixed
Amount of Power to a given External Circuit . 329
138. Importance of Low Resistance and High E.M.F for
Large Powers -333
139. Modifications Introduced into the Previous Results by
Limitation of the Maximum Current a Cell may
Produce . . 333
140. Efficiency ......... 335
141. Efficiency of Electric Transmission of Energy . . 339
142. Connection between Electrical Efficiency of Transmis-
sion and Ratio of the Power Received to the Maxi-
mum Power Receivable 343
143. Economy in Electrical Transmission of Energy :
Kelvin's Law 346
CHAPTER VIII
QUANTITY AND CAPACITY
144. Electric Quantity and its Measurement . . . 348
145. Ballistic Galvanometer ... . 349
146. Measurement of Quantity by Ballistic Galvanometer 353
147. Correction of Ballistic Galvanometer for Damping . 356
148. Determination of Decrement and Logarithmic Decre-
ment 357
149. Constant of a Ballistic Galvanometer . . . 359
150. Comparison of Quantities 361
151. Capacity 362
152. Condensers : Mechanical Analogies .... 363
153. Units of Capacity ; Farad ; Microfarad . . 365
154. Variation of the Capacity of a Condenser with the Area
of its Coatings and the Distance between them . 366
155. Relation between the Electrostatic Unit of Capacity
and the Farad 367
156. Capacity of Spherical and Plate Air Condensers in
Farads 368
157. Capacity of Cylindrical Condensers . . . .369
158. Specific Inductive Capacity 370
159. Dielectric Strength of Insulators .... 372
160. Resistivity of Insulators 373
161. Construction of Condenser of Large Capacity . .374
162. Condensers for Large P.Ds.. Leyden Jars . . . 376
xxii CONTENTS
163. Comparison of Condensers . ..... 379
164. Potential Divider . . . . . . 3 80
165. Combined Capacity of Several Condensers . . . 382
1 66. Charged Condensers are Stores of Electric Energy, not
of Electricity " . 384
167. Energy wasted in Charging a Condenser from a Source
of Constant P.D. 386
168. Absolute Measurement of Capacity .... 387
169. Measurement of Specific Inductive Capacity, and Resisti-
vity of Insulators 388
170. Standard Air Condensers ...... 392
171. Ratio of Electromagnetic and Electrostatic Units of
Quantity .... ... 394
172. Use of Condensers for Comparing E.M.Fs. of Cells or
other Current Generators . . . . . 397
173. Condenser Method of Measuring the Resistance of a
Cell ;...*. . 398
CHAPTER IX
POTENTIOMETER MEASUREMENTS
174. Poggendorff's Method of Comparing the E.M.Fs. of Cells
or Batteries 400
175. Principle of the Potentiometer 403
176. Calibration of Potentiometer Wire .... 404
177. Industrial Form of Potentiometer .... 406
178. Modern Form of Crompton Potentiometer . . . 408
179. Dial Potentiometer 409
1 80. Calibration of Voltmeter by Potentiometer : Volt (or
Ratio) Boxes 411
181. Standard Resistances for Current Measurements . 414
182. Calibration of Ammeters 416
183. Comparison of Resistances by Potentiometer . . 417
184. Measurement of Power 420
185. Advantages and Disadvantages of Potentiometer
Measurements . . . . . . . 420
CHAPTER X
INDUCED CURRENTS
186. Introductory 423
187. Direction of Induced Currents due to Magneto-Electric
Induction 424
1 88. Lenz's Law : Fleming's Rule 425
189. Relation between Quantity Induced and Resistance of
the Circuit 426
190. Determination of Constant of a Ballistic Galvanometer
by Earth-Inductor Method .... 429
191. Distribution of Magnetism in a Bar Magnet . . 43 *
192. Flux Density over Cross-Sections and over Surfaces of a
Magnet 432
193. Mutual Induction r .435
CONTENTS xxiii
SECTION PAGE
194. Unit of Mutual Induction: Henry .... 437
195. Self-induction 438
196. Induction Coil , 438
197. Induction of Currents in Parallel Wires . . .441
CHAPTER XI
MAGNETISATION OF IRON
198. Lifting Magnets 442
199. Relation between Lifting Force and Current-Turns . 442
200. Lifting Force and Flux Density : 446
201. Magnetic Saturation 450
202. Magnetic Field produced by Current in a Straight Con-
ductor 450
203. Magneto-Motive Force . . . . . 454
204. Testing Magnetic Properties of Iron by the Ballistic
Method 457
205. Permeability 460
206. Hysteresis of Iron 460
207. Remanent Magnetism : Coercive Force . . . 463
208. Loss of Energy due to Hysteresis. Mechanical Analogy 464
209. The Magnetic Circuit : Reluctance .... 467
Appendix I. — Short History of the Absolute Unit of Resistance,
and of the Electrical Standards of the Board
of Trade . . . . . . . 473
Appendix II. — Comparison of C.G.S. and British Systems of
Units . 5I4
Appendix III. — Relations between the Practical C.G.S. Electro-
magnetic and C.G.S. Electrostatic Units . .
Appendix IV. — Specific Gravities, Specific Resistances, and
Specific Conductivities of Mixtures of Pure
Sulphuric Acid and Distilled Water . .
Appendix V. — Showing the Dimensions of Wires according
to the British Standard Wire Gauge (S.W.G.)
as well as the approximate Relations between
Lengths, Resistances, and Weights of Pure
Copper Wire at a Temperature of 15° C. . ^5
Appendix VI. — Windings Tables —
(a) Ordinary Cotton Covered (single) . . . ^g
(b) Ordinary Cotton Covered (double) . . . CJQ
(c) Specially Fine Cotton Covered (single) . c2O
(d) Specially Fine Cotton Covered (double) . c2i
(e) Silk Covered (single) 222
(/) Silk Covered (double) ^
(g) Enamel Insulated -2.
(h) Enamel Insulated and Cotton Covered
(single) 525
(») Enamel Insulated and Cotton Covered
(double) 526
Appendix VII.— Table of Symbols 527
Index * ... 529
PRACTICAL ELECTRICITY
CHAPTER I
THE ELECTRIC CURRENT AND ITS MEASUREMENT
I. What is meant by an Electric Current, and by its Direction of Flow — •
2. Production of an Electric Current : Electric Circuit — 3. Cells
and Batteries — 4. Conductors and Insulators — 5. Properties of an
Electric Current — 6. Current Strength — 7. The. Strength of an
Electric Current : by which of its Properties shall it be Directly
Measured ? — 8. Definition of Unit Current ; Ampere — 9. Electrolysis,
Electrochemical Equivalent — 10. Definition of Unit Quantity
of Electricity; Coulomb — u. Definition of the Direction of the
Current — 12. Objection to the Usual Mode of Constructing Volta-
meters— 13. Description of Practical Forms of Sulphuric Acid Volta-
meters— 14. Relative Advantages of Voltameters and Galvanometers
— 15. Measurement of Current by Galvanometers ; Tangent Galvano-
meters— 16. Meaning of the Relative and the Absolute Calibration of
a Galvanometer — 17. Comparison of Tangent Galvanometer with a
Voltameter — 18. Absolute Calibration of Tangent Galvanometer —
19. Calibrating any Galvanometer by Direct Comparison with a
Tangent Galvanometer — 20. Graphically Recording the Results of
an Experiment — 21. Practical Value of Drawing Curves to Record
Results — 22. To Construct a Galvanometer Scale from which the
Relative Strengths of Currents can be at once ascertained.
i. What is meant by an Electric Current, and by its Direction
of Flow. — In the various industries in which electricity is em-
ployed, as in the telegraph, telephone, electric lighting, electro-
typing, electroplating, torpedo exploding, electric traction, the
electric transmission of power, and in the working of machinery
by the aid of electromotors, it is the so-called " electric current "
that is made use of. Hence a knowledge of the laws of this
electric current, a clear conception of its so-called properties,
combined with a practical acquaintance with the modes of
measuring it, must be of especial importance for a right under-
standing of the working of the apparatus employed in the above-
mentioned industries. Indeed, such knowledge is absolutely
necessary if the user of electrical apparatus is desirous of em-
ploying it to the best. advantage, of being able to correct faults
when they occur, as well as of effecting improvements in the
appliances themselves.
B
2 ;\ PRACTICAL ELECTRICITY
It/jstcasiottiaiy to: speak of an electric current as if it had an
independent1 existenorapart from the " conductor " through which
it is said to be flowing, just as a current of water is correctly
spoken of as something quite distinct from the pipe through which
it flows. But in reality we are not sure that this is the case.
Modern theory, however, suggests that electricity is atomic in
its nature and of two kinds, and that the two kinds pass in
opposite directions along the conductor.
So the student must not assume that the conventional
expression, " The current flows from the copper pole of a galvanic
battery to the zinc pole through the external circuit," implies
a certain knowledge of, the real direction of flow, any more than
the railway expressions, " up train " and " down train," mean
that either train is necessarily going to a higher level than the
other. In the case of a stream of water flowing along a river bed
we are quite sure that there is water in motion, and everyone
is agreed as to which way the water is flowing ; a cork or a piece
of wood thrown on the water indicates by its motion the direction
in which the water is moving.
Nor, again, must an electric current be supposed to be like
waves of sound travelling along, since, in this latter case, although
there is no actual travelling along of matter,
still the direction of motion of the wave of
sound is perfectly definite. Indeed, a wire
along which an electric current is flowing is
more like a wire at each end of which a
musical instrument is being played, so that
the sound is travelling in both directions
along the wire at the same time. In short,
the statement that an electric current is
flowing along a wire is only a short way of
expressing the fact that the wire and the
space around the wire are in a different state
from that in which they are when no electric
current is said to be flowing. So that when
a body and the space around the body possess
certain properties that they do not usually
possess, an electric current is said to be
flowing through that body.
2. Production of an Electric Current:
Electric Circuit. — Perhaps the simplest
method of producing an electric current is to
place a piece of copper and a piece of zinc in
simple1 ceifan4gckcuit. a jar of dilute sulphuric acid and join the
Fig. i. — Simple cell and
circuit.
PRODUCTION OF AN ELECTRIC CURRENT 3
two plates together by a piece of wire*, thus forming what is
called an " electric circuit " (Fig. i).
3. Cells and Batteries. — The jar and the dissimilar metals in
dilute acid shown in Fig. i constitute what is termed a " galvanic
cell," or more shortly a " cell " ; and a number of such cells
Fig. ib. — Five simple cells Connected in Series.
connected together form a " battery." Fig. la shows dia-
grammatically a circuit containing one cell, and Figs, ib and ic,
a circuit with a battery of five cells in series.
The peculiar properties exhibited by an electric circuit, such as
described in Section 2, rapidly become less marked, and in a
short time practically disappear. If, however, the jar be divided
into two parts by a porous partition between the plates, and
copper sulphate be substituted for sulphuric acid in the compart-
ment containing the copper plate (Fig. 2), the length of time
during which the properties are appreciable is very greatly
increased. Such an arrangement is called a " constant " cell
because the effects it produces are
much more constant than those of
the cell previously described.
Many forms of " constant "
cells have been devised and are
now used in preference to simple—"
cells.
4. Conductors and Insulators. —
When a wire is joined to the
two plates c and z (Fig. 2), the wire and the space around it
are found to possess properties which they did not previously
possess. This fact is expressed by saying that " an electric cur-
rent is flowing through the wire," and the wire is spoken of as
being a " conductor " of electric current. If the wire be cut and
Fig. ic. — Diagram of five simple cells
Connected in Series.
PRACTICAL ELECTRICITY
Porous
joartitrioti
Fig. 2.— "Constant" Cell.
the cut ends be kept apart in the air, the
peculiar properties disappear ; we then say
the current " ceases," or is " stopped."
As an air space between the cut ends of the
wire stops the current, the air is said to be a
" non-conductor " of electric current. If
the two ends of the cut wire be pressed to-
gether, or against a plate of clean metal, or
against another piece of wire, or be dipped
into mercury, the current again " flows,"
but if glass or dry wood, silk or cotton, or
oil, replace the metal plate, wire, or mercury
respectively, the current does not flow.
We can therefore say that some materials
are conductors and others non-conductors.
Non-conductors are called " insulators,"
and wires covered with non-conducting
coatings are spoken of as " insulated wires."
5. Properties of an Electric Current. — These properties are : —
(1) A suspended magnet put in nearly any position near a
conductor through which an electric current is flowing will
be deflected, showing that a force is exerted on the magnet
(Fig. 3). This force is mutual, so that if a magnet be brought
near any substance traversed by an electrical current, this sub-
stance will generally be acted upon by a force tending to move
it (Fig. 4). Also any piece of soft iron put near a conductor
carrying a current will become magnetised (Fig. 5). The action
in all these cases is just as if the body conveying the current had
become magnetic. This is further shown by the fact that any
two wires through each of which a current of electricity is passing,
act upon each other with a magnetic force in nearly every position
in which the wires may be placed relatively to one another
(Fig. 6).
(2) If the circuit
through which an
electric current is
flowing be partly
solid and partly
liquid, then the
liquid will generally
be decomposed into
two parts, one part
going to one side of
the liquid in the pig. 3.— Magnet deflected by Conductor Carrying Current
PROPERTIES OF AN ELECTRIC CURRENT 5
direction in^which the current may
be said to be flowing, and the othei
part going to the other side of the
liquid in the opposite direction to the
flow of the current (Fig. 7) .
(3) The body conveying the current
becomes more or less heated (Fig. 8).
In popular language the current is
said :
(1) To deflect the magnet, and
magnetise the iron.
(2) To decompose the liquid.
(3) To heat the body through which
it is flowing.
But as we have no evidence of the
current apart from the conductor
through which it is said to flow, it is
more accurate to speak of a current
being said to flow through a con-
ductor in which these effects are
found to be produced, than to say
that the current produces these
effects. The latter expression, how-
ever, for brevity's sake, is generally
adopted ; and, indeed,
the heat generated in
a wire conveying a
current has so many
analogies with the heat
produced in a pipe by
the friction of a stream of water passing through
it, that we can frequently assist ourselves
by thinking of an electric current as
a stream of matter passing
through the wire as water
would pass through
a pipe filled with
sponge, or loosely
packed with sand.
But the analogy,
like many other
analogies, must not
be pressed too far,
. „ ., . Fig. 5. — Iron Rod Picking up Nails when a Current Flows,
especially as there IS through a Wire Coiled round it.
Fig. 4. — A piece of Tinsel Coiling itself round a Magnet
when a Current Flows through the Tinsel.
6 PRACTICAL ELECTRICITY
this very great difference between a current of water flowing
in a pipe and a current of electricity in a wire, viz. that
in the former case no effects are produced external to the pipe,
whereas in the latter the whole space surrounding the wire is
affected. For example, if an electric current is flowing through
a conductor, a compass needle brought within two or three
inches of it is deflected. But suppose not merely is there a
current of electricity flowing, but also a steady stream of water
Fig. 6. — Two Coils Standing on Narrow Bases Falling Down when a Current
Flows through them in Opposite Directions.
passing through the interior of the conductor, the conductor being
in reality a pipe. The water stream, however, is a perfectly
steady one, therefore it makes no sound ; and supposing the
water has been previously brought to the temperature of the
pipe, the presence of the water inside the pipe cannot be detected
by the pipe feeling hotter or colder to the touch ; consequently,
it would be extremely difficult to detect this stream of water
by any test made outside the pipe.*
The Magnetic, Chemical, and Heating effects of a current are
utilised practically in a number of electrical instruments and
processes ; for instance :
* A "compo " gas pipe answers very well for this experiment.
PROPERTIES OF AN ELECTRIC CURRENT 7
Fig. 7. — Tube ABC contains solution of Common
Salt with a drop of Hydrochloric Acid, and is
coloured red with Litmus. When a current
flows Chlorine is liberated in limb A, which
bleaches the liquid, while Caustic Soda is
formed in limb c, making the liquid dark blue.
the effect which always occurs
when a current flows.
6. Current Strength. — The
magnitude of the effects pro-
duced in and near an electric
circuit can be varied in several
ways. As a rule, these are more
pronounced when the number or
size of cells employed is in-
creased, and we say that the
* It is desirable to show in opera-
tion to students as many as possible
of the instruments enumerated under
the three heads, Magnetic Property,
Chemical Property, and Heating
Property, but at this early stage it is
only necessary to describe the instru-
ments in so far as their operation
illustrates the respective property of
the current.
Magnetic Property. — Needle
telegraph,* the Morse instru-
ment, telephones, electric bells,
arc lamps, dynamo machines,
electromotors, and, in fact,
all instruments using electro-
magnets.
Chemical Property. — Electro-
plating, electrotyping, the ex-
traction of aluminium and other
metals from their ores, the pro-
duction of sodium and chlorine
from salt, the manufacture of
pure copper, the cleansing of the
mercury used in separating gold
from sand, etc.
Heating Property. — Elec-
tric welding, electric heat-
ing and cooking apparatus,
electric lamps, contri-
vances for lighting gas or
oil lamps electrically, fuses
for torpedoes, etc.
The heating effect of the
current is, as we shall see,
Fig. 8.— Glow Lamp.
8 PRACTICAL ELECTRICITY
increased effect is due to a " stronger " current, or to a current
of " greater strength." An electric current is thus said to be
strong or weak according as the magnitudes of the effects it pro-
duces under given conditions are large or small ; in other words
we take the amount of the effect produced as a sort of measure of
the current strength. As, however, the magnetic, chemical, and
heating effects of a current do not all alter in the same propor-
tions when the current is changed in any way, it is important to
consider which property should be taken as a direct measure of
the current.
In Section 5 we have stated that the production of heat always
accompanies the passage of a current, and it might seem that the
amount of heat produced in a given time ought to be taken as a
measure of the current. But in addition to the difficulty of
measuring the small amounts of heat produced by weak currents
the only way we have of measuring the amount of heat given to a
body is an indirect one, and consists in measuring its rise of
temperature by means of a thermometer. Further, a thermo-
meter measures merely rise of temperature, and not the amount
of heat, and as the rise of temperature depends on the mass and
nature of the material heated, and on the facilities for cooling
which exist, as well as on the strength of the current to be
measured, it is evident that this method of measurement is
neither simple nor convenient.
To ascertain which of the properties of a current can be
best employed for measuring its strength, an experiment may be
made with the following apparatus : —
A, B, c, D, E, F, G (Fig. 9) are instruments so arranged that the
same electric current will be sent through them all by the " bat-
tery," b b, on joining the wires P and Q. A is a " sulphuric acid
voltameter " consisting of two platinum plates dipping into
moderately dilute sulphuric acid in a vessel v, closed by an air-tight
stopper s. Through this stopper passes a glass tube, /, open at
both ends, with its lower end nearly touching the bottom of
v, and graduated at its upper part in fractions of a cubic centi-
metre or Cubic inch. B consists of two thin copper plates,
p p, partly immersed into a solution of copper sulphate (the blue
vitriol of commerce), and is called a "copper voltameter." c is
a coil of insulated wire with a magnet, m, suspended so as to turn
freely inside the coil, the whole arrangement forming what is
called a " galvanoscope." D is an " electromagnet " consisting
of a piece of iron of horse-shoe form round the ends of which are
coils of covered wire wound in opposite directions. E represents
an " electric fan " formed by metal blades mounted on the shaft
to PRACTICAL ELECTRICITY
of a small electromotor. F is a coil of bare wire immersed in
paraffin oil, the temperature of which can be measured by the
thermometer T, the arrangement being called a " calorimeter,"
and G is an electric lamp.
Connect the two wires P and Q, and attow the current to pass
for a convenient time through these seven pieces of apparatus,
then it will be found that : —
(1) The liquid has risen a distance d^ in the tube t of the vol-
tameter A, indicating that the passing of the current through the
liquid from one of the platinum plates to the other has caused
cl cubic inches of gas to be generated.
(2) One of the plates in the copper voltameter has increased
in weight by Wt grains.
(3) The magnetic " needle " m of the galvanoscope c has
all the time been kept deflected from its original position through
a number of degrees NJ.
(4) If at any time during the passage of the current the
" armature " a was placed carefully on the ends of the horse-shoe
electromagnet D it required a pull of w± Ibs., as measured by the
spring balance, to pull it off, when the handle at the top of the
apparatus was slowly turned.
(5) The fan will have made a certain number of revolutions Rr
(6) The mercury in the thermometer T of the calorimeter F
has risen through DJ
(7) The lamp G has been emitting light of a certain intensity
LI-
Next increase the strength of the current passing through the
apparatus, A, B, c, D, E, F, G, by increasing the number of cells
forming the battery b b, or in any other way, such as will be
described later on, and repeat the experiment for the same time
as before, then each of the effects previously observed with these
instruments will be increased, and instead of the results cv
Wlf NJ, wlt Rlf DJ, Lj, we shall obtain c2, W2, NJ, w2, R2, D£, L2.
But it will be found that the new values do not all bear the same
ratio to the corresponding old ones. For example, if c2 is twice
clt then N§ may be more or less than twice NJ, but will generally
be less than twice, while wz, R2, D£, and L2 may be found to be
much greater than twice wlf Rx, DJ, and Lx respectively. On the
other hand, if the strength of the second current be so chosen as
to make D£ exactly twice DJ, then generally it will be found that
w2 and L2 are rather more than twice wl and LI} while c2 and W2
are less than twice cx and Wx respectively. R2 may be either
less than or greater than twice Rx.
The needle m of the galvanoscope c, Fig. 9, is shown suspended
MEASURING EFFECTS OF A CURRENT 11
by a silk fibre, and the needle is deflected arid moves relatively to
the coil when the current passes. If the silk be replaced by a
fine wire fixed to the needle at its lower end and at its upper
end to a torsion head, the needle could be brought back to the
position it occupied when no current was flowing, by turning
the torsion nead, H Fig. 10, and thus make the relative positions
of needle and coil the same. The angle through which the torsion
head requires turning to bring the needle back, is a measure of the
moment of the couple exerted on the needle by the current-
carrying coil. Let these
angles be di and d\ in
the two experiments de-
scribed on page 10 ; we
shall then find that if c2
is twice clt then dz will be
twice dlf and W2 twice Wj.
If then we arbitrarily
define the strength of the
current as being directly
proportional to the gas
evolved in a given time
in the sulphuric acid
voltameter, we must con-
clude that if c2 is exactly
double Cj we have doubled
the current strength, that
the amount of copper
deposited is directly pro-
portional to the current,
and that the couple
exerted between a coil
and a magnet in fixed relative positions is directly pro-
portional to the strength of the current. But, on the
other hand, if we prefer to say that strength of current
is directly proportional to the angular deflection of the
needle m in the galvanoscope c, then we must conclude
that, as N£ is less than twice NJ, we have not quite doubled
the strength of the current ; whereas if we prefer to say
that current strength shall be regarded as proportional to the
force required to detach the armature a of the electromagnet
D, or, instead, proportional to the rise of temperature of the liquid
in the calorimeter F in a given time, or to the light given out by
the lamp G, then we must conclude that the strength of the
current has been more than doubled. Which of these is right
Fig. 10. — Torsion Galvanometer.
12 PRACTICAL ELECTRICITY
and which wrong ? So long as no one of the effects varies we
may be safe in concluding that the strength of the current is
constant, but if the different effects to which we have been re-
ferring vary from one time to another, then which of them shall
we take to represent by the magnitude of its variation the
change that has taken place in the current strength ?
In the case of measuring the velocity of a stream of water, or the
number of gallons of water per minute discharged by a river, no
two experimenters could differ much. One of them, by the employ-
ment of better constructed measuring instruments, or it may be
from having greater experience in making such measurements,
might get answers slightly different from, and more accurate than,
those obtained by the other experimenter. But they could not
have such totally different conceptions of what should be meant by
the velocity of the water in a particular part of the channel, or
of the total discharge in gallons per minute, that the results
obtained by one observer were, apart from all mere errors of
experiments, twice as great as those obtained by the other.
And this is because they would be dealing with the actual flow
of a material substance — water.
The flow of an electric current, however, being merely a con-
ventional method of expressing the fact that a conductor has
acquired certain properties that it does not usually possess, there
is no question of right or wrong, but only one of convenience,
in selecting whichever we choose of the so-called properties of
the current as the one we arbitrarily decide to employ as the
measure of the current strength.
7. The Strength of an Electric Current : by which of its Pro-
perties shall it be Directly Measured? — To assist us in deciding
whether the amount of the magnetic action, or of the chemical
action, or the amount of heat produced in a given time, shall
be arbitrarily taken as that magnitude to which the current
strength shall be defined as being directly proportional, we ob-
serve that of the seven pieces of apparatus A, B, c, D, E, F, G
employed in the previous experiment, A and B are the only two
which give results that steadily increase in the same proportion
when the current is increased ; but if c were replaced by a zero-
torsion instrument (Fig. 10) the results obtained with a third
piece of apparatus would increase in the same proportion as in
A and B. Consequently, while on the one hand, our estimate
of the relative strength of two currents would be quite different
according as we selected the angular deflection of the magnet m
(Fig. 9), or the force of detachment of the armature a to be the
direct measure of the current strength ; on the other hand we
DEFINING STRENGTH OF A CURRENT 13
should arrive at practically the same estimate whether we chose
to say that the current was directly proportional to the rate of
production of gas in the sulphuric acid voltameter A, or to the
rate of deposition of copper in the copper voltameter B, or to the
couple exerted between a coil and a permanent magnet in fixed
relative positions.
But in addition to this agree-
ment between the relative amounts
of different chemical actions pro-
duced by two currents there is
another equally important fact,
viz., that the rate at which a -par-
ticular chemical effect is produced
by a current is practically in*
dependent oj the size and shape of
the apparatus. Thus, suppose we
have two sulphuric acid volta-
meters, the platinum plates being
of totally different shapes and
sizes (Fig. n) ; two
copper voltameters
also of different
shapes and sizes
(Fig. 12), the copper
plates, for example,
being much larger,
and, either much
nearer together, or
much farther apart1
in the one than in
the other ; also two
galvanoscopes (Fig.
13) , which may look
very much like one another, but the bobbin of the instrument to
the right is wound with a few turns of thick wire, while that of the
other galvanoscope to the left is wound with many turns of
fine wire ; two electromagnets (Fig. 14), which differ from one
another in the same sort of way as do the galvanoscopes, and
two calorimeters (Fig. 15), the two instruments in each case
being selected so as to be distinctly different in size and form.
Then, if an experiment be made with each pair of apparatus,
a certain current being sent through both sulphuric acid volta-
meters for a certain time, and a current, which may or may not
be of the same strength as the former, through both the copper
Fig. ii. — Two Sulphuric Acid Voltameters having
Platinum Plates of Different Sizes and at Different
Distances Apart.
14 PRACTICAL ELECTRICITY
voltameters, etc., the following results will be observed : In the
two sulphuric acid voltameters quantities of gas equal in volume
at the same pressure and temperature, and, therefore, possessing
Fig. 12. — Two Copper Voltameters having plates of Different Sizes
and at Different Distances Apart.
the same mass, will be developed in the same time, in spite of the
platinum plates being of a very different size and at a very
different distance apart in the two voltameters.* Similarly, in
spite of the difference in size and form in the two copper volta-
meters, the increase in weight of the plate of the one will be
practically the same as the increase in weight of the corresponding
Fig. 13. — Galvanoscope to the Left Wound with Many Turns of Fine Wire ;
Galvanoscope to the Right with a Few Turns of Thick Wire.
plate of the other, unless the current be so strong that the
deposited copper falls to the bottom instead of forming an
* Equality of pressure may be obtained by using for the voltameters
two vessels of the same size as well as two tubes of the same bore, and
filling the vessels with the same quantity of dilute sulphuric acid of the
same specific gravity. In that case, if the level of the liquid in the two
tubes be the same to start with, the liquids will be found to rise at exactly
the same rate in them on the same current being sent through the two
voltameters.
DEFINING STRENGTH OF A CURRENT 15
adherent deposit. But in the case of the^two galvanoscopes, the
two electromagnets, and the two calorimeters, although the
current passing through the two apparatus in any one pair is the
same, the effects depend on the shape, on the size, and on very
many 'details in the arrangement, etc. Hence, to specify the
strength of a current by the magnitude of the deflection of the
needle of a galvanoscope, it would be necessary to state the
exact mode of constructing each part of the galvanoscope in
great detail, as well as the exact position of the instrument
relatively to neighbouring magnetic pieces of iron. Whereas,
to specify the strength of a current by the amount of gas pro-
Fig. 14. — Electromagnet to the Left Wound with Many Turns of Fine Wire;
Electromagnet to the Right with a Few Turns of Thick Wire.
duced in a given time in a sulphuric acid voltameter, or by the
amount of copper deposited in a given time on one of the plates
of a copper voltameter, neither the shape nor size of the plates,
nor the distance between them, need be taken into account within
wide limits.
In both the voltameters it is chemical decomposition that
takes place — in the former, this decomposition being the splitting
up of the liquid into gases ; in the latter, the splitting up of the
copper sulphate, and the deposit of copper on one of the copper
plates, together with a loss of an equal weight of the metal of the
other copper plate to give back to the solution the amount of
copper taken out of it. In c and D (Fig. 9) the effects produced
are both magnetic, but we have found that N£ does not bear to
NI the same ratio that w+ bears to w ; whereas in the case of the
16 PRACTICAL ELECTRICITY
voltameters we always find that c2 bears to c± almost exactly the
same ratio that W2 bears to Wt. Consequently, so far as we
have seen at present, the amount of chemical action produced
in a given time by a current appears to be a more direct measure
of its strength than the magnitude of some of the magnetic
effects produced, and is also proportional to the magnetic effect
between a coil and magnet in fixed relative positions.
Fig. 15. — Thermometer to the Left Surrounded with Many Turns of Fine
Wire ; Thermometer to the Right with a Few Turns of Thick Wire.
Let us examine this point still further. In Fig. 9 all the ap-
paratus is joined up " in series " — that is to say, the current
passing through any one instrument passes through every other.
But in Fig. 16 C2 and C3 are two sulphuric acid voltameters
" in parallel," and not in series, with one another. For the
current which comes along the wire W1 and passes through sul-
phuric acid voltameter Cj divides into two portions, one of which
passes through C2 and the other through C3 ; the two portions then
recombine and flow away together by the wire W2. Also from the
DEFINING STRENGTH OF A CURRENT 17
construction of the apparatus it will be seen that the rise of liquid
in the tube T1 measures the production of gas in the voltameter
C1} while the rise of liquid in the tube T2 measures the sum of
the quantities of gas produced in the voltameters C2 and C3
together. Now, experiment shows that, if precautions similar
to those referred to in the note on page 14 for using the apparatus
in Fig. ii be taken, the liquid rises at exactly the same rate in
the tube TX that it does in the tube T2. Consequently the rate of
production of gas in ct is equal
to the sum of the rates of pro-
duction in C2 and C3 together.
Further, whether clf C2, and
C3 be all sulphuric acid volta-
meters, or all copper volta-
meters, or all silver voltameters,
or, indeed, all voltameters of the
same character, it will be found
that, no matter what be the
shapes or sizes of the different
voltameters, and no matter
what be the areas of the
platinum, copper, or silver
plates immersed in the respec-
tive liquids, or the distances
apart of the plates, the amount
of chemical action produced in
a given time in Cj is
almost exactly equal to
the sum of the amounts
of chemical action pro-
duced in c« and
together. The plates,
in any one of the
voltameters, clf may
be large or small, near together or far apart — may be, in fact,
moved about while the chemical action is going on. The
current may be strong and the chemical action take
place rapidly, or it may be weak and the action proceed
slowly, and it may be varied while the action is progressing ;
but the same general result still remains true. Measure the
amount of chemical action that has taken place in C2 and m C3,
add the two together, and it Vill be found to be practically equal
to the action that has taken place in ct in the same time.
Now, when a river divides in consequence of the existence of
Fig. 16. — Voltameters c, and c, in Parallel with One
Another, but in Series with Voltameter c,.
i8 PRACTICAL ELECTRICITY
an island in mid-stream, we know that the number of gallons of
water flowing per minute on the two sides of the island must
together equal the total number of gallons per minute flowing in
the main stream, simply because the water which does not go
past one side of the island must go past the other ; and similarly,
if we are to look upon a current of electricity in the same way as
a current of water, we must expect that, when it divides into two
parts, the sum of these parts must always be equal to the whole,
whether the current which divides is a large one or a small one.
The experiment just described shows that, if we say that a current
is directly proportional to the rate at which chemical action is
produced in a voltameter, this statement will always be true,
whatever be the current in the main circuit ; but it will not
generally be true if we take any of the other effects occurring in
the instruments indicated in Fig. 9 (page 9) as a direct measure
of a current. Thus, in Fig. 16, if clf C2, C3 represent galvano-
scopes, such as c in Fig. 9, the deflection of the first will not
generally be equal to the sum of the deflections of the other two ;
and even if this were the case for one current in the main circuit,
it would not be the case for any other. Nor will any simple
relation be found to connect the deflection of the first instrument
with those of the other two, unless elaborate precautions be
taken in the construction of the apparatus.
8. Definition of the Unit Current ; Ampere. — We may therefore
define the strength of a current as being proportional to the amount
of chemical decomposition it can produce in a given time ; and
an unvarying current which, when passed through a solution of
nitrate of silver in water, deposits silver*' at the rate 0/0*00111800
of a gramme per second, is taken as the unit of current and called
one " international ampere," or, more shortly, one ampere.
The reason why the number crooiiiSoo is chosen is as follows:—
An ampere was originally defined as " one-tenthf of a C.G.S. (centi-
metre gramme second) unit of current." Now the C.G.S. unit of current
is defined as that current which, flowing through a conductor of i centimetre
length placed along the circumference of a circle of I centimetre radius, exerts
a force of i dyne% on unit magnetic pole at the centre.^
* Silver is used because it gives a heavier deposit than other metals,
and does not oxidise readily. The deposit can therefore be weighed with
greater accuracy.
f The ampere was taken as ^ of a C.G.S. unit because at the time the
" practical " system was adopted, the C.G.S. unit was considered too large
tor practical purposes.
J A dyne is a force approximately equal to ^|T of the weight of a gramme
in London.
§ Another way of expressing this, and one more easily realised ex-
perimentally, is that current which, flowing through a circular conductor of
i centimetre radius, exerts a force of 2-0- (6-283) dynes, on unit pole at the
centre ; for the circumference of a circle of unit radius is 2?r.
THE UNIT CURRENT; THE AMPERE 19
To understand this definition we must know what is meant by " unit
magnetic pole." When a bar magnet is dipped into iron filings, the filings
adhere to the magnet, especially towards its ends, and if the magnet is
long and thin, they are attracted only near the ends. The magnetism
thus appears to be concentrated towards the ends of the magnet, and these
ends are called " poles." If a pole of one magnet be brought near one
of the poles of another magnet, a force is exerted between them. Suppose
two long thin magnets exactly alike, and placed so that one pole of one
is near one pole of the other magnet, and that the other two poles are as
far apart as possible, then the force between the magnets will be due
almost entirely to the mutual action of the adjacent poles. If the magnets
be such that the poles exert a force of one dyne on each other when at a
distance of one centimetre apart, they are said to be " unit poles," or
" poles of unit strength." A unit pole is therefore one that exerts a force
of one dyne on an equal pole, when the two poles are one centimetre apart,
and a current which exerts a force of one-tenth of a dyne on such a pole,
under the conditions stated in the definition of the c.G.s. unit (page 18)
is found by experiment to deposit silver from a solution of silver nitrate
at the rate of 1-118 milligrammes per second.
We may here add that the direction of the force between two poles is
found by experiment to be in the line joining the poles, and that of the
force exerted by a current in a short conductor on a magnetic pole is at
right angles to the plane containing the conductor and the pole.
When the distance between two magnetic poles is changed, experiment
shows (see Sect. 250) that the magnitude of the force between them varies
inversely as the square of the distance, i.e., if the distance be doubled, the
force is reduced to one-quarter of its previous value. A similar law
between force and distance exists in the case of a short conductor carrying
a current and a magnetic pole.
5
Fig. 17. — Silver Voltameter for Measuring Currents of about One Ampere.
The metal deposited by the current does not adhere well to
the plate of a voltameter or " electrolytic cell" if the action pro-
ceeds too rapidly ; also errors will arise in the estimation of a
current by the electrolytic method, unless certain precautions
be carefully attended to. Thus, when measuring a current of
about one ampere with a silver voltameter, it is advisable to adopt
the following arrangement : — The " cathode," (sometimes spelt
" kathode") or plate on which the silver is deposited, should
take the form of a light bowl K (Fig. 17), not less than
20
PRACTICAL ELECTRICITY
10 centimetres* in diameter, and from 4 to 5 centimetres in depth,
and made of platinum, so that it may be easily cleaned with
nitric acid. The " anode," or plate from which the silver is
electrically removed, should be a disc of pure silver, A, of about
30 square centimetres in area, and from 2 to 3 millimetres thick.
Riveted to the anode is a strip of pure silver s, and by means of
the metal clamp c and nut N the anode is supported centrally
within the cathode bowl with its upper surface just below the
level of the liquid. The liquid usually employed is a neutral
solution of pure silver nitrate, containing about 15 parts by weight
of the salt to 85 parts of distilled water.
Fig. 1 8. — Desiccator used with the Silver Voltameter.
Electric contact is made between the wire W1 and the bowl by
means of three metal pins p, on which the bowl rests ; and the wire
W2 is electrically joined to the anode disc by the strip s being
held fast in the rnetal clamp c, to which the wire W2 is attached.
In addition to the surface of the anode plate being turned into
silver nitrate by the passage of the current, there is a tendency for
small bits of silver to become detached and to fall into the bowl,
thus making its weight too great. To prevent this, the anode
may be wrapped round with pure filter paper, secured at the
back with sealing-wax as shown at A, Fig. 17;
* One metre is 39-370 inches, therefore 10 centimetres correspond
with a little less than 4 inches. One square metre is 1,550 square inches
therefore 30 square centimetres is a little less than 4! square inches.
ELECTROCHEMICAL EQUIVALENTS 21
When making an observation, the current should be allowed to
pass for about half an hour, and be maintained as constant as
possible. A full description of the method of making a measure-
ment is given in Appendix I., page 490.
To obtain a uniform adherent deposit of silver, it is
desirable that the cathode should possess about 30 square
centimetres of surface for every ampere passing. Hence, if a
large current of several hundred amperes had to be measured
by means of a silver voltameter, the apparatus would necessarily
be large and costly. In the voltametric measurements of large
currents, therefore, it is usual to replace the platinum bowl and
the silver disc by copper plates, and the solution of silver nitrate
by one of acidulated copper sulphate.
The chief reasons for using the rate of deposition of silver in the
practical definition of the " international ampere " are (i)
currents can be measured with precision by the silver volta-
meter in any civilised country at moderate cost, the quantities
to be determined, viz., mass and time being susceptible of very
accurate measurement ; (2) the amount of deposit is indepen-
dent of the value of gravity, of temperature, of humidity, and
of atmospheric pressure to a very high degree.
Although the primary definition of the ampere is based on the
magnetic property of electric currents and the C.G.S. system of
mechanical units, its precise realisation necessitates the con-
struction of very accurate and costly instruments, a knowledge
of the strength of pole or moment of a magnet, and also of the
acceleration of gravity at the spot where the experiment is
carried out. Another method of determining the ampere is based
on the forces which exist between coils carrying currents, and a
very exa'ct measurement has been carried out by one of the
authors and Mr. F. E. Smith, of the N. P. L., using an apparatus
designed in 1898-99 at the Central Technical College (now the
City and Guilds (Engineering) College). Experiments of this
nature have of necessity to be made to realise the ampere as based
on the C.G.S. unit of current strength, and the results are usually
expressed in terms of the amount of silver deposited per second.
9. Electrolysis ; Electrochemical Equivalent. — If a number of
voltameters containing, for example, solutions of silver nitrate,
copper sulphate, zinc sulphate, etc., respectively, be placed
in series, and a current be sent through them for a certain time,
the weights of the metals deposited on the cathodes of the
respective voltameters, or the weights of the other constituents of
the respective salts set free at the anodes, are very approxi-
mately proportional to the chemical equivalents. Thus since
22 PRACTICAL ELECTRICITY
the atomic weights of silver, copper, and zinc are respectively
107-88, 63-57, and 65-37,* and, since silver is monatomic while
copper and zinc are diatomic, it follows that, as an ampere is the
current that deposits 0-001118 gramme of silver per second, the
weights of copper and zinc that will be deposited per second per
ampere are respectively about
— X ^ X 0-001118, or 0-00032.94 gramme, f
2 107°**
and
- X — 5_3Z x 0-001118, or 0-0003387 gramme.
The first quantitative experiments on " electrolysis" the name
given to electric decomposition, were carried out by Faraday in
1833, and although he found that the proportion of the weights
of different substances liberated by a given current flowing for
a certain time differed sometimes by as much as 2 per cent, from
the ratio of their chemical equivalents, he attributed this to
inaccuracy in his experiments. He, therefore, concluded that
the " electrochemical equivalents " of substances were directly
proportional to their chemical equivalents.
Among the many investigations that have been conducted for
comparing the rates of deposit of copper and silver the most com-
plete is probably that carried out by Prof. T. Gray. He found
that the amount of copper deposited per second per ampere
varied slightly with the size of the cathode and the temperature
of the copper sulphate bath. If, however, the anode and cathode
plates have each an area of about 50 square centimetres per
ampere passing, and if the solution in the bath be formed by
dissolving pure copper sulphate in distilled water until the
density becomes 1-18, and afterwards adding about i per cent, of
sulphuric acid, the weight of copper deposited per second per
ampere is very approximately 0-0003286 gramme, and is but little
affected by temperature. The difference between 0-0003286
and the theoretical value 0-0003294 arises mainly from the fact
that copper plates lose weight when immersed in acidulated solu-
tion of copper sulphate. To allow for this the experimental value
has been used in the calculations which follow.
It will be observed that the weight of silver deposited per
second per ampere in a silver voltameter is nearly four times as
great as the weight of copper deposited in a copper voltameter.
* These are " International Atomic Weights " (1909), based on that of
oxygen being 16-00 ; the atomic weight of hydrogen on this basis being
I -008.
\ See pp. 22 and 25 for experimental value 0-0003286.
ELECTROCHEMICAL EQUIVALENTS 23
This reason would alone render the silver voltameter much to
be preferred for the measurement of small currents, but for large
currents the cost of silver is excessive> so copper is employed.
A current of one ampere, when passed through a solution of
dilute sulphuric acid, decomposes about 0-00009334 gramme of
the liquid per second, The acid in the voltameter may be con-
veniently diluted with water until the specific gravity of the
mixture is about i»i, which corresponds with a mixture of about
15 per cent, by weight of pure sulphuric acid at 15° C.
The volume of mixed gas (oxygen and hydrogen) that is
produced per second by the decomposition, corresponding with
a current of one ampere, is about 0-1734 cubic centimetre, when
the temperature at which the gas is evolved is o° Centigrade, and
the atmospheric pressure that of 76 centimetres of mercury.
When the temperature is T° Centigrade, and the height of the
barometer h centimetres, the volume of gas evolved by one
ampere in one second is approximately —
0-1734 X 76 X (273+7°) ,.
— i^_j_ -- v /J — L cubic centimetres.
AX 273
Example i. — How many amperes would deposit 5 grammes of
copper in half an hour, the current being supposed constant ?
As 0-0003286 gramme is deposited in I second by I ampere,
5 grammes are deposited in i second by
- - — — amperes.
0-0003286
Hence 5 grammes are deposited in 30 X 60 seconds by
amperes.
0-0003286x30x60
Answer. — About 8-453 amperes.
Example 2. — How many grammes of copper would be deposited
by a steady current of 40 amperes acting for 5 hours ?
i ampere acting for i second deposits 0-0003286 gramme,
therefore 40 amperes acting for 60 X 60 X 5 seconds deposit
0-0003286 X 40 X 60 X 60 X 5 grammes.
Answer. — About 236-6 grammes.
Example 3. — How many amperes would deposit 9 grammes
of copper in 2\ hours, the current being constant ?
Answer. — About 3-043 amperes.
Example 4. — How many grammes of copper would be deposited
by a steady current of 1-5 amperes acting for 16 seconds ?
Answer. — About 0-007886 gramme.
24 PRACTICAL ELECTRICITY
Example 5. — How many grammes of dilute sulphuric acid
would be decomposed by a steady current of 12 amperes acting
for one hour ? Answer. — -About 4*032 grammes.
Example 6.- — A current is passed through two voltameters
in succession, one having silver plates and the other copper.
After the current has ceased a deposit of 2-03 grammes of silver
is found in the former voltameter ; how much copper has been
deposited in the latter ? Answer.— 0*597 gramme.
Example 7. — If the mixed gas produced in a sulphuric acid
voltameter beat 20° C., and the barometer stand at 77-5 centi-
metres, what volume of gas would be produced in half a minute
by a steady current of 18 amperes ?
i ampere in i second produces about
Q-I734X76X (273+20) cubic centimetres of
77-5X273
therefore 18 amperes in 30 seconds produce about
0-1734 X 76 X 293 X 18 X 30
cubic centimetres of gas.
77-5X273
Answer. — About 98-5 cubic centimetres of gas.
Example 8. — If the temperature of the mixed gas in a sulphuric
acid voltameter be 19° -5 C., and the height of the barometer 75
centimetres, what current would produce 50 cubic centimetres
of mixed gas in one minute ? Answer. — About 4-43 amperes.
Example 9. — A silver voltameter and a copper voltameter
are arranged like C2, C3, in Fig. 16, so that the main current divides
between them. A steady current of 3 amperes is kept flowing
in the main circuit for one hour, and it is then found that the
deposit of copper in the copper voltameter is 0-4 gramme. What
is the deposit of silver in the other voltameter ?
Answer. — About 10-71 grammes.
io. Definition of Unit Quantity of Electricity : Coulomb. — In
the preceding section we have seen that the amount of chemical
decomposition is proportional to the strength of the current,
and to the time the current flows ; it is therefore proportional
to the product of the current strength /* and the time t. A
similar rule holds in the case of the flow of water or gas, the
amount carried depending on the current and the time, current
being considered as the velocity of flow multiplied by the area of
the channel or pipe. The product of current (of water or gas)
* The letter / has been adopted internationally as the symbol for current
strength.
THE UNIT QUANTITY; THE COULOMB 25
and time is called the " quantity " of liquid or gas, and may
be expressed in gallons or cubic feet or other convenient units.
In the same way the product of electric current and time is
called " electric quantity," and when the current is one ampere
and the time one second, the quantity conveyed is called " one
coulomb." We may therefore define a coulomb as the quantity
of electricity conveyed by a current of one ampere flowing for one
second. For any other values of / and t we have
coulombs = amperes X time in seconds,
= It.
There is another unit of quantity in commercial use, viz.
the " ampere hour "* and as I hour is 3,600 seconds,
i ampere hour =3, 600 coulombs.
As the amount of chemical decomposition in a voltameter is
proportional to the current and the time, it is proportional to the
quantity of electricity which passes through the voltameter, and
we may express the electrochemical equivalents of substances
as so many grammes, or milligrammes, per coulomb .
For example we have : —
Electrochemical equivalent of silver = 1-118 mgs. per coulomb.
copper = 0-3286
„ „ zinc = 0-3387
water = 0-09334 „
hydrogen = 0-01044 „
oxygen = 0-08290 „
Voltameters are. in reality coulomb -meters, as the amount
of chemical decomposition depends on the number of coulombs
of electricity passed through them. Special forms of voltameters
are frequently employed by Electric Lighting Companies as
house meters, to register the quantity of electricity the " con-
sumer " has allowed to pass through his lamps. (See Sect. 128.)
Example 10. — How many coulombs pass through the volta-
meters mentioned in example I ? state also the quantity in am-
pere hours,
1st method :
Quantity in coulombs = current in amperes X time in seconds.
= 8-453 X number of seconds in half an hour.
= 8-453x30x60.
Answer = 15,215 (approximately).
• An ampere hour is the quantity of electricity conveyed by a current
of one ampere flowing for one hour.
26 PRACTICAL ELECTRICITY
Quantity in ampere hours = am peresx hours.
= 8-453 Xj.
Answer = 4-226 (approximately).
2nd method :
Massdeposited=numberof coulombs X electrochemical equivalent.
/. 5=number of coulombs X 0-0003286.
.'. Number of coulombs = 5-^-0-0003286.
= 15,215 (approximately).
Ampere hours = number of coulombs -=-3, 600.
_ 15,215
3,600
= 4-226 (approximately).
Example n. — Express the quantities of electricity used in
examples 2 to 8 in coulombs and ampere hours.
Answers to Example n. Coulombs. Ampere hours.
No. 2 720,000 20O
No. 3 . . „ . . . 27,389 7-608
No. 4 24 0-006
No. 5 43,200 '12
No. 6 • 1,816 0-504
No. 7 540 0-15
No. 8 . . . . . . > 266 0-0772.
ii. Definition of the Direction of the Current : Ions.— The next
thing to define is the direction of the current, which, as already
explained, can only be done in a conventional way. In the case of
a sulphuric acid voltameter, we have hitherto only spoken of the
total quantity of gas given off at both platinum plates, but if
these gases be collected in separate tubes, as can very conveniently
be done in the Hoffmann's voltameter (Fig. 19), then it is found
that at one of the plates P oxygen gas o is given off and at the
other plate hydrogen H is liberated, and the current is said to
travel through the liquid towards the plate at which the
hydrogen is given off, or, in other words, the current flows
through the liquid with the hydrogen. Hence in the Hoffmann's
voltameter, shown in Fig. 19, the current would be said to flow
through the liquid in the short horizontal tube, from right to left.
The gases are evolved exactly in the proportions in which they
have to be combined together to form water — viz., two (or more
accurately 2-002 at 15° C.) volumes of hydrogen and one of
oxygen.* So that the electrolytic action effected by sending a
* That the gases are hydrogen and oxygen can be proved by the fact
that on turning the stop-cocks s, s, the one gas H when lighted will burn
with a pale blue flame, and the other o will ignite a glowing piece of wood.
DEFINING DIRECTION OF A CURRENT 27
current from one platinum plate to andther in dilute sulphuric
acid is exactly the same as if the water had simply been decom-
posed.
If an acid, a silver, a copper, and a zinc voltameter be all
joined together, so that the same current passes through them,
then it will be found that the hydrogen in the first, the silver
in the second, the copper in the third, and the zinc in the fourth,
all travel in the same direction in the circuit ; so that if through
the liquid in an acid voltameter the current
be said to go in the direction in which the
hydrogen travels, then through the liquids in
a silver, a copper, and a zinc voltameter,
it must be said to go in the direction in which
the silver, the copper, and the zinc travel.
Or generally the current in a voltameter may be
said to travel with the metal from the anode
towards the cathode, hydrogen behaving in this
respect, and, as is well known, in other
respects, like a metal.
The components into which an " electrolyte "
is decomposed by the passage of a current
are called " ions," and the ion which travels
with the current is called the " electropositive
ion," while the one which travels against the
current is called the " electronegative ion"
Other names for these ions are cation and
anion, meaning the ions which appear at the
cathode and anode respectively.
With the definition given above of the
direction of a current, we find that if a com-
pass needle, n s (Fig. 20), be pivoted so as to
turn in a plane at right angles to the plane
of the paper, and a current flow along any
wire, A B c D, which is in the plane of the
paper, then the north-seeking end* of the compass needle will
* The " north-seeking " end oi a magnet is the one that points towards
the geographical north. The simple expression " north " end is confusing,
since in England it refers generally to the end of a magnet that points to the
north, while in France it refers to the end that points to the south, the
French using that definition because that end is attracted by the earth's
magnetism situated in the southern hemisphere, and the unlike ends
attract one another. Calling the ends of magnets " red " and " blue " is
equally confusing, as some people use one of these two colours, and others
the other colour, to indicate the. same end. As, however, the north-
seeking end of a magnet is usually marked by instrument makers with a
scratch or a cut, it would probably be best to call the " •north-seeking "
28 PRACTICAL ELECTRICITY
come towards the observer if the current flow round the wire in the
direction indicated by the continuous arrow — that is, counter-
clockwise ; whereas the south-seeking end of the needle will come
towards the observer if the current flow in the direction of the dotted
arrow— that is, clockwise.
Similarly, if A B (Fig. 21) be any bit of wire in the plane oi the
paper, the north-seeking end of the needle (n, say) will come
towards the observer if the current
flow along this bit of wire, A B, in
such a direction that A B may be
regarded as forming part of a
counter-clockwise circuit round
the needle.
Therefore, in the upper three of
the illustrations of Fig. 22, the
end n will come towards the
Flg- 20* observer, while in the lower three
it will be the end s that will come out towards the observer.
Or, again, if a wire conveying a current be coiled round a piece
of iron shown end-on to an observer, then the end of the iron
nearest him will act as the north-seeking end of a magnet when the
current appears to the observer to flow round the wire in a counter-
clockwise direction. If the observer now look at the other end of
the bar, he will of course see the
south-seeking end, and in his new
position the current will now appear
to him to flow round the wire in
the same direction as that in
which the hands of a clock go (or
clockwise). ^ The relative magnetic
polarity of the iron bar and the
direction of the current, as indicated
by the arrows, are shown in Fig. 23. Flg* 2I-
The magnetic polarity of the end of an iron bar round which
a current is flowing does not depend on whether the current is
flowing from the left to the right-hand end of the bar, as in the
first of Fig. 23, or from the right to the left-hand end, as in the
last of Fig. 23 ; but merely on the direction the current flows
round the bar. Now, in spite of the difference of the winding of
the wire on the first and last of Fig. 23, it will be found that in
both cases, if the bar be looked at end-on from the right, the
and "south-seeking" ends of a magnet the "marked end" and "unmarked
end" respectively. In this work where the words "north end" or "north
pole" are used they are to be understood to mean "north-seeking.
DIRECTION OF A CURRENT
29
Fig. 22.
current is clock-
wise, whereas if
the bar be looked
at end-on from
the left the cur-
rent is counter-
clockwise.
Perhaps the
simplest method
for remembering
the connection between the magnetic polarity of an iron bar and
the direction in which a current circulates round it is that, if a
current circulates round the bar in the direction in which the
handle of a corkscrew (Fig. 24) is turned when the corkscrew is
screwed down or up, the point of the screw will move towards
the north-seeking magnetic end of the iron bar.
Example 12. — A compass needle is supported under a telegraph-
wire running north and south. How will the needle deflect
if a strong current flow through the wire towards the south ?
Answer. — The north-seeking end of the needle will turn towards
the east.
Example 13. — A flat vertical conductor is fastened against a
wall, and in front
is suspended a mag-
netic needle pivoted
so as to turn on a
vertical plane par-
allel to the wall.
The north - seeking
end of the needle is
weighted so that the
needle stands ver-
tically when no
current is flowing.
Which way must a
current flow in the
conductor to make
the upper end of
the needle point to
the right ?
A nswer. — Down-
wards.
Example 14. —
Fig. 23. Draw an arrow on
3o PRACTICAL ELECTRICITY
the movable card of a compass, so that when the compass is
placed above a horizontal conductor conveying a strong current
the arrow will indicate the direction of the current.
Answer. —
12. Objection to the Usual Mode of Constructing Voltameters. —
The sulphuric acid voltameters, as usually pictured in books,
which are still the only forms obtainable at some shops, are
extremely unsuitable for practi-
cal use, as it is troublesome, after
the tubes in which the gas is
collected are full of gas, to fill
them with liquid again for a new
experiment.* The apparatus
shown in Fig. 19, page 27, is very
convenient when it is required to
collect the oxygen and hydrogen
separately, but it has the incon-
venience that, the platinum
plates being small and far apart,
it requires the employment of
several galvanic cells to make
the gas come off quickly. For,
although the quantity of gas pro-
duced in a given time by the
same current is practically in-
dependent of the shape and size
of the plates, the ease with which
this current can be generated
depends very materially on the
size of the plates and their dis-
tance apart, and if we wish to
* The improved forms of volta-
meters described in Section 13 have
been adopted by many instrument-
makers since the first appearance of
this book.
FORMS OF ACID VOLTAMETERS
produce chemical decomposition quickly, we ought to have the
plates large and very near together, and the liquid employed ought
to contain something like 33 per cent, of strong sulphuric acid by
weight, the mixture having a specific gravity of about 1-25 at
15° C.* Such a mixture conducts electricity more readily than
solutions of other strengths.
13. Description of Practical Forms of Sulphuric Acid Volta-
meters.— In Fig. 25 is shown a very convenient form of voltameter,
designed by Prof. Ayrton, consisting of a glass vessel closed at
the top with an indiarubber stopper I and containing moderately
dilute sulphuric acid. The two
platinum plates p are held to-
gether by indiarubber bands,
but prevented from touching one
another by small pieces of glass
tubing put between the plates
at the top and bottom, or to save
the expense of thick platinum
plates, two pieces of thin plat-
inum foil may be used, stuck at
the bottom with bicycle or other
suitable cement, to a piece of
glass tube, the weight of which
causes the two pieces of foil to
hang vertically, and therefore at
the same distance apart all the
way down. Wires coated with
gutta-percha to prevent their
being corroded by acid being
spilt over them, or better still, platinum wires go from
the plates, one to the " key " K (which is raised up
above the general level of the apparatus to prevent its being
corroded by drops of acid), and the other wire to one of the
terminal binding screws seen in the figure. On pressing down
K, the current produced by a generator attached by wires to the
two binding screws, seen at the right-hand side of the figure, is
allowed to pass through the apparatus. The gas which is
generated is unable to escape from the vessel when the pinch-cock
c is closed, and accordingly forces the liquid up the graduated
tube t. This tube passes air-tight through the indiarubber
stopper I, reaches nearly to the bottom of the vessel, and termin-
ates at the upper end in a thistle funnel F, so that if the current
is by accident kept on for a longer time than is necessary to cause
• See Appendix IV,
Fig. 25. — Ayrton's Form of Sulphuric Acid
Voltameter.
PRACTICAL ELECTRICITY
the liquid to rise to the top of the graduated tube, the liquid
collects in the funnel instead of spilling over. This tube is also
sloped so that the rise of liquid in the tube may increase the
pressure of the gas in the upper part of the voltameter as little
as possible.* The second tube might be simply terminated with
a piece of indiarubber tubing closed with a spring pinch-cock, c, on
opening which the gas is allowed to escape, and the liquid runs
back out of the tube /. If this is done suddenly, however, there
is a tendency for small particles of the liquid to be jerked out
of the lower tube. To prevent these particles being thrown on
to the stand of the apparatus, the tube is carried up, and its end
bent over into the thistle funnel F.
Instead of observing the distance the liquid travels up the
graduated tube t (Fig. 25) in a given time, we may notice the time
it takes to travel from a certain fixed mark at one end of the
tube to another fixed mark at the other. In other words, instead
of measuring the volume of gas produced in a given time, we may
measure the time taken to produce a
given volume. And since for different
currents the times taken for the same
volume of gas to be produced must
be inversely as the volumes of gas
produced in the same time, we can
deduce the current by employing a
tube which has not been subdivided
into equal volumes, but only has two
marks on it. With this method of
( using a voltameter to measure
^currents there is no necessity for
the tube to be long, since it can be
Fie. 26.— Mather's Form of Sulphuric conveniently expanded into a bulb
B (Fig. 26), and great sensibility can
be combined with compactness by the bore of the tube being
made small at the places where the reference marks m2 and
ml above and below the bulb, are made. The wires pass through
the indiarubber stopper inside glass tubes to ensure that all
the current passes through the liquid.
The spring pinch-cock should not be left squeezing the
indiarubber tube of the voltameter (Figs. 25 and 26) when the
instrument is out of use, for continued pressure on the sides of the
* If the vessel be full of liquid so that there is no gas between the top
of the liquid and the indiarubber stopper I at the commencement of the
experiment, the error arising from the compression of the gas produced
by the rise of liquid in the tube t may be neglected.
SULPHURIC ACID VOLTAMETERS
33
tube causes it to acquire a permanent set and prevents it from
regaining its circular form when the pinch-cock is removed.
Another form of voltameter, devised by J. A. McMichael, Esq., is
shown in Fig. z6a. Connection with the platinum plates is made
through wires sealed into glass tubes containing mercury, which
are seen projecting just above the top of the rubber stopper. The
measuring tube on the right is graduated in ampere-minutes, and
by passing a current for a period of one minute through the
voltameter its strength can be read off directly in amperes.
14. Relative Advantages of Voltameters and Galvanometers. —
One great advantage that voltameters possess over galvano-
meters is that a given current produces the same rate of chemical
decomposition at any place on the earth's surface, this rate being
quite independent of the force of gravity, or of the earth's mag-
netism, both of which differ in intensity at different places. The
indications of galvanometers, and of most other forms of current
measurers, are influenced by gravitational or magnetic forces,
and so do not possess the same immunity from local conditions
as voltameters. For these reasons the electrochemical definition
of the ampere is now employed for international purposes.
The disadvantages of employing a volta-
meter for the practical measurement of
currents are (i), that it requires a strong
current to produce any visible decomposition
in a reasonable time ; and (2), that a measure-
ment of time is necessary. Even the current
of one ampere, which is about six times that
used in an ordinary 8-candle incandescent
lamp, would require nearly three hours to
decompose one gramme of dilute sulphuric
acid, whereas the weak currents used in
telegraphy, and, still more, the far weaker
currents used in testing the insulating char-
acter of specimens of gutta-percha, india-
rubber, etc., might pass for many days
through a sulphuric acid voltameter without
causing any noticeable amount of chemical
decomposition. Indeed, not to mention the
enormous waste of time, and the difficulty
of keeping the current strength which
it was desired to measure constant
all this time, the leakage of the gas Fig. 26«.— McMichaei's Form
, . , „ , of Acid Voltameter grad-
Which WOuld take place at all parts OI uated in ampere-minutes,
the apparatus that were not hermetically 1 a "
34 PRACTICAL ELECTRICITY
sealed,* would render such a mode of testing quite futile.
Hence, although the voltametric method is a fairly direct
way of measuring a current strength, and is one of the
most accurate ways of determining the strength of current;
exceeding a few tenths of an ampere, that can be kept
constant for half an hour or so, still the very fact that the
amount of chemical decomposition produced in a given time by
a certain current is independent of the shape or size of the instru-
ment, makes it very difficult to increase its sensibility Con-
sequently some other apparatus must be employed for practically
measuring small currents, and the law of the apparatus — that is,
the connection between the real strength of the current and the
effect produced in the apparatus — must either be experimentally
ascertained by direct comparison with a voltameter, or an
instrument constructed so that the current can be calculated
from its dimensions and the C.G.S. unit of current strength denned
in Section 8. When the law of the apparatus has been found,
it is said to be " calibrated."
But if we are going to compare together the indications of
two instruments produced by various currents, the second in-
strument cannot be much more sensitive than the first; what
advantage, therefore, can arise from employing an instrument as
unsensitive as a voltameter ? This leads us to the fact that it
is very much more difficult to increase the sensitiveness of volta-
meters than of " galvanometers."^ We might increase the
magnitude of the indications of a voltameter, such as that shown
in Fig. 25, by using a tube / of very small bore, or by putting
several such voltameters in series, and collecting the gases given
off by each into one vessel ; but we cannot by either of these
means succeed in constructing a voltameter which possesses
anything like the sensibility that can be very easily given to a
galvanometer.
The indications of any measuring instrument may be increased
in three distinct ways. As an illustration, let us consider an
ordinary spring-balance, like the one attached to the apparatus
D in Fig. 9, page 9. We may, in the first place, use a microscope,
or we may fit the balance with a wheel and pinion, or employ
* A glass vessel is said to be hermetically sealed when any opening
that previously existed in it has been closed by heating the glass round
the opening until it becomes soft and sticky, and pressing the edges together.
f While a " galvanoscope " is the name given to an instrument used for
ascertaining whether a current is flowing, or merely which of two currents
is the stronger, a " galvanometer " is the name given to an instrument by
means of which the relative strengths of currents can be compared. Any
galvanoscope when so calibrated becomes a more or less sensitive galvano-
meter.
VOLTAMETERS AND GALVANOMETERS 35
some other magnifying arrangement to render the extension of
the spring more apparent; or the electromagnet may be so
constructed, either by employing more iron or by putting more
convolutions of wire round its limbs, so that the pull on the
" keeper " or armature a (Fig. 9), caused by passing a given
current round the coils of wire, is increased ; or, lastly, we may
use a weak spring in the balance, so that, for a given pull on the
keeper, the movement of the index may be large.
Each of these three methods can be applied with great success
to galvanometers. In the first place, the sensitiveness may be
increased by using a long pointer, and the pointer may be made
light, and therefore easily moved, by forming it of a very fine
glass tube, or of a narrow strip of some light substance like alu-
minium. But the best of all methods, and therefore the one
employed with very sensitive galvanometers, consists in using
a ray of light several feet long, but, of course, quite weightless,
reflected from a small mirror attached to the needle, thus
making what is called a " reflecting galvanometer." The sensi-
bility of a galvanometer can also be made large by winding the
bobbins with very many turns of very fine wire (see Sect. 35) ;
also by placing the bobbins very near the suspended needle.
Friction can be diminished by suspending the little magnet with
a thin fibre of untwisted silk. And lastly, by employing a very
weak " controlling magnet " or by putting it at some distance
from the galvanometer, the " torque"* required to turn the needle
can be reduced to a very small amount, and therefore a consider-
able deflection can be produced by an extremely weak current.
And so successful have been the various attempts to increase
the indications of galvanometers that it is now possible to measure
accurately an electric current which is so small that it would
have to flow for a million years through a voltameter before it
produced as much chemical action as a current of one ampere
could produce in a single hour.
Now, experiment shows that a galvanometer of a particular
shape and size, and with a definite magnetic needle, acted on by
a definite controlling force, produced, say, by the earth's magnetism,
or by some fixed permanent magnet, has a perfectly definite law
connecting the magnitude of the deflection with the strength of the
current producing it, although the absolute value of the current
in amperes necessary to produce any particular deflection can be
increased, or diminished, by using thick wire and few turns,
* Torque is the tendency that any system of forces has to cause a body to
turn, so that torque bears the same relation to turning that a force has to
motion in a straight line.
36 PRACTICAL ELECTRICITY
or fine wire and more turns, to make a coil of the same dimensions.
If, for example, with a particular gauge of wire employed to
fill up the bobbin it requires 2f times as many amperes to produce
a deflection of 40° as it requires to produce a deflection of 20°,
then if a much finer gauge of wire be employed to fill the bobbin
there will still be required 2| times as many amperes to produce
a deflection of 40° as are required to produce a deflection of 20°.
But in the second case y^^ of an ampere may be all that is
required to produce the 20° deflection, whereas five amperes may
be required to produce the same deflection in the first. The
law of the instrument remains the same, although its sensibility
has been increased 5,000 times by using finer wire to wind on
the bobbin.
Thus, while we may take advantage of the absolute character
of the amount of chemical action to furnish us with our " standard
current meter," we can avail ourselves of the variation that can
easily be made in the deflection of a galvanometer needle cor-
responding with the same current, to furnish us with instruments
of greater and greater degrees of delicacy.
15. Measurement of Current by Galvanometers : Tangent
Galvanometers. — As galvanometers are capable of being used
over such a wide range of current strength, it is advisable to
study them somewhat closely at the present stage. A very
useful form of instrument is shown in Fig. 27, page 37, consisting
of a small magnetic needle n s suspended by a fibre of unspun
silk at the centre of a comparatively large, circular coil of wire.
A thin glass pointer, pp, attached to the needle, moves over a
graduated scale, which is fixed to a disc of looking-glass to avoid
" parallax " in reading the deflections. When no current passes
through the coil, the magnet n s behaves like a compass needle
and sets itself in a direction nearly north and south, and if turned
in any other direction by the finger or other means, will promptly
return to the north and south position when freed. The force
(or rather torque) which is operative in causing this return to the
zero position, is called the " controlling force," and is usually
due to the earth's magnetism ; in such cases the needle is said to
be controlled by the earth's magnetic field. The space in the vicinity
of any magnet where a force would be exerted on a magnetic
pole if such were present, is spoken of as the magnetic field of
the magnet, and the field is said to be strong or weak according
as the force exerted on a unit pole is large or small. In fact,
the strength of a magnetic field at any point in space, is, by defini-
tion, measured by the force in dynes exerted on a unit magnetic pole
placed at that point. The sense of the field is taken as that in
TANGENT GALVANOMETERS
37
which a north-seeking pole placed at trie point would tend to
move, and may be found practically by a small compass needle
with its centre at the point considered. Over considerable
distances the earth's field (where undisturbed by masses of
iron, or other magnets, or by electric currents), is uniform in
strength and direction, and urges the north-seeking pole of a
magnet northwards and its south-seeking pole southwards.
G
B
Fig. 27. — Tangent Galvanometer ;
the smaller diagrams A, B and C show various modes of supporting the Fibre.
With A the needle can be moved sideways by sliding the roller R in the spring clips s, s, and can
be raised or lowered by turning this roller. With B the pin p is held by a single screw s
instead of between two brass plates, as shown in the complete galvanometer. With C the
pin P is held by a set screw s in a support made with a ball top B. This fits in the hole h in the
plate i and forms a ball and socket, so that the needle can be accurately centred. The ball
and socket joint is clamped to the semicircular support A with the screw s.
In most places on the earth's surface, the direction of the earth's
magnetic field is neither horizontal nor vertical, its true direction
being shown by a magnetic instrument called the " dipping
needle." The angle between a horizontal line in the magnetic
meridian* at any place, and the position taken up by the axis of
a dipping needle, is called the " angle of dip " at that place. For
* The vertical plane in which a freely suspended magnet sets itself
at any place, is called the magnetic meridian at that place.
38 PRACTICAL ELECTRICITY
galvanometric purposes it is customary to consider the earth's
magnetic force resolved into two components, the horizontal
component and the vertical component respectively, and the
former is frequently employed as the controlling field in galvano-
meters.*
The magnitude of the earth's horizontal component is different
at different places on the earth's surface, and also changes daily
and alters from year to year. At any one place, however, the
daily and yearly changes are comparatively small. For the
year 1918 its mean value in London was approximately 0-1846,
the average yearly change and daily variations amounting to
about i and 2 parts in 1000 respectively, so that for many pur-
poses we may take the horizontal component of the earth's
field in undisturbed areas as approximately constant.
If, therefore, an instrument such as that shown in Fig. 27 be
placed in an undisturbed area, then when a current passes
through the coil the needle will be influenced by two magnetic
fields, one due to the earth's horizontal field and the other due
to the current, the field of which is at right angles to the plane
of the coil (see Fig. 37). The needle will be deflected, and take
up a position of equilibrium along the direction of the resultant
field, the position of which can be found by the parallelogram of
forces Let NQ (Fig. 27^) represent in magnitude and direction the
earth's controlling field, and NP the field at n (Fig. 27), due to the
current in the coil, then the diagonal NR of the parallelogram NQRP
represents in magnitude and direction the resultant of the two
fields, and the magnet n s will set itself in this direction. The
current causes the magnet to move through the angle QNR,
from its zero position, and this is called the angle of deflection.
If the lines NQ and NP are perpendicular, i.e., if the needle n s
lies in the plane of the coil when there is no current passing, then
we have Np = QR = NQ tan QNR
Now NP represents the deflecting field, which is, by definition,
proportional to the strength of the current, and may be written
N p = k /, where k is a constant and / the current in amperes,
and N Q represents the horizontal component of the earth's
magnetic field (usually denoted by H). We may therefore
write the equation as
k I == H tan d, where d = angle QNR.
or 7 - — tan d, (i)
k
* In some cases, permanent magnets are used to produce controlling fields
stronger or weaker than those due to the earth alone.
RELATIVE AND ABSOLUTE CALIBRATION 39
and since H and k are constants, we sefr that the current is pro-
portional to the tangent of the deflection in the instrument used as
described. For this reason galvanometers having large plane
coils and small needles are called Tangent Galvanometers.
From what has been said above it will be seen that the magnetic
needle n s will set itself along the direction of the resultant NR,
whether the needle be a strong magnet or a weak one ; in other
words, the deflection of a galvanometer needle controlled by a con-
stant magnetic field is independent of the strength of the deflected
magnet. This is true of any form of galvanometer having a
moving needle, magnetically controlled. It is, however, desirable
to use strongly magnetised needles, for
by doing so the forces operative are in-
creased, thus diminishing any error
that may be introduced by torsion of
the suspending 'fibre, or by friction at
the pivots ; and the motions of the
pointer are quickened.
16. Meaning of the Relative and the
Absolute Calibration of a Galvanometer.
— Two distinct things are required to
be known with reference to a particular
galvanometer — first, the law connecting Fig. 27*
the various deflections with the relative
strength of the currents required to produce them ; secondly,
the absolute values of the currents — that is, the number of
amperes required for the same purpose — or, what is sufficient if
the first has been ascertained, the number of amperes required to
produce some one deflection. The first is sometimes called the
" relative calibration," the second the " absolute calibration "of
the galvanometer.
A galvanometer with its bobbin wound with thick wire may be
compared directly with a voltameter, and the relative calibration
of the galvanometer determined ; then if the same space on the
bobbin be wound with any other gauge of wire the relative calibra-
tion of the galvanometer will be the same, and therefore known,
provided that nothing but the winding has been altered. Or if
a galvanometer wound with thick wire be compared with a
voltameter, and its absolute calibration determined, and if,
further, the law of change of sensibility with gauge of wire has
also been ascertained experimentally 'then the absolute calibration
of the same galvanometer, when wound with any gauge of wire,
filling the same space, will be known without further experiments,
provided that only the winding has been changed.
40 PRACTICAL ELECTRICITY
17. Comparison of Tangent Galvanometer with a Voltameter. —
The apparatus shown in Fig. 28 can be used to show by experi-
ment that the tangent of the deflection of a galvanometer
of the form shown, is proportional to the rate at which chemical
decomposition occurs in a voltameter in the same circuit ; and
in this way we may prove that the electromagnetic definition of
current strength is relatively consistent with the chemical de-
finition. A voltameter, v, is connected up with the galvanometer,
G, and a " set of resistances," each consisting of a coil of wire with
its ends connected with two successive terminals, tlt t2, etc.
These coils are wound on bobbins, and are placed underneath
the base board to which the whole of the apparatus is fixed,
and by means of which it can be bodily carried from place to
Fig. 28. — Comparison of Tangent Galvanometer with a Voltameter.
place ; (from the laboratory to the lecture-room, for example, for
demonstration to a class). The magnitude of the current is
altered by joining the wire, w, to the various terminals, tlf t2, t3,
etc., on the base board. T, x are the main terminals, or binding
screws, to which the wires coming from the current generator
are attached.
It may be noticed that in the particular experiment shown in
Fig. 28 it is quite unnecessary to know the length or gauge of the
wire that has been wound on the various bobbins ; nor is it at all
necessary that all the coils should be made of the same length or
thickness of wire, since, whatever resistance be inserted in the
circuit, the current that passes through the voltameter is the same
as the current that passes through the galvanometer, so that the
variation in strength of the current is known from the voltameter
observations, and not from the length of wire that has been
introduced into the circuit. Indeed, the resistances in this
experiment may be dispensed with altogether when there is any
GALVANOMETER AND VOLTAMETER 41
other easy mode of altering the current Strength by using, for
example, different numbers of " cells " or a different kind of
battery to produce the current, but in practice this result is
generally most easily attained by the use of a set of resistance
coils.
The comparison might be performed by observing for a
number of different currents, the rise of the liquid in the
graduated tube of a voltameter such as that shown in Fig.
25, in a given time, and the corresponding steady deflec-
tion of the needle, or of the pointer, of the galvanometer. But
more accurate observations can be made if, instead of observing
the different lengths of the tube through which the liquid rises
in a given time corresponding with the different currents used,
the times be noted during which the liquid rises through a given
volume — viz., that between the two marks mlt m2, of the volta-
meter tube (Figs. 26, 28). A calculation can then be made of
the rate at which gas is evolved by the current, and from this the
strength of the current in amperes can be found. Thus, let v
be the volume, in cubic centimetres, of the bulb B (Figs. 26, *
28) between the marks mlt w2, and suppose the strength of the
current be such that it takes t seconds for the liquid to rise from
the mark ml to the mark m2, then the number of cubic centimetres
of gas generated by the current in every second is
v.
— cubic centimetres.
t
An actual experiment carried out by first-year students at the
City Guilds College, gave the results tabulated below, the volume
of the bulb being 9^6 cubic centimetres : —
TABLE I.
Observed quantities.
Deduced from observed quantities.
Time in
seconds (t)
to fill bulb.
Steady de-
flection (d) of
galvanometer
in degrees.
Cubic cms
of gas per
second.
Tangent of
deflection.
Ratio,
cubic cm
tan d.
v
0-187*
(A mpere) .
573
260-5
177
122
*3
15
30-3
40-4
50-9
61
0-0168
0-0369
0-0542
0-0787
0-116
0-268
0-584
0-851
I-230
1-804
0-0625
0-0631
0-0637
0-0640
0-0641
0-090
0-197
0-290
0'42I
0*619
From column 5 it will be seen that the rate of production of gas
(i.e., the relative current strength) bears a nearly constant ratio
42 PRACTICAL ELECTRICITY
to the tangent of deflection of the galvanometer, the greatest
variation from the mean being ij per cent., and this is within the
possible error of experiment. It is therefore evident that the
voltameter and tangent galvanometer are in substantial agree-
ment with each other.
Columns 2 and 3 of the table give a relative calibration of
the galvanometer. As the numbers in column 3 are proportional
to -, (being equal to — ,) it is evident that column 2 and a column
giving the values of — would also constitute a relative calibration
of the instrument, and for this purpose the volume of the bulb
need not be known.
18. Absolute Calibration of Tangent Galvanometer. — The above
experiment also enables the relation between the current strength
in amperes, and the deflection of the galvanometer (or the
tangent of its deflection) to be determined, and in this way
gives the absolute calibration of the instrument. For example,
one ampere liberates in a sulphuric acid voltameter 0-1734 cubic
centimetre of gas per second, at standard temperature and
pressure, and if we suppose the actual temperature to be 18° C.
and the height of the barometer 750 millimetres, it follows
from the expression given in Section 9 that the volume liberated
per ampere second under these conditions is
4- 18
0-1734 XX±-= 0-I87.
Hence the current is — - — amperes.
0-187^
Calculating out the values of — - — for each observation in the
0-187*
table, we obtain the strength of current in amperes used in each
experiment. These are tabulated in column 6. Columns 2 and 6,
therefore, give the absolute calibration of the galvanometer.
Another important quantity relating to the particular galvano-
meter as used in the experiment, can be deduced from the above
TT
table, viz., the value of — in equation (i), Sect. 15. This quantity
fv
is often called the constant of the tangent galvanometer, under the
conditions in which it was used, and is such that the current in
amperes producing any deflection equals the constant of the instrument
multiplied by the tangent of the deflection. The same thing may
CALIBRATION OF GALVANOMETERS 43
be expressed by saying that the " constant " of a tangent gal-
vanometer is equal to the current in amperes which will produce
a deflection of 45°, since tan 45° =i. In the case under considera-
tion the constant is 0-351, so that the instrument could now
be used to measure strength of currents in amperes ; the con-
trolling field H remaining constant.
19. Calibrating any Galvanometer by Direct Comparison with
a Tangent Galvanometer. — Knowing the law connecting the
deflections of a tangent galvanometer and the currents producing
them, viz., current :: tan of deflection, we may now use this
instrument as a " standard " with which to compare other gal-
vanometers, and thus determine the relation between deflections
and current strength in cases where the construction of the
instrument makes it impossible to predict the relative calibration.
To do this the instrument to be calibrated is connected up in the
same circuit as the tangent galvanometer, so that the same
current passes through both, and simultaneous readings of the
two instruments taken. The current is then changed to another
value and a second pair of readings observed ; this is done for
several other different currents, such as will give deflections
distributed over the whole scale of the instrument to be calibrated.
Apparatus for making an experiment of this kind is shown in
Fig. z8a, where a rough and portable galvanometer, D (sometimes
called a "detector"), is coupled in circuit with a tangent gal-
vanometer, G. The current is varied by sliding two zinc rods in
or out of a V-shaped tube containing a solution of zinc sulphate.
Table II. gives the results obtained in such a calibration, and
constitutes a " relative calibration table " for the detector D.
TABLE II.
Deflection of galvano-
meter to be calibrated.
Deflection of Standard
G al vanometer .
Tangent of deflection
of Standard Galvano-
meter (or relative
current strength).
o-o degrees
3-8 „
10-0
15-0
20-5
36-5 »
50-0
65-0
79'°
O'O degrees
21-3 „
30-1
387 „
42-3
52-2
55-8 „
61-6
697
O-OO
o-39
0-58
0-8o
O'QI
1-29
1-47
1-85
270
44 PRACTICAL ELECTRICITY
The method of calibrating a galvanometer described above is
suitable for use when the standard instrument and the one to be
calibrated are approximately equal in sensitiveness. Cases,
however, frequently occur where one galvanometer gives a large
deflection for a current which produces only a small deflection on
the other one. When this happens part of the current may be
diverted from the coil of the more sensitive instrument by con-
necting a wire from one of its terminals to the other, and the length
of this wire may be adjusted so that the two galvanometers give
about equal deflections. The galvanometer whose terminals are
so connected by wire external to the instrument is said to be
" shunted," the external wire being called " a shunt" If the
shunt be arranged so as to produce no magnetic effect on the
Fig. z8a. — Calibrating a Detector by Comparison with a Tangent Galvanometer.
needle of the instrument, and this is quite easy to do, the relative
calibration is not affected by the shunt, although the absolute cali-
bration may be much altered by it. For example, if the un-
shunted galvanometer requires I amperes to produce a deflection,
d°, the shunted instrument will require a larger current, /',
to give the same deflection, and the ratio of 7 ' to / will be larger
the shorter the length of wire (of given size and material), used
as the shunt.
20. Graphically Recording the Results of an Experiment. —
The results of the experiment given in the above table are best
recorded graphically by points on a sheet of squared paper.*
* Prior to the commencement of the courses at the Finsbury Technical
College, in 1879, squared paper was practically used in England only for
the recording of results of original experiments. And as these results,
rather than the training of the experimenter, were the most important
part of the investigation, the paper was very accurately divided, and sold
at a high price totally out of the reach of students. It became, therefore,
USE OF SQUARED PAPER
45
This has oeen done in Fig. 29, where the points a, b, c, d, e, f, and
g represent the numbers in the first and last columns of the table.
In plotting a " calibration curve " the distances of the points
a, b, c, etc., from the line o Y (Fig. 29) measured parallel to o x
necessary to have squared paper specially made0 cheap, and at the same
time sufficiently accurately divided for students' purposes ; and such paper,
machine-ruled, can now be obtained at less than a farthing per sheet, or at
about one-thirtieth of the cost of the older squared paper.
46 PRACTICAL ELECTRICITY
should be taken to represent the deflections of the galvanometer
calibrated, and the distances of the same points from o x, measured
parallel to o Y, the corresponding values of current strength.
It may be asked how distances along a line can represent the
angular deflections of a galvanometer, or the strengths of currents
producing such deflections. What is meant is this : the line
o x is subdivided into a number of equal divisions by the ruling
of the squared paper ; one, or any convenient number, of these
subdivisions is taken arbitrarily to stand for i°, then any de-
flection, say of d°, is represented by d times the number of
divisions arbitrarily chosen to stand for i°. Similarly one, or any
convenient number of divisions along o Y is taken arbitrarily
to stand for one unit of current strength, and n times this number
of divisions will represent a current strength of n times the
unit chosen. In Fig. 29 one division along o X stands for i°
deflection of the galvanometer calibrated, and 10 divisions along
o Y is taken to represent unit current strength, the current which
produces a deflection of 45° on the tangent galvanometer being
taken as unity. Curves representing other quantities may be
drawn in the same way. For example, the height of the baro-
meter from hour to hour, the variation of the price of some
commodity from day to day, or the depth of water from point
to point along some section of a river, can be readily shown on
squared paper, and generally a kind of picture illustrating how
one thing varies relative to another may be given by such curves.
In selecting the scale to which a curve is to be plotted, that
is, determining what number of divisions along o X or along o Y
should be taken to stand for i° deflection, or for unit current
strength respectively, we should be guided by the consideration
that the resulting curve should represent the experimental num-
bers quite as correctly as, or preferably, rather more correctly
than, the accuracy to which the readings of the instruments were
taken. For example, if the deflections of the galvanometer
to be calibrated can be read, say, to J of i degree, the scale along
o x should be such that the length representing J° can be seen on
the squared paper quite easily, in order that there may be no
difficulty in reading the curve to J°. Similar considerations
will show what scale along o Y should be chosen. Care should
be taken to select convenient scales so that the numbers may be
plotted easily, and when plotted, be easily read.
Squared paper is usually divided decimally, that is, the
cardinal lines are spaced 10 divisions apart ; it is therefore
desirable to choose the scale, where possible, so that one division
represents one unit of the quantity to be plotted, or 10 units, or
CALIBRATION CURVES 47
a tenth of a unit, or some decimal multiple or sub-multiple of
10. This cannot always be done, and scales of two or five
divisions per unit, or two or five units per division, or decimal
multiples, or sub-multiples of these may then be used without
much inconvenience ; and the aim in choosing scales should be
to make the curves as easy to read as possible, consistent with
their being read with the necessary precision.
The larger the scales to which a curve is plotted the more
accurately it can be read, but in many cases the use of a very
large scale is objectionable, for where this enables the points
to be plotted with a far greater accuracy than was attained in
making the observations, the large scale merely magnifies the
errors of observation, and points which, if plotted to a reasonable
scale, would lie in proximity to a curve, appear to be dotted
about in an irregular manner like the stars of a constellation.
Having chosen suitable scales and plotted a sufficient number
of points from the experimental results, a curve, as regular as
possible, should be drawn through the points. This can be done
by bending an elastic strip of wood so as to pass as nearly as
possible through all the points plotted and using this as a ruler
to draw the curve. Unless the experiment has been performed
with great accuracy — to attain which requires care and practice —
it must not be expected that a curve so drawn will pass through
all the points. Some of them, such as b, Fig. 29, are sure to be a
little too low, meaning that either the deflection of the standard
galvanometer had been read too low, or that of the galvanometer
calibrated had been read too high. Other points, such as e, will
be too high on the paper, owing to errors of reading in the
opposite directions, or it may happen that the observations
corresponding to b and e have been correctly made, and mistakes
made in the plotting. To avoid errors of the latter kind all
plottings should be checked,.
21. Practical Value of Drawing Curves to Record Graphically
the Results of Experiments. — It may be asked, But is it not
possible that the points b and e, although not on the curve,
may be quite correct ? The answer is, No, because experience
makes us quite sure that the connection between the deflection
of the galvanometer G and the current strength must be a con-
tinuous one, and, therefore, that the points correctly representing
the true connection must all lie on an elastic curve, or on such a
curve as can be obtained by bending a thin piece of wood or steel,
and, consequently, that if no mistake has been made in plottmg
the points b and e, some mistake must have been made in taking
the observations. But what is even more important, we are also
48 PRACTICAL ELECTRICITY
sure that the points b' and e' on the curve, obtained by drawing
lines through b and e respectively parallel to OY, give far more
accurately the relative strengths of the currents producing
respectively the two deflections in question, than the currents
obtained directly from the experiment itself. Drawing the
curve, then, corrects the results obtained by the experiment. But it
does something more than that — it gives, by what is called " inter-
polation" the results that would have been obtained from inter-
mediate experiments correctly made ; that is to say, it tells us what
would be the relative strengths of the currents that would pro-
duce deflections intermediate between the deflections that were
actually observed. For example, suppose it be required to know
the strength of current which will produce a deflection of 41 J°,
for which deflection no experiment has been made, compared with
that which will produce a deflection of, say 28 J°, for which
deflection also no experiment has been made, then all that is
necessary is to draw a line parallel to OY*, through the point A
in ox corresponding with 41^°, similarly to draw a line parallel
to. OY, through the point B in ox, corresponding with 28J°,
and read off the lengths of the lines between ox and the points
p and Q, where they cut the curve, then the strength of the current
which produces the deflection 41^° on this particular galvano-
meter bears to the strength of the current that produces the de-
flection 28J° the ratio of the length AP to the length BQ.
If the curve is an absolute and not merely a relative calibration
curve, then the scale on which it is drawn will be known and
therefore the number of amperes corresponding with either AP
or BQ.
The method of plotting the results of experiment: on squared
paper, and drawing a curve through them to graphically record
the result, has a third important use in that, just as a map gives
a better idea of the shape of a country than pages of description,
a curve enables us to see at a glance the general character of the
result obtained.
For example, suppose that the results obtained in some particu-
lar calibration of a galvanometer are : —
Deflection. Relative Strength of Current.
10 24
I7-3 4i-5
22-8 547
29-5 . . . . . . . . 70-8
37-4 §97
* Or imagine such a line drawn.
USE OF SQUARED PAPER
49
no exact notion of the law of the galvanometer can be obtained
by a glance at these figures ; but if they be plotted on squared
paper a straight line passing through zero (Fig. 290) is obtained,
and from this we see at once that this particular galvanometer
has, in some way or other, been so constructed that the angular
deflection of the needle is directly proportional to the strength
of the current.
In the great majority of cases the angular deflection of the
needle of a galvanometer is not proportional to the current
strength, and a calibration curve is then needed to show the
connection between
them. After a little
experience the eye
becomes accus-
tomed to the peculi-
arities of curves,
and a glance at the
calibration curve is
then sufficient to
convey much infor-
mation about the
instrument to which
it refers. It is
always difficult for
anyone to grasp the
meaning of a table
of figures, even if it
be as simple as that
just given, but the
curve which repre-
sents them is much
more readily under-
stood, and its chief characteristics can also be more easily
remembered.
The curves are rendered more expressive if they are always
plotted so that the horizontal distances, or " abscisses," represent
the values of the thing easily observed — for example, the angular
deflections of the needle of a galvanometer, the hours of the day,
the days of the week, etc. ; and the vertical distances, or
" ordinates," represent the values of the variable quantity
which it is desired to record — for example, the relative
strengths of the current producing these observed deflections,
the heights of the barometer, the price in pounds of some
commodity, etc.
current.
g CO CO C
3 0 0 C
/
/
.
7
s60
/
/
/
/
r
S20
10
0
;
/
/
2j
10 20 30 40
Galvanometer Deflection,.
Fig. 2911.
50 PRACTICAL ELECTRICITY
It might at first sight appear to be a matter of indifference which
of the two quantities was plotted horizontally ; so also the
north on a map might be at the bottom or at either of the two
sides. But, just as convention has led to maps being always
drawn with the north at the top and with the east t« the right
hand, so by common agreement the values of the previously
unknown quantity are plotted vertically, and the values of the
quantity which is assumed to vary regularly in a known manner
are plotted horizontally. Hence
in graphically recording the tem-
perature at different hours of the
day, temperature is plotted verti-
cally and time horizontally ; or in
drawing a curve to indicate the
depth of the Atlantic at different
points between England and
America, depth is plotted verti-
cally, and distance measured,
from either England or America,
along the surface of the sea
Fig. 30. — Protractor used in subdividing •u~_',,~,,.4.~Ti.r,
a Galvanometer Scale. horizontally.
22. To Construct a Galvano-
meter Scale from which the Relative Strengths of Currents
can be at once Ascertained. — Galvanoscopes, and even cheap
galvanometers, are frequently constructed with scales divided
simply into degrees, so that it is generally impossible by the
mere inspection of the deflections produced by different
currents to determine the exact relative strengths of these
currents. If a calibration curve has been drawn from the results
of previous tests, the relative strengths of any currents can, of
course, be ascertained by using the curve to interpret the meaning
of the galvanometer deflections. Constant reference, however,
to a calibration curve or to a table of values, leads to much waste
of time, and therefore, when a galvanometer is to be used under
the same general conditions, it is better to construct a scale with
the graduation marks so drawn that the relative strengths of
currents are directly proportional to the deflections they produce
as measured by the numbers on this specially constructed
scale.
Such a scale may be made as follows : — Ascertain from the
calibration curve (Fig. 29) the angles in degrees measured along
o X, which correspond with currents whose relative strengths are,
say, o-i, 0-2, 0-3, 0-4, etc., and make a table of them as
follows : —
DIRECT READING SCALE
Relative current strength,
o-o
o-i
0-2
0-3 ..
0-4 .. .,
etc
etc.
Inflection in degrees,
o-o
0-7
i'3
2-5
4-0
etc
etc.
Then, by means of a protractor (Fig. 30), set off these angles
on a blank scale, making each 5th mark longer than the inter-
mediate ones, and numbering each 5th (say) as indicated in
Fig. 31, which shows the resulting scale. As the scale is intended
merely as a relative one, the relative current strengths may be
multiplied by 10, or by any other convenient number, to avoid
decimals on the scale. If more than 30 divisions were wanted
on the scale the spaces could be further subdivided by eye, or
3i. — Dkect Reading Scale.
the relative current strengths in Fig. 29 could be multiplied by
a number which would make the current strength corresponding
to the highest point of the scale to be used, equal to the number
expressing the number of divisions desired. If this multiplier
was not a convenient whole number it would be advisable to
plot a new curve, using as current strengths the original values
52 PRACTICAL ELECTRICITY
multiplied by the number in question, and then make a new
table for use in marking off the scale.
It will be noticed that the divisions on the scale (Fig. 31)
are crowded together at the beginning and end, and spaced
farther apart in the intermediate portions, and on referring to
the curve (Fig. 29) we see that the crowded portions of the
scale correspond with the steep parts of the curve, and the longer
divisions with the part having the least slope. The shape of the
calibration curve therefore enables us to see what would be the
character of a direct reading scale constructed from it.
Example 15. — From the calibration curve shown in Fig. 29
find the relative values of the currents required to produce
deflections of 5°, 30° and 70° respectively, taking the current
which gives a deflection of 5° as unity.
"Answer. — i : 2-39 : 4-5.
Example 16. — It is required to use currents 2, 3 and 4 times as
large as that which produces a deflection of 8° on the instrument
whose calibration curve is given in Fig. 29. What deflections
of the galvanometer correspond with these currents ?
Answer.— 35°, 63°, 75-5°.
CHAPTER II
MAGNETIC FIELDS
23. Magnetic Fields: Magnetometer — 24. Lines of Magnetic Force —
25. Strength of Magnetic Poles — 250. Hibbert's Magnetic Balance —
256. Balance for Finding Strength of Pole — 26. Magnetic Moment
— 27. Absolute Measurement of Magnetic Field and of Magnetic
Moment — 28. Mapping Magnetic Fields — 29. Comparing the Relative
Strengths of different parts of a Magnetic Field by the Vibration
Method — 30. Comparing the Relative Strengths of different parts
of a Magnetic Field by the Magnetometer Method, — 300, Difference
of Magnetic Potential ; Equipotential Surface.
Addendum to Chapter II. : Electric Lines of Force and Electrostatics,
23. Magnetic Fields. — As the action of most galvanometers
depends on the magnetic fields produced by the electric current
passing through their coils, it is desirable to study them in some
detail.
Two of the characteristics of magnetic fields at any selected
point are direction and strength. We have already mentioned
(see Section 15) that the direction of a field is determined by
that direction in which a small compass needle sets itself when
placed at the point in question, and also that the strength of a
field at a point is measured by the number of dynes exerted on a
unit magnetic pole placed at the point. Although it is possible
to measure the strength of a magnetic field in the way just stated,
this is by no means the most convenient way. Nearly all
measurements are most easily made by comparing the thing to be
measured with another thing of the same kind, this latter being
taken as the unit or standard.. For example, in measuring the
length of a room we compare its length with that of a foot rule
or something equivalent to this. The foot rule is thus taken as
the unit of length, and the ratio of the length of the room to that
of the foot rule, is the number expressing the length of the room
in feet.
To be of real use, the foot rule in the above example must be
of fixed length, it should not be made of material easily stretched
or compressed, or such as is greatly affected by atmospheric
temperature, pressure, or humidity. In other words, its length
53
54 PRACTICAL ELECTRICITY
must be constant. Similarly, in the case of magnetic fields, if we
have, or can produce, a field whose strength is constant, we may
take this field as our standard field and measure other fields by
comparing them with this standard field.
As previously stated in Section 15, the earth's magnetic field,
at a given point in an undisturbed area, is very nearly con-
stant, and for many measurements its horizontal component
is taken as a standard field, the strengths of other magnetic
fields being measured by comparison with it. Probably the
simplest way of making this comparison is to arrange that the
direction of the field to be measured is horizontal, and at right
angles to the earth's horizontal component, as was done in the
case of the tangent galvanometer (Sect. 15). The field to be
measured is thus superimposed on the earth's field at right angles
to the latter, and will deflect a small magnetic needle, placed
at the point at which the field is to be measured, from the magnetic
meridian through an angle depending on the strength of the super-
imposed field. If, therefore, we measure the deflection of the
small magnet which is caused by the superimposed field, we can
find its strength by the triangle of forces, for, as magnetic fields are
measured by the mechanical forces they exert on a unit magnetic
pole, all propositions relating to the composition of forces are
applicable to magnetic fields.
Let the line o H (Fig. 32) represent in direction and magnitude
the strength of the earth's hori-
zontal component (H), and o F,
that of the field (F) to be measured,
the angle H o F being a right angle,
then the diagonal o R will repre-
sent the resultant of o H and o F,
and from the above figure we
32. have
o F = H R, 1 .
= o H,tan a,%
or F = H tan a. (2)
In words we may say that when the superimposed field is
horizontal and at right angles to the earth's horizontal component,
the strength of the superimposed field is equal to the strength of
the earth's horizontal component multiplied by the tangent of
the deflection produced by the superimposed field.
It is not essential that the directions of the two fields to be
compared be at right angles to each other, but this relative
direction gives a simple formula, (2) above, and nearly maximum,
MEASURING MAGNETIC FIELDS
55
deflection. If the directions be inclined at any angle
obtuse (Figs. 33 and 330), we have
acute or
Sin HOR
sin HRO
sin HOR
(in both figs.)
sin ROF
sm a
or
F =
sin 08 - a)'
H sin a
sn
where HOR = a,
- «)' (3)
The method of measuring magnetic fields just described above,
is called the magnetometer method, and is of great convenience and
utility. In fact, in all galvanometers which have moving needles
controlled by a constant magnetic field and deflected by the
current to be measured, this method is made use of.
Any freely suspended magnetic needle arranged so that its
deflection can be observed may be used as a magnetometer. A
simple application of it is illustrated in Fig. 34, in which
the strength of field atji point along the axis of a magnet is being
measured. Fig. 35 shows an actual magnetometer in which the
magnet m is provided with a long pointer moving over a large
graduated circular scale, and a straight scale 140 centimetres
long, on which the distances of the ends of magnets from the
needle can be read off. By observing the deflections produced
by a given magnet M placed at several distances from the needle,
and plotting a curve between distances and tangents of the
corresponding deflection, the manner in which the strength of
56 PRACTICAL ELECTRICITY
field of the magnet varies with the distance along its axis can be
readily shown.
A reflecting magnetometer, by means of which more accurate
observations can be made, is shown in Fig. 350. The small
N
Fig. 34. — Principle of the Magnetometer.
magnetic needle, to which a mirror is attached, is suspended in
the support A and its deflections measured on the scale c. A
stand B supports M so that its axis produced passes through the
centre of the suspended needle.
24. Lines of Magnetic Force. — If a bar magnet be placed below
a sheet of glass and fine iron filings are sifted over the plate, and
the plate tapped, the filings will arrange themselves in a pattern
such as shown in Fig. 36. On inspection it will be seen that the
Fig. 35- — Magnetometer with Pointer.
filings set themselves in lines, and the direction of one of these
lines passing through a given point is approximately that in
which a very small compass needle would come to rest if placed
with its centre at the point considered. The lines, therefore,
show the direction of the magnetic force at various points in the
field of the magnet, and are consequently called " lines of magnetic
force," or more shortly, " lines of force." They are the lines along
which the magnetic force acts.
LINES OF MAGNETIC FORGE 57
Since a compass needle, or a magnetic pole, placed at any
point in the vicinity of the magnet producing the field, shown
in Fig. 36, would experience a force, a line of force may be said to
pass through this point, and as the number of such points is
Fig- 35«.— Reflecting Magnetometer.
infinite we might say that in a given magnetic field there exists
an infinite number of lines of force. It is therefore impossible to
draw a diagram representing all the lines of force of a magnet, and
even if such a diagram could be drawn it would tell us merely the
direction of the field at any and every point, but nothing about
the strength of the field at various points. A convenient conven-
tion, however, exists, according to which only a few of the lines of
Fig. 36.— Lines of Force with a Bar Magnet.
force are drawn, and tfre proximity of the lines in the vicinity of
any point indicates the strength of field at that point. For
example, the lines may be drawn so that the number passing
through an area of one square centimetre, normal to the lines of
force at any point, is numerically equal to the force which unit
magnetic pole would experience if placed at that point ; in this case
the density of lines (number per square centimetre) in the vicinity
of a given point would express tie strength of the field at that
58 PRACTICAL ELECTRICITY
point. Diagrams drawn according to this convention have a
quantitative, as well as a qualitative meaning, and are much
more instructive than charts showing direction only. Fig. 37
is a diagram of the magnetic field produced by a current passing
through a circle of round wire, c c c, made on this plan, whilst
Fig. 38 shows a diagram such as would be obtained by the iron
filings method. The former gives more information of value
than the latter. From Fig. 37 it will be seen that the lines are
very close together in proximity to the wire, especially near the
Fig* 37» — Lines of Force due to a Current in a Circular Coil (to scale).
inner circumference, whilst near the centre of the circle they are
farther apart and approximately parallel to the axis of the coil.
This shows that the field is strongest near the wire, and very
nearly uniform in the immediate vicinity of the centre. It will
also be noticed that all the lines of force which are completed in
the diagram form closed curves. This is an important property
of ah* lines of magnetic force.
In a uniform magnetic field the force exerted on a given mag-
netic pole is the same at every point of the field ; . and when such
a field is represented by lines drawn in the way described, the lines
are parallel to each other, and at equal distances apart.
If we consider the magnetic field produced by an isolated
LINES OF FORGE
59
pole of unit strength,* we know that the strength of field at a
distance of i centimetre from the pole is unity (for at this distance
the force exerted on another unit pole is I dyne), and is the same
at every point on a spherical surface of I centimetre radius
concentric with the pole. Such a field would be represented by
drawing radial lines from the pole, equally distributed all round
the pole, the number being such that the density of lines over
the surface of the sphere is one line per square centimetre of area.
As the surface of a sphere of unit radius is 4 TT, we can say that
the number of lines of force emanating from unit magnetic pole
is 4 TT, or 12-56 approximately.
25. Strength of Magnetic Poles. — We have already defined
a magnetic pole of unit strength (see Section 8) as one
* Although an isolated pole cannot be obtained, the pole of a very long,
thin, straight magnet is an approximation thereto.
6o PRACTICAL ELECTRICITY
that acts on an equal pole at a distance of one centimetre from
it with a force of one dyne. If one of the unit poles be re-
placed by another magnetic pole which exerted a force of two
dynes on the unit pole, such pole would be said to be of strength 2.
Similarly, if the other unit pole were now replaced by a pole
of strength 2 the force would again be doubled. We therefore
see that the force exerted between two poles at unit distance
apart is equal to the product of the strengths of the two poles.
This may be expressed thus : — the force between two magnetic
poles of strengths m and m' respectively, when placed at a distance
of one centimetre apart— m m' dynes.
If the distance between two given poles be altered, experiment
shows that the force between them varies inversely as the square
of the distance (the law of inverse squares), so we may write the
complete law of force between magnetic poles as :— force between
two magnetic poles of strength m and m' respectively, when placed
at a distance d centimetres apart
mm'
= —jr dynes- (4)-
This law is exactly the same as the Newtonian Law of
Gravitation. The same law also holds for the force between
two quantities or charges of electricity (see page 81), and for
this reason a magnetic pole of strength m is sometimes said to
possess, or have a charge of, m units of magnetism.
By assuming m' in the last equation to be unity, we see that
the force exerted on unit pole by a pole of strength m placed
at d centimetres away is
m
-p dynes,
and as the force on unit pole measures the strength of the
magnetic field it follows that an isolated pole of strength m pro-
m/
duces a field of strength — at a point d centimetres from the
pole.
WThen two poles acting on each other are of the same kind,
say both north-seeking, or both south-seeking, the force is found
to be one of repulsion, whilst if they are dissimilar poles they
attract each other. The two kinds of magnetism, therefore,
have opposite properties and may be spoken of as -f or —
magnetism respectively, north-seeking magnetism being con-
sidered -f. A north-seeking pole of strength m acts on a north-
seeking pole of m' strength with a force
LAWS OF MAGNETIC FORGE
-\-m mf m mr
~W ;
and that between two south-seeking poles is
61
whilst a north-seeking pole m and a south-seeking pole — m'
exert a force of
m (—m)' . —m m'
~^~ ~^r-
A positive sign for the force, therefore, is associated with a
repulsion, and a negative sign indicates an attraction.
39-— Hibbert's Magnetic Balance.
250. Hibbert's Magnetic Balance. — The law of inverse squares
may be proved in a simple manner by means of the Hibbert's
Magnetic Balance, shown in Fig. 39. In this instrument a long,
thin, magnetised steel rod s n is suspended so that it can swing
in a vertical plane, and balanced so as to rest horizontally when all
other magnets are far removed from it. Another magnet nf s' is
fixed horizontally to a slider s, capable of being moved up and
down, its position being indicated by the vertical scale shown.
When ri is placed aSove n a force of repulsion is exerted between
them, and the end n is depressed. By placing a small weight
on the lower magnet between o and s, and adjusting it, the hori-
zontal position of s n can be restored. When this has been done
approximately, the needle n' s' is moved in the direction of its
own axis into the position which produces maximum force
between n and nf. The weight is again adjusted until s n
is level, and the horizontal distance h of the weight from o
62
PRACTICAL ELECTRICITY
(which can be read off on the horizontal scale) is proportional to
the force exerted between n and n'. By bringing n' s' lower
down the distance between n and n' is decreased, the force
between them is increased, and to obtain balance the weight
has to be moved farther from the centre, say to Ar Calling
d and dl the distances between n and nr in the two cases described
above, the following relation will be found to hold, viz. :—
hd2 = h^j2 (approximately).
h d*2
"• £-*•
The results of an actual experiment carried out in the way
described above, are given in the following table : —
Values of d.
Values of h.
Values of hd *
9 cms.
8-5 cms.
688
10 „
6-86 „
686
II „
57 «
690
12 „
4'8 „
692
13 M
4'i »
693
14 M
3-55 »
694
From this table it will be seen that the law stated is very nearly
true. An exact agreement between the products h d2 is not to
be expected, since the force exerted between s and s' have not
been taken into account, and the distances h and d cannot be
measured very accurately.
256. Balance for Finding Strength of Pole.— The strength of
magnetic poles of long, thin magnets can be determined by a
balance such as is shown in Fig. 40. It is convenient to use
three magnets of similar shape for this purpose. Call them
I., II., and III. respectively, and the strengths of these poles
mlt m2, and mB. Place I. on the pan hanging from the beam,
and counterpoise it when the other magnets are far removed ;
then put II. vertically above or below I. with similar poles
adjacent to each other. Balance the instrument by moving the
rider R and call the equivalent weight placed at the end of the
beam c. Replace II. by III. and let the weight be denoted by b.
Next replace I. by II. and call the weight a.
Thus c = 2 m± m2 — d2 approximately,*
b = 2 m± m3 — d2,
a = 2 m2 m3 — d2,
* The equation is only approximately correct, because all forces except
those between adjacent poles have been neglected,
STRENGTH OF MAGNETIC POLES
from which we get
63
or
Similarly m2
c b = 4 m^ m2 m3 -f- d*,
d
fab
NIC
From the above it will be seen that if the strength of pole of one
magnet (say mj is already known, then a single observation
will enable m2 to be found.
Fig. 40. — Balance for Finding Strength of Magnets.
26. Magnetic Moment. — The two poles of any magnet are found
to possess equal quantities of magnetism, but of opposite sign ;
and if the magnet be short or be shaped so as to bring the two
poles very near together, the magnetic field produced at a distant
point by one pole is very nearly equal and opposite to that
produced by the other pole, and so the resultant field at the dis-
tant point is very small, in spite of the fact that the strength of
each pole may be considerable.
64
PRACTICAL ELECTRICITY
On the other hand, a magnet, whose poles are of the same
strength as those of the magnet considered above, but are
situated far apart, say near the ends of a fairly long, straight
bar, would produce at a distant point a magnetic field of strength
much greater than that given by the short magnet. The effect
produced by a magnet depends, therefore, not only on the
strength of its poles but also on the distance of those poles apart :
in fact the effect is proportional to the strength of the poles,
and also proportional to the distance between the poles. The
product of the strength of the poles of a magnet and the distance
between the poles is consequently an important magnitude, and
has been called the moment of the magnet, or more shortly, its
magnetic moment. Magnetic moment may be defined by the
equation M—m I, where m is the strength of each pole and /
the distance between the poles.
A fairly direct way of measuring the moment of a magnet
in mechanical units, is to suspend it by a torsion wire in a uniform
magnetic field of known strength, such as that produced by the
earth in an undisturbed area, and observe the torque* required
to maintain the magnet in a
position perpendicular to the
meridian (Fig. 41). Calling the
torque T and the strength of
field H, we have
. T = M H,
T
and M — -— .
H
If the known field be of unit
strength, the value of H above
is i, and the equation reduces
to M = T, from which we see
that the moment of a magnet
is measured by the torque it
Fig. 41. — Torsion Apparatus for Measuring
Magnetic Moment.
* The "constant" of the torsion wire, i.e. the torque required to
produce a twist of unit angle in the wire (i radian), can be found by
well-known mechanical methods. One of these is the vibration method,
in which a non-magnetic bar or disc of known moment of inertia K, is
suspended from the wire, and the time, T seconds, of torsional vibration
determined. Calling c the "constant," the relation
T --
enables c to be calculated, for c = — ™ K.
MEASURING MAGNETIC MOMENT 65
exerts when placed perpendicular to the Jines of force in a field
of unit strength.
Another method of measuring magnetic moment is by magneto-
meter, as indicated in Figs. 34, 42, and 43.
D
Fig. 42. — Measuring Magnetic Moment by Magnetometer.
Let a small magnet n s (Figs. 42 and 43) be placed at p, distance
D from the middle of the magnet N s, whose moment is to be
determined, and suppose the strength of the field at p, when N s
is taken away, is H, and its direction x Y perpendicular to P Q.
Also let / be the distance between the poles of N s. The pole
near N produces a field at P in the direction P Q of strength
2, (see section 25), and the pole near s produces a field at p
in the direction of p Q' (opposite to p Q) of strength =
The deflecting field acting on n s is the difference of these,
viz. :—
F =
m
m
2 D ml
2D M
since M = m I.
Fig. 43. — Measuring Mag-
netic Moment by Mag-
netometer.
66
PRACTICAL ELECTRICITY
Now a deflecting field F acting on a needle controlled by a
field H at right angles to F will produce a deflection a, where
F = H tan a, (see section 23).
2 D M
therefore
or
M =
= /i tan a,
tan a,
(/ \2
^ j is small compared with D2 the expression
reduces to
M
HD*
tan a (approx.).
(5)
If the magnet N s be placed with its centre on the line x Y
(Fig. 42), and perpendicular to this line, as shown in Fig. 43,
and the deflection produced be a', then
M = H \D* + (-} \z tan. a',*
V2/ /
which reduces to
M = H D3 tan a', (approx.).
when ( - ) is small compared with D2.
(6)
27. Absolute Measurement of Strength of Magnetic Field and
of Magnetic Moment. — As will be
seen from Sections 15 and 26, the
measurement of currents and of
magnetic moments there described
depends on the knowledge of the
strength of the magnetic field con-
trolling the needle. It is, therefore,
desirable to know how such strengths
may be measured. The usual way
of doing this is, in outline, as
follows : —
Suspend a bar magnet of moment
M (as yet unknown) by a torsionless
fibre and determine its periodic time
of vibration, T, when placed in the
magnetic field H, to be measured.
w
Fig. 45.
* The student should work this out as an exercise on the principles
involved.
MEASURING MAGNETIC MOMENT 67
When the magnet is at an angle a to the magnetic meridian
XY (Fig. 44), the moment of the forces tending to bring it back to
the meridian is m H I sin a, / being the distance between the poles
of the magnet. Similarly, in the case of a simple pendulum
(Fig. 45), the restoring moment, when the angular displacement
from the vertical is a, is W l^ sin a. The law of control is there-
fore of the same form in the two cases, and the time of vibration
of the suspended magnet can be deduced by analogy from that
of the simple pendulum.
In the latter case, as is well known (for small oscillations),
Tl = 27c A/ — , where g is the acceleration of gravity.
s
This may be written (by multiplying numerator and denomina-
tor under the root by
where u represents the mass of the pendulum.
It will be noticed that u l-f is the moment of inertia K± of the
pendulum, and ug its weight W, hence
The period of small oscillations of a simple pendulum is therefore
equal to 2 it multiplied by the square root of the ratio of its moment
of inertia to the controlling moment which would exist if the dis-
placement from the vertical were go9.
Applying this rule to the suspended magnet we get
~lK
K being the moment of inertia of the magnet, which can be
calculated from its mass, size and shape.
This experiment enables the product M H to be determined,
for from the last equation
':.: ;. • (7)
By using the same bar magnet to deflect a needle n s, as in
M
Fig. 43 (say), we can find the ratio — , for
M = H Z)3 tan a' (approx.) [See (6) section 26] ;
i.e. ~=D* tan a' (approx.).
ti
68 PRACTICAL ELECTRICITY
Dividing M H by — we get
— , (approx.). (8)
from which equation H may be calculated. The same experiment
also enables the magnetic moment M to be found, for
MHx =
H
Z)3 tan a'. (9)
This experiment determines both the moment of a magnet and
the strength of a magnetic field in absolute measure, and is
of great importance in magnetic measurements. (For details
of the. conditions and corrections necessary for obtaining very
accurate values, a work dealing with terrestrial magnetism may
be consulted, such as Stewart and Gee's Practical Physics.)
28. Mapping Magnetic Fields. — In Figs. 36 and 38 we have
seen that the lines of force of a coil carrying a current, or of
a magnet, can be shown by iron filings.
There are various easy ways of fixing these curves marked
out by the iron filings, and so enabling a record to be kept of the
" lines of force," from which we can at once see the position in
which a little compass needle will place itself when put anywhere
in the magnetic fieM. One of the simplest is to use waxed
paper instead of the glass of section 24 ; then, after the filings
have been lightly scattered and the paper gently tapped in
order to assist the filings in taking up their proper positions,
to warm the paper with the flame of a Bunsen gas-burner
moved quickly over it. The wax is thus melted, and the filings
stick to it when it becomes cool and hard again.
Figs. 36, 46, 47, and 48 show the lines of force obtained with
a straight magnet, with two straight magnets placed end on
with poles of the same name near one another, and with two
horse-shoe magnets. The horse-shoe magnets have fitted to
them curved pole pieces of soft iron, and with the second one
(Fig. 48) there is in addition a cylinder of soft iron placed between
the pole pieces to render the lines of force more or less radial, a
result of great value in certain cases (see Moving Coil Ammeters,
Sec. 43).
The direction of the lines of force may also be traced out by
using a small compass needle, n s (Fig. 49) ; for at any particular
spot where this little compass may be put the needle places itself
MAPPING MAGNETIC FIELDS
69
so that its axis is a tangent to the line of force at that spot.
A sheet of paper having been placed on the horizontal table,
and fixed by means of the spring clips, the little compass is placed
at some particular spot, and as soon as the needle has come to
rest a point is marked
with a pencil close to each
end of the needle, n,
The compass is then re-
moved and these two
points joined with a
straight line ; next, the
compass is placed a little
farther on, so that the n
end of the needle is close
to the point formerly
occupied by its s end. A
second short line is now
drawn joining points 2
and 3, and thus by draw-
ing a number of such
adjacent short lines we
have a line of force
marked out by a large
number of its chords.
The compass method
of tracing out lines of
force is, of course, a much
more lengthy one than
that of using iron filings,
but it gives far more ac-
curate results, since the
friction resisting the com-
pass needle taking up the
AUn nn Fig. 46.— Lines of Force with Two Bar Magnets ; Like
AISO, Un- Poles near one another.
right position is very ^
small compared with that ^
between the filings and ^ \ ^S||8
,, , . ° ,, *&z&-K.
the paper on which they
are scattered,
less the filings be scat-
tered extremely sparsely, the magnetism induced in them
sensibly disturbs the magnetic field, so that they indicate, not
the magnetic field due to the coil alone, but the magnetic field
due to the coil as disturbed by the presence of a large number of
little magnets.
PRACTICAL ELECTRICITY
47.— Horse-Shoe Magnet with Curved Iron Pole Pieces,
Fig. 48— Horse-Shoe Magnet with Carved Iron Pole Pieces ; the Magnet
has also an Iron Cylinder between the poles.
MAPPING MAGNETIC FIELDS 7*
Further, it is important to remember, when mapping out a
field due to a magnet, or to a coil carrying a current, and especially
when the delicate compass method is employed, that the result
can only be correct when no other magnet is near enough to pro-
duce a disturbance. Close to the magnet, or coil, under test the
disturbance will be small, unless the disturbing cause be very near
or very powerful ; but at some distance from the magnet, or
coil, under test the force which is being examined is itself so small
that its direction and magnitude may be seriously altered, unless
care be taken to eliminate all disturbing magnetic actions such as
Fig. 49.— Mapping Out the Lines of Force with a Compass Needle.
that of the earth, etc. To test whether this condition is fulfilled
remove the magnet whose field is to be examined away to some
distance, or stop the current passing through the coil, if it be the
magnetic field due to a coil that is being investigated, and
examine whether the compass needle, when placed anywhere in
the area under examination, shows no tendency to .place itself
in one position more than another — that is, shows that it is not
acted on by any directive force.
In order to arrive at this state of things it is clear that the earth's
magnetic force, which is present everywhere, and the magnetic
action set up by any iron pipes, rails, etc., in the neighbour-
hood, must be neutralised by magnets or currents judiciously
disposed.
In obtaining the lines of force seen in Fig. 49, no precaution was
taken to neutralise the disturbing action of the earth's field, the
72 PRACTICAL ELECTRICITY
direction of which is shown by the arrow. Hence the lines of
force in the further parts of the figure are twisted somewhat in a
northerly direction, while in the nearer portion they are bent
southwards, the effect of which is clearly seen at the left-hand
lower corner. %
If the main disturbance be that due to a uniform magnetic
field such as is produced by the earth, a very convenient method of
neutralising it over an area of two or three square feet consists in
placing a uniform sheet of copper just on or under the area in
Fig. 50. — Arrangement for Neutralising a Uniform Magnetic Field.
question and sending a current through it in such a direction and
of such a strength as to set up a uniform magnetic field exactly
equal and opposite to the disturbing one.
To avoid the use of a strong current, which would be necessary
if we desired to employ a large current sheet, a set of strips
Sj, s2, etc., of copper (Fig. 50) may be joined up in series, the whole
current passing through them all in succession.*
29. Comparing the Relative Strengths of Different Parts of a
Magnetic Field by the Vibration Method. — Not merely does the
position of rest of a pivoted compass needle show the direction of
the tangent to the line of force at the particular point, but the
square of the number of vibrations made by the needle in a given
time, when Set swinging, gives a measure of the strength of the
magnetic field at that point. This follows from the formula (7) ,
Sect. 27, viz. : —
M H = ~j^~ ; which may be written
* When the disturbance is due to the earth's field alone, the current
must flow from west to east beneath the paper, and if the sheet is laid
on the table beneath the paper, or at any rate is not more than an inch or
two below it, the current strength must be about 073 ampere per inch
width of sheet.
STRENGTH OF MAGNETIC FIELD
73
_
M T*
or ti =
where n is the number of vibrations in unit time.
For a given needle, which is not put into so powerful a field
that its strength is altered, the quantities K and M are fixed ;
consequently such a needle may be used to measure the relative
strengths of different parts of a field.
If the magnetic field which is to be explored be a some-
what strong one, it will be difficult to time accurately the
rapid vibrations of an ordinary compass needle. It is better,
therefore, to increase its moment of inertia by adding mass to its
two ends, which can be conveniently done by selecting two
leaden shot of about equal size, making a cut in each, and slipping
one over the point of the needle at each end. The needle is then
balanced by moving one or other of the shot nearer to, or farther
from the centre of the needle, and the shot can be secured in
position by slightly squeezing them with a pair of pliers. A
compass needle with weighted ends — the whole, however, much
enlarged — is seen in Fig. 51.
When such a weighted needle is used to explore the field
produced by a current flowing round a large circular coil, like
that seen in Figs. 38 and 49, it is found that at all points distant
from the centre of the
coil by not more than
about one- tenth of its
radius the number of
vibrations per minute
made by the needle is
practically the same,
And, since the map-
ping of the lines of
force shows that within
this little region round
the centre of the coil
the lines of force are straight and all perpendicular to the
plane of the coil, we see that within this region the magnetic
field due to the current flowing round the coil is a nearly uniform
one. Consequently if a needle not longer than about one- tenth
of the diameter of the coil be suspended at the centre of the coil,
and if the controlling force be that due to the earth or to a
distant magnet, the needle will be acted on by two nearly uniform
Fig. 51.— Weighted Compass Needle for Measuring the
Strength of a Magnetic Field (about two-and-a-half
times full size).
74
PRACTICAL ELECTRICITY
magnetic fields, and, from what has been already said, it will
place itself along the resultant of these two fields.
30. Comparing the Relative Strengths of Different Parts of
a Magnetic Field by the Magnetometer Method. — The magneto-
s
meter may in several cases be conveniently used for finding the
relative strengths, or the absolute strengths, of different parts
of a magnetic field, more especially when such field is due to a
coil or magnet which can be readily moved. In Fig. 52 is shown
an application of this method for finding how the strength of
field varies in the plane of a coil carrying a current ; and also for
STRENGTH OF MAGNETIC FIELD 75
determining how the strength of field varies at different points
along the axis of such a coil ; a current of constant strength
flowing through the coil in each case. The magnetometer, con-
sisting of a short needle, to which a long pointer is attached, is
suspended from a rod r, by means of a silk fibre, and is contained
in a sector shaped box, g g, having a scale at the bottom, on which
the deflections of the needle may be read. The box g g is raised
above the base, so that the level of the needle is at the same
height as the centre of the coil c c. A board, B B, carrying the
coil is provided with pins, which enable the coil to slide either
along a groove e in the base parallel to the axis of the coil, or
along another groove e' at right angles to the former, which
ensures that the centre of the needle remains in the plane of the
coil. A galvanoscope, G, is connected through a key, K, with
the coil c c, and also with a wire w w of variable length ; by these
means the current passing from a battery through the coil c c
may be maintained constant.
Before making an experiment, the coil c c is placed so that the
magnetic needle is at the centre of the coil, and the whole base
turned to bring the plane of the coil into the magnetic meridian.
When in this position the pointer on the needle should read zero
if no current be flowing. On passing a current through the circuit
the needle will be deflected, and the magnitude of this deflection
can be adjusted to a convenient value by altering the length of
wire, w w, included in the circuit. Maintaining the current
10
y
06
O 4 6 12 16 20 24 28 32
Distances of centre of needle from plane of coil in centimetres
Fig. 53. — Variation of Strength of Field along Axis of Coil.
constant at this value by aid of G, the deflection d produced by
the current when the needle is at the centre of the coil should
be observed. The strength of field at the centre will then be
#tan d, where H is the strength of the earth's field at
76
PRACTICAL ELECTRICITY
the point occupied by the needle ; (Formula (2) Sect. 23).
On moving c c a few centimetres to the right in a direction
perpendicular to its own plane, the deflection will be found
to diminish, and by making several observations with the coil
at different distances from the needle, the relation between the
strength of field at points along the axis of a coil carrying a
current, and the distance of the points from the plane of the
coil can be readily determined. The figures given in the follow-
ing table show the results obtained in an actual experiment.
They are plotted in Fig. 53.
Axial distance of centre of
needle from plane of coil.
Deflection of
Needle.
Tangent of
Deflection.
Htan d.
O
40-4
0-851
0-154
4
393
0-818
0-147
8
34*0
0-674
0-122
12
29-0
o-554
0-0996
16
21-8
0-400
0-072O
20
16-7
0-300
0-O54O
24
12-9
0-229
0-04I2
28
9-3
0-164
O-O295
32
7'5
0-132
0-0238
In the above experiment the coil c c was a circular one of 20
centimetres radius, the current employed 4-9 amperes (approxi-
mately), and the value of H, the earth's horizontal component,
0-18 dynes per unit pole.
It is possible to calculate from the definition of current strength
and the law of inverse squares, the relation between strength
of field and distance along the axis of a circular coil, which was
found experimentally above.
. Let OB (Fig. 54) represent the axis of a single turn coil c c', whose
plane is supposed to be perpendicular to the paper, and P a point
on the axis at a distance x from o. Consider a short length,
/, of the coil at c perpendicular to the paper, and let a current of
strength / amperes be flowing. The force exerted on a unit
pole at P due to the current / in this short conductor of length
/ would, by definition, be/= cp2, and its direction PQ at right
angles to CP. Similarly, a length / at c', the opposite end of a
diameter, would act on unit pole at P with equal force (since
FIELD ALONG AXIS OF CIRCULAR COIL 77
c'p = CP), and the resultant of these two forces, / and /', would
be given by p R both in magnitude and direction.
But PR = 2 / cos a,
_2_ Il_ r_
~ IO CP2 CP*
_ 7,1 Ir
~ 10 CP3 '
so that the force exerted on unit pole by the current in two short
lengths / of conductor at opposite ends of a diameter of the circle
is directed along the axis of the circle, and of magnitude
multiplied by the sum of the lengths of the short conductors. This
is true of any pair of short conductors at opposite ends of a
Q,
Fig. 54. — Geometrical Construction for Finding the Strength of Field at a point on the
Axis of a Circular Coil.
diameter, and as the whole circle can be supposed to be divided
up into such pairs, the force exerted by the current in the whole
circle will be got by writing 2 TT r, the length of conductor form-
ing the circle, instead of 2 / in the expression above. Hence the
total force F is given by
P = 2 TT r ~
10 CP3
2 TT
10
do)
78
PRACTICAL ELECTRICITY
This may also be written : —
10 r
where p is the angle CPO (Fig. 54), the angle subtended at p by
the radius co.
Working out the values of F for the distances % used in the
experiments recorded in Fig. 53, we get the values of F to be 0-154,
0-148, 0-123, °'097> 0-073, 0-055, 0-040, 0-030, 0-023 respectively,
which agree practically within the errors of observation with the
•V*
4
J25 ioo 1-5 5-0 25 O 25 5-o 7-5 10-0
" Distances from centre of coil (centimetres)
Fig. 55.— Variation of Strength of Field in Plane of CoiL
12-5
corresponding values of H tan d given in the table above. At
the centre of the coil we have x = o, and the formula reduces to
_._ 2 TC /
10 r (n)
The same apparatus can, by moving the coil along the groove
e', be used to find how the magnetic force varies along a diameter
of the coil. Fig. 55 shows the results obtained in this way.
From the curve it will be seen that the force is practically uni-
form near the centre of the coil. It is possible to determine
from first principles the relation between the force and the dis-
tance from centre of the coil, but the calculation is not sufficiently
simple to be given here. An approximation which may be used
near the centre is : —
where b is the distance of the point considered from the centre,
DIFFERENCE OF MAGNETIC POTENTIAL 79
300. Difference of Magnetic Potential : Equipotential Surface. —
As a magnetic pole is acted on by a mechanical force when situated
in a magnetic field, mechanical work will, in general, be done when
the pole is moved from one point of the field to another, and
the value in ergs of the work done in moving a unit pole from
one point to another point is called the difference of magnetic
potential between the two points. If no work be done in moving
Fig. 5 5 a.— Lines of Force and Equipotential Surfaces (dotted) due to Circular Current
the pole from one point to the other the two points are said to
be at the same magnetic potential, or to lie on an equipotential
surface. It will be evident that if the pole be moved at right
angles to the lines of force no work will be done, so any surface
which is everywhere perpendicular to the lines of force of a mag-
netic field will be an equipotential surface. Such surfaces can
be drawn to represent a magnetic field quantitatively in a manner
analogous to that employed with lines of force, Section 24. For
8o PRACTICAL ELECTRICITY
example, if the surfaces be supposed drawn through points on a line
of force whose distances apart are such that one erg of work would
be done in moving unit pole from one point to the next, then the
series of surfaces would indicate quantitatively the nature of the
magnetic field, for the direction of the field at any point would
be normal to the equipotential surface passing through that
point, and the proximity of the surfaces drawn as above des-
cribed would show the strength of the field, this strength being
expressed by the reciprocal of the distance apart of adjacent
equipotential surface in the neighbourhood of the point con-
sidered. Fig. 55« shows the lines of force and also the sections
of equipotential surfaces, in dotted lines, due to an electric
current flowing in a circular coil. It will be noticed that
where the lines of force are nearest together the equipotential
surfaces are nearest to each other, so both systems of lines
indicate the character of the magnetic field.
When the first of the points considered is supposed to be
at an infinite distance from the magnet or coil producing
the field, the work done in bringing unit pole from the first
point to the second is called the potential of the second point.
Every point in a magnetic field may therefore be said to
have a potential, and the difference of magnitude of this
quantity for any two points considered is the difference of
potential between the two points, for no matter what path
be taken in moving unit pole from one point to the other,
exactly the same amount of work must be done against the
magnetic forces. An exact analogy exists in the work done
against gravity in moving a given mass from a point at one level
to a point at another level ; this is quite independent of the path
traversed. If the given mass be unit mass, the work done will be
a measure of the difference of gravitational potential between the
two points. No work is done against gravity by moving a mass
from one point to another on the same contour line, (on a map,)
because all such points are at the same gravitational potential, for
contour lines represent the intersections of level surfaces with the
earth's surface. These contours usually differ in height above
Ordnance Datum by definite amounts, say I ft., 10 ft., or 100 ft.
according to the nature and extent of country shown on the
map ; they are nearest together where the slope of the land
is steepest, and far apart in parts nearly level. Their closeness
therefore, indicates the gradient of the land, or the gravitational
potential gradient, as it may be called, in the same way as the
closeness of the equipotential surfaces in the map of magnetic
field show the magnetic potential gradient.
ELECTROSTATICS 81
ADDENDUM TO CHAPTER II.
Electric Lines of Force and Electrostatics. — When two dissimilar sub-
stances, such as silk and glass, or ebonite and cat's fur, are rubbed
together and separated, they possess the property of attracting light
bodies, such as pith balls, pieces of paper, etc., and of attracting each
other ; the new condition of the glass and silk is described by saying
they are electrified, or have electric charges on them. A conducting body
supported by an insulator, say silk, ebonite, sealing wax, etc., if touched
against the rubbed glass and taken away, exhibits similar properties to
those shown by the glass, and the body is said to have been electrified
by contact with the glass, and to possess an electric charge. Similarly,
an insulated conductor touched against the silk would become electri-
fied, and if placed near the body which had touched the glass the two
would attract each other. But if both the conductors were electrified
by contact with the glass, they would repel each other ; they would
also repel if both were electrified by touching the rubbed silk. These
phenomena are usually regarded as showing that two kinds of
electricity exist (called respectively vitreous and resinous, or positive and
negative) and that bodies charged with the same kind repel each other,
whereas bodies charged with opposite kinds attract each other. It is also
found that exactly equal amounts of the two opposite kinds are produced
whenever electrification occurs, just as equal amounts of magnetism of
opposite kinds always exist in a magnet.
The phenomena exhibited by electrically charged bodies are very similar
to those possessed by the poles of magnets, and conceptions of lines of electric
force, electric fields, electric moments, etc., analogous to the corresponding
magnetic quantities have been developed. The system of measurement
of these electrostatic quantities is exactly like that used for magnetism.
For example, unit charge, or unit quantity* (in electrostatic measure)
is defined to be such that the force of repulsion between two unit charges
at unit distance is unity, viz. : i dyne, in the C.G.S. system ; similarly, the
force exerted between two quantities q and q' at distance d is equal to
$-4 , an expression identical in form with the one for magnetic forces in
formula (4) Section 25. The strength of an electric field is measured by
the force in dynes exerted on unit charge placed in the field, just as the
strength of a magnetic field is measured by the force in dynes on unit pole.
The direction of the field is taken to be that of the force exerted on a vitreous
(or positive) charge.
Diagrams representing the directions of the electric forces in the vicinity
of electrified bodies can be drawn just as magnetic fields are represented,
and equipotential services everywhere at right angles to the lines of force,
delineated.
If a single charged sphere be placed in a large enclosure the lines of
electric force in the vicinity of the sphere will be radial (this follows
from considerations of symmetry), and the equipotential surfaces concen-
tric spheres. Further, the force exerted on a small charge at any external
* This unit of quantity is, of course, quite different from the coulomb
defined, in Section 10. In fact, i coulomb is found, by experiment, to be
approximately equal to 3,000,000,000 i.e. (3 x io9) electrostatic units, and
i electromagnetic unit of quantity (io coulombs) is, therefore, equal to
3 x io10 electrostatic units (Section 171).
It is interesting to notice that 3 x io10 expresses the velocity of light in
centimetres per second, and electromagnetic theory indicates that the
ratio of the two units of quantity should be numerically equal to the
velocity of light, A proof of this is beyond the scope of an elementary
work.
82 PRACTICAL ELECTRICITY
point by the charged sphere can be shown to be the same as if the whole
charge was concentrated at the centre of the sphere, for in this latter case,
assuming the spherical surface removed, the directions of the lines of
force must also be radial, and the equipotential surfaces concentric spheres.
Consequently the force on a unit charge at a distance d from the centre
of a sphere charged with quantity q will be
q x i q
-V '•'• jr. '
and the distance between adjacent equipotential surfaces must be such that
the product of this force into the distance is unity.* Calling this dis-
tance d't we must have
T' x * = '• « * = di~
From this we see that as d increases d' increases in a duplicate ratio, so that
the equipotential surfaces become further apart as the distance from the
sphere increases.
If we suppose this sphere to be surrounded by a hollow concentric sphere
of conducting material, the lines of electric force will still be radial, f The
" difference of potential " between the spheres will, by definition, be equal to
the work done on unit charge when moved from one surface to the other.
By summation (or by integration, or by plotting a curve between force and
distance from the expression / = |-a, and finding its area) this work can
be shown to be q [ --- ), where rl and rt are the radii of the inner and
Vl ^2'
outer spheres respectively. If rt be infinite, i.e., the inner sphere be
alone in space, the above expression becomes - ; this result is generally
r\
expressed by the statement that the electric potential of a sphere in space
is equal to the charge on it, divided by its radius.
To the ratio — . - the name capacity is given, so we see
potential difference
that the capacity of a sphere of radius r^ surrounded by another concentric
one of radius rt is given by the expression — - — — , and for a sphere
rt — fj
isolated in space the capacity is r lt i.e., equal to the radius of the sphere.
An arrangement of two concentric conducting spheres insulated from
each other, or in fact any two conducting surfaces adjacent to each other,
is termed a condenser. If we consider two adjacent surfaces which are
in definite relative positions, the capacity of the arrangement will be
constant, if the insulating medium (or dielectric as it is termed) between
the surfaces remains unaltered. The quantity of electricity on each sur-
face will therefore be proportional to the P.D. { between them, and as the
force between quantities in fixed relative positions depends on the product
of these quantities, we see that the force exerted between two surfaces
of a condenser is proportional to the square of the potential difference
between them. This fact provides us with a method of measuring potential
differences (see Section 48).
Experiment proves that the capacities of condensers depend on the insu-
lating substance between the surfaces, and the ratio in which the capacity
of a given condenser is changed by substituting any substance for air is
called the specific inductive capacity of the substance.
* More accurately expressed by saying that the line integral of the
force from one surface to the other is unity.
f For the symmetry is not disturbed by its presence.
J P.D. is an abbreviation for potential difference.
CHAPTER III
GALVANOMETERS, ELECTRODYNAMOMETERS, AND AMMETERS
31. The Tangent Galvanometer — 32. Adjusting the coil of a Tangent
Galvanometer — 33. Tangent Scale — 34. Tangent Law — 35. Variation
of Sensibility of a Tangent Galvanometer with Number and Size of
Turns — 36. Value in Amperes of the deflections of a Tangent Galvano-
meter controlled by the Earth's Field — 37. Pivot and Fibre Suspen-
sions— 38. Sine Galvanometer ; Sine Law — 39. Electrodynamo-
meters — 40. Construction of Proportional Galvanometers — 41.
Galvanometers of Invariable Sensibility — 42. Permanent Magnet
Ammeters — 43. Moving Coil Ammeters ; Single Pivot Galvanometer
— 44. Soft Iron Ammeters, Spring and Gravity Control — 45. Hot
Wire Ammeters.
31. The Tangent Galvanometer. — In the previous chapters
we have seen how currents may be measured by comparing the
strengths of the magnetic fields they produce with another mag-
netic field of constant strength, and are now in a position to deal
with current -measuring instruments more fully. As mentioned
in Section 15, the relation between the angular deflections of a
magnetic needle and the currents which produce them becomes
very simple in the case where the two magnetic fields are uniform
and at right angles to each other, the tangent of the deflection
is proportional to the current passing round the coil of the galvano-
meter. This law holds for an instrument when the following four
conditions are fulfilled : —
(1) The needle is controlled by a uniform magnetic field of
constant strength.
(2) The diameter of the coil is large compared with the length
of the needle.
(3) The needle is suspended sufficiently near the centre of the
coil for the field which is produced by the current passing
round the coil to be a uniform one in the neighbourhood
of the needle.
(4) 'The axis of the needle is parallel to the plane of the coil
when no current is passing.
When these four conditions are all fulfilled the calibration
curve of the galvanometer, when tested by comparison with a
83
84
PRACTICAL ELECTRICITY
voltameter, as described in Section 17, will be found to be of
the shape shown in Fig. 56 ; and if any three points, p, Q, R, be
taken on this curve, it will be found that the lengths A p, B Q,
c R, parallel to o Y, bear to one another the ratios of the tangents
of the angles represented by o A, o B, and o c respectively.
Such a galvanometer (seen in detail in Fig. 27) is, there-
fore, called a " tangent galvanometer," and it may be henceforth
used without refer-
ence to any volta-
meter for the com-
parison of current
strengths, as they
will be simply pro-
portional to the
tangents of the angles
through which the
magnetic needle is
deflected.
32. Adjusting the
Coil of a Tangent
Galvanometer. — We
have next to con-
sider how we can
adjust the coil of a
galvanometer so as to
be sure that its mean
plane is parallel to
the axis of the needle when no current is passing. Owing
to the coil having a certain breadth, it is sometimes impossible
to see the needle when looking down on to the coil ; indeed,
it is for this reason that the long light pointer attached
to the needle is placed at right angles to the needle. It
would not be right to assume that because the instrument
has been so turned that the pointer points to the zero
on the scale, therefore the plane of the coil is parallel to the
magnetic axis of the needle, for even if the scale has been attached
to the instrument so that the line of zeros is at right angles to the
plane of the coil, it does not follow that the pointer itself is at
right angles to the needle. The two may even have been placed
at right angles to one another by the maker, and yet the pointer
may have been bent subsequently, so that they are not at right
angles when used ; or no experiment may have been made by
the maker to test this, as he is aware that the user will probably
make a test and adjust the pointer for himself.
Fig. 56. — Calibration Curve of a Tangent Galvanometer.
TANGENT GALVANOMETER 85
The test for parallelism of the axis oHhe needle with the mean
plane of the coil may most simply be made as follows : — Turn
the instrument until the pointer points to o°, send any convenient
current through it, and observe the deflection, then reverse the
direction of the current without altering its strength,* and observe
the deflection on the other side of the scale. If these deflections
are exactly equal, then the plane of the coil is parallel to the axis
of the needle when the pointer points to o°, and the instrument is
properly adjusted. But if one deflection is, say, 47° to the left,
and the other, say, 44° to the right, the pointer is not at right
angles to the magnetic axis of the needle, supposing, of course,
that the scale has been so fixed that the line of zeros is exactly
at right angles to the plane of the coil. Next, turn the
instrument a little about its centre in the direction opposite to
that in which the needle moved when the greater deflection was
obtained. The pointer will now, of course, not point to zero ;
let it stand at i° to the left. Again send a current, first in
one direction, obtaining a reading, say, 46° to the left, and in
the other direction, when it gives a reading of, say, 45° to the
right. Now remembering that the pointer started from i°
to the left, the true deflections of the needle are respectively,
46°— 1°, or 45° to the left, and 45°+ 1°, or 46° to the right.
Hence, the fault is now on the other side, or the left deflection
is smaller than the right, and we have, consequently, turned
the instrument too much. Turn, therefore, the coil round a
very little in the opposite direction, so that when no current is
passing through the instrument the pointer stands at, say, J° to
the left, and send as before reverse currents of equal strength,
obtaining readings, 454° to the left and 44 J° to the right, which,
corrected for the initial zero error, correspond with equal deflec-
tions of 45° to either side.
The instrument will now be correct when it is so placed that
for no current the pointer stands at J° left, and it can be so used,
but not, however, with the tangent scale described in the next
section. To enable us to employ the side of the dial graduated
* This may be done by causing the current to pass through any
galvanoscope, the law of which may be quite unknown ; and taking care
that the deflection of the needle of this galvanoscope after the current has
been reversed is the same in direction and in amount as it was before the
current through the galvanometer was reversed, for if we leave the
current through the galvanoscope unchanged in direction when its direc-
tion through the galvanometer is reversed in the experiment, it will not be
necessary to know that the coil and needle of this auxiliary galvanoscope
are symmetrical, or that the strength of a current producing a deflection
to the right is the same as that of a current producing the same deflection
to the left.
86
PRACTICAL ELECTRICITY
in tangents, as well as to avoid having to remember the J° left
error, do not alter the position of the instrument, but bend the
pointer until it points to o° for the same position of the instrument
in which it previously pointed to J° left. The instrument will
now behave as a correct tangent galvanometer when the pointer
stands at o° for no current.
33. Scale for a Tangent Galvanometer. — The scales of tangent
galvanometers are frequently simply divided into degrees, and
references have constantly to be made to a table of tangents to
enable the galvanometer to be used. A better plan is to divide
the scale, not into equal divisions, but into divisions the lengths
of which become smaller and smaller as we depart from the zero
or undeflected position of the needle, in such a way that the
number of divisions in any arc is proportional, but not necessarily
equal, to the tangent of the angle corresponding with that arc.
Or the scale may, as
shown in Fig. 57, be
divided into degrees
on one side, and on the
tangent principle on
the other.
Such a tangent scale
can be most easily
constructed in the fol-
lowing way : — Draw a
tangent FAF (Fig. 58)
to a circle, and starting
from the point of con-
tact A of this tangent
line with the circle,
mark off A B, B c, c D,
etc., in both directions
all equal to one another. Then join the centre o of the circle
with the points B, c, D, etc., by straight lines cutting the circle
in the points i, 2, 3, etc. ; then the numbers I, 2, 3, 4, etc.,
will be respectively proportional to the tangents of the angles AOI,
AO2, A03, etc.
For tan AOI = — ;
o A'
Fig. 57. — Scale for a Tangent Galvanometer,
tan AO2
tan A03
A C _2 A B
O A* O A*
AD 3 A B
; = ; and so on.
o A o A
TANGENT GALVANOMETER
87
Beginners are apt to think that, because the divisions on such
a tangent scale are very much crowded together in the higher
part of the scale, the value of a current can be more accurately
ascertained by taking a reading on the degree side, and then
finding the value of the tangent in a table of tangents, than by
reading it off on the tangent scale. But this seemingly greater
accuracy is
quite delusive,
since what has
to be ascer-
tained in either
case is the tan-
gent of the
angle, not
merely the
angle, and al-
though on the
degree side of
the Scale the FiS- 58- — Constructing a Scale for a Tangent Galvanometer.
angle can be
read much more accurately than can be its tangent, or a number
proportional to its tangent, on the other side, this only indicates
.that the error of a tenth of a degree in a large angle, although a
much smaller proportional error than a tenth of a degree in a
smaller angle, produces a far greater proportional error in the
tangent. For example, if 2O°-i be read instead of 20°, the error
made in reading the angle is 5^, whereas if 85°-! be read instead
of 85°, the error is only BJo, or less than a quarter of the
preceding error. But the tangents are in the first case 0-3659,
and 0-3640, the error in the tangent, therefore, is gif^y, or
about 1^2, whereas the tangents in the second case are n-66
and 11-43, so that the proportional error is if|g, or about ^,
which is nearly four times as great as before. Hence in this
case, when the proportional angular error is diminished to
one quarter, the corresponding proportional error in the
tangents is increased four times. The crowding together of the
divisions on the tangent scale at the higher readings is, therefore,
a correct indication of the inaccuracy likely to occur in taking
readings in that part of the scale.
It can be shown that if one current strength has to be measured
by a tangent galvanometer, the result, other things being the
same, will be most accurate when the deflection produced is
45° ;* or if two currents are to be measured, their ratio will be
* The student should prove this as an exercise.
PRACTICAL ELECTRICITY
most accurate when the deflections they produce are as nearly
as possible at equal distances on the two sides of 45°.
We may here recall attention to the fact that the deflec-
tion produced by a given current passing through a tangent
galvanometer is not altered by varying the strength of the mag-
netic needle of the galvanometer, or by varying its length, provided
that the needle is not made so long as to render the tangent law
untrue for the particular galvanometer. For altering the strength
of the needle alters the deflecting and the controlling forces in
exactly the same proportion, so that
the direction of the resultant of these
two forces remains unchanged. So,
also, altering the length of the
needle does not change the direction
of the resultant force. Hence the
advantages gained by using a strongly
magnetised needle are, first, that it
moves more quickly to the deflected
position when a current is sent
through the galvanometer, and re-
turns more quickly to the zero when
the current is stopped ; secondly, that the friction at the pivot on
which the needle turns, or the torsion of the silk fibre supporting
the needle, introduces less error in a measurement.
34. Tangent Law. — The conditions under which the tangent
law is true, may be stated most generally thus :—
If any body N N' (Fig. 59), turning about an axis at o, be acted
on by two forces whose directions lie in a plane at right angles
to this axis and intersect at a point N, the tangent of the angle
made by N o with one of the forces P, will be proportional to the
magnitude of the other force Q when : —
(1) The controlling force, P, is constant in magnitude, but not
necessarily in direction.
(2) The deflecting force, Q, acts at right anghs to the controlling
force.
In the tangent galvanometer these conditions, as already
explained, are necessarily satisfied by the construction of the
apparatus without any adjustment being necessary when the
deflecting force is varied in magnitude. So in the apparatus seen
in Fig. 60, where both the controlling and the deflecting force are
produced by weights, the above conditions will also be auto-
matically fulfilled for any position of the rod N N', if N N' be short,
and if the pulley p be placed far away from N N' in such a position
that the thread k k is horizontal. Any weight w' put into the
WHEN THE TANGENT LAW IS TRUE 89
scale pan plus the weight of scale pan, will therefore be propor-
tional to the tangent of the angle which N N' makes with the
direction of the controlling force.
This tangent is proportional to the length z R if the scale s s be
horizontal and initially adjusted so that its zero line z coincides
with the pointer attached to N N' when the only force acting on
N N' is that due to w, the controlling force. For the required
tangent is the ratio of z R to o z, ando z is, of course, a constant.
With the apparatus illustrated in Fig. 61, which is a more
accurate, but at the same time a more expensive one than that
shown in Fig. 60, the pulley p is comparatively near the rod N N'.
Fig. 60. — Simple Mechanical Apparatus for Testing the Tangent Law.
Hence an adjustment is necessary to keep the thread k k always
horizontal, that is, at right angles to the direction of the control-
ling force. This adjustment is made by turning the tangent screw
T, and the simplest way of insuring that the pulley p has been raised
or lowered sufficiently to keep the thread k k horizontal, when the
rod N N' is deflected, is to commence the experiment by turning
the levelling screw s, until the level L shows that the bar b b
is perfectly horizontal ; then, after putting each of the different
weights w' into the scale pan, to turn the screw T until th$
thread k k is seen to be parallel to one of the edges of bar b b.
As N N' is not symmetrical above and below the axis o in the;
apparatus shown in Fig:. 61, and, therefore, is not self-balanced,
90 PRACTICAL ELECTRICITY
we must, before any measurements are commenced, screw the
counterpoise weight c in or out, until the rod remains balanced
in any position when the controlling and the deflecting forces
are both naught. These forces are easily made naught by resting
the weight w on the block of wood B, and by taking the thread
k k off the pulley p and resting the scale pan on the base -board of
the instrument. The scale s s is adjusted as before, so that when
the controlling weight w alone acts on N N', the zero line of the
scale coincides with the position taken up by the pointer, only
this adjustment can now be made very accurately by using as
the pointer the wire stretched along the centre of the moving
Fig. 6r.— Improved Mechanical Apparatus for Testing the Tangent Law.
arm, and ensuring coincidence by observing when the image of
this wire seen in the mirror which is attached to the scale, coincides
with the zero line.
The controlling and deflecting weights may of course be inter-
changed, in which case the rod N N' will remain horizontal instead
of vertical, when the controlling force alone acts on it, and the
tangent of the angle, which is proportional to the magnitude of
the deflecting force, will be measured on a vertical scale.
35. Variation of the Sensibility of a Tangent Galvanometer
with the Number of Windings, and with the Diameter of the
Coil. — A tangent galvanometer, whose bobbin contains only
one turn of wire, is not suitable for measuring very weak currents,
as it is not sufficiently sensitive. In order to obtain a delicate
tangent galvanometer, the bobbin must be wound with many
turns of fine wire, and the greater the number of turns employed,
TANGENT GALVANOMETER SENSIBILITY 91
the smaller will be the current needed to produce a given deflection
on the instrument. The exact way in which the sensibility of a
tangent galvanometer is dependent on the number of windings
may be experimentally tested by means of the apparatus shown
in Fig. 62, and this may also be used to ascertain the variation
in sensibility pro-
duced by changing
the diameter of the
bobbin on which the
wire is wound.
The main portion
of this apparatus
has already been de-
scribed (Section 30),
but in Fig. 62 a
smaller coil c c is
shown, whose mean
diameter is exactly
half that of the lar-
ger one. This small
coil is mounted on a
board b b, and, when
placed in position
on B B, is concen-
tric and coplanar
with the larger. On
the larger bobbin
c c are wound two
distinct coils of in-
sulated wire, both of
the same mean dia-
meter, oneconsisting
of twelve convolu-
tions and having its
ends attached to
two of the binding
screws, i, 2, the
other of four con-
volutions and hav-
ing its ends attached to the other two binding screws, 3, 4.
If the binding screw 2 at the end of the first coil be joined
by a piece of wire, as shown in the figure, to the binding
screw 3 attached to the beginning of the second, the current
will go 12+4, or sixteen times round the bobbin; whereas
92 PRACTICAL ELECTRICITY
if the wire connect the end of the first coil, 2, with the
end of the second, 4, and the current enter and finally leave
the bobbin by the two binding screws i, 3, attached re-
spectively to the beginnings of the two coils, then the current will
go twelve times round the bobbin in one direction and four times
in the other, or practically 12 — 4, or eight times round the bobbin.
Now, experiment shows that if a current of constant strength*
be passed successively first four, then eight, then twelve, then
sixteen times round the bobbin, and if this is kept in a fixed
position during the experiment, the tangents of the corresponding
deflections produced will be as four to eight, to twelve, to sixteen,
that is, simply proportional to the number of times the current
passes round the bobbin. This proves that the sensibility of
a tangent galvanometer is proportional to the number of turns
of wire used on its bobbin.
We may next proceed to investigate the effect of the size of the
bobbin by experiments made on the small coil c c. The diameter
of this coil is only one half that of c c, and there are four convolu-
tions of wire wound upon it. When experiments are made it is
found that, if the two bobbins c c and c c are placed so as to be
in the same plane, and so as to have their centres coincident with
that of the suspended magnet, the tangent of the deflection
produced by any current flowing round the smaller one is twice
as great as the tangent of the deflection produced by the same
current flowing four times round the larger bobbin ; and also,
if the same current pass four times round the smaller bobbin
in one direction, and eight times round the larger in the opposite
direction, that no deflection is produced whatever the current
may be.
From the above observations we learn that the tangent of the
deflection produced by a current, that is, the sensibility of the
instrument, is directly proportional to the number of convolutions
of wire, and inversely proportional to the diameter of the coil.
In order, therefore, to get a sensitive instrument we should
use coils of small diameter, and wound with many turns of wire,
and it might be imagined that a tangent galvanometer intended
for the measurement of very weak currents should be made in
this way. As a matter of fact, however, the coils of good tangent
galvanometers are always large in diameter compared with the
length of the suspended needle ; and the number of turns of wire
used in winding is always limited by the consideration that the
depth and width of the channel in which the wire is wound must
not exceed a certain fraction of the diameter of the coil. These
* The current may be kept constant, as described in Section 30.
TANGENT GALVANOMETER SENSIBILITY 93
restrictions are only imposed in order to ensure the fulfilment of
the tangent law, and need not be considered when there is no
necessity for the tangent of the galvanometer deflection to be
strictly proportional to the current.
An instrument which is to be used as a tangent galvanometer
must, however, be so constructed that all the conditions mentioned
in Section 31, earlier, as necessary to ensure the fulfilment of
the tangent law, are complied with. Now when the needle in the
box g g, Fig. 62, is deflected, its poles move away from the coil
c c, and the force exerted by the current in this coil is less, after
the needle has moved, than before. The tangent law will not
hold good unless the change produced in this way is small enough
to be neglected. In order to test this point, the apparatus shown
in Fig. 62 is arranged so that each of the coils c c, c c, can be
moved either in its own plane or perpendicular to its plane,
as described in Section 30.
Experiments such as those described in Section 30 show that as
the bobbin is moved the deflection alters, and that the change pro-
duced for the same amount of motion is proportionately greater
for the small bobbin c c than for the large one c c. For example,
when the coil was moved parallel to itself, and so that its axis
passes through the centre of the needle, we found that the tangent
of the deflection of the needle for a given current was propor-
tional to
r*
where r represents the mean radius of the coil and x the distance
from the mean plane of the coil to the centre of the needle. Now
it is clear from this formula that for a given change in x there will
be a greater change in the value of this fraction the smaller r is.
It thus becomes apparent that any error due to want of proper
centring of the needle of a tangent galvanometer, or to the
actual movement of its poles when it is deflected, must prove far
more serious when the bobbins are small than when they are
large ; and for this reason instruments in which the tangent law
is to be accurately relied upon are constructed with large bobbins.
Example 17. — A tangent galvanometer wound with 50 con-
volutions of wire gives a deflection of 45° when a current of 0-05
ampere passes. What current flowing in a coil of 2 turns of the
same diameter would produce the same deflection ?
Answer: — 1-25 amperes.
Example 18. — In the above example find the current required
94 PRACTICAL ELECTRICITY
to produce a deflection of 25°, (a), with the 50 turn coil, (b), with
the 2 turn coil.
(a) Let / be the current in amperes required.
7 tan 2°
Then
J. llV-^-Ll
' -
0-05 tan 45°
/. / = 0-0233 ampere.
ta" *°
(6, -
1-25 tan 45°
.*. / = 0-583 ampere.
Example 19. — A current of o-i ampere passes through a coil of
20 turns the mean diameter of which is 12 inches. What must
be the size of a coplanar concentric coil of 5 convolutions carrying
0-25 ampere, which would produce an equal magnetic force at its
centre ?
Let d be the diameter required:
Then, since the magnetic force is :: to the current and to the
number of convolutions, and inversely as the diameter, we have
o-i x 20 0-25 x 5
12 d
. d _ 0*25x5x12
O'l X2O
= 7-5 inches.
Example 20. — A tangent galvanometer is made with two coils of
equal diameter, the first consisting of 500 convolutions of wire,
the second of one convolution. If a current of 0-25 ampere
Sent through the first cause a deflection of 45°, what current sent
through the second in the opposite direction, while the same
current was still flowing through the first, would cause the de-
flection to become one of 10° ?
Let / be the unknown number of amperes :
500 x 0-25 —ix/ _ tan 10°
Then
500 x 0-25 tan 45°
Answer. — 103 amperes.
Example 21. — A tangent galvanometer with its needle sup-
ported independently of the coil (as in Fig. 62) gives a certain
deflection for a current of / amperes, when the needle is at the
centre of the coil. Through what distances must the coil be
moved along its own axis if a current of io/ amperes is to give
the same deflection ? (Radius of coil, io centimetres.)
TANGENT GALVANOMETER EXAMPLES 95
Let the distance required be x centimetres ; the question then
requires that the strength of field at a distance x along the axis
of the coil when a current 10 / is passing, must be equal to that
at the centre of the coil when a current / flows through it.
Making use of the formula deduced in Section 30, viz. : —
2 .
F = -- , we have
10 V
), for
For a current of 10 / the force at a distance x along the axis
Force at centre, (x = o), for current I = - — |-
Equating the two we get
lOf'
or 10 r» = y+*;
squaring both sides and extracting cube root
or x* = r
* = r
— 19-1 cms.
From this we see that by moving the coil so that the needle is
19-1 centimetres from its centre (a distance nearly equal to the
diameter of the coil), the sensibility of the instrument is reduced
to ye of its former value.
Example 22. — From the curve given by the graph, Fig. 53,
find the distances along the axis of the coil at which a magnetic
needle must be placed so that the sensibility of the galvanometer
so formed may be reduced in the proportions \, £, and J respec-
tively, the sensibility with the needle at the centre of the coil
being considered unity. Find also these distances by calculation.
Answers. — From curve, 15-2, 24-8, 34 (approx.).
By calculation, 15-3, 247, 34-6.
o6 PRACTICAL ELECTRICITY
36. Values in Amperes of the Deflections of a Tangent Galvano-
meter controlled only by the Earth's Magnetism. — The sensibility
of any galvanometer depends not merely on the coil, but also
on the strength of the controlling field. If this controlling field
be altered by bringing up a magnet, then even if the magnet be
so placed that the position of rest of the needle for no current
be unchanged, still the force, and therefore the current required
to turn the needle through a given angle will be altered. For let
the controlling force N P be increased to N P' (Fig. 63) so that the
zero position of the needle is the same, but the needle is held
in that position with a greater force, then in order that the angle
P'NR' may remain of the same value as before, the deflecting
force P R must be increased to P'R', that is, in the same pro-
portion as the controlling force. If the current has the same
value as before, so that P' R" is equal to P R, then the angular
deflection of the needle instead of being P N R' will be reduced
to P' N R". Even if the controlling field be merely that due
to the earth, this will alter from place to place, and from year
to year ; so that a tangent galvano-
meter requiring a current equal to I
ampere to produce a deflection of 45
degrees in some particular town, will
generally need a somewhat different
current to produce the same deflection
if moved to another town, and even if
kept in the same position the absolute
calibration will be found to gradually
alter with time.
When the needle of a tangent
galvanometer is supported in such a way that it turns in a hori-
zontal plane, and when the controlling force is entirely produced
by the " horizontal component of the earth's magnetic force" the
following formula connects the current I in amperes, passing
through the coil, with the deflection d in degrees, the radius r
of the coil in centimetres, and the number of convolutions n of
wire on the bobbin
T .
/ =- -tan d, (12)
2-x n
where H is the strength of the horizontal component of the
earth's magnetic field at the place where the galvanometer
is situated. This follows from the formulse given in Sections
15 and 30, combined with the fact just proved, that the sensibility
of a tangent galvanometer is proportional to the number of
convolutions of wire on its bobbin.
INTENSITY OF EARTH'S FIELD
97
The quantity - in the above expression for /, is constant
for a given time and place in an undisturbed area, and may be
written as klf the formula then becoming
I =
tan d.
(13)
In Table III. is given the average values of H at Greenwich
Kew, Valencia, Stonyhurst and Eskdalemuir, for the years 1914
to 1919, and also the values of kl when r is measured in centi-
metres and in inches respectively. From 1910 to 1913 the
mean values of H at these stations remained practically unaltered,
ard those for 1918 and 1919 are equal except at Stonvliurst.
where a new magnet was set up in the interval.
TABLE III.
Value of ki, when r is in
Pl___
Vpnr
Volno nf H
r Jace.
i enr.
value 01 /7.
Centimetres.
Inches.
1914
0*1852
0-2948
0*7489
1915
0*1851
0*2946
0*7484
Greenwich
1916
0*1849
0*2943
o*7474
1917
0-1848
0*2942
07471
1918
0*1846
0*2939
0*7463
1919
0-1846
0-2939
07463
1914
0-1849
0*2Q43
0*7474
i9T5
0-1846
0-2939
07463
Kew
1916
0-1846
0*2939
0*7463
1917
o 1841
0-2935
o*7454
1018
0-1843
0-2934
' 07451
1919
0*1842
0-2932
0-7448
1914
0-1789
0*2847
0*7233
*9r5
0-1787
0*2844
07224
Valencia
1916
1917
0-1787
0-1786
0-2844
0*2843
07224
0*7221
1918
0*1784
o -2840
07213
1919
0-178;
0-2840
0*7213
1914
0-1735
0*2762
070x3
1915
0-173*
0*2761
0-7009
Stonyhurst
19.6
1917
0*1734
0-1734
0*2760
0*2760
07003
07009
1918
01733
0-2758
0*7004
1919
0*1729
0-2752
0-6991
1914
0*1680
0*2674
0*6792
i9T5
0*1679
0*2673
0-6789
Eskdalemuir
1916
1917
0*1676
0*1673
0-2668
0-2663
0*6776
0-6764
7918
0-1671
0*2661
0-6756.
1919
0*1671
0*2661
0-6756
When the controlling force acting on the needle of a tangent
galvanometer is due to the presence of a distant magnet, placed
H
98 PRACTICAL ELECTRICITY
so that the needle is parallel to the plane of the coil when no cur-
rent passes, the preceding formula holds true, but the constant,
klt must be determined experimentally.
If the value of k^ for the earth's field alone be accurately known
for the particular place and the particular time, then the value of
kl for any other controlling field may be ascertained by employing
the principle described in Section 29. Remove all magnets,
pieces of iron, etc., so that the needle of the tangent galvano-
meter is acted on by the earth's field alone, and count the number
of oscillations, nlf say, that the needle makes in any convenient
interval of time. Replace the controlling magnet, or magnets,
as desired, and again count the number of oscillations, n2, say,
that the needle makes in the same time, then the kl for the
earth's field alone must be multiplied by n^/n^ to obtain the
kl to be used in the preceding formula for the particular com-
bination of controlling magnets in question.
Example 23. — How many amperes would deflect the needle
of a tangent galvanometer 60° in the year 1914, the controlling
force being the horizontal component of the earth's magnetism
at Greenwich, and the galvanometer having a coil 5 inches in
radius, wound with six convolutions of wire ?
The number of amperes is — ?3 ^ X
Answer. — 1-079 ampere.
Example 24. — Through what angle would 0-598 ampere
deflect the needle of a tangent galvanometer with a bobbin
7 inches in radius, wound with five convolutions of wire, in the
year 1918, the controlling force being the horizontal component
of the earth's magnetism at Kew ?
.,tan ^
0-7451 x 7
= 0-5731,
d = 29°24 Answer. — 29^4.
Example 25. — If the horizontal component of the earth's
magnetism in 1914 at Stonyhurst be the controlling force in a
tangent galvanometer, the bobbin of which is n inches in dia-
meter, how many convolutions of wire must be wound on in
order that a current of 0-964 ampere may give a deflection
of 45° ?
Answer. — 4 convolutions.
PIVOT AND FIBRE SUSPENSIONS 99
Example 26. — If the horizontal component of the earth's
magnetism in 1917 at Kew be the controlling force in a tangent
galvanometer, the bobbin of which is wound with eight con-
volutions of wire, what must be the radius of the coil in order
that a current of 0-384 ampere may give a deflection of 50° ?
Answer. — 3-45 inches.
Example 27. — About how many times the horizontal compo-
nent of the earth's magnetism must the controlling force be in a
tangent galvanometer, having a coil 5 inches in radius wound
with six convolutions of wire, in order that a current of 20
amperes may cause a deflection of 45° ?
Answer. — About 32 times.
Example 28. — The needle of a tangent galvanometer when
acted on by the earth's field alone makes one oscillation in 1-3
second, whereas, when the controlling magnet is placed in
position, it makes one oscillation in 0-433 second. If the coil be
15 centimetres in radius, and be wound with twenty turns of
wire, what current will produce a deflection of 30° in 1918 at
Greenwich ?
Answer. — 1-14 ampere.
Example 29. — Find the mean diameter of a single turn tangent
galvanometer coil, such that one C.G.S. unit of current (10 am-
peres) will produce a deflection of 45°, the needle being controlled
by the earth's horizontal field at a place where #=0-1852.
Answer. — 67-8 centimetres.
26-66 inches.
It is not necessary that the coil of a tangent galvanometer
should be circular, but in order to obtain the straightness of
the lines of force in the neighbourhood of the axis, as seen in
Figs. 38 and 49, and not merely for points actually on the
axis, of which we could only avail ourselves by using an infinitely
short magnet, the diameter of all parts of the coil must be large.
Hence, if an elliptic or other non-circular coil were used, its
smallest diameter would have to be large, and consequently its
largest diameter unnecessarily large.
37. Pivot and Fibre Suspensions. — There are two principal
methods of supporting the needles of galvanometers. These
are illustrated in Fig. 280. In D the little magnet has a jewel in
its centre, and rests on a sharp pivot, as in an ordinary pocket
compass ; whereas in G the needle is supported by a fine fibre of
unspun silk, the upper end of which is fastened in one of the
ways illustrated in Fig. 27, so that it can be lowered on to the
100
PRACTICAL ELECTRICITY
card on which the scale is marked, when the instrument is being
carried about, and raised again so as to be in the centre of the
coil when the instrument is in use. The fibre suspension in-
troduces far less resistance to the motion of the needle than the
best jewel and pivot ; but with a fibre suspension it. is generally
necessary that the instrument should have levelling screws,
such as are seen attached to G, Fig. 280, and that it should be
levelled before being used.
Fig. 64.- Section of Galvanometer with Silk Fibre Suspension, Pivoted for
Turning round its Centre.
A galvanometer needle should therefore be supported by
a pivot when the instrument has to be moved about, and used
quickly in different positions. But when the galvanometer is
employed in a fixed position, and great accuracy is desired, the
needle ought always to be suspended by a fibre of unspun silk.
38. Sine Galvanometer. — As the tangent galvanometer re-
quires a coil of large size compared with the length of the needle,
the form is not well suited for instruments of very great sensibility.
There is, however, another kind of galvanometer which is free
from this defect, viz., the sine galvanometer. In this type of
instrument there is no restriction as to size or shape of coil, the
only conditions being that the controlling field be constant and
uniform, and that the coil and needle always occupy the same
relative position when the readings are taken. In Section 6,
it was shown that when the needle and coil are in the same
relative positions, the couple exerted between them is propor-
tional to the rate of chemical decomposition, and therefore to
SINE GALVANOMETER:
the current strength. When a needle is deflected by a current
in a coil, and the coil turned to follow up the needle until the
relative position of the two is a definite one, the torque exerted
on the needle by the current, when equilibrium exists, being equal
to that exerted on the needle by the controlling field, is propor-
tional to the sine ot the angle of deflection. Obviously, the
current strength is therefore proportional to the sine of the
angle through which the needle is deflected from the magnetic
meridian.
A
Fig. 65. — Apparatus for Mechanically Testing the Sine Law. Adjustment made
by Altering the Direction of the Deflecting Force.
Fig. 64 shows a section of a galvanometer arranged so that it
can readily be turned about its centre for making relative measure-
ments of current strength by the sine method, and in Fig. 65 is
illustrated an apparatus for mechanically testing the sine law.
Here a rod, N N', representing a needle, is pivotted at o and
counterbalanced by a nut c on the screwed end of N N'. From
the lower end, N, hangs a weight, w, and to the same point is
attached a thread, k k, supporting a scale pan and weight w'.
An arm, o D, pivotted at o has another arm, E, clamped to it
by a nut n, and E carries a pulley, P Q, over which k k passes.
The arm o D is fixed to a tangent wheel and can be turned about
o by the screw T. At the lower end of o D is a piece of mirror
glass, G, with a scratch on it ; a pointer on the lower end of
N N' can be sighted and the arm o D adjusted until the pointer
is directly opposite the scratch, by turning the screw T. A
PRACTICAL ELECTRICITY
horizontal scale, s s, with a mirror behind it, enables distances
from a vertical plane through the axis o to be measured ; these
distances being proportional to the sines of the angles of deflection
of N N' from the vertical position. To make an experiment the
weights w and w' and scale pan are removed, N N' balanced
by the counterpoise c ; the weight w is then put on and the
scale s s adjusted until its zero is directly behind the thread
supporting w. The thread k k is then put over the pulley P Q,
and a weight w' placed in the scale pan. The arm o D is now
adjusted so that the mark on G is
directly opposite the pointer on N, by
means of the tangent screw T. On
taking different values of w' and the
corresponding readings 5, it is found
that the two quantities w' and s are
proportional, i.e., the weight w' is pro-
portional to the sine of the angle
through which the rod N N' is deflected
by w'.*
Proportionality between w' and 5 will
be found to exist, whatever the direc-
tion N k of the deflecting force relative
to the rod N N', provided this be
unaltered during a set of experi-
ments. The ratio of w' to s will,
however, alter when this direction is
changed.
39. E!ectrodynamometers. — Another form of current measur-
ing instrument for which the law connecting the deflection and
strength of current is known, is the electrodynamometer. It
consists essentially of two coils of wire carrying the same current,
and the force, or torque, exerted between the coils depends on
the strength of the current passing. As we have already seen
(Sect. 24), a coil carrying a current creates a magnetic field in its
neighbourhood, just as a magnet does ; we may, therefore, regard
such a coil as a magnet, and two adjacent coils having currents
passing through them will usually exert a force on each other.
If the coils are kept in the same relative position, the magnitude
of this force will be doubled if the strength of current in either
coil be doubled, and if the current in both coils be doubled, the
force will be quadrupled. When the two coils are in series with
each other, doubling the current in the circuit will double it in
both coils, and hence make the mutual force four times as great
* w' here includes weight of scale pan.
Fig. 66. — Simple Electro-
dynamometer.
ELECTRODYNAMOMETERS
103
We may therefore conclude that the force exerted between the two
coils of an electrodynamometer, whose coils are in a fixed relative
position to each other, is proportional to the square of the strength
of the current flowing through them.
An electrodynamcmeter of a simple form is shown in Fig. 66,
whilst Fig. 67 illustrates an instrument^ used in practice. ID
both instruments one of the
two coils is suspended by a
silk thread, and the fixed
relative position of the
stationary and moving coils
is brought about by means
of a spiral spring shown at N
Fig. 66. This spring is at-
tached to the torsion head T
at its upper end, and to the
suspended coil E F G at its
lower end, and by turning T
the pointer P fixed to the
moving coil can be brought
to the zero mark on the
scale shown in plan in Fig.
68. When so adjusted the
relative position of
the stationary and
suspended coils is
perfectly definite.!
Stops s s prevent P
moving far away
from the required
position. Usually
the planes of the two coils are perpendicular when P is at
zero.
Mercury cups, m mf, Figs. 66 and 67, are used for leading
the current to and from the moving coil, the path of the
current, starting from the left hand terminal, being as
follows : — Through the fixed coil, A B c D to the mercury
cup m, then through the moving coil, E F G, to the mercury
cup m' and the right hand terminal. When a current passes
through the instrument a couple exists between the coils,
tending to place the moving coil parallel to the fixed one.
This turns the moving coil away from zero in a counter-
clockwise direction, and by turning the head T clockwise, the
spring exerts a torque in the opposite direction, which can be
Fig. 67. — Siemens Electrodynamometer.
io4
PRACTICAL ELECTRICITY
adjusted so as to exactly balance the couple due to the current
in the coils. The torque of the spring is proportional to the
angle through which its upper end is twisted, so that the angle
of torsion measures the square of the strength of the current.
We may, therefore write
72 ::a
where a is the angle T is turned through.
or I2 = k* a,
/ = kV*.
(14)
k being called the constant of the electrodynamometer, and
which may be determined by comparison with a voltameter
or an absolute tangent galvano-
meter. When an electrodynamometer
is intended to measure very small
currents, say less than J an ampere,
mercury cups are not necessary, as
flexible wires can be substituted.
Such an instrument is shown in
Fig. SSb.
Example 30. — An electrodynamo-
meter is used to measure the relative
values of two currents which neces-
sitate rotations of the torsion head
of 25° and 144° respectively ; find
the ratio of their strengths.
Let /! and 72 denote the strength of the two currents, then
I2 =
= 5 k,
= 12 kt
12
Example 31. — A current of 25 amperes gives a deflection of
140° on an electrodynamometer ; what current will produce a
deflection of 320° ?
100
Answer. --- ==- = 37-8
V?
amperes
Example 32. — A current which deposits copper at the rate of
1-97 grammes per minute produces a deflection of 225° on an
electrodynamometer; find the constant of the instrument.
Answer. — 6-6.
PROPORTIONAL GALVANOMETERS 105
The Siemens electrodynamometer shown in Fig. 67 has one
moving coil and two fixed ones of different numbers of turns,
and by this device the range of current which the instrument is
capable of measuring is considerably extended. The outer or
thick fixed coil has 4 turns, and the inner or thin fixed one about
60 turns. They are connected together and to the moving
coil at D, and their free ends joined to the left and right hand
terminals respectively. When it is desired to employ the thick
coil the middle and left-hand terminals are used, and for the thin
coil, middle and right.
In using an electrodynamcmeter care should be taken to place
it so that the plane of the moving coil is perpendicular to the
magnetic meridian, otherwise the reading of the instrument will
be influenced by the earth's magnetic field,, and a current through
the dynamometer in one direction will give a different reading
from that produced by an equal current in the other direction.
Example 320. — The constants of the two windings of the
dynamometer in Fig. 67 are 3-40 and 0-920 for the " thick "
and " thin " coils respectively. What deflections will be caused
by a current of 16 amperes, passed (a), from middle to left-hand
terminal, and (b), middle to right-hand terminal.
Answer. — (a) 22-1 divisions.
(b) 302 divisions.
40. Construction of Galvanometers in which the Angular
Deflection is directly Proportional to the Current. — We have
already seen (Section 15) that the current is proportional to the
tangent of the deflection of the galvanometer needle, when neither
the magnitude nor direction of the controlling force is altered as
the needle moves into a new position on being deflected, and
when, in addition, the direction of the controlling force is at right
angles to the direction of the force with which the current passing
round the coil acts on the needle.
In order, therefore, that the angular deflection may be directly
proportional to the current, we must either cause the needle on
being deflected to move into a position in which the current
passing round the coil acts more powerfully on it, or into a
position in which the controlling force becomes weaker • or we
may arrange that both these results may be produced.
The first condition may be obtained in a rough way by employ-
ing the very defect of construction previously referred to in the
adjustment of the tangent galvanometer, which made the de-
flection on one side of the zero larger than that produced by the
same current on the other — viz., not putting the coil so that its
io6
PRACTICAL ELECTRICITY
plane was parallel to the suspended magnet when no current was
passing through the coil. The needle, when deflected to that
side on which the greater deflection is obtained, will, instead of
moving from a stronger to a weaker part of the magnetic field
Fig. 69. — Walmsley and Mather's Proportional Galvanometer.
produced by the current, move at first into a stronger part, and
then afterwards into a slightly weaker part. The effect of this
arrangement is to make the proportional law connecting current
and deflection approximately true for a much larger deflection
from the undeflected position of the needle than if we com-
menced with the needle parallel to the plane of the coil for no
current. But this arrangement has the disadvantage that it
can only be used for currents deflecting the needle to one side
of the scale, for, if the current be
flowing in the opposite direction, the
defect of want of proportionality
between current-strength and deflec-
tion will be increased.
This plan, by means of which the
proportionality on one side of the
scale is sacrificed to increase that on
the other, has been employed by
one of the authors (W. E. A.), and
later on by MM. Carpentier and
Deprez, and others, for making
proportional galvanometers.
Another device for causing the strength of the deflecting
field to increase as the needle deflects, is employed in the galvano-
meter originally devised by Professor Walmsley and one of the
authors (T.M.), and in use for many of the experiments of the
first-year students at the City and Guilds College. This instrument,
as illustrated in Figs. 69 and 6ga, consists of two coils shaped as
Fig. 69*.— Walmsley-Mather
Galvanometer.
PROPORTIONAL GALVANOMETERS 107
shown, and fixed so that they are separated by a distance a little
less than the length of the needle. The galvanometer is placed
so that when no current is passing through the coils the needle
hangs symmetrically between them, and when the controlling
field is a uniform one, the current is directly proportional to the
angular deflection up to 45° or 50°.
Even although the controlling magnet of a galvanometer
be rather near the needle, the controlling field may be regarded
as an approximately uniform one if the deflections of the needle
be all very small. Similarly for very small
deflections the deflecting field may be re-
garded as approximately uniform what-
ever be the shape and size of the coil or
of the needle. If then, in addition, the
controlling magnet be so placed that when
no current is passing the needle makes
about the same small angle with the plane
of the coil on one side of it that it makes
with that plane on the other side, for the
greatest deflection employed, the distribu-
tion of the forces will be as in Fig. 70,
where N p represents the magnitude and
direction of the controlling force, P RJ the
magnitude and direction of the deflecting
force for current I, P R2 for current 2, P R3
for current 3, etc., P R2 being twice p Rlf
p R3 three times P Rj, etc.
Therefore the angular deflections of the needle for currents
i, 2, 3, etc., are P N Rlf P N R2, P N R3, etc., and, as these angles
are all very small, and the base lines are proportional to the
currents, it follows that the angular deflections are also propor-
tional to the currents. Indeed for very small deflections this
result will be nearly true, whether the angle NPX is a little less
than, or a little more than, or exactly equal to a right angle ,
that is, whatever be the angle the needle makes with the plane of
the coil provided that this angle is small.
41. Galvanometers of Invariable Sensibility. — Now that
measuring currents in amperes has acquired the same sort of
practical importance as weighing coals in tons or finding the
number of cubic feet of gas passing through a pipe, it is necessary
to have galvanometers which are portable, and whose indications
are not affected by moving the galvanometer from one place to
another, or by placing it near an iron pipe, a fire-place, or
even near the powerful electromagnets of the dynamo machines
io8 PRACTICAL ELECTRICITY
which are employed for the mechanical production of electric
currents.
An instrument of this type should be " direct-reading " ; that
is, the deflection of the pointer must indicate at once the current
in amperes, for in commercial work there is no time to refer
to a table of values, not to mention the risk that would be intro-
duced by a table of values belonging to some other instrument
being used by mistake.
Such instruments, by means of which the current can be read
off at once in amperes without any calculation or reference to any
calibration curve, are called " ammeters,"* and since about the
year 1880 so much attention has been given to the design and
construction of this class of electrical meter that it is now pos-
sible to measure a current with as much accuracy as a leg of
mutton can be weighed in a pair of scales, or with a spring-
balance, and with even greater facility.
The controlling force must necessarily be exerted in such
a way that it is the same wherever the ammeter is placed ; indeed,
many ammeters are so constructed that, the controlling force is
not changed by laying the instrument on its side, or in any other
position, so that a current can be read off equally well whether
the ammeter is lying on a table, hung up on a wall, held in the
hand, or used on board a ship rolling in a heavy sea.
There are three distinct ways in which the controlling force is
exerted in ammeters.
(1) By means of a powerful permanent magnet placed inside
the instrument and rigidly fixed to it.
(2) By means of a spring.
(3) By means of a weight.
The first two methods have the advantage that with their use
the moving part of the ammeter can be balanced like a wheel in
a watch, so that the instrument can be made to read correctly in
any position ; the former of these two has also the further
advantage that as the control exerted by a powerful magnet close
to the needle is very large, outside magnetic disturbances have
little effect. But while a magnet or a spring can be made con-
stant enough in its action for many practical purposes, its
variation with time is of course greater than that of a weight,
hence the third method of control is the one adopted when
accuracy is of more importance than portability.
In the earlier editions of this book several ammeters were
described, and their advantages and disadvantages compared.
But the methods of constructing the coils and needles, and the
* Abbreviation for ampere-meters.
AMMETERS 109
various devices that are now adopted ii> applying the controlling
force in one or other of the three ways just referred to have
become so numerous, that anything like a complete description
of all the types of ammeters now in use, and an examination of
their relative advantages would alone fill a good-sized book.
A mmetvrs, besides differing in the methods used for exerting
the controlling force, also differ in design, depending on whether
the instrument is intended to measure currents of very different
values, or only currents all of about the same value. In the
former case the design should be such that the scale is equally or
nearly equally divided, so that there is about the same distance
between any adjacent pair of division marks, while in the latter
the scale should be very " open " ; that 'is, the division marks
should be widely separated at the one part of the scale which is
in constant use, and crowded together at those parts which
correspond with currents which rarely have to be measured.
Instruments with this latter type of scale are especially employed
when an ammeter is used to measure the currents supplied to
single lamps, or groups of lamps.
In Section 5 we saw that when a conductor conveying a
current is placed near a magnet there is a force exerted between
the conductor and the magnet, tending to make them move
relatively to one another. The force acts in such a direction
that a wire carrying a current tends to move perpendicular to
itself and perpendicular to the lines of force due to the magnet.
It is only when the wire lies along the lines of force that the
action between it and the magnet is naught, however strong
be the current and however powerful the magnetic field. With
any other position of the wire relatively to the direction of the
magnetic field there is some force, and this force has its greatest
value for a given length of conductor carrying a given current,
and placed in a field of given strength, when the conductor is
perpendicular to the lines of force.
By employing a very powerful magnet the force exerted on a
wire, even when conveying a feeble current, can be made con-
siderable, and this action has been employed by Maxwell, Lord
Kelvin, Deprez, d'Arsonval, Weston, and others, to obtain
galvanometers which are not only very sensitive, but the indi-
cations of which are very little affected by extraneous magnetic
disturbance.
42. Permanent Magnet Ammeters. — The earliest ammeter,
having an equally divided scale so that the deflection in degrees
was directly proportional to the current, was the " permanent
magnet ammeter " devised by Professor Perry and one of the
no
PRACTICAL ELECTRICITY
Fig. 71.— Ayrton and Perry's Permanent Magnet
Ammeter. Latest Form.
authors in 1880. The coil was wound on the two halves of a
flat brass tube A A (Fig. 71), shown unwound in the figure, and
inside this tube, at its centre, there was pivoted a small soft-iron
needle, shaped like a long ellipsoid, n n (Fig. jia) and controlled
by a powerful permanent
magnet, M M (Fig. 71).
The weight of the pointer
p and any dissymmetry
s cf the needle was ac-
curately counterbalanced
by a small weight, w,
hence no controlling force
was introduced by gravity,
and the instrument could
be used equally well in
any position.
On a current flowing
round the coil there was
exerted a greater or less force tending to place the axis of the
needle along the axis of the brass tube A A (Fig. 71), while the
controlling magnet M M exerted a force tending to place the axis
of the needle along the line joining the tips of the soft-iron pieces
PP; the needle therefore set itself in the direction of the
resultant magnetic f: eld.
In addition to giving the pole pieces the shape seen in the
figure, the wire was heaped up somewhat near the ends of the coil,-
therefore not merely did the
controlling force diminish as
the needle deflected, but
the deflecting force, for a
given current, also increased.
Thus, as explained in Section 40, there were
two causes tending to make the angular deflec-
tion vary in direct proportion to the current
flowing, and, when sufficient care was exercised in
winding the coil, a straight line calibration could
be obtained.
When, however, these instruments began to be
manufactured in large numbers, the labour of**?. 7i«. — Needle
JT • j.1- • J- X AT. -11 A - -i , M Staff« and Pointer
modifying the winding of the coil by trial, until of Ay.to,. ar.i
direct proportionality was obtained, became too -Sagiet Ammeter?
great, and, instead of depending only on the shape
of the pole pieces, of the needle, and of the coil for obtaining a
straight line law, two soft -iron cores screwing into the ends of
PERMANENT MAGNET AMMETERS in
the brass tube were added, and by screwing these cores in more
or less the rate at which the deflecting force (for a given current)
varied with the position of the needle could be altered.
Later on a third plan was employed. The soft-iron pole pieces
were themselves made adjustable, and to prevent the controlling
force produced by the pole pieces, when withdrawn, falling off too
rapidly as the needle deflected, the ends of the poles were made
concave instead of convex as before. These movable poles intro-
duced the power of making another adjustment in addition to that
effected by screwing the soft -iron cores previously mentioned in
or out of the coil, and by means of these two adjustments not
merely could the angular deflection be made nearly proportional
to the current, but the deflection for the same current could be
increased- or diminished. Thus the sensibility of the instrument
could be adapted to suit an engraved direct-reading scale, instead
of each scale having to be engraved somewhat differently to suit
the sensibility of the instrument.
Another method of adjusting the straight line calibration,
carried out by Mr. Esson, is shown in Fig. 71. Here the small
screws s s which pass right through the pole pieces, could be
advanced or withdrawn so as to alter the shape of the magnetic
field controlling the needle, and therefore make the controlling
force fall off more or less rapidly as the needle was deflected.
By means of the brass nuts N N the needle and pointer could be
moved relatively to the pole pieces into a stronger or weaker part
.of the magnetic field due to the magnet, and thus the sensibility
of the instrument could be adjusted.
The Ayrton and Perry permanent magnet ammeter had an
important advantage over the various types of soft -iron needle
ammeters that are at present constructed, in that the deflection
of the pointer indicated not merely the strength of the current
but also its direction ; for in certain cases, such as the charging
of "secondary" cells, the supply of current to " arc lamps" &c.,
a knowledge of the direction of the current is as important as
the measurement of its strength.
To cause the deflection of the pointer to be exactly the same
on the two sides of the zero when a current was reversed, an
adjustment was necessary, but this was easily effected in the
Esson method of construction by turning the coil about the brass
screw which held it to the bar at the back of the magnet, this
screw being placed so that a line drawn through it passed through
the centre of the needle.
By employing a very short needle, and a very light pointer
made of thin aluminium, corrugated to give it mechanical
H2 PRACTICAL ELECTRICITY
strength, the combination seen in Fig. jia had only a small
moment of inertia. This, combined with the very strong per-
manent magnet producing the control, rendered the instrument
very quick in action. Therefore, instead of the needle being set
swinging, and only coming to rest after some time, when a change
suddenly occurred in the current, the needle moved sharply into
its new position, and all such changes, even if quickly produced,
were accurately indicated.
The promptness of action of the permanent magnet ammeter,
its extreme freedom from extraneous magnetic disturbance,
its power to indicate the direction of the current as well as its
strength, and the fact that this form of ammeter could be used
in a horizontal or vertical position, or even on board a rolling
ship or on a rapidly -moving train, led to many thousands of them
being employed, in spite of the fact that their sensibility gradually
became greater as the permanent magnet grew weaker.
43. Moving Coil Ammeters. — The permanent magnet ammeter
described in the previous section had a very strong controlling
field so as to render its readings practically independent of its
position in the earth's field, or of stray magnetic fields in the
vicinity of dynamos ; and to make it quick and precise in action.
This, however, had its disadvantages, for the use of a strong
controlling field reduced the sensibility of the instrument and
necessitated the use of a strong deflecting field, so that a con-
siderable number of turns of wire wound very near the needle were
required to produce the deflection. A little consideration will
show that a modification by which the magnetic field of the
permanent magnet could be utilised as the deflecting field
instead of a controlling field, would make a strong magnet
advantageous, for then an increase in the strength of the magnet
would produce greater sensibility, and at the same time render
the instrument less liable to error from disturbing fields. As
mentioned in Sect. 41, instruments of this class (named " moving
coil galvanometers ") have been employed by Maxwell, Lord
Kelvin, Deprez, d'Arsonval, and others. Of late years great
developments have been made in measuring instruments of this
type, and the moving coil galvanometer, or ammeter, is one of
the commonest forms in commercial use.
A very convenient, portable, and accurate moving coil ammeter
was brought out by Mr. Weston, of Newark, America, in 1888,
and a view, about two-thirds full size, of the working parts of a
recent type, arranged for reading milliamperes or thousandths of an
ampere, is shown in Fig. 72. The coil c c is wound on a rectangu-
lar metal frame, and is pivotted between jewelled centres, one oi
MOVING COIL AMMETERS 113
which is seen at j ; it can turn in a narrow air gap between the
pole pieces and cylindrical iron core of a magnet, like the
one shown in Figs. 47 and 48, the movement being controlled
by spiral hair springs, s s, of non -magnetic material, which also
serve to lead the current into and out of the coil.
Fig. 72. — Working Parts of Weston Mil. Ammeter.
Figs. 73 and 730 illustrate a form of moving coil instrument
made by Messrs. Nalder Brothers and Thompson, of London.
The latter figure shows separate views of (a) , the magnet with pole
pieces, (b), the cylindrical core and its support, (c), the coil with
control springs, pivots and pointer attached, and (d), the brass
bar which carries the top jewel, in which the pivot on the pointer
end of the coil works, whilst in the former figure the parts are
assembled, but the cover removed to show their relative positions,
and also the scale, reading from o to 150 milliamperes.
To obtain a strong magnetic field in which the coil turns, the
air gap is made as short as possible, consistent with freedom
of motion, and the coil is made very light in order to prevent
damage to pivots by wear and transit, and also to keep its in-
ertia small. By these means the movements of the coil when
the current through it changes, are made quick and decisive,
and oscillations of the coil about its new position of equilibrium
are checked by currents induced, in the metal frame or " former "
on which the coil is wound.
As the strength of the magnetic field in the air gap is very
114
PRACTICAL ELECTRICITY
nearly uniform, the turning moment exerted by the coil is
practically proportional to the strength of current passing through
it, and, as the control springs exert a torque approximately
proportional to the angle of twist, a given change in current
produces the same change of deflection whatever the initial
position of the coil ; consequently the divisions of the scale are
uniform. Further, as the coil and pointer are carefully balanced,
the ammeter can be used in any position.
Fig. 73.— Nalder Bros. & Thompson Moving Coil Milliampere Meter.
The spiral springs which lead the current to and from the coil
are of necessity kept of small cross-section, otherwise they would
be too rigid, the control exerted would be excessive, and the
instrument would be insensitive. This limits the strength of
current, which can be led into the coil to a fraction of an ampere,
for with larger currents the springs woul<J become heated and
change their elasticity. Consequently, to measure large currents
with instruments of this class, they must be "shunted" as
described in Section 19, so that only a fraction of the whole current
passes through the moving coil. In many cases this is a great
convenience, for the shunt and ammeter may be a considerable
distance apart without the necessity of incurring great expense
in thick copper wires. For example, an instrument showing the
MOVING COIL AMMETERS
strength of current sent out from an electric lighting station may
be placed in the engineer's office and the " shunt " near the
dynamos, the two being connected by comparatively Email and
therefore inexpensive wires.
Fig. 73«. — Parts of Nalder Bros. & Thompson Instrument.
A simple form of moving coil ammeter, constructed by
Messrs. Paul, is seen in Fig. 74. It consists of a deep cylindrical
permanent magnet, M, with a very narrow air gap. In this
gap is suspended, by means of a very thin strip of phosphor
bronze, a coil wound in accordance with the principle developed
by the authors for obtaining the greatest deflecting torque
with a given strength of field,
a given current, a given number
of windings of wire on the coil,
and a given moment of inertia.
The coil, which is shown full
size in Fig. 74^, has no station-
ary iron core in its centre, as in
the Weston or Nalder instru-
ment, but the bundle of wires
which form one side of the coil
c are nearly in contact with the
bundle of wires forming the
other side c, so that the cross-
section of the coil has the form
of two circles almost touching
one another. The coil is con-
tained in a thin tube, T T, made of silver, partly to protect it
from mechanical injury and partly in order that the instrument
may be rendered dead beat by the eddy currents, which are in-
duced in the good conducting silver tube when it swings in the
Fig. 74. — Ayrton and Mather's Moving Coil
Ammeter, about one-fourth of the full size.
PRACTICAL ELECTRICITY
magnetic field, damping the motion of the tube and quickly
bringing it to rest.
One of the terminals of the ammeter xx (Fig. 74) is connected,
by means of the spiral of wire s with the top of the phosphor-
bronze strip, and the bottom of this strip is gripped in the screw-
clip sx (Fig. 74<z) , to which one end of the wire wound on the coil
c c is soldered. The other end of the coil of wire is soldered
to a similar screw-clip, s2, at the bottom of the silver tube, one
or both of these clips being insulated from
the tube itself by a collar of ebonite, E,
and in the lower clip s2 is gripped the
upper end of a spiral made of extremely
fine phosphor bronze, the lower end of
this spiral being attached to the terminal
T2 (Fig. 74).
Or instead of using the screw-clips, Sj
and s2 (Fig. 740), the bottom of the
straight strip of phosphor bronze, which
supports the coil, and the top of the
phosphor-bronze spiral, which is used to
make electric connection with the lower
end of the coil, may be soldered in position.
The pointer p p (Fig. 74^) is made of a
narrow tube formed out of very thin
aluminium sheet, the ends of this tube
being squeezed flat in a vertical plane at
the place where it projects over the scale
so that the deflection may be accurately
read. The best way of fastening the
Fig. 74«-~ Moving Coil, full J °
size, of Ayrton and Mather's pointer to the Silver lube IS as lOllOWS t
Under the top clip s1 there is screwed a
piece of aluminium A B /, cut out of somewhat thicker aluminium
sheet than that used for making the pointer, and shaped as shown.
The narrower end of this piece is rolled up into a little tube, tt
into which the end of the tubular pointer is fixed with varnish.
On removing the glass shade G G (Fig. 74) which covers up
the ammeter and protects it from dust and draught, the pointer
can be accurately adjusted to zero by turning the nut N.
The final adjustment of the sensibility of a permanent magnet
instrument can be conveniently made by slightly altering the
strength of the field in the neighbourhood of the coil. This can
be easily done by diverting more or less of the lines of force
through a piece of iron, the number so diverted being varied by
altering the distance between one of its ends and one pole of the
SINGLE PIVOT GALVANOMETER 117
Fig* 75- — Paul's Single Pivot Galvanometer.
magnet, with an adjusting screw, the other end of the iron being
permanently in contact with the other pole of the magnet ; such
a device is called a " magnetic shunt."
Fig. 75«.— Plan of Single Pivot Galvanometer.
Single Pivot Moving Coil Galvanometer. — Another form of
moving coil instrument intended for measuring very small
it8
PRACTICAL ELECTRICITY
currents, say millionths of an ampere, is shown in Figs. 75, 750,
75&, and 750. The former figure gives a general view of the
galvanometer, whilst 75*2 and 756 are respectively plan and
section. In this instrument the coil H is circular and surrounds a
soft iron sphere, E (Figs. 750, 756, 750), the magnetic field in which
Fig. 756. — Section of Single Pivot Galvanometer.
the coil moves being produced by a ring shaped magnet, D. The
coil is pivotted at K, the centre of the sphere (Fig. 75^) ; a verti-
cal radial spindle seen between H and K is attached to the coil
at its upper end, and at its lower end carries a pivot resting on
the jewel. A pointer L, fixed to the bottom of the coil, moves over
a scale M, on which the deflection of the instrument may be
read. Current is led to and from the moving coil by spiral
springs, shown above and below the coil in Fig. 75c, which also
serve to control the
movement.
The device of
pivotting the coil at
the common centre
of coil and sphere
ensures that the coil
shall swing clear of
the core and mag-
net, even if the
instrument is not
quite level ; it also
prevents accidental displacements, and the use of a single pivot
instead of two considerably reduces the friction.
44. Soft Iron Ammeters : Spring and Gravity Control. — Per-
manent magnet ammeters, especially moving coil ones, require
very careful workmanship in their construction, and are in
consequence somewhat expensive. Instruments whose action
depends on the magnetisation of soft iron when a current passes
Fig. 75 c. — Core, Coil and Pole Pieces of Single Pivot
Galvanometer.
SOFT IRON AMMfcTfiRS
119
found it can be made comparatively cheaply. In some forms the
deflecting force is due to the repulsion of two pieces of iron
magnetised in the same direction by the current to be measured,
and in others the attraction of magnetised iron is utilised. In
either case the controlling force may be furnished by a spring >
or by gravity acting on a weight fixed to the axis which carries
the pointer. The latter form of control is commonly used in
instruments which are fixed in position, whilst for portable
Figs. 76 and y6a. — Nalder Gravity Control Ammeter, two-thirds of full
ammeters, or ammeters for use on shipboard, spring control
is generally adopted.
A repulsion type instrument with gravity control, as made
by Messrs. Nalder Brothers & Thompson, Limited, is shown
in Fig. 76, and the working parts to a larger scale in Fig. 760.
The pointer P is fixed to an axle, E F, which is pivotted at its
ends, and a weight w tends to hang vertically and keep the
pointer at zero. To the pivotted system an iron rod, or bundle
of iron wires, A B, are attached, and lie parallel to a bundle of
wires, c D, fixed to the framework in which the axle is pivotted.
lao PRACTICAL ELECTRICITY
When an electric current passes round the coil, say in a clockwise
direction, the ends A and c of the iron wires will be south seeking
poles, and the ends B and D, north seeking ; the two electro-
magnets will therefore repel each other with a force depending
on' the strength of the current, and A B being movable, it will
be pushed away from c D, thus causing the pointer to move in a
counterclockwise direction. The displacement will go on until
the controlling moment exerted by the weight w balances the
moment due to the
forces of repulsion,
when the pivotted
system will be in
equilibrium, and the
position of the
pointer, if the in-
strument has been
properly graduated,
will indicate the
strength of the cur-
rent passing through
the coil. To pre-
vent undue vibra-
tion of the pointer
an air dash pot,
Fig. 766.— Damping Device in Nalder Ammeter. P' ^^ box-shaped
vane v, are arranged
as shown in Fig. j6b, in the recent form of instrument, and the
resistance to motion, due to displacement of air by the movement
of v, soon brings the system to rest.
Reversing the current through a soft iron ammeter of the
kind described above does not reverse the deflection, for although
the polarity of the pieces of iron will be reversed by this change,
the force between them is still one of repulsion, so that the
direction of deflection of such instruments is the same whether
the current passes in one direction or the other. This is true of
all soft iron ammeters as of electrodynamometers. Another
property common to these instruments is the nature of the scales,
for when the pointer is in any given position the deflecting
moment is approximately proportional to the square of the
strength of current, and owing to this fact the divisions are
usually crowded together near the zero and open out higher
up the scale.
The current which produces full deflection of the pointer
of an ammeter can be varied to suit actual requirements by
SOFT IRON AMMETERS
121
altering the winding of the coil. When large currents are to be
measured only a small number of turns are necessary, whereas
if a small current is to be determined, the number of turns on the
coil must be large. In fact, with a given size of coil and given
working parts, the product amperes multiplied by turns is approxi-
Figs, 77 and 770;. — Evcrshed Gravity Control Ammeter, two-thirds
of lull size.
mately constant.* Small variations can be made by altering the
controlling force.
Another form of soft iron ammeter is shown in Figs. 77, 770,
and is made by Messrs. Evershed & Vignoles. Limited. The
moving part, or needle, A B, Fig. 770, is a half cylinder of
sheet iron mounted concentric with the staff s s, which is
pivotted at its ends, and controlled by the weight w. The staff
passes along the axis of a brass tube, x T, Fig. 77, the back end of
which carries the back jewel, and around the outside of this
* To this product the name ampere-turns is given.
122 PRACTICAL ELECTRICITY
pig. 76. — Perspective View of New Evershed and Vignoles Instrument.
Fig. 7&a. — Section of Evershed and Vignoles Instrument.
SPRING CONTROL AMMETERS 123
tube is wrapped a triangular shaped piece of soft sheet iron
shown at c D. When a current passes round the coil the needle
A B moves towards the narrower part of c D against the action
of the control weight w. By fixing c D to T T at different posi-
tions circumferentially, the shape of the calibration curve, and
therefore the nature of the scale of the instrument, can be
varied to suit different requirements. For example, the scale
may be one of nearly equal divisions, say from 5 of the highest
Fig- 79- — Hartmann and Braun Hot Wire Ammeter.
reading, or may have divisions near together at each end and wide
apart at some particular place.
The latest form of Evershed & Vignoles' instrument is illustrated
in Figs. 78 and 780. It has an oval shaped coil c c with a narrow
internal cavity, c, into which a volute shaped piece of sheet iron,
F F, is attracted when a current circulates in the coil. The iron is
fixed on a pivotted staff s, to which the pointer P, damping vane
v, and several balance weights are attached, and the movement
is controlled by a spiral spring Q. Behind the scale plate is a
sector shaped box, B, in which the vane v moves, and the air
friction caused by air displacement effectually checks oscillations
of the system without introducing errors due to solid friction.
The winding of the coil is, of course, arranged to suit the current
to be measured.
124
PRACTICAL ELECTRICITY
45- Hot-Wire Ammeter.- -Instead of making use of the
magnetic property of electric currents, this form of instrument
utilises the heating effect of a current, for its indications depend
on the expansion of a wire which is heated by the passage of the
current. As the expansion of metals, for mcderate changes of
d temperature , is extremely
— ^rr----1 "* small, some method of mag-
nifying the extension is
necessary. In the instru-
Fig. 80. — Sagging Wire. i ' T?'
this is done by aid of " sagging wires" and depends on the
fact that a nearly straight wire fixed at both ends and kept
taut by a force at right angles to the wire applied near the
middle point, alters its sag, for a given change of length of the
wire, by an amount greater than the change of length. Suppose
a small force,/ (Fig. 80), to act on the nearly straight wire w w,
fixed at Q and Q', the point R will be displaced a little from
the straight line Q Q' ; and this displacement is called the " sag "
of the wire. If now the wire w w increases in length by a small
amount, /, due to heating, say, the wire will take up the position
Q R' Q', the distance R R' is the change of sag due to this change
of length, and is greater than /. This arrangement gives one
magnification, and in the actual instrument double magnification is
obtained, as shown in
Fig. 8oa. Here the
force / (Fig. 80) is pro-
duced by the tension
of a second wire, w',
w', fixed at K and at-
tached to w w at R,
and which is kept taut
by a silk thread s pass-
ing round a pulley, p,
and attached to a
stretched spring, s,
anchored at L. When
a Current IS passed Fig gofl_ Diagram of « Sagging Wire " Magnifying System.
from Q to Q' through
w w, it is heated, and expands and sags, thus causing w' w' to sag
and allow the spring s to contract and turn the pivotted pulley p
and pointer p in a clockwise direction. The amount of movement
of the pointer depends on the extension of w w, and therefore on
the heating produced by the current, so the scale over which the
pointer moves can be graduated to read the current directly.
HOT WIRE AMMETER 125
Fig. 79 shows an elevation of a modern form of hot wire in-
strument, the lettering of which corresponds with that of Figs.
80 and Sou, but the spiral spring shown in Fig. Soa is replaced by
a flat spring s in Fig. 79 The wire w w is made of platinum-
iridium, and is carried by an iron plate 1 1 and a piece of nickel
steel, N, fixed to 1 1. This arrangement is used to prevent
changes of sag taking place when the instrument as a whole
changes its temperature, for the proportions of iron and nickel
steel* are chosen so that the coefficient of expansion of
the combination is equal to that of the wire ww. To prevent
the pointer oscillating much about its position of equilibrium, an
aluminium sector, A, is attached to the axis carrying the pointer,
and this passes between the poles of a permanent magnet, M.
When the sector is moving in the magnetic field electric currents
are produced in the metal which tend to stop the motion, and
by this means the pointer is brought quickly to rest.
A hot wire instrument, such as shown above, is only suited for
measuring small currents, say up to about 0-2 ampere (200
milliamperes) , because the wire w w must be thin in order to be
kept taut by a small side-pull. For larger currents " shunts "
are required, such as described in Section 19, in order that
only a fraction of the whole current passes through w w. It is
also customary, in large current instruments, to arrange that
two or more parts of the wire w w are electrically in parallel.
Until quite recently platinum -silver was used for the working
wire w w, because of its comparatively large coefficient of expan-
sion, but its relatively low melting point and small tensile
strength proved serious disadvantages. Platinum-iridium is
far superior in both respects, and permits of the wire being safely
heated to a far higher temperature, thus obtaining increased
elongation, and at the same time reducing the errors caused by
external changes of temperature.
* Nickel steel has a very small coefficient of expansion.
CHAPTER TV
DIFFERENCE OF POTENTIAL, AND RESISTANCE
46. Difference of Potentials — 47. Potential of the Earth Arbitrarily
called Nought ; Positive and Negative Potentials — 48. Measurement
of Potential Difference — 49. Electrometer — 50. Ohm's Law — 51.
Resistance — 52. Ohm : Unit of Resistance — 53. Resistance Coils
and Resistance Boxes — 54. Volt — 55. Ohm's Law Applicable to
Complete Circuits ; E.M.F. — 550. Electromagnetic Definition of
E.M.F. — 56. Current Method of Comparing P.Ds. — 57. Reason
for Using High Resistance Galvanometers for P.D. Measurements,
and Low Resistance Galvanometers for Current Measurements — 58.
Voltmeter — 59. Resistances of Ammeters and Current Voltmeters —
60. Ammeters used as Voltmeters — 61. Moving Coil Voltmeter —
62. Calibrating a Deflectional Voltmeter — 63. Voltmeters used as
Ammeters — 64. Gold Leaf Electroscope — 65. Sensibility of Gold-Leaf
Electroscopes.
46. Difference of Potentials. — When a current of electricity
is flowing through a wire, it has the same strength at all cross-
sections of the wire. If, for example, the wire be cut anywhere
and a galvanometer be put in circuit, the galvanometer will
always show the same deflection while the same current is
flowing ; or if several galvanometers, or ammeters, be placed at
different parts of the same circuit, each instrument will be found
to indicate the same current. In the same way, in the case
of a water-pipe, the quantity of water passing every cross-section
of the pipe per second is exactly the same as soon as the flow of
water becomes steady. Just at the commencement, when, for
example, some water has entered at one end of the pipe, and none
has flowed out at the other — when the pipe is filling in fact —
the flow at different cross-sections may be different ; so also, in
many cases, just at the moment after completing an electric
circuit, the current will differ at different cross-sections. But as
soon as the flow in each case becomes a steady one this difference
disappears, and the strength of the water current — that is, the
number of gallons of water passing per minute (not, of course,
the velocity of the particles of water) — is the same at all parts of
the pipe, even if the pipe be broad at some points and narrow
at others. In the same way the strength of the electric current
126
DIFFERENCE OF POTENTIAL OR P.O. 127
flowing through a single circuit is " imiform "* at all parts
of the circuit, independently of the thickness of the conductor,
and of the material of which it is made.
But although the stream of water is the same at all parts of the
pipe, the pressure per square inch of the water is by no means the
same, even if the pipe be quite horizontal and of uniform cross-
section. This pressure per square inch of the water on the pipe,
which is the same as the pressure per square inch of one portion
of the water on another portion adjacent to it, becomes less
and less as we proceed in the direction of the flow. It is, in
fact, this difference of pressures, or " loss of head " as it is some-
times called, that causes the flow to take place against the friction
of the pipe, the difference of pressures at any two points, in the
case of a steady flow through a horizontal pipe of uniform sectional
area, being balanced by the frictional resistance of that length of
pipe for that particular rate of flow.
Quite analogous with this, there is, in the case of an electric
current flowing through a conductor, a " difference of potentials "
between any two points in the conductor, and this difference of
potentials, or " potential difference " (or " P.ZX" as it may be
shortly called), is needed to overcome the resistance of the con-
ductor, or opposition that it offers to the passage of an electric
current through it. In fact the analogy between difference of
potentials and difference of fluid pressures is so marked that the
name " pressure " is now frequently used to stand for difference
of potentials.
The pressure per square inch of the water at any point of a
tube conveying a stream can be ascertained by attaching a verti-
cal stand-pipe to the tube, and observing to what height the water
is forced up in this stand-pipe, and if at a number of points,
PJ, P2, P3, P4, P5, P6 (Fig. 81), in a glass tube, 1 1, conveying a stream
of water, a series of vertical glass stand-pipes, S1s2 . . . se, be
fixed, the height to which the water is forced up in them will
show the distribution of pressure along the tube. If the tube
it be straight and of uniform cross-section, and if the flow be a
steady one, the tops of the water columns in the stand-pipes will
be found to lie all in one straight line, Qi Q2 • . . Q6 ," therefore,
if the length PX P2 of the uniform tube be equal to the length
P4 P5, the difference between PX QL, the height of water in the
stand-pipe Sj, and P2 Q2, the height of water in the stand-pipe
* Uniform refers to space, constant to time. The height of the houses
in a street is generally not uniform, but it is constant so long as there
is no change made in the height of the houses. If water be run out of a
cistern the level at all parts of the surface of the water is uniform, but it
is not constant, since it steadily falls as the water runs out.
128
PRACTICAL ELECTRICITY
s2, is exactly equal to the difference between P4 Q4 and P5 Q5.
Also,< if the length Px P4 be three times the length P4 P5, the
difference between PI QL and P4 Q4 is equal to three times the
difference between P4 Q4 and P5 Q5. Or, in other words, when
there is a steady flow of liquid through a uniform tube, the difference
o/ pressure between any two points is proportional to the distance
Fig. 81.— Apparatus for Testing the Distribution of Water Pressure.
between these points. And this is true whatever the inclination
of the tube 1 1 to the horizontal, provided that the tube is
straight and of uniform cross-section everywhere.
If the tap T! and the screw pinch -cock s1 be fully open, and the
screw pinch-cock s2 be fairly open, the stream of water through
the tube 1 1 will be rapid, and the slope of pressure — that is, the
line Qx Q2 .. . . Q6 joining the tops of the columns of water in
the stand-pipes — will be steep. If now the pinch-cock s2 be
screwed up a little so as to impede the passage of the water, the
flow will be decreased, and the slope of pressure R± R2 . . . R6 will
be less inclined to the horizontal than Qj Q2 . . . Q6.
As the pinch-cock s2 is more and more screwed up, the pressure
line will become more and more horizontal until, when the
flow is entirely checked, the line Hx H2 . . . H6 joining the tops
of the columns of water in the stand-pipes becomes quite hori-
zontal and at the same level as the water in the cistern cx.
From this we see that the pressure is the same at all points along
the horizontal pipe PX P6, through which no flow is taking place,
so there is no difference of pressure between any two points
WATER ANALOGY OF ELECTRIC FLOW 129
along the pipe ; and as " difference of potential " is analogous to
fluid pressure, we conclude that there is no P.D. between points
in an electrical conductor through which no current is passing,
or, in other words, " all points of an electrical conductor on which
electricity is at rest are at the same potential."
It will be noted that if there be any flow, the level of the water
in the first stand-pipe Sj is less than that in the cistern itself,
which is seen through a little glass window at the right of the
cistern Cj. This is on account of the resistance offered to the
flow by the tap TJ and by the indiarubber tube TX t. Similarly,
if the pinch-cock sl be screwed up so as to check the flow
between P3 and P4, there will be a sudden drop in pressure
between P3 and P4, so that the tops of the water columns in the
standpipes will now be in two different straight lines, ux U8 U3
and U4 U6 U6, parallel to one another, but the latter U4 u, U6,
much lower than the former.
As the pinch-cock s± is screwed up more and more the lines
ux U2 U3 and U4 us U6 will become more and more horizontal,
but at a greater distance from one another, until, when sx is
entirely closed, the former line will coincide with Hx H2 H3, while
the latter will sink down to the level of the tube P4 P6 P6 itself.
In a very similar way the " electric pressure " (or " potential"
as it is usually called) at different points of a wire conveying a
current, can be measured by apparatus which we shall presently
describe, and if a number of measurements be made of the
potential at different points of a circuit conveying a current, it
will be found that the results are smaller and smaller as we
proceed in one direction ; and, further, if the conductor be all of
uniform gauge and material, and the electric current be a
steady one, it will be found that the P.D. between any two points
is proportional to the length of the conductor between these
points (see Section 49). f
Electricity is put in motion, and a current of electricity is
produced, as a consequence of the potential varying from place
to place, just as a current is produced in water when subjected
to pressures which are not uniform. In order to produce and
maintain a current of either water, or electricity, work of some
kind has to be done. Thus in Fig. 81 the current of water in
the tube 1 1 will gradually diminish as the water passes from the
upper to the lower reservoir, and will cease entirely as soon as
the reservoir q is empty. In order to maintain the current it is
necessary to provide some means of keeping up the level of water
in the upper reservoir, and the simplest method of doing so is by
means of a pump working at such a rate that water is raised from
I
130
PRACTICAL ELECTRICITY
the lower vessel C2 to the upper one ct just as fast as it flows from
G! to C2 through the tube 1 1. Exactly analogous with this pump
in the water circuit is the " voltaic cell," or the " dynamo machine,"
or other " current generator," in the electric circuit. A current
generator does not create electricity any more than a fire-engine
Supply
Overflow
Fig. 82.
to
-Alternative Arrangement of the Cistern for the
Apparatus in Fig. 81.
creates water, it merely
sets it in motion, and
in either case work has
to be done in keeping
up the flow (see Chap-
ter VII. on Electric
Energy and Power).
In the case of the
water flow we may
commence by filling
the reservoir c± and
maintain the level by
allowing water to flow from the cistern of the building into the
reservoir cx as fast as it flows out. Or, to save trouble, we may
let the water run into the reservoir rather faster than it flows
out through the tube tt, and allow the surplus to flow out
through an 'overflow pipe o (Fig. 82). With the latter arrange-
ment the level of the water in the reservoir cx will remain auto-
matically constant whatever be the flow through the tube 1 1,
provided, of course, that the tap T2 be opened enough to cause
the flow from the house cistern into the reservoir to be never less
than the flow out through the tube 1 1.
If the substance
flowing were a gas,
the distribution of
pressure could not be
measured by stand-
pipes, since if the pipes
were open at the top,
the gas would flow out ;
or the outside air
would flow in, and, if
the pipes were closed they would all be filled with the gas itself,
or with a mixture of gas and air.
The distribution of pressure along a pipe, p p (Fig. 83), convey-
ing a stream of gas might be measured relatively to the outside
atmospheric pressure by means of " manometers" Mlf M2, M3
attached at the points P1,-P2, P3 of the pipe, the difference of level
of the liquid on the two parts of the tube of each manometer
Fig. 83. — Apparatus for Testing the Distribution of Gas
Pressure relatively to the Atmospheric Pressure.
DISTRIBUTION OF GAS PRESSURE 131
measuring the excess of the pressure of- the gas at that part of
the pipe over the atmospheric pressure. Or, if we desired that our
measurements should be independent of the atmospheric pressure
and merely indicate the pressure at various parts of the pipe
relatively to the pressure at one point P4, then the manometers
might be arranged
as in Fig. 84, in
which case the "
difference of level
of the liquid in the
curved tube of any
one manometer, M2
(say), would show
how much the
nrP^lirf1 of thp P~a«; Fig- 84. — Apparatus for Testing the Distribution of Gas Pressure
relatively to the Pressure at One Point of the Pipe.
at the point P2 of
the pipe p p exceeded the pressure at the point P4. Perhaps
the most convenient way would be to construct the apparatus
as seen in Fig. 85, since then the pressure of the gas at any point
P2 relatively to the pressure at P4 would be at once seen from the
distance the top of the column of liquid in the tube M2 (Fig.
85) was below the horizontal line H H ; and the difference of
pressure of the gas at any two points P2 and P3 would be therefore
measured by the difference in the depths below the horizontal
line H H of the tops of the liquid columns in the manometers
M2 and M3.
If the pressure of the atmosphere surrounding the apparatus
in Fig. 83 were changed, then, although the flow of gas along the
Fig. 85. — Simpler Apparatus for Testing the Distribution of Gas Pressure
relatively to the Pressure at One Point of the Pipe.
pipe p p might remain exactly the same, as well as the pressures
at its two ends, the difference of level of the liquid in each of the
manometers in this figure would change. But the level of the
132 PRACTICAL ELECTRICITY
liquid in the manometers in Fig. 85 is wholly independent of the
outside atmospheric pressure, and depends solely on the length,
cross-section, shape, and internal character of the pipe p p, on
the rate of flow, and on the nature of the fluid flowing through
the pipe. These manometers tell us nothing about the absolute
pressure of the gas at the different points of the pipe through
which it is flowing, but only the pressures relatively to the pres-
sure at the point P4.
47. Potential of the Earth arbitrarily called Nought ; Positive
and Negative Potentials. — So in the same way the electric po-
tential of a point in a wire through which a current is flowing is
usually measured relatively to that of some other point of the wire.
And even when one point of the wire is connected with the earth
and the potentials of different points of the wire are measured
above or below the potential of the earth, which is arbitrarily called
nought, it is still but a relative measurement, for in thus taking
the potential of the earth as the potential level to measure from,
no assumption is made as to the earth having no electricity on it.
Measuring electrical potentials relatively to that of the earth
is, therefore, like measuring heights above the Trinity water-
mark, or measuring longitude east or west of Greenwich, the
place which is arbitrarily said to have zero longitude.
A similar convention is followed in the measurement of tem-
perature, for in the centigrade scale the temperature of melting
ice is called o°, while in the Fahrenheit scale the zero is a tem-
perature much below this, and one which is roughly that of a
mixture of ice and salt. Now, although Fahrenheit is said to
have called this temperature zero because he had an idea that
it was the lowest temperature that could be produced artificially,
no such assumption is at present made in calling this particular
temperature o° F.
In addition to a P.D. being said to exist between two points in
a conductor through which a current is flowing, any two con-
ductors are said to differ in potential when there is a tendency for
electricity to pass from one of them to the other, just as the
contents of two gas-holders would be said to differ in pressure
if a tendency for the fluid to pass from one to the other existed,
or the tendency for the fluid to pass into the atmosphere was
different in the case of one, from what it was in the other. This
tendency may manifest itself in four ways : —
(i) By the production of a current (lasting, it may be, for only
the fraction of a second) when the two conductors are touched
together, or when they are electrically connected by means of a
wire, or other conductor ;
EFFECTS OF POTENTIAL DIFFERENCE 133
(2) By a " brush discharge " or an " electric spark " passing
between the conductors when they are near together, and when
the P.D. between them is high ;
(3) By small light bodies, such as grains of dust, pieces of
paper, pith, etc., being attracted backwards and forwards between
the conductors ;
(4) By the conductors trying to approach one another, as if
there were an attraction between them.
When different pieces of electrical apparatus are enclosed
in a metallic box (a not infrequent arrangement) the potential
of the box itself is usually called nought, and the potentials of
the different bodies inside it are measured relatively to that
of the box by the methods subsequently described. This
box in such a case is sometimes called, in electrical language,
the " earth," but it must not, therefore, be inferred that there
is any metallic connection between the box and the ground ;
the box and all the apparatus inside it might, indeed, be up in
a balloon, and still the joining of some part of the internal
apparatus to the metallic box by wire might be called " earth-
ing " that piece of apparatus.
A conductor is said to have a " positive potential " when on
earthing the conductor a current flows from the conductor
to the earth, and a "negative potential" when the current
flows in the opposite direction. Also when two conductors,
A and B, are in such a condition that, if joined with a con-
ductor, a current would flow from A to B, then, irrespectively
of the actual signs of the potentials of A and B, as denned in the
last sentence, the potential of A is said to be " higher " than that
of B ; (see Section n for the definition of the direction of a
current). Further, if two bodies, whether conductors or not,
differ in potential, a positively electrified body, placed in their
neighbourhood, tends to move away from the body having the
higher potential towards the other body having the lower potential.
48. Measurement of Potential Difference. — The question now
arises, How are we to measure potential differences ? i.e.,
How are we to determine whether one P.D. is two or three times
another P.D. ? This may be answered in the same way as the
similar question discussed in Section 7, which relates to the
measurement of current strength, for the magnitude of one of
the effects exhibited by bodies at different potentials may be
chosen as a measure of the P.D. between them. But as electrical
phenomena are manifestations of energy (which, according to the
law of conservation of energy cannot be created or destroyed,
but only changed in form), it is desirable that the choice be such
134 PRACTICAL ELECTRICITY
as will make the relations between the electrical quantities and
the mechanical quantity Energy (or work) as simple as possible ;
for this purpose the property of attraction (Chapter II., page 81)
is the most convenient one to choose.
If a quantity of water of volume Q flows through a uniform
horizontal pipe, A c B, Fig. 850, the energy lost by the fluid as it
passes from A to B (or the work done in overcoming friction in the
intervening portion
of the pipe), is equal
to Q multiplied by
the difference of
A! ^ C_| B1 pressure between A
Fig. 85*. and B. A current
of electricity can
also do work, as is shown by the ventilating fan, Fig. 9,
shown being driven electrically, and by the heating effect of a
current, and the analogy between hydraulic and electrical work
will be preserved if the measure of P.D. is such that the work done
by a quantity of electricity q passing from a point A' to a point
B' in a wire A' c' B' is equal to the product of q and the potential
difference between A' and B'. This requirement is satisfied if
we take as our definition the following : — The force of attraction
between two bodies in definite relative positions, but at different
potentials, is proportional to the square of the P.D., or in other
words, the potential difference between two bodies in definite relative
positions is proportional to the square root of the force of attrac-
tion between them. A similar definition might have been chosen
for current strength, for as already shown in Section 39, the force
between two coils carrying the same current is proportional to
the square of the strength of the current, and therefore the current
strength is proportional to the square root of the force.
In the electrodynamometer, altering the current strength
alters the magnetic condition of both of the coils in the same
ratio, and in consequence alters the force between them in the
duplicate ratio ; so also in an instrument for measuring the force
of attraction between two conductors at different potentials,
altering the P.D. between them alters the electrical condition
of both, and thus changes the force between them in a duplicate
ratio.
49. Electrometer. — The forces between conductors at dif-
ferent potentials are called electrostatic forces, because they are
believed to be due to quantities of electricity at rest on the
surfaces of the conductors, and instruments for measuring these
forces are called " electrometers"
ELECTROMETERS
135
Electrometers, like galvanometers, are of two kinds, those in
which the measurement is made by noting how much a needle is
deflected against the action of a controlling force, and those in
which we observe by how much the controlling force must be
increased to resist the motion of the needle and keep it in a fixed
position. The latter or zero type of electrometer has an advan-
tage over the former, in that it enables the simple definition of the
Fig. 86. — Ayrton and Mather's Zero Electrometer, or Zero Electrostatic
Voltmeter, one-third of the full size.
measurement of difference of potential given above to be made
use of in practice.
The electrostatic forces between bodies at different potentials
are very small in magnitude, unless the potential differences are
very large, and it is only within comparatively recent years
that instruments for measuring the forces produced by P.Ds.,
such as are used for electric lighting in houses and for telegraphic
purposes, have been constructed. A zero electrometer devised
by the authors is shown in Figs. 86 and S6a ; the moving part
N, or needle* as it is called, takes the form of two thin narrow
• A magnetised sewing-needle having been originally used for the
suspended magnet in a galvanometer, the name needle came gradually
to designate the little magnet in a galvanometer, whether it was long
136
PRACTICAL ELECTRICITY
pieces of aluminium a, a (Figs. 86 and S6a), joined together at
the top and bottom by cross pieces, b, b, and supported by means
of a thin strip of phosphor bronze from a head H, carrying an
index c, which can be turned round over a graduated dial. The
conductors, 1 1, or the " inductors " as they are called, into
which the two parts a, a of the needle are attracted, are shaped
as shown, and, by means of a pointer p, carried from the bottom
of the needle, the position of the needle can be observed. As
usual, parallax is avoided by observing the reflection of this
pointer p in a piece of looking-glass g
fixed to the base of the instrument.
Any P.D. set up between the needle and
the inductors is then measured by turning
the head H until the pointer p (carried by
the needle) is brought into the same
position that it occupied when the needle
and the inductors had the same potential ;
the angle through which the index c has
been turned is noted, and its square root
taken. For this is the angle through which
the strip carrying the needle has been
twisted, and, therefore, this angle measures
the moment of the force, or the torque,
that has been exerted on the needle.
The terminals TX T2 are connected respectively with the
needle N and the inductors 1 1, and equality of potential of these
two bodies can be secured by connecting these terminals together
with a piece of wire, thick or thin. For if there be any difference
of potential, a momentary current will flow through this wire
which will annihilate the P.D.
Further, il the terminals be joined respectively by wires with
any two conductors A and B, momentary currents will flow,
and the potentials of the needle and inductors will become
respectively the same as those of A and B. In fact, we may
say generally, that if any number of conductors be touched together,
or be joined by wires, and if no current be flowing between any
of the bodies, the conductors and wires are all at the same potential.
To be strictly correct, this general proposition requires that all
the conductors should be made of the same material, and be at
the same temperature.
and pointed like a sewing-needle, or short and blunt. And now the
expression needle is employed for the suspended movable part of an
electrical measuring instrument, even when the shape of the moving
system in no way resembles that of a sewing-needle, as in the electro-
meter shown in Fig. 86,
Fig. 86«.— Details of Needle
and Inductors, rather larger
than full size.
ZERO ELECTROSTATIC VOLTMETER
This last proposition can be stated briefly and completely
thus : — the potential of all parts of a conducting system composed
of the same material at the same temperature and on which electricity
is at rest, is uniform.
In order to ensure that the electric force exerted on the needle
shall be wholly due to the P.D. between it and the inductors, and
that no part of this force shall be caused by the attraction of
external bodies, the in-
terior of the glass shade
is coated with a conduct-
ing transparent varnish
devised by the authors,
the composition and
action of which are ex-
plained later in Section
64.
The spindle of the
needle in the electrometer
(Fig. 86) moves in guides
top and bottom, the upper
guide being clearly seen
in Fig. 860, which shows
the top of the needle and
of the inductors rather
larger than full size ;
hence the instrument may
be turned upside down,
or carried about without
its being necessary to
clamp the needle, and
without there being much
risk of breaking the thin
phosphor - bronze strip
supporting it.
If, in addition to sending a steady stream of water through
the tube it, shown in Fig. 81, the water in the tube be now
used as a conductor and a steady electric current b^ sent
through it, the various P.Ds. between the pairs of points
P! and P2, P2 and P3, etc., can easily be measured with the elec-
trometer just described by simply dipping wires, attached
respectively to the terminals of the electrometer, into the water
in the various pairs of standpipes sx and S2, S2 and S3, etc.
For, since there is no electric current in the water in a
stand-pipe itself, there can be no P.Ds. between the different
Fig. 866.— Ayrton and Mather's Zero Electrostatic
Voltmeter (Later Form).
138
PRACTICAL ELECTRICITY
parts of the water in the same stand-pipe ; hence the water in
the stand-pipes can be used simply as extensions of the wires
attached to the terminals of the electrometer. When the screw
pinch-cock s1 is fully open, so that the tube 1 1 is throughout of
uniform bore, it will be found that the P.Ds. between the different
pairs of points are related to one another in exactly the same
way as are the differences between the water pressures for the
same pairs of points.
Thus the distribution of potential along a uniform conductor
conveying a steady electric current is exactly analogous with the
distribution of fluid pressure along a uniform tube, through which
flows a steady stream of liquid.
50. Ohm's Law. — But if instead of measuring the P.D.'s
between different points along a conductor through which flows
a steady current we measure the P.D.'s between two fixed points
in a given conductor through which different currents are flowing,
then the P.D. does not vary with the current in the same way
that the difference of pressure between two points in a given
tube varies with the stream of fluid flowing through it. Let us
consider the second case first : — Keep the level of water in the
reservoir cx (Fig. 81) constant in the way already described, open
the screw pinch-cock s2 a certain amount, the screw pinch-cock
s1 being fully open, and, when the stream has become steady,
measure with a graduated glass the number of cubic centi-
metres of water that flow through the tube tt per second, also
the difference of pressure between two fixed points in the tube
PX and P6 for example. Next open the pinch-cock s2 a little
more, and again measure the number of cubic centimetres of
water per second that flow out of the tube, as well as the difference
between the height of the water in the stand-pipes sl and s6.
If such measurements be made for several different steady rates
of flow, numbers like the following will be obtained, and when
plotted they give the curve seen in Fig. 87, concave to the axis
along which difference of level is plotted.
Difference of Level
Flow in Cubic
Ratio of Difference
in Centimetres.
Centimetres per second.
of Level to Flow.
6-9
I-2O
5-75
12-4
, -" 2'OO
6-20
18-7
2-78
6-73
24-0
3'39
7-08
29'5
4-09
7-21
. 36-2
4-76
7*60
4*i
5'26
8-00
FLUID AND ELECTRIC FLOW DISSIMILAR 139
If the numbers in the third column were all the same it would
tell us that the ratio of the difference of level to the number of
cubic centimetres flowing per second — that is, the ratio of pressure
to current — was a constant for a given pipe. In that case the
points on the curve in Fig. 87 would all lie in one straight line,
and to double, treble, quadruple the current would require
Curve connecting Rate oj Flow of
»>§ Water with Loss of Head.
V. 5
|4
|3
1
3
2,
5 10 15 £0 £5 50 55 40
Difference, of level in centimetres
Fig. 87.
49
exactly double, treble, quadruple the pressure. But the numbers
in the third column steadily increase as the current increases,
and if we examine the numbers in the first two columns we
find that to increase, for example, the flow from i'20 to 4-76 cubic
centimetres per second — that is, to make the current not quite four
times as great— r-we have to increase the difference of level from
6-9 to 36-2 centimetres — that is, to increase the pressure more
than five times.
The quantity of water, therefore, that flows per second through
a given pipe does not increase as rapidly as the difference of
pressure between two fixed points in it, or in other words, we must
more than double, treble the difference of pressure to produce
twice, three times the flow, even although the tube through
which the water flows remains absolutely unchanged. It
might, therefore, have been expected that the same sort of
inequality would be found in the ratio connecting the P.D.
between two fixed points in a conductor and the current flowing
through it.
140 PRACTICAL ELECTRICITY
But that is not the case, for it the conductor K (Fi~s. 88
and 880) remains at the same temperature, and be not changed
in any way, experiment shows that the P.D. between two fixed
points, KJ, K2 in it, measured by the electrometer E (in the way
already described in Section 49), is directly proportional to
the current flowing through this conductor, the currents being
measured relatively to one another by any suitable galvano-
meter G,* for which the law connecting current and deflection
Fig. 88. — Apparatus for Testing Ohm's Law.
has been obtained by a relative calibration, as described in
Section 19.
For carrying out these tests the current can be conveniently
produced with a battery, B B, of what are known as " dry cells "
or of " accumulators " (for both of which see later Sections) ;
and its strength can be varied by altering the number of cells
employed. This alteration in the number of cells that are
used in the different tests, .can be easily effected by means
of the mercury switch -board, s s, seen in front of the battery of
cells in Fig. 88.
* For the details of the construction of the galvanometer illustrated
in Fig. 88, see Section 43.
A zero electrodynamometer (Fig. 886) may with advantage be sub-
stituted for the galvanometer G, for then the current will be propor-
lional to the square root of the reading of the dynamometer just as the
P.D. is proportional to the square root of the reading of the zero electro-
meter. With these instruments, as their laws are known from first
principles, no preliminary calibration is necessary.
VERIFICATION OF OHM'S LAW
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142
PRACTICAL ELECTRICITY
This experimental re-
sult, that the ratio of the
P.D. to the current, if
steady, is absolutely con-
stant for a given conductor
at constant temperature, is
known as " Ohm's Law,"
since it was first published
by Ohm in 1827, although
not exactly in the form
here given. And it is
important to notice that
all experiments that have
— _ been made to test its
Fig. 88a.— Diagram of Circuit for Testipg Ohm's Law. ,
accuracy, even when made
with the most sensitive instruments yet constructed, have failed
to detect any inaccuracy in this law.
It is sometimes stated that Ohm's law is self-evident, but
that misconception has arisen first from the law being so extreme-
ly simple, and secondly from its wide use in electrical calculations
having gradually led people to imagine that no connection
between P.D. and current for a given conductor, other than
direct proportionality, could
exist.
51. Resistance. — Since the
ratio of the P.D. to the current
has a constant value for each
conductor this ratio has been
called by a special name — the
" resistance " of the conductor,
and gradually people have
grown to think and speak about
the electric resistance of a wire
as being a definite property
which belongs to the wire like
its length and its cross -section.
If, however, the ratio of P.D.
to current had been no more
constant for a given conductor
than is the ratio of pressure
to flow for a given tube carry-
ing a liquid stream, it is
practically certain that this
Conception WOUld not have
UNIT OF RESISTANCE: THE OHM 143
come into existence. Therefore the mere statement that a
definite wire has a definite resistance 'is in itself an assertion,
although not of course a proof, that Ohm's law is true.
The analogy between the distribution of water pressure and
of electric potential is a very useful one for students to use, as
it enables them better to grasp the meaning of electric potential ;
but, like many analogies, it must not be carried too far ; for not
merely, as we have seen, is the ratio of difference of pressure to
the quantity of a fluid flowing per second not constant for a given
pipe, but any bend made in a straight pipe, even when the cross-
section of the pipe is in no way decreased, causes a diminution
in the flow for the same difference in pressure between its two
ends ; whereas bending a wire through which a steady electric
current is flowing, has no effect on the electric stream. Even
a sudden expansion in a pipe, that is an enlargement of the bore,
for a short distance checks the fluid stream, whereas if the cross -
section of a conductor be made larger for a short portion of its
whole length, either no change whatever is observed in the current,
or the change, if noticeable, is always an increase and never a
diminution in the steady current flowing.
52. Ohm : Unit of Resistance. — To compare the resistances of
two conductors we might connect them together in such a way
that the same current passed through both, and then find the
ratio of the P.D. between the ends of one of them to the P.D.
between the ends of the other, by the zero electrometer shown in
Fig. 86 or 866 ; this ratio of the P.Ds. would give the ratio of the
resistance of the two conductors. But if we wish to express the
resistance of a single conductor numerically, we must choose
a " unit " of resistance, in terms of which other resistances may
be stated. Various units of resistance, differing slightly from
one another, have been adopted from time to time, and called
the " ohm," but the value that was definitely recommended
to the Board of Trade in 1892 by the Committee appointed to
advise them, was defined thus : " The resistance offered to an
unvarying electric current by a column of mercury at the tempera-
ture of melting /c0 14-452 1* grammes in mass of a constant cross-
sectional area, and of the length of 106-3 centimetres may be adopted
as one ohm " ; and at the Electrical Congress held in Chicago
* This mass of mercury at o° c. is required to fill a tube 106-3 cm- l°ng,
whose cross-section is one square millimetre, very approximately, so it is
nearly exact to say that a column of mercury 106-3 centimetres long and one
square millimetre in cross- section, has, at the temperature of melting ice, a
resistance of one ohm. The mass of mercury rather than its cross-section
was specified in the recommendation because it is much easier to weigh the
mercury accurately than to measure the cross-section of a small tube to the
same degree of accuracy.
144 PRACTICAL ELECTRICITY
in 1893, this value was unanimously accepted by the Chamber
of Delegates, composed of members nominated by the Govern-
ments of the United States, Great Britain, France, Italy, Germany,
Mexico, Austria, Switzerland, Sweden, and British North
America.
Finally, for the purpose of distinguishing this unit of resistance
from any other, it was decided to call it by the name of the
" international ohm." The definition of " the international ohm "
given above, with the addition of two zeros after the figure 3 in
106-3*, thus making it 106-300, was agreed upon by the
delegates to the International Conference on Electrical Units
and Standards, held in London in October, 1908.
In choosing this particular unit the objects aimed at were : —
(i), To obtain a unit of convenient magnitude for practical
purposes which could be accurately made in any part of the
world, and so be suitable as an international unit', and (2), to
make the relation between mechanical energy and electrical energy
a simple one. For example, the unit of energy (or work) in the
centimetre, gramme, second (C.G.S.) system of mechanical units
is the erg, which is the work done by a force of one dyne acting
through a distance of one centimetre. This unit is a very small
one, so a large multiple of it, io7 ergs, (10 million ergs), is fre-
quently used as a practical unit of energy and called a joule.
One joule of energy represents a definite amount of heat (the
mechanical equivalent of heat being known). We have already
seen that when an electric current passes through a conductor
heat is generated in that conductor, the amount depending
on the strength of the current, the time the current flows, and the
resistance of the conductor. * Now the practical units of current
and time have already been agreed upon, viz., the ampere and
the second, and if the unit of resistance be chosen so that a
current of I ampere flowing through that resistance generates
in one second an amount of heat equal to the heat equivalent
of i joule, the relation between mechanical and electrical energy
will be a simple one. The resistance of a column of mercury, such
as specified in the above definition, satisfies this condition with a
fair degree of accuracy, and hence is adopted as the practical
unit of resistance. Other methods of determining the ohm,
independent of heat measurements, have been devised and used ;
some of which are referred to in Appendix I., where a brief
sketch of the history of the British system of electrical units,
now the system of the world, is given. It should be read by
* To give greater precision in the specification of the length of the
column.
PRACTICAL UNITS OF RESISTANCE 145
all those who are interested in seeing how the interdependence
of electrical theory and practice, each on the other, has led to the
building-up of a complete system of electrical standards, now
accepted by all nations as a common heritage.
Eventually the international ohm will be so generally used
that no other unit of resistance will be met with, and probably
the adjective international will then be dropped. For some
years, however, the " B. A. unit of resistance,"* the " legal ohm "
(so called because it was legalised in France) f, and the " inter-
national ohm " must be carefully distinguished from one another.
Their relative values are given in the following table : —
TABLE IV.
RATIOS OF THE PRACTICAL UNITS OF RESISTANCE.
international
international
ohm
ohm
=
1
1
•0024
•0136
legal ohm.
B.A. unit.
legal ohm
legal chm
=
0
•9976
•01 12
international
B.A. unit.
ohm.
B.A. unit
B.A. unit
=
0
0
•9866
•9889
international
legal ohm.
ohm.
53. Resistance Coils and Resistance Boxes. — Coils of wire
whose resistances are equal to that of the mercury column men-
tioned above, are constructed in large numbers by instrument
makers, and multiples and submultiples are also made. They
are frequently grouped together in boxes called resistance boxes,
just as multiples and submultiples of a gramme or a pound may
be grouped to form boxes of weights.
The principle employed in constructing multiples of the unit
can be explained by reference to Fig. 88. If a connection be
made with the conductor K at a point K3 intermediate between
the terminals iq and K2, and the P.D. between iq and K3, K3 and
K2 and between iq and K2 be measured by the zero electrometer
when a steady current passes through the conductor, the latter
P.D. is found to be equal to the sum of the two former P.D.s,
and as the current is the same in the two parts of the conductor
* The B.A. (British Association) unit of resistance was the result of an
early attempt to obtain a unit satisfying the requirements named at the
beginning of the previous paragraph. (See Appendix I.).
| This unit was defined as the resistance of a column of mercury one
square millimetre in cross-section and 106 centimetres in length, at a
temperature of melting ice. As made in England the " legal ohm " was
about one part in 2,000 too large. France has now adopted the inter-
national ohm, so the " legal ohm " will soon become obsolete. The B.A.
unit, however, w still largely used in telegraphic work.
K
146 PRACTICAL ELECTRICITY
we see that the resistance of the whole is equal to the sum of the
resistances of the two parts, when the parts are connected in series.
From this it follows that two unit coils connected in series form
a combination whose total resistance is two units, and a coil
which has a resistance equal to that of the two would be called
a two-unit coil. In a similar way coils of three, four, or any
number of units may be obtained. Coils of half a unit can be
constructed by making
two equal coils such
that when connected
in series, they have a
resistance equal to that
of a unit coil. Another
way is to connect two
unit coils in parallel ;
a coil whose resistance
is equal to that of this
combination would be
Fig. 89.-Resistance BOX. a half -unit coil. For
a certain P.D. main-
tained between the terminals of the two -unit coils in parallel
would cause equal currents to pass through each, the total
current would therefore be double the current in one, and
the ratio of P.D. to current for the combination would
be half the magnitude of the corresponding ratio for either
of the separate coils, so the resistance of the combination
would be half the resistance of either coil, and consequently
have a value of half a unit. Similarly three unit coils in parallel
would have a resistance of one -third of a unit, and n units in
parallel, a resistance of i/nth of a unit. A coil whose resistance
is equal to that of five unit coils in parallel will have a value of
one-fifth of a unit, i.e., two-tenths of a unit, and two such coils
in parallel would have a resistance of one-tenth of a unit. From
this it will be understood how multiples and submultiples of unit
resistances may be derived.
A resistance box with coils of o-i, 0-2, 0-3, 0-4, i, 2, 3, 4, 10,
20, 30, 40 ohms (total in ohms) is shown in Fig. 89, the interior
being arranged as indicated in Fig. 890. The coils are made of
wire w1, w2, wound on bobbins B B fixed to an ebonite slab E, and
the ends of the wires are soldered to other wires, w w w, screwed
to brass blocks, c1 c2 c8, secured to E. Taper metal plugs, p1,
p2, with ebonite tops, can be inserted between adjacent blocks
(see P2), and when so inserted current can pass from c2 to c3
directly without going through the wire w2. The coil between c1
RESISTANCE COILS AND BOXES 147
and c3 is then said to be cut out of circuit, or short-circuited. Such
an arrangement forms a very convenient means of altering the
resistance of a circuit, and is much used in practice.
To minimise the current which may pass across the surface of
the ebonite from one brass block to the next when the plug is
removed, the blocks are undercut as shown in Fig. 8ga, so that the
ebonite between the blocks may be more readily cleaned, and also
to increase the length of ebonite surface between adjacent blocks.
It will be noticed that the wire on the bobbins B, B, Fig. 8qa,
is doubled back on itself (see near tops of bobbins), and any
current passing through the wire will flow round the bobbin an
equal number of times in opposite directions, so that the magnetic
effect of the turns in one direction will be neutralized by the
opposite magnetic effect of the turns in the other direction. If
this were not done, a resistance box, when being used, would act
like a box of electromagnets, and prove very inconvenient in
the vicinity of sensitive galvanometers. Coils made in this
fashion are said to be wound " non -inductively."
When a resistance coil has to carry a large current, the heat
produced by the current may cause a large rise of temperature,
and in such cases it is de-
sirable to use coils of un-
covered wire hanging freely in
the air. To use coils doubly
wound like those seen in Fig.
89^ would be very difficult with
bare wires hanging in the air,
for there would be great danger
of the convolutions of the bare
wires which constitute one half
of the coil touching those of
the other half. If this oc-
curred, the resistance of the
coil would, of course, be
altered, and, in addition, since the potentials of the adjacent
parts of the two halves where the current enters and leaves the
coil would differ considerably if the current were strong, there
would be considerable risk of sparking if a contact occurred.
To overcome this difficulty, and still to obtain a magnetic
balance, the arrangement seen in Fig. 896 may be employed.
Each coil consists of two spirals, of the same resistance and
containing the same number of convolutions, joined up in
parallel, but one coil is wound right-handed fashion and the
other loft-handed. The current, therefore, divides into two
148 PRACTICAL ELECTRICITY
equal parts, which, circling round the two coils in opposite
directions, neutralize one another's magnetic effects. With this
device the points of the two coils which are adjacent have
practically the same potential ; therefore no serious change will
be caused in the parallel resistance, nor will injurious sparking
occur, if by chance the two coils swing into contact.
Resistance boxes of many ranges and sizes
are made in great numbers ; one of the com-
monest contains a unit coil, and multiples
of the unit, tens, hundreds and thousands,
making totals of about 10,000 or 11,000
ohms. In former years the usual values of
coils in such a box were i, 2, 2, 5, 10, 10, 20,
50, 100, 100, 200, 500, 1000, 1000, 2,000, 5,ooo,
making a total of 10,000 ; but now the sub-
divisions, i, 2, 3, 4, 10, 20, 30, 40, 100, 200,
30O, 4oo, iooo, 2,ooo, 3,ooo, 4,ooo, giving a
total of 11,110, is preferred by many workers.
Large currents. Boxes containing subdivisions of an ohm
down to o'l or 0*01 are fairly numerous, as also are high resistance
boxes with coils from iooo or 10,000 to totals of 100,000 or
1,000,000 ohms.
54. Volt. — Since the ratio of the P.D. maintained between
the terminals of a conductor to the current that flows in it is
constant, it follows that the P.D. that must be maintained at the
terminals of a resistance of one international ohm when a current
of one ampere passes through it must have a perfectly definite
value. This value is taken as the practical unit of P.D., and called
the " international volt."
If, instead of basing our unit of P.D. on the international
ohm, we base it on the B.A. unit of resistance or on the legal
ohm, then we obtain the "B.A. volt" and the "legal volt."
And the equations connecting the values of the three units are
exactly the same as those connecting the three ohms, viz. : —
TABLE V.
RATIOS OF THE PRACTICAL UNITS OF P.D.
international
international
volt
volt
=
1-0024
1-0136
legal volt.
B.A. volt.
legal volt
legal volt
=
0-9976
1-01 12
international
B.A. volt.
volt.
B.A. volt
B.A. volt
=
0-9866
0-9889
international
legal volt.
volt.
PRACTICAL UNITS OF P.O. 149
There is, however, but one ampere we need consider here, viz.,
that defined in Section 8.
Example 33. — With a P.D. of 100 international volts main-
tained between the terminals o£ a glow lamp a current of 0-3
ampere passes through it, what is the resistance of the lamp ?
Answer. — 333-3 international ohms.
Example 34. — If the P.D. be reduced to 98 international
volts and the resistance of the filament remain as before, what
current will pass through it ?
Answer. — 0-294 ampere.
Example 35. — A telegraph line between London and Birming-
ham has a resistance of 950 B.A. units ; what will be the P.D.
between its ends when a current of 0-025 ampere is passing through
it ? Answer. — 23-75 B.A. volts, 23-43 international volts.
Example 36. — By how much per cent, does the international
volt exceed the B.A. volt ?
Answer. — 1-36 per cent.
Example 37. — A P.D. of 7 international volts is maintained
between the terminals of a resistance of 2,475 legal ohms, what is
the current that passes ?
Answer. — 0-002835 ampere.
Example 38. — If a wire has 235 B.A. units of resistance, what
is its resistance in international ohms ?
Answer. — 231-85 international ohms.
Example 39. — If a wire of uniform cross-section has a resistance
of 54 B.A. units at a certain temperature, by how much per cent,
must its length be reduced so that it may have a resistance of 50
international ohms at the same temperature ?
Answer. — 54 B.A. units equals 54 x 0-9866, or 53-276, inter-
national ohms, therefore the length must be reduced by
3-276
-— -?, or by about 6-15 per cent., in order that the wire may
have a resistance of 50 international ohms.
Example 40. — What resistance in B.A. units are respectively
equal to 100, 200, 300, 400, and 500 international ohms ?
Answer. — 101-36, 202-72, 304-08, 405-44, 506-80 B.A. units.
55. Ohm's Law applicable to Complete Circuits : E.M.F. — In
Section 50 we have shown that the law holds for the part of
the circuit between the terminals iq and K2 of the conductor
K, Fig. 88. The circuit there used is represented diagrammati-
cally by Fig. 88a, where B B is the battery, G the galvanometer
150 PRACTICAL ELECTRICITY
K the conductor, and E the electrometer; and the relation
arrived at was
P.D. between terminals of K
= a constant.
current through K
To this constant the name resistance was given, and the magnitude
of this constant denoted the resistance of the conductor K,
i.e. the resistance of the part of the circuit between the points
KJ and K2. But the current flows through the complete circuit,
comprising, battery, galvanometer, and connecting wires, and the
conductor K ; and as the battery, galvanometer, and connecting
wires are conductors (since they permit the current to flow
through them) they also have resistance, and the magnitude of
the pressure which causes the flow through the whole circuit
must be greater than that which causes the flow through K only,
and greater in the proportion of the resistance of the whole
circuit to the resistance of K. If we call the pressure which causes
the flow through the whole circuit the electromotive force of the
battery (abbreviated E.M.F.), we have the relation,
E.M.F. of battery resistance of whole circuit
P.D. between terminals of K ~ resistance of K
or
E.M.F. of battery P.D. between terminals of K
Resistance of whole circuit ~ Resistance of K
but the resistance of K has been defined as
P.D. between terminals of K
current through K.
hence the above may be written :
E.M.F. of battery . .
•= — : 7 — : — : — -. r = current through the circuit,
Resistance of whole circuit
the current being the same in all parts of the circuit, or
E.M.F. of battery
/; — — - — -. r- = Resistance of the whole circuit.
Current through the circuit
We have thus a relation which may be expressed in symbols by
| = R, or E = IR, or / = f, (15)
/ K
the latter being the usual way of writing Ohm's law as applied
to complete circuits.
If the terminals of the electrometer in Fig. 880 were connected
with the ends of battery B B, instead of the terminals KJ K2 the
measurement made by the electrometer would give the potential
E.M.R ELECTROMAGNETIC DEFINITION 151
difference between the battery terminals. Calling this potential
difference V, the resistance of the circuit outside the battery Rlt
and the resistance of the battery R2, the total resistance of the
circuit will be R± + R2, and we have
y = Rlt and j = Ri + #2>
or V = IRlt and E = I (Rl + # 2)>
= IRi + IR2,
= V +IR2. (16)
Hence the E.M.F. of the battery is equal to the potential difference
between its terminals, plus the product of the current passing and
the internal resistance of the battery.
If the current be stopped by interrupting the circuit outside
the battery, / will be zero and the product I R2 = o, so under
these conditions
E = V, or, in words,
the E.M.F. of a battery is equal to the potential difference between
its terminals when no current is passing through it.
55«. Electromagnetic Definition of E.M.F. — E.M.Fs. (or
P.Ds.) are generally regarded as the causes of electric currents
flowing in complete circuits, or the causes of the tendency for
flow to take place in incomplete circuits.
There are several ways, other than by batteries, of producing
E.M.Fs., chief amongst these being the cutting of lines of force
by conductors, which forms the basis of a definition of E.M.F.
much used in practice, viz., electromagnetic E.M.F. is measured
by the rate of cutting of lines of magnetic force, or, in other words,
unit E.M.F. is generated in a conductor when it cuts lines of
magnetic force at the rate of one line per second.
That the above definition is consistent with that of P.D.
previously given in Section 48, may be seen from the following
considerations.
The absolute unit of current was defined as that current
which flowing through a conductor of unit length bent into
an arc of unit radius exerts unit force on unit pole at the
centre (see Section 8). We have also shown (Section 24)
that at unit distance from unit pole the strength of mag-
netic field is unity, and that the density of magnetic lines
over a surface perpendicular to their direction will be one
line per square centimetre. Now, by Newton's third law of
motion, "action and reaction are equal and opposite," so that
if the conductor carrying the current exerts a force of one
152 PRACTICAL ELECTRICITY
dyne on the unit pole, the unit pole will exert an equal force
on the conductor. The conductor will therefore be subjected to
a force of one dyne in a direction perpendicular to the plane
containing the conductor and the pole, and the work done in
moving the conductor through a mean distance of one centimetre
against this force will be one erg. At the same time the con-
ductor will have swept over an area of one square centimetre
of the surface of a sphere of unit radius surrounding the pole, and
will therefore have cut an amount of magnetic flux represented
by one line of force.*
As the force acting, and therefore the work done, will be
proportional to the current flowing, and to the distance the
conductor moves, we conclude that the work is proportional to
the product of the current and the number of lines of force cut
by the conductor, and if suitable units be taken we may write
W = !'$, where W is the work done in ergs, /'the current in
C.G.S. units, and $ the number of lines cut.
What becomes of the work done in moving the conductor ?
Experiment shows that the current in the circuit is slightly
increased whilst the movement is taking place and additional
heat is produced in the circuit equivalent to the work done.
This change of current must be due to some cause, and as the
only change made in the circuit is the movement of the con-
ductor, we attribute the change of current to this movement, and
say that the movement generates an E.M.F.
The equation W= I'3> may be written
$
W= I't -
. ' -«* • : , •- .
in which Q = I't, is the quantity passing in time t ; and in order
that the analogy between electrical and hydraulic work mentioned
in Section 48 may be maintained, viz., work = quantity x pressure,
$
the pressure must be represented by — . Hence the pressure
<£
(or E.M.F., as it is called in this case) is equal to —, i.e., equal
$
to the rate of cutting lines of force. Writing — = E we have
W = I'tE
or Y=rE' (I7)
* Since the number of lines of force emanating from unit pole is 417-,
Section 24, and the area of the surface of a sohere of unit radius is also 471-.
COMPARING P.Ds. AND RESISTANCES 153
from which we see that the rate at Mich work is done in an
electrical circuit is equal to the product of the current and the
pressure.
If the rate of working in a circuit in which unit (C.G.S.) current
is flowing be one erg per second, the pressure (or E.M.F.) must be
unity in C.G.S. measure. This condition therefore fixes the
magnitude of the C.G.S. electromagnetic unit of E.M.F. The
magnitude so derived is far too small for practical purposes, so
the unit adopted in practice (the volt) is one hundred million
times as large as the C.G.S. unit, or
1 volt = 108 C.G.S. electromagnetic units of E.M.F.,
so an E.M.F. of one volt is produced when io8 lines of force
are cut per second.
In the foregoing reasoning we have considered unit length of
conductor and unit magnetic field, but it will be understood that
in a given magnetic field the force exerted on a conductor lying
perpendicular to the field will be proportional to the length of the
conductor, and also to the strength of the field as well as propor-
tional to the strength of the current flowing in the conductor.
The number of lines of force cut during a given movement in a
uniform field will also be proportional to the length of the con-
ductor and to the strength of the magnetic field so that the
equation W = I' $ (18)
is true for any length of conductor and any magnetic field,
all the units being C.G.S. units. It is of fundamental importance
in electrical engineering.
56. Current Method of Comparing P.Ds. and Resistances.—
From Ohm's law it follows that the current flowing through anj?
conductor at constant temperature is directly proportional to
the P.D. between its terminals. Such a conductor may be a
coil of a galvanometer, or it may consist of a galvanometer G
together with a wire w (Fig. 90) in series with it. And no matter
how the shape of the
circuit composed of G and
w may be altered, pro-
vided that the joint re-
sistance of G and w
together is not Changed, Hg, 90.— Galvanometer with Added Resistance for
., • Measuring Potential Differences.
the current passing
through the galvanometer will be directly proportional to the
P.D. which is maintained between TX and T2, the terminals of
the arrangement. If then the galvanometer has been calibrated
relatively for current, it is calibrated for the relative measurements
154 PRACTICAL ELECTRICITY
of any P.D. which may be set up between TJ and T2 by con*
necting the terminals with any conductors, or points in the
same conductor, between which a P.D. exists.
In place then of employing the zero electrometer (Fig. 86),
we may use the combination of galvanometer and auxiliary
resistance w to compare, for example, the P.D. between the
points A and B (Fig. 91) with the P.D. between the points c and
D in the conductor A B c D conveying a steady current. For the
P.Ds. in question will be
simply proportional to the
two currents that flow
Fig- 91. through the galvanometer
when the terminals TJ,
T2 (Fig. 90) are connected respectively first with the points A
and B and then with the points c and D, provided these P.Ds.
are not altered by the points being connected with T1 and T2.
Further, since the resistance of a conductor is the name given
to the ratio of the P.D. between its ends to the current that
flows through it, and, since the current that flows through A B
is necessarily the same as that flowing through c D, when
arranged as shown, it follows that —
resistance of A B _ potential difference between A and B
resistance of c D potential difference between c and D
therefore
resistance of A B_current when Tt and T2 are joined to A and B
resistance of c D~~current when TJ and T2 are joined to c and D'
the current in each case being the current through the galvano-
metric arrangement (Fig. 90), the above proviso being understood.
Consequently, if the value of one of the resistances A B or c D
be known in international ohms, the value of the other in inter-
national ohms can be at once found by the method of testing
just described.
57. Reason for Using High Resistance Galvanometers for
P.D. Measurements, and Low Resistance Galvanometers for
Current Measurements. — When using a galvanometer for the
comparison of two P.Ds., or for the comparison of two resistances
by the method described in Section 56, it is not necessary that
the galvanometer should be calibrated absolutely in amperes,
for, as we have just seen, all that is required to be known is the
ratio of the currents that produce different deflections, not the
actual value of these currents in amperes. But there is one
condition in connection with the galvanometric arrangement G w
P.O. AND CURRENT MEASUREMENTS 155
(Fig. 90), that it is most important to fulfil, viz., that the applica-
tion of the terminals Tp T2 to the points A and B or to the points
c and D shall not alter the distribution of potential that previously
existed in the conductor A B c D. In fact, the test must not alter
the thing tested, an all-important rule to remember in experi-
menting.
Whenever a galvanometer, properly constructed and calibrated,
is introduced into any circuit the galvanometer measures the
current flowing after the galvanometer has been inserted,
but this is not necessarily the same as the current that flowed
before the galvanometer was inserted. These two currents will
only be the same in value when the resistance of the galvanometer is
small compared with that of the rest of the circuit, and when the
other conditions remain unchanged. It will, therefore, be only
under these special circumstances that the deflection of a galvano-
meter will measure the current that passed through the circuit
before the circuit was disturbed by the insertion of the galvano-
meter into it.
Similarly, whatever be the resistance, small or large, of a
galvanometer, or of a galvanometric arrangement G w (Fig. 90) ,
provided that this resistance remains quite constant, the relative
P.Ds. between two pairs of points A and B, c and D, can be
accurate!}7 compared by means of this galvanometric arrangement ;
only it must be carefully remembered that the P.Ds. that are
thus compared are the values existing after the joining of the
terminals TI, T2, to the points A and B, or to the points c and D,
and not the values of these P.Ds. before the application of the
measuring instrument. And it will be only when the resistance
of G and w combined is very large compared with the resistance
of the conductor A B, and also with the resistance of the
conductor c D, that the application of the galvanometer will
produce no appreciable disturbance in the distribution of poten-
tial along the conductor A B c D.
Therefore for P.D. measurement it is desirable that the gal-
vanometer G and the auxiliary conductor w should together have a
high resistance, and that the required sensibility of the galvanometer
should be attained by winding the galvanometer with a large number
of convolutions of fine wire*
58. Voltmeter. — A " voltmeter " is an instrument which enables
the P.D. between its terminals to be read off directly in volts.
Whether the voltmeter be of the electrostatic type and its action
depend on the attraction of electrified bodies, or whether it be
* Fine wire should be used in order that a large number of turns may
be put near the needle.
156 PRACTICAL ELECTRICITY
of the galvanometer form and the P.D. be indirectly measured
by the current it produces through a fixed resistance, it is
obviously necessary that the sensibility of the instrument
should not be affected by moving it from place to place. In
fact, a voltmeter must possess the constancy of an ammeter, with
the addition that its resistance must be constant, and any
ammeter of practically constant resistance when graduated to
indicate the P.D. between its terminals in international volts
instead of the current passing through it in amperes, becomes
a voltmeter.
The electrometer described and illustrated in Section 49
gives the same reading for the same P.D. between its terminals
if the instrument be levelled each time after being moved. Its
relative calibration is, of course, known, since our fundamental
definition of the relative value of P.Ds. is based on the use of
this electrometer. If, then, we ascertain the angle alf through
which the index c has to be turned to bring the pointer p to
the zero position when a known P.D.* say V^ international
volts, is*set up between the terminals Tl and T2 of the instrument,
the P.D. in international volts F2 corresponding with any other
angle a2, through which the index c must be turned to bring p
to zero is known from the equation —
or F2 =
y«i
— i— being a constant for the particular instrument.
A P.D. whose value is known in international volts can be
applied to the terminals xlt T2 of the electrometer (and so the
constant — ^ can be experimentally found) by connecting TA and
Va,
T2 to the ends of a conductor, c (Fig. 94), whose resistance R in
international ohms has been ascertained, and through which flows
a current of / amperes, as measured by the ammeter A. For this
P.D. is equal to / x R international volts.
* A.P.D. of known value may be obtained by passing a current whose
strength is measured by an ammeter through a resistance, whose value
in terms of the international ohm can be determined by the method of
Section 56.
ELECTROSTATIC VOLTMETER
The constant — ^is about 2*37 for the zero electrometer illus-
Vaj
trated in Fig. 86, and that is to say that the index c has to be
turned through about 360° to bring the pointer p to zero when
a P.D. of 45 volts is maintained between the terminals of this
instrument.
The dial at the top of the electrometer is initially graduated
in degrees or other divisions of equal value. But after the
constant of the instru-
ment has been experi-
mentally determined, in
the way just described,
this degree scale may
conveniently be replaced
by one graduated in
square roots with which
the P.D. can be read off
directly in international
volts. The electrometer
then becomes a direct-
reading " electrostatic volt-
meter " of the zero type.
Another form of electro-
static voltmeter intended
for measuring compara-
tively small P.Ds. is
shown in Fig. 92. The
instrument is of the de-
flectional reflecting type.
Instead of the needle
being brought back to
the zero position before
taking a reading as
described in Section 49, it is allowed to deflect until the torsion
of the fine wire suspending the needle balances the attraction
between the needle and the inductors ; the magnitude of the
deflection is measured by the movement of a beam of light
reflected from a small mirror attached to the needle, which
forms a " spot " on a fixed scale. This latter device enables
very small deflections to be measured, for the reflected beam
turns through an angle equal to double the angular move-
ment of the needle, and acts like a massless pointer of length
equal to twice the distance between the mirror and scale.
Fig. 92. — Ayrton and Mather's Reflecting Electrostatic
Voltmeter (case removed).
i5« PRACTICAL ELECTRICITY
On this account reflecting instruments are used in many kinds
of delicate measurements.
59. Resistances of Ammeters and Current Voltmeters.— From
what has been said in Section 57 it will be understood
that the resistance of an ammeter should be small compared
with that of the rest of the circuit in which it is to be
used, and that the resistance of a voltmeter should be
large, compared with that of the circuit on which it is
employed. The magnitudes of the resistances, however, are
purely relative ; for use on a circuit of very low resistance
a voltmeter of only a few ohms would be quite suitable,
whereas for high resistance circuits, instruments having many
thousands of ohms resistance would be necessary to measure
P.Ds. with reasonable accuracy. Similarly, ammeters for
measuring currents flowing in high resistance circuits, such as long
telegraph lines, may be many ohms in resistance, and yet not
cause much change of current when inserted, whilst ammeters
for use in low voltage circuits conveying large currents (hundreds
or thousands of amperes) must have extremely low resistance,
only a few millionths of an ohm, if their insertion into the circuit
is not to change the current appreciably.
60. Ammeters used as Voltmeters. — If an ammeter with its
scale graduated in volts instead of (or in addition to) its being
graduated in amperes has a low resistance, it will be suitable for
measuring any small P.D. that may exist between two points
separated by a very small resistance. For example, it may be
used to measure the P.D. between two points near together in a
thick copper electric -light main through which a current is
flowing, or to measure the P.D. between the terminals of a gal-
vanic cell of very low internal resistance. On the contrary, if
the resistance of the instrument alone, or the resistance of the
instrument and its auxiliary wire w, combined, (Fig. 90), be
high, it may be used to test a larger P.D. between two points
separated by a larger resistance ; for example, the P.D. between
the positive and negative electric -light mains in a house.
Beginners sometimes feel mystified that the same instrument
is sometimes employed to measure a current and at other times a
P.D. ; that in the former case, when it is called an ammeter,
it may be "short-circuited" with impunity, but must not be
disconnected, whereas when it is called a voltmeter it may be
disconnected but on no account may it be short-circuited. c
The difference arises not from any intrinsic dissimilarity
between an ammeter and a current voltmeter, but from the differ-
ent ways in which the instruments are employed, An ammeter
AMMETERS USED AS VOLTMETERS 159
is put into the main circuit in series with, the rest of the apparatus,
as is the galvanometer shown at G in Fig. 88, and the ammeter
A in Fig. 94, whereas a voltmeter is placed as a branch
circuit in parallel with the part of the circuit, the P.D. between
the terminals of which is to be measured ; for example, the zero
electrostatic voltmeter E in Figs. 88 and S8a, and the voltmeter
v in Fig. 94. If the voltmeter be of the current type, then both
it and the ammeter simply measure a current directly, but the
current that the instrument G in Fig. 88 and A in Fig. 94
measures is the current flowing through the main conductor, K
in Fig. 88 and c in Fig. 94 respectively, whereas the current
that the voltmeter v, Fig. 94, measures is the current that the
P.D. between the terminals of the main conductor c will send
through a resistance which is quite external to the main circuit,
viz. the resistance of the voltmeter itself.
If the resistance of an ammeter be but a small fraction of the
resistance of the rest of the circuit in which it is placed, the cnly
result of short-circuiting the ammeter by bridging its terminals
with a short piece of thick wire, is to electrically remove the
instrument from the circuit, for the current remains practically
unchanged in strength, and practically the whole of it now passes
through the short circuit : whereas in short-circuiting a voltmeter
we short-circuit all that part of the circuit with the terminals of
which the voltmeter is connected, and thus cause a great, and
possibly a dangerous, increase in the current in the remainder of
the circuit. For example, the short-circuiting of an ammeter which
is used to measure the electric -light current passing through a
house will simply cut this particular ammeter out of circuit,
whereas short-circuiting the voltmeter, which is placed across the
house mains for measuring the P.D. supplied to the house,
would momentarily extinguish all the lamps in the neighbourhood
and compel the electric current-generating-station to produce
an enormous current. Almost instantaneously either the piece of
wire used to make the short circuit would itself be burnt up, or
one of the "fuses," the name given to the pieces of easily-fusible
metal placed in the circuit to diminish the damage caused by
such accidents, would itself be volatilised by the excessive
current.
On the other hand, disconnecting one or both of the voltmeter
wires from the main circuit stops, of course, the current through
the voltmeter itself, but produces practically no effect on the
main current, whereas disconnecting the ammeter stops the
main current altogether, unless the ammeter has been short-
circuited before being disconnected.
160 PRACTICAL ELECTRICITY
61. Moving Coil Voltmeter. — The moving coil ammeter, de-
scribed in Section 43, lends itself extremely well for use as
a portable voltmeter in consequence of its freedom from out-
side magnetic disturbance, its quickness of action, its capability
of being used in any position, and its great sensibility, so that
the resistance of the coil and of the auxiliary wire w (Fig. 90)
combined, can be very high. Indeed, in some moving coil
voltmeters, intended to measure a maximum P.D. of about 140
volts, the resistance of the moving coil is about 100 ohms, and
that of the auxiliary stationary wire about 16,000 ohms, which
is a resistance far higher than that of any other type of volt-
meter of the same range and quickness of action. The instru-
ment, however, can only be employed to measure small currents,
which is a disadvantage when it is desired to use it directly
as an ammeter, but this becomes an advantage when the
instrument is used as a voltmeter, since the smaller the current
taken by a voltmeter, other things being equal, the better the
voltmeter, for the smaller is the disturbance of the circuit caused
by applying the voltmeter.
62. Calibrating a Deflectional Voltmeter. — If the law of the
instrument be unknown as well as the P.D. in volts that pro-
duces any particular deflection, we can calibrate the instrument
throughout the scale in volts in one or other of five distinct ways.*
1. Place the voltmeter v to be calibrated in parallel with a
zero electrostatic voltmeter E and apply different P.Ds. between
the common terminals of the two instruments. Measure each
P.D. in international volts by means of the electrostatic volt-
meter and observe the corresponding deflection on the deflectional
voltmeter.
2. If the voltmeter to be calibrated has a very much longer,
or a very much shorter, range than the voltmeter with which it
is to be compared
— for example, if
the one reads from
o to 500 interna-
tional volts, while
the other reads
from o to 60 inter-
national volts —
Fig. 93.— Comparing Two Voltmeters of Very Different then W6 may prO-
ceed as follows : —
Place two conductors A B, c D (Fig. 93) in series, and, by
using the method previously described in Section 56, or a
* For methods in which a standard cell is employed see Chapter IX,
CALIBRATING VOLTMETERS 161
modification of it, determine the resistance of the two conductors
in series A D relatively to that of one of them, A B. For example,
let it be found that the resistance of A D is ten times that of A B.
The actual resistance of the conductors need not be known, but
we must make sure that the resistance of the low reading volt-
meter shunting A B is large compared with that of A B.
Attach the terminals of
the voltmeter of the shorter
range to the points A and B
respectively, and the ter-
minals of the other volt-
Fig. 94.— Calibrating a Voltmeter by using an meter to the points A and D
Ammeter and One Known Resistance. , . , -> , .. . -.
respectively. Send different
currents of suitable, but not necessarily of known, values through
the conductor A D. Observe the corresponding readings of the
two voltmeters, and remember that the P.D. between the points
A and D is always ten times the corresponding P.D. between the
points A and B.
3. Join the voltmeter v (Fig. 94) to be calibrated to the termi-
nals of a conductor c whose resistance R is known in international
ohms. Send different currents in succession through this con-
ductor, and measure the currents with the ammeter A. Observe
the deflections of the voltmeter which correspond with each of
the currents, 7lf /2, 73, etc., amperes, and note that they are
produced by P.D.s of ^R, I2R, I3R, etc., international volts.
If the voltmeter v be an electrostatic one, so that no current
whatever passes through it, the deflection of the ammeter A
will measure the true current passing through the conductor c.
If, however, v be a voltmeter that takes a current, then it must
not be forgotten that the current passing through the ammeter
is the sum of the currents passing through the conductor c and
through the voltmeter. The
error introduced by assuming
that the ammeter measures
simply the current passing
through c will be the smaller
the leSS is the resistance Of C Fig. 94*-— Calibrating a Voltmetei by using an
, . . J . . J . Ammeter and One Known Resistance,
compared with that of the
voltmeter. It will be better, therefore, that c should have
a comparatively small resistance, and that the necessary P.D.
should be produced between its terminals by sending a strong
current through it.
If, however, there be a risk that such a current will warm the
conductor c and so change its resistance, then it is better to join
L
162 PRACTICAL ELECTRICITY
up the apparatus as in Fig. 94^. In that case the resistance that
must be used in calculating the P.Ds. set up between the terminals
of the voltmeter v is R + Ra international ohms, where R, as before,
is the resistance of the conductor c, and Ra is the resistance
of the ammeter A. So that when the currents are /-,, 72, 73,
etc., amperes respectively, the P.Ds. are I^R + Ra), I2(R + Ra),
I3(R + Ra), etc., international volts. The connection shown in
Fig. 94 may be used without introducing error, if for R the
resistance of c in parallel with that of the voltmeter be taken.
4. Let BJ, B2, B3, etc. (Fig. 95), be binding screws attached to
different points of a conductor which may be composed all of one
wire of uniform cross-section, or of different pieces of wire of
any cross-sections joined up to one another in series. Compare
Fig. 95. — Calibrating a Voltmeter by using Several Known Resistances in Series, with
One Known Current passing through them.
the resistances of the sections with one another by the method
described in Section 56, and compare the resistance of some
one of the sections with a standard international ohm, or
with some conductor whose resistance is known in international
ohms, then the resistance of each of the sections BX B2, B2 B3,
B3 B4, etc., will be known. Let these resistances be respectively
Rlt R2, R3, etc., international ohms.
Send a current through the conductor BX B2 B3, etc., and keep
the current quite constant at some convenient numbej of amperes,
as measured by the ammeter A. Then the P.D. between any pair
of the binding screws attached to different points of the conductor
is known in international volts ; for example, the P.D., between
binding screws B! and B4 is / (Rl+ R2+ Rs) international volts.
By connecting, therefore, the terminals of the voltmeter to be
calibrated with each of the pairs of binding screws in succession
a series of deflections is obtained, the P.D. corresponding with
each of which is known in international volts.
5. If the voltmeter be a galvanometric one it may be calibrated
by measuring its resistance Rg*, ascertaining the currents Ilt I2,
* For the sake of brevity the word international will, throughout the
remainder of this book, be omitted before the words volt and ohm, but it
is to be understood that in all cases where no prefix is mentioned the word
international is implied.
VOLTMETERS USED AS AMMETERS 163
73, etc., in fractions of an ampere that produce the deflections
dlt d2, d3, etc. These deflections will then correspond with
P.D's. of IiRg, /2^» I*Rg> etc-> volts maintained between the
terminals of the voltmeter, or with P.D's. of /j (Rg + Rw),
I2 (Rg + Rw), 73 (Rg + Rw), etc., volts maintained between the
terminals TI and T2 (Fig. 90) where Rw is the resistance of the
auxiliary wire w placed in series with the galvanometer. ,4
Example 41. — An ammeter of 17 ohms resistance has been
graduated to read milliamperes (thousandths of an ampere)
directly. What external resistance must be added to the instru-
ment so that the same scale will measure P.Ds. directly in volts ?
If a resistance of 1000 — 17 i.e. 983 ohms be added to the gal-
vanometer, a P.D. of V volts, maintained between the terminals of
the ammeter and resistance combined, will send V milliamperes
through the arrangement, and will, therefore, produce a deflection
of V on the scale. Answer. — 983 ohms.*
Example 42. — A voltmeter having 2,475 ohms resistance has
been calibrated to read off volts. It is desired that a deflection of
d divisions shall correspond with a P.D. of 5 d volts instead of d
volts. What external resistance must be added to the voltmeter
to obtain the result ? Answer. — 4 x 2,475 or 9,900 ohms.
63. Voltmeters used as Ammeters. — Any voltmeter, whether
electrostatic or of the current type in combination with a constant
resistance, can be used and graduated as an ammeter. For,
consider the arrangement No. 3, Section 62, used for calibrating
a voltmeter, and illustrated in Fig. 94. With every current which
is measured in amperes with the ammeter A there is a certain
deflection of the voltmeter. If, then, these deflections be marked
not in volts but with the numbers of amperes as measured with
the ammeter, the reading on the scale of v will at any time
give the current in amperes passing through it and the conductor
c together, when the two are used in combination as shown. The
* This question may also be worked out as follows : — Let R be the
resistance required, then a P.D. of V volts will cause a current of
X 1000 milliamperes to flow, i.e., IOO° V milliamperes; and as
A + 17
the arrangement is to read directly in volts, this current must give a
deflection of V. But a deflection of V means a current of V milliamperts
. 1000 ' y _ y
1000.
and R = 1,000 — 17 = 983.
164 PRACTICAL ELECTRICITY
graduation of the voltmeter scale in amperes will not, however
be correct if the voltmeter be used as a shunt to some other con-
ductor having a different resistance from that of c.
The device just mentioned enables a moving coil instrument,
such as was described in Section 61, through which only a small
current can be passed, to indirectly measure any current no matter
how large. In such a case, and generally when the voltmeter
used as an ammeter is to be portable, the conductor c may be
placed inside the case of the voltmeter.
It is to be noticed that the combination of voltmeter and
conductor c, of fixed resistance, can be graduated and employed,
as an ammeter, whatever the relative resistances of the voltmeter
and the conductor may be. One important advantage, however,
is gained by making the resistance of c very low compared with
l-ig. 96. — Shunted Voltmeter used as Ammeter.
the voltmeter, and that is the facility for altering the sensibility
of the arrangement.
For, suppose that the conductor c of Fig. 94 takes the form
of a short, wide strip (Fig. 96), having therefore a very low
resistance, and that the voltmeter joined up as a shunt to it has
a resistance of Rv ohms, large compared with that of the strip ;
further, suppose that a current of 7 amperes sent through the
arrangement as measured by the ammeter A deflects the pointer
of the voltmeter to the end of its scale.
Next, let a resistance of Rvohms be put in series with the
voltmeter (Fig. 96), then it will require twice the P.D. to be
maintained between the points x and Y to produce the same
deflection as before on the voltmeter. Therefore it will require
twice the current to flow through the strip, and, since by hy-
pothesis the resistance of the voltmeter is very high compared
with that of the strip, the current passing through the voltmeter
is inappreciable compared with that flowing through the strip.
Therefore twice the current flowing through the strip means
practically twice the current in the main circuit H j. In other
words, by adding to the voltmeter branch a resistance of Rv ohms
we have halved the sensibility of the arrangement which is, used,
EXAMPLES 165
as an ammeter, for measuring the current in the main circuit H j.
And, generally, if a resistance of n R ohms be added to the volt-
meter branch, the current in H j that produces any particular
deflection of the voltmeter will be n -}- I times the current
required to produce the same deflection when the voltmeter
terminals are joined direct to the points x and Y.*
Example 43. — A strip of platinoid of resistance 0-017 onm
is shunted with a galvanometer of 305 ohms' resistance in series
with a variable resistance. The galvanometer is of such sensi-
bility that a P.D. of 0-5 volt causes a deflection of 270 scale
divisions when the resistance in series with the galvanometer is
1000 ohms. If the scale is a proportional one, what must be the
resistance in series with the galvanometer in order that when
10 amperes pass through the strip the deflection shall be 100
scale divisions ?
When 10 amperes pass through the strip the P.D. between
its terminals is 10 X 0-017, or °'I7 vo^- Therefore the current
*
that this P.D. produces through the galvanometer is - - —
where R is the resistance in ohms to be put in series with the
galvanometer. But by hypothesis a current of - — - , or
305 + 1000'
0*0003833 ampere produces a deflection of 270 scale divisions,
and therefore, since the scale is a proportional one, a current of
- X 0-0003833, or 0-000142, ampere will produce a deflection
of 100 scale divisions. Hence
0-17
= 0-000142
305 +R
or R = 892 ohms.
Answer. — 892 ohms.
Example 44. — Calculate for the strip and galvanometer re-
ferred to in the previous question the resistances that must
be placed in series with the galvanometer in order that 20, 30,
and 50 amperes through the strip may produce 100 divisions'
deflection.
Answers. — 2,089; 3,286; 5, 680 ohms respectively.
* If Rs be the resistance of the shunt between the points x and Y, then
adding a resistance Rv + Rs to the galvanometer circuit will halve the sens-
ibility of the arrangement, whatever the relative values of Rv and Rs, and in
general adding resistance n (Rv + Rs) to the galvanometer circuit will reduce
the sensibility to of the original value. This should be proved as
an exercise by the student after reading Chapter VI.
166 PRACTICAL ELECTRICITY
64. Gold-Leaf Electroscope.— If we desire to measure the
P.D. between two insulated bodies which have been electrified
by touching them, for example, one with a rubbed piece of ebonite,
and the other with a rubbed piece of glass, it would be impossible
to employ any form of current voltmeter. For no matter how
fine or how long were the wire used in winding the galvanometer,
or how large was the resistance of the added wire w (Fig. 90),
the flow of electricity which enabled the P.D. to be indicated
would at once destroy the very P.D. we desired to measure.
An electrostatic voltmeter must, therefore, be employed in such a
case, but as there is no difficulty in producing a P.D. of many
hundreds of volts by means of rubbed ebonite or rubbed glass,
the voltmeter may, for many purposes, be of a much rougher
kind than the one already described.
When it is only required to know whether one potential is
higher, or lower, than another, or whether the potential of a body
is plus or minus, that is to say, whether a positive current would
flow from the body to the ground, or from the ground to the
body, if they were connected together by a wire, such a qualitative
test can be conveniently made with a " gold-leaf electroscope."
This instrument, as formerly constructed, had a variety of
faults, but the illustrated description that was given, in the
earlier editions of this book, of the proper way to construct a
gold-leaf electroscope, has induced some manufacturers, at any
rate, to cease reproducing instruments possessing the glaring
defects of the older types. In the present edition of the book
it will be, therefore, sufficient to describe the way in which a
gold-leaf electroscope may be satisfactorily constructed.
A glass shade G G (Fig. 97) rests on a wooden base, and is
covered inside with the conducting varnish devised by the
authors*, or with strips of tin- foil T, placed only just
* When the metallic foil is stuck on the glass shade, as indicated in
Fig. 97, so that the moving system can be fairly well seen at a distance
through the openings between the s1rips, the screening action, although
considerable, is by no means complete, and when the area of the
metallic coating becomes sufficiently large compared with the area of the
glass as to render the screening practically perfect, there is considerable
difficulty in seeing the moving system sufficiently well to enable small
changes in the deflection to be observed at a distance.
The authors, therefore, experimented on methods of coating the whole
of the interior of the glass shade with a transparent varnish that should
be sufficiently conducting to act perfectly as an electrostatic screen,
and yet hard enough that the inside of the glass could be cleaned when
desired without risk of the varnish being rubbed off. And this, they find,
can be satisfactorily accomplished in either of the following ways : —
Method No. i. — Dissolve £ ounce of transparent gelatine in i ounce of
glacial acetic acid by heating them together in a water bath at 100° C.
To this solution add half the volume of dilute sulphuric acid, which has
GOLD-LEAF ELECTROSCOPE
167
so far apart as is necessary to enable the gold leaves to be
easily seen. These strips of tin-foil are bent round the bottom
of the glass shade, and connected electrically with a brass
ring, which encircles the outside of the bottom of the glass
shade. 'To this ring three horizontal brass lugs are attached for
enabling the shade to be screwed to the wooden base, and to
one of them is fixed a binding screw,
s, for holding any wire which we
wish to electrically connect with the
tin- foil coating. Inside the glass
shade G, G, thin rods of good insu-
lating glass gg are cemented into
two short l>rass tubes, or sockets,
fixed to the base, and the glass rods
are joined together at the top by
being cemented into a little metallic
tube / 1, carrying the thick wire w w,
and the gold leaves L. This wire w
passes through the top of the instru-
ment without touching it, and may
carry at its top a little knob or a
little binding- screw, v is a glass
vessel containing lumps of pumice-
stone soaked in strong sulphuric
acid, which absorbs any water vapour
in the interior of the electroscope, Fis- 97>~AyrE°1eC?roico^y s Gold"Leaf
and thus keeps the glass rods g g dry.
'. When the instrument is not in use the little metal plug or
stopper p (which is made to slide a little stiffly on the wire w
been prepared by mixing i part of strong acid with 8 of distilled water by
volume, and apply the mixture while still warm to the glass shade, which
should be previously polished and warm. When this film has become
very nearly hard apply over it a coating of Griffith's anti-sulphuric enamel,
the chief ingredient of which is resin dissolved in fusel oil.
Method No. 2. — Thin the gelatine solution, prepared in the manner
previously described, by the addition of acetic acid (say, 2 volumes of -acid
to i of solution), and after polishing the glass, float the thinned solution
over the glass cold. Drive off the excess of acetic acid by warming, allow
the glass to cool, and repeat the floating process, say, twice. Thin the
anti-sulphuric enamel by the addition of ether, and float it over the gelatine
layer applied as just described. Expel the ether by heating, and apply
a second layer of this thinned anti-sulphuric enamel.
It is advisable to varnish the inside of the glass shades or glass fronts,
not merely of electrostatic voltmeters, in one of the ways just described,
but of current voltmeters, ammeters, or indeed of any instrument where
the electrification of the glass produced by cleaning it on a dry day might
cause a deflection of the pointer of the instrument — a cause of error that
has been noticed with electrical measuring instruments placed in hot dry
engine-rooms of electric-light stations.
168 PRACTICAL ELECTRICITY
by the hole in the stopper being lined with cork) should always
be pushed down, and the hole at the top of the instrument thus
closed to keep out dust and damp. If this precaution be carefully
attended to on every occasion that the electroscope is left unused,
even for a short time, and the surface of the glass rods g, g, be initi-
ally carefully cleaned, the insulation of the instrument will
remain so good, even for a year after the acid has been put on to
the pumice stone, that an electric charge given at any time to the
gold leaves will remain practically undiminished by leakage
during an hour even on a very damp day.
With a given gold-leaf electroscope the divergence of the gold
leaves depends simply on the P.D. between the gold leaves L and
the tin-foil coating T. For the gold leaves constitute a flexible
needle corresponding with N in Fig. 86, and the tin-foil coating
is the stationary inductor (called I in the same figure) to which
the gold leaves are attracted with a force depending on the
P.D. between them and the tin-foil coating. This attraction
causes the leaves to diverge, and to be, therefore, lifted ; the angle
of divergence for any particular P.D. being such that the attractive
forces exactly balance the controlling forces introduced by the
weight of the leaves which are slightly displaced from the vertical
position. A gold-leaf electroscope is, therefore, a " deflectional
gravity-voltmeter. ' '
65. Sensibility of Gold-Leaf Electroscopes. — As already ex-
plained, gold-leaf electroscopes are frequently used merely as
qualitative instruments, but, employing method No. 2, Section
62, a gold-leaf electroscope may be calibrated, if desired, by
comparison with the zero electrostatic voltmeter (Fig. 86).
The law connecting the divergence of the leaves with the P.D. set
up between them and the case depends on three things (i) the
length of the leaves (2) , the weight per square inch of the leaf, and
(3) the size of the case. If the length of the leaves and the size
of the case be fixed, it follows, from our original definition of what
is meant by one P.D. being twice another, that the P.D. required
to produce any particular divergence is simply proportional to the
square root of the weight of the leaf per square inch.
Specimens of gold leaf from different gold-beaters appear
to vary as much as 20 per cent, in the weight per square inch, but
the lighter the leaf the lower will be the price, provided that it
is not much below 40 shillings per book of 1000 leaves, in which
case cheapness may result from the impurity and not from the
thinness of the gold. At 40 shillings per thousand sheets of 22
carat gold, the sheets being 3j inches square, the weight per
square inch is about 0-013 grain. With leaves, each 2j inches
SENSIBILITY OF ELECTROSCOPES 169
long, cut from this quality of material and suspended in a con-
ducting case 4| inches internal diameter, a divergence of about
56° is obtained for a P.D. of 1000 volts, set up between the leaves
and the case. Reducing the length of the leaves to ij inches
increases the divergence for the same P.D. to 60° and in addition
it renders the various divergences between the leaves in degrees
more nearly directly proportional to the P.D. in volts.
The calibration curve can also be rendered much more nearly
a straight line by increasing the diameter of the case, but this
has the counterbalancing effect of diminishing the sensibility for
the same leaves, as may be seen from the following table :—
LEAVES EACH ij INCHES LONG. P.D. OF 1000 VOLTS MAINTAINED
BETWEEN LEAVES AND CASE.
Internal Diameter of
Case in Inches.
Divergence between Leaves
in Degrees.
41
6
8
10
60°
54°
48°
44-5°
Plotting a curve to represent the above four pairs of values
and continuing the curve forwards, it is seen that the divergence
rapidly approaches 40°, which means that however large may be
the diameter of the conducting case the divergence will be about
40° when a P.D. of 1000 volts is maintained between this case
and a pair of leaves each i J inches long cut from a 40 -shilling
book of 22-carat gold leaf.
With the leaves each ij inches long the case can be made as
narrow as 4! inches in diameter and still nearly direct pro-
portionality of P.D. and divergence be obtained up to 70°
whatever be the weight of the leaves. This is the size of leaf and
case, therefore, that may be conveniently adopted, and the con-
stant of instruments so constructed will vary from about 6°
per 100 volts to 6° per 225 volts, as the material used in making
the leaves costs 40 shillings per 1000 sheets, or a few pence when
the material is " Dutch metal."
For measuring P.Ds. of 2,000 volts, or higher, such as are now
maintained between the underground mains with certain electric
light systems, the leaves may be conveniently made out of lead
foil instead of gold-leaf.
CHAPTER V
GALVANIC CELLS
66. Chemical Action in a Simple Voltaic Element: Polarisation — 67.
Daniell's Use of a Depolariser : Two-Fluid Cell — 68. Local or Pre-
judicial Action — 69. Gravity Daniell's Cells — 70. Minotto's Cell —
71. Resistance of Daniell's Cells — 72. Grove's and Bunsen's Cells —
73. Potassium Bichromate Cell — 74. Storage or Secondary Cell — 75.
Leclanchg Cells — 76. Dry Cells— 77. Hellesen and Dania Dry Cells—
78. G. E. C. and Obach Cells— 79. Blue Bell and Columbia Cells— 80.
Extra-Sec and Inert Cells— 8 r. Edison-Lalande Cell— 82. Standard
Cells, Clark's and Weston's Cells— 83. Calculation of the E.M.F. of a
Cell from the Energy Liberated by the Chemical Action.
66. Chemical Action in a Simple Voltaic Element. — A simple
voltaic element is illustrated in Fig. i, and a battery of five
elements in Fig. ib. When the terminals of such a cell are
joined by a conductor a current flows through the conductor,
hydrogen is given off at the copper plate and the zinc plate is
gradually dissolved, forming zinc sulphate. The zinc in dis-
solving in the sulphuric acid liberates some of the energy used
in reducing the metal from its ores and part of this energy ap-
pears in the electric circuit usually in the form of heat produced
in the wires through which the current flows. In fact a cell may
be regarded as a sort of furnace in which the zinc takes the place
of coal as fuel, the chief difference being that the fuel is burnt
at a low temperature instead of a high one.
In primary batteries the plate which 'is dissolved when the
current flows is called the " positive plate," for the current passes
through the liquid from this plate to the one unattacked, this
being called the " negative plate." As, however, the current in
the outer circuit passes from the unattacked plate to the one
dissolved, the terminal on the former is called the " positive
terminal," and that on the latter the " negative terminal."
The gradual replacing of the sulphuric acid in the liquid by
zinc sulphate lowers the E.M.F. of the cell ; but a more serious
falling-off of the E.M.F. of a simple voltaic element, when sending
a current, arises from the polarisation which is caused by some of
the hydrogen gas which is liberated at the copper plate sticking
170
POLARISATION IN SIMPLE CELLS
Fig. 98,
to it and setting up an opposing or back E.M.F., in consequence
of the tendency of the hydrogen to recombine with the SO4,
or with the oxygen from which it has been separated.
That the E.M.F. of a copper- zinc-dilute-sulphuric-acid cell
falls when the cell is allowed to send a current can be tested by
using a high-resistance voltmeter, v, and a suitable ammeter, A,
placed in an external circuit whose resist-
ance, R, can be varied (Fig. 98). We
commence by making R infinite, so that
the reading of the voltmeter gives the
E.M.F. of the cell (Section 55). Next,
R is made to have some convenient
constant value, and the reading of the
ammeter watched ; gradually this will be
found to fall, the fall being fairly rapid if
the value of R be not large. The reading
of the voltmeter also falls, and, since the value of R is constant,
the ammeter and voltmeter readings fall at the same rate, each
instrument telling us the same thing by the falling-off of its
deflection — viz., that either the resistance of the cell has
increased or its E.M.F. has diminished.
If, however, we now again make R infinite, we can ascertain
which of these two causes it was that made the current fall
off ; for, if the diminution of the current was due to an
increase in the resistance of the cell only, then on making R
infinite the reading of the
high-resistance voltmeter
v will be the same as it
was originally ; whereas,
if the diminution ob-
served in the current was
wholly, or partly, caused
by a falling-off in the
E.M.F. of the cell and not
entirely by an increase
in its internal resistance,
the voltmeter will read lower when R has been made infinite
at the end of the experiment than it did when R was made
infinite at the beginning. And this result is found to occur.
To ascertain at which of the plates of the cell the opposing
E.M.F. is set up, we may use the cell seen in Fig. 99, consisting
of two copper plates, cx and C2, and two zinc plates, zl and Z2,
dipping.into dilute sulphuric acid. If the plates are all quite clean
and no current has passed between any pair of them, the two
Fig. 99.— Cell arranged for Experiments on
Polarisation.
172
PRACTICAL ELECTRICITY
copper plates will be practically the same, so that if they be
joined together metallically through even a delicate galvano-
meter, G, no current will be observed, or if there be any current,
arising from some minute difference in the two copper plates, it
will be but a very slight one. And so with the two zinc plates,
on joining them together through a delicate galvanometer, no
current, or only a very small current, will be observed.
If now, however, one of the copper plates, Cj, and one of the
zinc plates, ZA, be used to send a current for a short time through
some conductor, and then, after breaking the circuit, the two
copper plates Cj and C2 be joined through the galvanometer, G,
Fig. 100.
No Current
CL,ean
Copper.
Fig. 101.
Clean I (Clean
Zinc 1 i Zinc,
Strong Current Strong Current Weak Current
Fig. 102.
Fig. 104.
it will be found that a polarisation current flows for a short time
from C2 to c1 through the external circuit, as if ct, the copper
plate, that has been used, were positive, or like a zinc plate
relatively to C2, the unused copper plate. Similarly, if the two
zinc plates, instead of the two copper plates, be joined together
through the galvanometer, a current will flow through the ex-
ternal circuit from Z1} the zinc plate that has been used, to Z2,
the clean zinc plate ; but this polarisation current will be very
small compared with the one obtained on joining the two copper
plates. Indeed, it is so small that we may say without appreci-
able error that the diminution of the current in a simple voltaic
element is due to polarisation at the copper plate. These tests and
the results obtained are shown symbolically in Figs. 100 — 104.
It is to be noticed that while the " primary current " flows
from Zj to cx through the liquid, and the " secondary current "
DANIELL'S TWO-FLUID CELL 173
flows from c1 to C2 or from Z2 to zx through the liquid, the hy-
drogen gas in all three cases moves in the direction of the current,
the result obtained with the sulphuric acid voltmeter (see Section
n).
67. Daniell's Use of a Depolariser : Two-Fluid Cell. — Numer-
ous devices were tried to prevent the hydrogen gas sticking to the
negative plate ; Smee, for example, used a roughened platinum
plate instead of copper, the roughening being for the purpose of
enabling the hydrogen bubbles to become detached. But no
great improvement was introduced until Prof. Daniell, in 1836,
hit on the idea of surrounding the negative plate with a " de-
polariser " to prevent the hydrogen gas liberated from the dilute
sulphuric acid reaching this plate. Instead of putting both the
copper and the zinc plates in the dilute sulphuric acid, he sur-
rounded the copper plate with a solution of copper sulphate,
the two liquids being prevented from mixing together by a
porous diaphragm placed between them as shown in Fig. 2.
As before, the dilute sulphuric acid, acting on the zinc
plate, forms zinc sulphate and liberates hydrogen gas, but
this hydrogen gas arriving at the copper sulphate solution forms
sulphuric acid and deposits metallic copper on the copper
plate. Omitting, for simplicity, the water used to form the
solutions as well as the water of crystallisation of the copper and
zinc sulphate crystals, this chemical action may be represented
as follows : —
Before sending the current p
A(Cu)+/(CuSO4) .2 w(H2SO4)+w(Zn),
After sending the current
(k + 1) (Cu) + (l—i] (CuSO 4) g (SO 4H 2) + (m— i) (H 2SO 4) +
£ (ZnS04) + (w— i)Zn,
k and n being any arbitrary quantities of copper and zinc used to
form the copper and zinc plates, / and m any arbitrary quantities
of the copper sulphate and the sulphuric acid employed in the
two portions of the cell, and the arrow showing the direction of
the current in the cell itself. Substituting the atomic weights
for the various substances employed, and remembering that the
complete formulae for crystals of copper and zinc sulphate
are respectively CuSO4 + 5H2O and ZnSO4 + 7H2O, we find that
for every gramme of zinc that is dissolved off the zinc plate
about 4-4 grammes of zinc sulphate crystals are formed, about 3-8
174 PRACTICAL ELECTRICITY
grammes of copper sulphate crystals are decomposed, and about
0-96 gramme of copper added to the copper plate of a Daniell's cell.
Hence, since we know that about 0-0003286 gramme of copper
is deposited per second per ampere in a copper voltameter
(Section 9), it follows that in each hour for each ampere flowing
through a Daniell's cell about 1-18 grammes of copper is de-
posited, about 1-22 grammes of zinc is used up, about 4-62
grammes of copper sulphate is consumed, and about 3-0 grammes
of zinc sulphate is formed, which latter will become 5-4 grammes
when crystallised out, since the complete formula for zinc sul-
phate is ZnSO4 + 7H2O.
Therefore, in twenty-four hours, for each ampere flowing
through a Daniell's cell about I ounce of copper is deposited,
about 1-03 ounces of zinc
is used up, about 3-94
ounces of copper sulphate
are consumed, about 2-55
ounces of zinc sulphate
are formed, which be-
come 4-54 ounces when
crystallised out.
In the preceding no
allowance is made for
materials wasted on ac-
count of local action.
When a current is pro-
duced by a Darnell's cell,
1 copper is deposited on the
copper plate, copper sul-
phate is used up, the
Fig. zos.-Porous Pot Daniell's Cell. sulphuric add remains
unchanged in quantity, zinc sulphate is formed, and zinc is used up.
If, however, the copper sulphate solution is too weak, the
water is decomposed instead of the copper sulphate, and hy-
drogen is deposited on the copper 'plate. This deposition of
hydrogen lowers the E.M.F., and care should, therefore, be taken
to keep up a sufficient supply of crystals of copper sulphate.
Daniell originally used a membranous tube made of ox gullet
as his porous separator, but this was shortly replaced by a
"porous pot" made of unglazed earthenware, indicated by p
in Fig. 105, which illustrates a common form of Daniell's cell.
The zinc may be in the form of a rod, z, placed in the dilute
sulphuric acid which is put inside the porous pot, or in the form
of a hollow cylinder surrounding the porous pot, in which case
DANIELL'S CELL 175
the dilute sulphuric acid is, of course, placed outside the porous
pot and the solution of copper sulphate inside. The former
arrangement is the more usual. Electric connection is made
with the zinc by means of a copper wire, w, cast into it. The
copper plate c, which is usually cut out of sheet copper, is placed
in the solution of copper sulphate, and the whole is contained in a
glass, or glazed and highly vitrified stoneware jar, j. Electric
connection is made with the copper plate by means of a copper
wire insulated along its length with gutta-percha or indiarubber,
and having its end riveted, or soldered, to the top of the copper
plate. If solder be used the joint should be covered over with
wax, pitch, or other adhesive matter to prevent the copper sul-
phate coming into contact with the joint. For if this were to
happen the copper and solder being in metallic contact with one
another, and also both coming into contact with the solution of
copper sulphate, they would together form a little short-circuited
cell, galvanic action would take place and the solder would be
rapidly eaten away.
The E.M.F. of a Daniell's cell varies from about 1-07 volts to
1-14 volts, depending on the density of the copper sulphate solu-
tion and on the amount of zinc sulphate present in the dilute
sulphuric acid. As the copper sulphate is used up, and as the
density of the copper sulphate solution is thereby diminished,
when no steps are taken to maintain it constant, the E.M.F. of the
cell falls. It also falls because the sulphate of zinc, which is
formed by the eating away of the zinc rod, or plate, dissolves in
the dilute sulphuric acid. The cell has, therefore, its highest
E.M.F., 1-14 volts, when we start with the sulphate of copper
solution saturated and no sulphate of zinc yet formed 'and dis-
solved in the dilute sulphuric acid. The falling off of the E.M.F.
due to the weakening of the copper sulphate solution can be pre-
vented by having crystals of the sulphate placed in the liquid to
maintain the saturation, but we cannot so readily withdraw the
sulphate of zinc from the dilute sulphuric acid. Hence, if we
desire that the E.M.F. shall remain constant while the Daniell's
cell is sending a current, it is better to start with both solutions
saturated., The resistance of the cell will be higher and its
E.M.F. lower than when dilute sulphuric acid is used, but this
lower value of about i-io volts will be maintained nearly con-
stant while the cell is sending a current.
68. Local or Prejudicial Action. — If a piece of chemically
pure zinc be placed in strong, or in dilute, sulphuric acid, no
chemical action takes place, and no chemical action occurs if
a piece of copper or carbon also be introduced into the liquid,
176 PRACTICAL ELECTRICITY
provided that the zinc be not touched inside or outside the liquid
by the other solid. If, however, the conducting solid be now
touched against the zinc, either inside or outside the liquid, there is
a rapid evolution of hydrogen bubbles from the solid, and the zinc
is turned into zinc sulphate. We have, in fact, a short-circuited
cell consisting of an oxidisable metal — zinc — in contact with a less
oxidisable substance — copper or carbon — and both the oxidisable
and the non-oxidisable substances in contact with the liquid.
Now, ordinary commercial zinc has impurities in it, such as lead,
iron, and graphitic matter, so that when commercial zinc is
placed in dilute acid a number of short-circuited galvanic cells
are formed by the zinc, impurity, and liquid in contact, hydrogen
gas is rapidly evolved, the zinc is speedily converted into zinc
sulphate, and the energy that would otherwise be available for
generating a useful electric current is frittered away in the heat
produced by all these " local currents." It is, in fact, this
" local action " which enables the chemist to make hydrogen gas
by placing scraps of commercial zinc in dilute sulphuric acid.
With a cell, on the contrary, we desire that the zinc shall only
be used up when a useful electric current is produced— that is, a
current that passes through the wire joining the zinc and copper
plates outside the liquid. Or, in other words, we desire that
no chemical action shall take place when the terminals of the cell
are insulated from one another. We must either, therefore,
employ chemically pure zinc, or in some way prevent local action
taking place with commercial zinc. The normal price of such
zinc is about 2d. a pound, while that of redistilled chemically
pure zinc is from 35. 6d. to los. a pound, the labour of effectively
removing all the impurities from the zinc costing many times
as much as the zinc itself. To employ such zinc for ordinary
cells is, therefore, out of the question, and is indeed unnecessary,
since Sturgeon showed in 1830 that local action can be nearly as
well prevented by coating the surface of the zinc with an " amal-
gam " of zinc and mercury, or " amalgamating " the zinc, as it is
shortly called, as by employing the purest redistilled zinc.
To " amalgamate " a piece of zinc dip it into dilute sulphuric
acid to clean its surface, then rub a little mercury over it by
means of a piece of rag tied on to the end of a stick, and lastly,
leave the zinc standing for a short time in a dish to catch the
surplus mercury as it drains off.
The action of the amalgamated zinc is not well understood ;
some consider that amalgamating the zinc prevents local currents
by the amalgam mechanically covering up the impurities on the
surface of the zinc and preventing their coming into contact with
LOCAL ACTION
177
the liquid. By others it is thought that amalgamating the zinc
protects it from local action by causing a film of hydrogen gas to
adhere to it. This theory is based on the fact that while no action
takes place when amalgamated zinc is placed in dilute sulphuric
acid at ordinary atmospheric pressure, the creation of a vacuum
above the liquid causes a rapid evolution of hydrogen, which,
however, stops on the readmission of the air.
The addition of a very small amount of zinc to mercury causes
the mercury to act as if it were zinc alone, arising perhaps from
the amalgam having the effect of bringing the zinc to the surface.
A second prejudicial effect is produced by the copper sulphate
diffusing through the porous partition, coming into contact with
the zinc, and being changed into zinc sulphate, the copper which
is thus displaced from the sulphate being deposited on the zinc
in a metallic form, or as black cupric oxide, CuO, with the evolu-
tion of hydrogen. This impairs the action of the cell, as the zinc
partially coated with cupric oxide acts more like copper, and less
like zinc, than if it were not so coated ; the E.M.F. of the cell is,
therefore, lowered. Diffusion can be retarded by constructing
the porous partition so that it is only slightly porous, but this has
the disadvantage of causing the cell to have a high resistance.
A formation of metallic copper is also
produced in the pores of the porous
partition at any spot where the zinc
rod comes into contact with it, and, the
copper so deposited being in metallic
contact with the zinc rod, while both
are in contact with the liquid, the
arrangement forms a short-circuited
cell, leading to rapid waste of the
battery material, growth of the metallic
copper in the pores of the partition,
and probable disintegration of the wall
of the partition itself. To avoid this
the partition must be rendered non-
porous, by being dipped into paraffin
wax melted in warm oil, at any point
where it is likely to be touched by the zinc. For example, the
bottom of the porous pot p (Fig. 105), on which the zinc
rod rests, should be so treated before the cell is put together.
The tendency of the copper sulphate solution to diffuse to the
zinc plate, and the possibility of retarding this by diminishing
the porosity of the partition at the expense of increasing the
resistance of the cell, necessitates our considering, when we make
Fig. 106. — Meidinger Cell.
PRACTICAL ELECTRICITY
•a Daniell's cell, whether low resis-
tance or constancy and portability
are desired. And as examples of the
two opposite types of Daniell's cells
we may instance the " gravity
Daniell's cell " and the " Minotto's
cell."
69. Gravity Daniell's Cells. — Figs.
106, 107, and 108 show three forms
of Daniell's cells in which no porous
partition is employed, the copper
sulphate and the zinc sulphate solu-
Fig. 107.— caiiaud Ceii. tions being kept separated solely by
the action of gravity ; and as the zinc sulphate solution is the
lighter of the two, it is therefore put at the top. Fig. 106
shows a type of the " Meidinger " cell, in which the copper plate,
e (Fig. 106), is put inside a small inner glass tumbler, d d, so that
the particles of zinc which may become detached from the zinc
plate shall fall clear of the copper plate and be prevented from
coming into contact with it. In this type of Meidinger cell the
crystals of copper sulphate are in a glass tube, h, with only a
small hole at the bottom ; while in another type crystals are
contained in an inverted flask
open at the neck. Contact is
made with the copper plate by an
insulated copper wire, fg (Fig.
106), and the zinc plate, z z,
which is in the form of a cylinder,
is supported on a shoulder, b b,
formed by a contraction at b b of
the lower part of the outer glass
vessel, A A. The " Callaud " cell
(Fig. 107) is a simplification of
the Meidinger, being without the
reservoir for the copper sulphate
crystals and the small glass
tumbler to hold the copper plate.
The zinc cylinder hangs from the
upper edge of the jar.
Figs. 108, loSa, and 1086 show a
form of gravity Daniell used by the
Exchange Telegraph Company,
and made under the direction
of the late chief engineer to the
Fig. 108— Exchange Telegraph Co. s
Gravity Danieii.
Flan of copper electrode
Fig. io8a.
GRAVITY DANIELL'S CELLS
Company, Mr. F. Hig-
gins. It resembles the
Callaud cell in some
respects, but possesses
the advantage of
cheapness in construc-
tion, and exposes large
areas of surface of
the electrodes to the
liquids, thereby reduc-
ing the internal resist-
ance of the cell. A
gravity Daniell's cell
must, of course, not be
moved about, or if
moved great care must be taken to avoid the two liquids being
mixed together. To prevent the copper sulphate wandering to
the zinc plate, it is well to allow the cell to send a weak
current through an external circuit of considerable resistance
even when the cell is not in ordinary use.
70. Minotto's Cell.— In the " Minolta's "* cell the porous pot
is replaced by a layer of sand or sawdust of comparatively
high resistance, and it is constructed as snown in Fig. 109. At
the bottom of a glass, or glazed and highly vitrified stoneware, jar,
j, there is placed a disc of sheet copper, c, to which is riveted one
end of an insulated copper wire, which passes up through the
cell. Above this plate are placed some moist crystals of copper
sulphate, c s, and on the top a piece cf thin canvas, c, separating
the copper sulphate
from the layer of
sand or sawdust s.
On the top of the
sawdust rests the
zinc plate z, separa-
ted from the sand
or sawdust by an-
other piece of can-
vas, c. The cell is
completed by pour-
ing in some solution
of zinc sulphate, so
as to cover the zinc
Often wrongly spelt
i8o
PRACTICAL ELECTRICITY
Fig. 109.— Minotto's Cell.
disc, but not so much as to reach up
to the brass binding-screw B, cast
into the top of a little column of
zinc, forming part of the zinc disc.
Before putting in the sand or saw-
dust it should be soaked in a
solution of zinc sulphate and
squeezed partially dry, because if
put into the cell quite dry a long
time must elapse before the liquid
will soak through the sand or saw-
dust, and until this happens the cell
cannot come into action.
It is better to employ sand in
stationary Minotto's cells, as it
sinks down as the copper sulphate is consumed, but if the cells
have to be moved about then it is more convenient to use
sawdust.
71. Resistance of Daniell's Cells. — The resistance of a cell
varies with —
(1) The area of the plates immersed in the liquids ;
(2) The distance apart of the plates ;
(3) The composition of the liquids ;
(4) The thickness and constitution of the walls of the porous
partition.
A convenient apparatus for experimentally proving these
statements is shown
in Figs, no and
iiofl, and a diagram
of the connection in
Fig. iio&. The liquid
is contained in a
long wooden trough
rendered water-
tight by a lining of
gutta-percha or
Griffith's anti-sul-
phuric enamel, and
the copper and zinc
plates c and z are
supported by stout
Wires W W Sliding II0._celr arranged for proving that the E.M.F. is
in SCreW Clamps S S, Independent of the distance apart of the Plates and of
- , . , J , the Areas Immersed in the Liquids, and that the Resist-
OI WniCn Can DC ^nce depends on dimensions,
RESISTANCE AND E.M.F. OF CELLS 181
moved away from the other so as
to increase the distance between
the plates. The plates can also
be raised and lowered so that the
area of plate immersed can be
varied, and by shaping the plates
with a tag at their lower ends,
as seen in Fig. noa, this change
in immersed area can be made
very great. Plate z dips into a
porous pot containing zinc sul-
phate or dilute sulphuric acid at Fig. Irofl.
the right hand end of the trough,
and by using several different pots their influence on the
resistance of the cell may be studied.
Calling the resistance of the cell R&, its E.M.F. E, and
the readings of the ammeter and voltmeter 7 and V^ res-
pectively, then the equation E — Vl + IR2 of Section 55 may
be written —
V —V
— —-j — -1 (very approximately), (18)
where V0 is the reading of the voltmeter when the current 7 is
reduced to zero by making the external resistance R infinite, and
Fx is the reading when a current 7 amperes is flowing.
The same apparatus may also be used for showing that the
E.M.F. of a cell is independent of its size or shape, and depends
only on the nature of the materials used in constructing it, for by
making the external resistance R, Fig. no&,
infinite, and using a voltmeter of very high
resistance (say 10,000 ohms), we have
E = F0, very approximately,
and experiment shows that under these con-
ditions the reading of the voltmeter is the
same whether the plates be near together or
far apart, or whether they be fully immersed
or only just in contact with the liquid.
With a porous pot Daniell's cell, about 7 inches high, of
the relative dimensions shown in Fig. 105, the resistance may
be as low as J of an ohm when the solution in which the zinc
i8a
PRACTICAL ELECTRICITY
plate is immersed is dilute sulphuric acid of a specific gravity of
about 1-15* at 15° C. and the porous pot has a very open grain.
Such a cell must, however, be taken to pieces when not in use.
If it has to be put on one side for only an hour or two, it will be
sufficient to lift the porous pot with the contained zinc rod bodily
out of the cell, and to place it
in another empty jar, or stand
it in a dish while out of use.
The porous pot Daniell's cells
in the Muirhead type of battery
seen in Fig. in may have a
resistance of as much as 10
ohms apiece. Such cells were,
however, frequently used in
telegraph offices on account of
the ease with which they can
be coupled in series by means
of the composite copper and
zinc plates, and of the facility
with which such a battery can
be carried about. For, in ad-
dition to the cells being kept
in place by the wooden box,
the composite copper and zinc
plates serve as clips to keep
the porous pots in position,
and so prevent them shaking
about in transport.
One of the composite plates
is shown, in Fig. ma, flat as
received from the manufac-
turer, z being the zinc plate,
c the copper plate, and c a
copper strip, one end of which
is cast in the zinc plate and the
other riveted to the copper plate. The dotted lines in Fig. ma
show the plates, with the strip bent, ready for insertion into the
cells. Cells of this type can be left joined up for several weeks,
water and crystals of copper sulphate being added from time to
time as required.
Gravity Daniell's cells have been constructed by Lord Kelvin
so as to have a resistance of less than o-i ohm apiece. This
* For the percentage of sulphuric acid in solution corresponding with
various specific gravities, see Appendix IV.
RESISTANCE OF CELLS
183
result is attained by making the zinc and copper each in the
form of a large plate, the plates being placed horizontally one
above the other at a short distance apart. On the other hand,
Minolta's cells have frequently resistances of 20 or 30 ohms
each, this high resistance being of little importance when the
cells are employed to send a current through a large external
resistance, compared with the constancy that is attained by
employing a partition of sawdust some inches thick. Indeed, it
Fig. ma. — Composite Copper and Zinc Plates for Muirhead's Telegraph Battery.
(Flat, as received from manufacturers, in dotted lines with connecting
strip bent, for insertion in battery.)
is only necessary to pour a little water into such cells every few
days to make up for that lost by evaporation, in order that they
may be used for many months without any other attention
being given to them.
The resistance of a Daniell's cell, like that of liquids generally,
diminishes with increase of temperature ; hence, as its E.M.F.
is practically independent of changes of temperature, the current
sent by a Daniell's cell through a constant external resistance
increases as the temperature rises.
Example 45. — Calculate the weight of zinc sulphate formed
during 2| hours in a Daniell cell when a steady current of 0-5
ampere passes through it, assuming that no zinc is consumed by
local action. Answer. — 376 grammes.
Example 46. — In the last question it is found that 6-47 grammes
of copper sulphate have been used up. Calculate how much
per cent, of the copper sulphate has been wasted through local
action. , Answer. — n-6 per cent.
72. Grove's and Bunsen's Cells. — In the " Grove's " cell
a zinc plate is placed in dilute sulphuric acid, as in the
184
PRACTICAL ELECTRICITY
Darnell's, but the copper plate is re-
placed by one of -platinum and the
copper sulphate solution by strong nitric
acid, HNO3, which is generally said to
act as the depolariser. The Bunsen's
cell differs from the Grove's only in
the use of carbon in place of platinum.
These cells are shown in Figs. 112 and
113 respectively. Both cells give a
high E.M.F. 1-9 to 1-95 volts, and have
low internal resistances, so they may
be used for producing fairly large cur-
rents. During working, the cells give
off dark brown fumes of nitric peroxide,
NO2, and should be placed in the open
air or under a chimney.
The chemical action in a Grove's cell may be represented as
follows, omitting the water used to dilute the sulphuric acid : —
Before sending a current q
Fig. 112. — Grove's Cell.
£(Pt) +/(HNO 3)
After sending a current
m(H 2SO 4) +
k(Pt) + (l— 2)(HN03)
+2(N02)+2(H20)
(m— i)(H2S04) + (ZnS04)
A Grove's or Bunsen's battery must be taken to pieces
at the end of each day's use, since the mixing of the
liquids through the
walls of the very
porous pots used to
separate them, would
render the battery
practically useless the
next day. The porous
pots should be placed
in water after use, so
that all the zinc sul-
phate solution may be
dissolved out of the
pores of the earthen-
ware, for, otherwise,
when the pots are
dried the zinc sulphate Fig. 113— Bunsen's ceii.
CHROMIC ACID CELLS i»5
solution will crystallise in the pores and cause the pots to fall to
pieces.
Example 47. — If 4 Ibs. of zinc have been consumed in a Grove's
battery, how much sulphuric acid has been used up, assuming
that no local action has taken place ? Answer. — 6 Ibs.
Example 48. — 25 Grove's cells in series are sending a current
of 8 J amperes ; in what time will 2 Ibs. of nitric acid* be consumed ?
Answer. — I hour n minutes approximately.
73. Potassium Bichromate Cell. — This is a form of cell
devised by Prof. Poggendorff, in which the depolariser is
chromium trioxide (CrO3), popularly called chromic acid, since
chromium trioxide dissolved in water has a strong acid reaction.
But, as the chromium trioxide used formerly to be prepared, by
the user of the cell, by acting on potassium bichromate, K2Cr2O7,
with strong sulphuric acid, the cell is frequently called the
" potassium bichromate " cell. Now, however, crystals of
chromium trioxide containing 5 per cent, of water of crystallisa-
tion can be purchased ready prepared, and when these are used
the cell may be shortly called a " chromic acid " cell.
The cell is constructed in two forms, one without and one with
a porous pot, seen in Figs. 114 and 1140 respectively. The plates
employed are of carbon K, and amalgamated zinc, z (Fig. 114),
two carbon plates being generally used with the former type of
cell to diminish its resistance. The zinc plate z is supported by
the rod a, and should be pushed into the liquid only when the
cell is required to give a current, and with-
drawn directly the current is interrupted, other-
wise an insoluble chromium salt forms on the
surface of the zinc and interferes with the action
of the cell.
The chemical change which takes place when
a current passes through a single fluid chromic
acid cell, containing chromium trioxide dis-
solved in dilute sulphuric acid, is as follows : —
Before sending a current,
A(C) + J(CrO 3) + w(H 2SO 4) + w(Zn).
After sending a current,
<:
ft (C) + (l-a) (CrO 3) + (Cr l3SO 4) + (m-6) (H 2SO 4)
+6(H2O)+3(ZnSO4) + (w— 3)(Zn). out Porous Pot.
* The strong nitric acid may be assumed to contain 65% HNOS.
i86
PRACTICAL ELECTRICITY
Fig. 114*. — Fuller's Mercury Bichromate Cell
In the type of potassium
bichromate cell, having a
porous pot, the zinc, (Fig.
114^), is frequently cast in
the form of a thick cylinder
attached to a stout copper
wire, carrying the binding
screw, and both the zinc and
the wire are well amalga-
mated, or the rod is coated
with gutta-percha to insulate
it. In the porous pot con-
taining the zinc, there is put
a quantity of mercury to
maintain the amalgamation,
and either dilute sulphuric
acid or a solution of common
salt, NaCl. Sodium bichromate,
may be used with advantage
instead of potassium bi-
chromate. This cell has an E.M.F. of about two volts, and is
suitable for producing a fairly strong current for a short time.
When much used the cell becomes saturated with the potassium
and chromium sulphates, and a double salt, chrome alum,
KgCr^SO^ crystallises out and sticks so firmly to the bottom
of the cell that it is somewhat difficult to remove.
Example 49. — A single fluid potassium bichromate cell is
used to produce a current of i ampere for 10 hours. How much
sulphuric acid is consumed in the preparation of the necessary
amount of chromium trioxide and in the working of the cell, and
how much zinc sulphate and water are formed ? Allow 33 per
cent, additional for waste.
Answer. — Sulphuric acid, about 2 oz.
Zinc sulphate „ 1-4 oz.
Water ,, 0-27 oz.
Example 50. — What is the mean value of the current that a
chromic acid cell has been producing for 4 hours if the zinc, which
originally weighed 8 oz., has been reduced to 7! oz. ? Also, how
much sulphuric acid and chromium trioxide have been used up ?
Answer.- — Current, about 2-9 amperes.
Sulphuric acid, about 1-5 oz.
Chromium trioxide crystals, about 0-53 oz.
Example 51. — How much zinc, sulphuric acid, and chromium
trioxide would be consumed in a chromic acid cell having an
STORAGE CELLS
187
E.M.F. of 1-8 volt and an internal resistance of 075 ohm, it
used for 3 hours to send a current through an external resistance
of i J ohms ?
Answer. — Zinc, about 0-103 oz.
Sulphuric acid, about 0-308 oz.
Chromium trioxide crystals, about O'li oz.
74. Storage or Secondary Cell. — When strong and steady
currents are required, it is now customary to use storage cells,
which consist usually of lead plates in sulphuric acid ; such a cell
is shown in Fig. 115. The plates are generally made in the form
of grids or grooved sheets, those intended for positive plates*
being covered with red lead (Pb3O4) made into a paste with
sulphuric acid, and the negative ones with litharge (PbO) paste.
When an electric current is passed through diluted sulphuric
acidt from the plate pasted with red lead, to that pasted with
litharge, the red lead is
oxidised to lead peroxide
PbO2, and becomes of choco-
late colour, whilst the lith-
arge is reduced to spongy
metallic lead and becomes
slaty grey ; the cell is then
said to be " charged" and
will act as a current gen-
erator until the peroxide and
spongy lead are transformed
into lead sulphate (PbSO4).
The current on discharge
flows from the peroxide
plate to the spongy lead
plate through the outer cir-
cuit. By causing a current to :
pass through the cell in the
opposite direction it be-
comes " recharged." Second-
ary cells or " accumulators "
as they are sometimes called,
are now used in thousands for many purposes, such as electric
lighting and traction, propulsion of submarines when submerged,
electric cabs, delivery vans and trucks, gas, oil, and petrol engine
* The naming of the plates of secondary cells differs from that of primary
cells, for in secondary cells the plate to which the current passes through
the liquid when the cell is discharging is called the "positive " plate.
f Acid of density about i'z is usually employed.
Fig. i. T5.— Storage or Secondary Cell.
188
PRACTICAL ELECTRICITY
POLE
FILLING
APERTURE
CELL COVC*
ignitions, electro-plating, and for working telegraph and telephone
systems. They possess the advantages of high E.M.F. (about
2 volts), and very low internal resistance, and after being well
charged will produce strong currents of constant strength for long
periods. To prevent rapid deterioration of secondary cells of
the form above described it is desirable that they be recharged
before the P.D. has fallen below 1-8 volts.
Within recent years a new form of storage cell has been de-
veloped by Mr.
Edison, specially
intended for " accu-
mulator traction,"
in which the grids
are of nickel-plated
steel and the elec-
trolyte a solution of
potassium hydrate
(density 1-21), with
a small amount of
lithium hydrate.
The active material
on the positive plate
is nickel hydrate, in
which flakes of pure
nickel are embedded
to increase the con-
ductivity, and the
perforations on the
negative plate are
filled with a mixture
Fig. lisa — Edison Nickel-iron Alkali of iron Oxide and
Accumulator.
mercury.
Fig. 1150 shows the arrangement adopted ; the positive plates
are made of numerous perforated tubes of nickel-plated steel
containing the nickel hydrate and flaked nickel, whilst the nega-
tive plates are formed of numerous flat pockets, with finely per-
forated sides containing the iron oxide. The can, or container,
is made of corrugated nickel-plated steel, and is therefore light
and strong.
The changes in P.D. which occur during charge and discharge of
secondary cells are shown in Fig. 1156, the upper two curves of
which refer to a lead-acid accumulator, made by the Electrical
Power Storage Company, and the lower curve to a nickel-alkali
or Edison cell of similar ampere-hour capacity. It will be noticed
LECLANGHfe CELLS
189
that the lead cell has an average discharge P.D. of about 1-94
volts, whilst that of the nickel cell is about 1-2 volts.
75. Leclanche Cells. — In the primary cells we have hitherto
dealt with, the liquid acting on the positive plate is an acid
and the depolariser a fluid, but an important type of cell was
devised by Leclanche in 1866, in which the liquid acting on the
zinc, or positive plate, was a neutral liquid, viz. a solution of
ammonium chloride, popularly called sal ammoniac, NH4C1,*
2-4
6O
Fig.
8O IOO 12O
Ampere hours
340 160 ISO 200
1156.— Charge and Discharge Curves of Acid and Alkaline Storage Cells
of about equal capacities, t
and the depolariser was a solid, manganese peroxide, MnO2,
packed with bits of gas carbon round the carbon or negative plate.
The " Leclanche " cell is, therefore, a single-fluid cell, the porous
pot seen in Fig. 116, which illustrates one of the earlier forms of
this type of cell, being used merely for the purpose of keeping
the mixture of manganese peroxide and broken gas carbon in
contact with the carbon plate ; and, to prevent the mixture
being shaken out of the pot, it is closed at the top with pitch.
A small vent hole is left in the pitch to allow air and gas to
escape from the porous pot. The zinc is made in the form of a rod
* This salt can now be obtained in the form of large pellets, or buttons ;
they are sometimes called " Voltoids."
t The quantity of electricity (expressed in ampere hours) which a fully
charged cell will produce before re-charge is necessary, is called the
"capacitv" of the cell.
PRACTICAL ELECTRICITY
with a copper wire cast into the top of
it, and the rod rests in a recess in the
corner of the glass jar made to receive it.
Electric contact with the carbon plate
is sometimes made by means of a lead
cap cast on to it, firm connection being
made between them by the lead run-
ning into two small holes drilled side-
ways through the top of the plate, and
thus riveting the cap on the plate. To
prevent the liquid creeping up by capil-
lary action between the top of the
carbon plate and the lead cap, where
it would form a salt of lead and intro-
duce a high resistance between the two,
the top of the carbon plate, after the
holes have been drilled in it, is heated for
one hour in paraffin wax at a temperature of 110° C., and thus
rendered non-porous. Improved methods of making contact
with the carbon electrodes are shown in Figs. 119 and 120 ; carbon
heads are formed on the plates during manufacture and holes
provided in the heads to receive the metal terminals.
The chemical action of the Leclanche cell is as follows : —
Before sending a current,
A(C) -f ;(MnO2)-fw(NH4Cl)-j-«(Zn).
After sending a current,
Fig. 1 1 6.— LeclanchS Cell with
Porous Pot.
(w— 2) (NH4Cl) + (ZnCl2) + («— i) (Zn).
Manganese peroxide is therefore re-
duced to manganese sesqui-oxide, Mn2O3,
sal ammoniac and zinc are used up,
water and zinc chloride, ZnCl2, are
formed, and ammonia gas, NH3, is
given off. Substituting the atomic
weights we see that for every 50
grains of zinc used up about 82 grains
of sal ammoniac are consumed, and
about 134 grains of the manganese
peroxide are reduced to managnese
sesqui-oxide. If, however, too little
sal ammoniac be present, zinc oxide,
or zinc oxychloride, is formed instead _.
•;. ., , ,. Fig. 117.— Leclanche Agglomerate
of zmc chlonde, and the solution Celi-
LEGL4NCHE CELLS
191
becomes milky ; hence, when this happens, more sal ammoniac
should be added.
The E.M.F. of a Leclanche cell is about 1-5 volts, but in the
case of the porous pot form (Fig. 116) the E.M.F. falls rapidly
when the cell is used to send a strong current. It will, how-
ever, regain its value if the cell be left for some time unused, and
it does not sensibly diminish when the cell is put on one side,
even for some months. Hence, while the Leclanche cell is much
inferior to the Daniell's for the purpose of sending a steady current
for an hour or two, it is much superior to the Daniell's cell for pro-
ducing intermittent currents at any
time during the course of a year or
more — for example, such currents as
are employed for the ringing of
electric bells, for house telephones,
and for railway signalling.
The objections to this simple form
of Leclanche cell, in addition to its
rapid polarisation, are — (i) the use
of the porous pot, which increases
the resistance of the cell ; (2) the
evaporation of the liquid indicated by
the liquid filling only half the cell in
Fig. 116 ; (3) the eating away of the
zinc rod which occurs at the surface
of the liquid, thus rendering the rod
useless before the lower part is used
up ; and (4) the creeping of the salts,
this latter defect being, however, «*•
partly counteracted by dipping the
top of the porous pot and of the glass j ar as well as the upper part
of the carbon plate into melted ozokerite, or, best of all, into
paraffin wax melted in warm oil. Various modifications of the
Leclanche cell have been introduced to overcome the first two
defects. M. Leclanche in 1871 dispensed with the porous pot by
replacing the mixture of manganese peroxide and gas carbon with
a solid agglomerate composed of 40 parts of granulated manganese
peroxide, 52 of granulated carbon, 5 of gum shellac, 3 of potassium
sulphate, and a small quantity of sulphur. This mixture is heated
to 1 00° C, and pressed into moulds under great pressure : the
sulphur volatilises and leaves the blocks in a porous condition, so
that the liquid can soak into them. The negative plate is formed
by binding a block of the agglomerate, a, on each side of the
carbon plate with indiarubber bands (Fig. 117).
Fig. 119.— "Carsak Cell." (General
Electric GJ )
PRACTICAL ELECTRICITY
Other modifications of the Le-
clanche cells are shown in Figs. 118,
119, and 120, which represent " Six
block agglomerate," " Carsak," " and
" Lacombe central zinc " cells respec-
tively. Cells of the latter type are
by some firms termed " Car porous "
elements. In the " Carsak " cell
(Fig. 119) and other forms of " Sack "
cell the chief objections to the ordi-
nary porous pot form are eliminated
by using a large hollow cylinder of
zinc instead of a rod, substituting
powdered MnO2 for the granular
variety, and replacing the porous pot
by a wrapping of canvas or sacking.
These features reduce the internal resistance of the cells con-
siderably and greatly increase the current they can produce.
The curves in Fig. 121 give the results of tests made on three
types of Leclanche cell, when the outside resistance in each case
was maintained constant at 10
ohms, the plan adopted by the
Post Office for testing cells, and we
see that the current under these
circumstances fell to half its value
in seven, fifteen, and twenty days
respectively with the ordinary
porous pot Leclanche cell, the " Six
block agglomerate " cell, and the
" Carsak " cell. It will be ob-
served that in the first two cases
the polarisation is somewhat rapid
at the beginning and especially
rapid in the case of the ordinary
porous pot Leclanche cell.
With the " Carsak " cell the
fall of current is much more
gradual. It should, however, be
mentioned that the " life " (i.e.,
the time in which the current the
cell sends through an external
resistance of 10 ohms falls to half
its initial value) of many forms of
agglomerate block cell is con-
Fig. 1 20. — Lacombe Central Zinc Cell.
Perforated Carbon Cylinder with head ;
B, Cylinder of Porous Porcelain ; c,
Glass Base uniting A and B ; M, Mix-
ture of Ca-rbon and Manganese Di-
oxide (Depolariser) ; N, Insulator ; o,
Binding Screw ; R, Washer ensuring
contact ; s, Solution of Sal Ammoniac ;
z, Zinc Rod.
LECLANCHE CELLS
D
siderably less than that of porous « pot cells made by the
same firm. Using powdered, instead of granulated manganese
peroxide, increases the " life " of a cell.
Example 52. — If 2 Ibs. of zinc have been consumed in a
Leclanche battery, how much sal ammoniac has been utilised
in the same time ? Answer. — About 3-3 Ibs.
Example 53. — Compare the rates of using up manganese per-
oxide and sal ammoniac in a Leclanche cell.
Answer. — Approximately, as 163 to 100.
Example 54. — What is the cost of the material consumed in 6
Leclanche cells in series when developing a current of o-i ampere
for three hours a day for 200 days,
if 10 per cent, of the material used
is wasted through local action ?
Take the price of zinc as 2^d. per
lb., of sal ammoniac 455. per cwt.,
and of manganese peroxide as 145.
per cwt.
Answer. — Cost of —
zinc, 3d. ;
sal ammoniac, 8Jd. ;
manganese peroxide, 4|d.
A new form of Leclanche cell,
devised by M. Fery, which dis-
penses with porous pot and solid
depolariser, is shown in Fig. 1200.
The carbon c takes the form of a
hollow cylinder with slotted sides,
and the zinc z is a thick plate
resting on the bottom of a glass
jar v ; a wooden cross s separates the carbon from the zinc. The
oxygen of the air, some of which is dissolved in the electrolyte,
acts as depolariser. On closing the circuit, hydrogen forms on the
lower end of the carbon tube, as this end, being nearest
the zinc and therefore carrying most of the current, and the
hydrogen with the carbon and the dissolved oxygen (which
is most concentrated near the surface of the liquid) forms
a local gas battery which causes the hydrogen to recombine.
Carbon of a definite porosity is found to give the best
results. In this cell sal-ammoniac is regenerated during
working, and thus enables cells of large ampere-hour capacity
to be produced.
76. Dry Cells. — The commonest form of primary cell used at
present is a variety of Leclanche called the " Dry cell." The
N
Fig. izoa. — Section through Fery's
modified Leclanche" Cell.
194
PRACTICAL ELECTRICITY
name, although a convenient one, is not correct, for a really " dry"
cell would produce no appreciable current. A certain amount
of moisture must be present, otherwise the resistance of the cell
would be extremely high. Moisture is also necessary to the
S9J9cllU1Pl||IUI UJ
chemical action involved. Many attempts have been made to
construct a cell which could be turned upside down or used in any
position without interfering with its action. Volta constructed
a battery of zinc and copper plates with pieces of moist cloth in-
DRY CELLS 195
serted between them. Zamboni used discs of paper covered on
one side with tin and on the other with manganese peroxide ; but
batteries of this type, although they could produce a large E.M.F.
when a sufficiently large number of elements was employed, were
only able to furnish an extremely small current in consequence
of their large internal resistance. Wolf, Keisen, and Schmidt
tried to make a " dry cell " of moderate resistance by mixing
sawdust with cellulose. Desruelles filled a Leclanche cell with
asbestos fibre and spun glass ; Pollak employed a gelatine
glycerine ; but the first to construct a dry cell which could be
successfully used to produce an appreciable current was Gassner
in 1888.
The " Gassner' s " dry cell was a form of Leclanche cell, the
plates being formed of carbon and zinc, the latter being made in
the shape of a pot to contain a. jelly which surrounded the carbon
rod. This jelly was composed of sal ammoniac, zinc chloride
and oxide, calcium sulphate, and water, the zinc oxide being
possibly added to give porosity to the jelly. The E.M.F. was
about 1-3 volts, the internal resistance of different cells of the
same size was very different, and the resistance of any one cell
varied in an irregular way during working. The cells polarised
rapidly when used, and were also liable to short-circuit internally.
Nevertheless, their compactness, portability, freedom from all
creeping of the salts, and the fact that they did not dry up, led
people to consider whether cells constructed somewhat on the
principle of the Gassner dry cell might not be manufactured so
as to be commercially useful.
In most modern dry cells a carbon rod or plate surrounded
by a depolariser, consisting mainly of manganese peroxide, carbon
and graphite, forms the inner electrode, and a zinc pot or case
containing a semi-solid electrolyte containing sal ammoniac and
zinc chloride forms theouterone. Their E.M.F.s are about 1-5 volts
and their resistances average from about o-i to 0-5 ohms, according
to size. They may be classified in four main types as follows : —
(1) Those in which the electrolyte is in the form of a jelly
between the depolariser and the zinc.
(2) Those in which the electrolyte is a nearly solid paste
usually containing plaster of Paris.
(3) Cells having the electrolyte held in an absorbent paper or
pulp lining to the zinc container, and into which the depolariset
is packed.
(4) " Desiccated " or " Inert " cells which are made active
when required by the addition of water.
Examples of type (i) are the Hellesen cell described in Sect. 77
196
PRACTICAL ELECTRICITY
and the well-known Dania cell made by the Atlas Battery Co.
Of type (2) the " G. E. C." cell, the " E. C. C." cell, and the
" Obach " cell are makes in common use. Type (3) includes the
Blue Bell and Columbia cells of American make. Cells of type
(4) have been devised to minimise the chief defect of ordinary dry
cells, viz. : serious deterio-
ration when kept in stock
for any considerable time,
._. § particularly in hot places.
In these cells the electro-
lytic materials take the
form of dry powders, which
__ 2t remain unaltered and in-
active until water is added
when the cells are required
for use. Some of the best
known makes are the
' B" Extra Sec " cell made by
the General Electric Co.,
the Dura, made by Messrs.
Siemens Bros., the Inert,
made by the India Rubber
and Gutta Percha Co., the
H2O, the Reliable, and
W. O.* cells made by the
Edison Swan Co., the Asso-
ciated Battery Co., and the
Atlas Battery Co. respec-
tively.
Dry cells are now manu-
Fig. xa2.— Heiiesen Dry Ceil. f actured in huge numbers
for use in electric torches. Usually two or three are placed
side by side connected in series and put in a cardboard case, the
combination forming a " refill."
77. Heiiesen and Dania Dry Cells. — In the Heiiesen dry
cell, which was introduced into England by Messrs. Siemens
Bros., about 1890, a carbon rod c, Fig. 122, is surrounded by a
black paste D, composed of manganese peroxide, graphite,
and ammonium chloride wrapped in calico and tied with string.
Outside this is a white paste, E, of ammonium chloride, water,
flour, and plaster of Paris. These materials are contained in a
round zinc pot z placed in a square millboard case B, the corners
of which are packed with sawdust s d. The tops of the depolariser
* Signifying "water only."
V- "O.K. C." Cell.
z -
G.E.G. AND OBAGH CELLS
and excitant are covered by a layer* of
plaster of Paris, P, and the whole sealed
with a bituminous compound s. An air
tube A is provided to carry off any gas
that may be generated in the cell.
In the Dania cell the carbon is sur-
rounded by the depolarising paste
enclosed in a form of sack, and the
gelatinous electrolyte is placed between
the sack and the zinc. Numerous wooden
pegs driven into the sack and projecting
therefrom prevent contact between sack
and zinc. A layer of insulating material
separates the sack from the bottom of
the zinc container, which latter is sur-
rounded by a cardboard case. A layer
of sawdust rests on an annulus of paraffined paper, and the
cell is sealed, except for a vent, by a bituminous compound.
78. G. E. C. and Obach
Cells.— In the G. E. C. cell,
shown in Fig. 123, the carbon
c is in the form of a flat plate
and is surrounded by de-
•d polarising paste F of carbon
and manganese peroxide.
Between the cylinder F and
the zinc cylinder z is a white
electrolytic paste G of plaster
of Paris and sal ammoniac.
The pastes are covered with
sawdust s over which is a
bituminous seal H.
The Obach cell, made by
Messrs. Siemens Bros., is
shown in Fig. 1230. Here
a carbon rod c is surrounded
by a depolarising paste D
containing about 55 per
cent, manganese peroxide,
44 per cent* plumbago, and
i per cent, of gum ; it rests
on an insulating layer inside
the zinc container z and the
electrolytic paste E, com-
3«.— Obach Dry Cell.
198
PRACTICAL ELECTRICITY
Carboa
Fig. 123*.—" Blue Bell " Cell.
posed of 85 per cent, plaster of Paris, 15 per cent, flour moistened
with sal ammoniac, is poured between the depolariser and
the zinc. Sawdust or ground cork resting on a canvas disc
c d is placed between the pastes and
the seal s. A cardboard box B surrounds
the zinc and seal.
79. Blue Bell and Columbia Cells. —
The Blue Bell cell, made by the
Western Electric Co., is largely used
in the telephone work of the firm.
The carbon is of fluted cross section
(Fig. 1236), and rests on several layers
of pulpboard E. The zinc container
z is lined with absorbent paper G
Section of which is folded over the top of the
depolariser paste F after the latter
has been rammed in between the
carbon and the paper lining. The
latter is, of course, moistened with sal
ammoniac solution. A layer of sand s
is placedbetween the paste and the seal H.
A section of the Columbia cell is given in Fig. 1230, the lettering
of the parts corresponding with those in Fig. 1236, excepting
that the layer of sand s in the Blue Bell
cell is replaced by layers of sand and sawdust
shown at sx and S2 respectively, and a corru-
gated cardboard collar S2 between them.
The air space in S2 gives room for expan-
sion of the electrolyte caused by the passage
of large currents. The letter G in Fig. I23C
indicates absorbent pulpboard.
80. Extra-Sec and Inert Cells. — These cells
are good examples of type 4, viz., Desic-
cated Cells. The Extra-Sec cell, made by
the General Electric Company, is very
similar in construction to the Dania, de-
scribed in Section 77. Instead, however,
of the electrolyte being in a gelatinous form
between the sack and the zinc, this space is Fig. ™y.— Columbia Ceil,
partly filled with the electrolytic salts in a
dry state, together with gum in powder form, the whole
material being specially treated by a method which makes
it non-hygroscopic. A small ebonite tube passing through
the seal of the cell communicates with the space above
EDISON-LALANDE CELLS
199
mentioned, and through this tube the space is filled with water
when it is desired to make the cell active. The water
dissolves the salts and forms with the gum an electrolyte
of the gelatinous type.
The Inert cell, made by the India Rubber and Gutta Percha
Company, is shown in section in Fig. 124. A carbon rod A is
surrounded by the depolarising
mixture contained in a sack B
bound up with string ; a rubber
band c separates the sack from
the zinc container D. Between
B and D is rammed the electrolyte
E in the form of dry powder.
A cardboard case F surrounds
the zinc. G indicates two cork
covers through which the stopper s
H and vent tube j pass. K and
L are the positive and negative
terminals respectively, and the
space s is filled with water when
the cell is required for use.
81. Edison-Lalande Cell.— This
cell, shown in Fig. 125, consists
of plates of black oxide of copper
and zinc immersed in a solution of
caustic potash, a layer of heavy
oil being poured over the solution to
prevent evaporation and " creep-
ing." No local action or polari-
sation takes place in this cell ; under normal conditions it is
an easy matter to set it up to give any required number
of ampere hours, and to so proportion the constituents
that they are all exhausted at the same time. This is
a matter of considerable importance where closed circuit
working is employed, as in some systems of telegraphy
and in " alarm " circuits. Although the E.M.F. of the
Edison-Lalande cell is low (0-75 volt), its resistance is also
low, and the cell is capable of producing large currents.
A strong solution of caustic potash, one to three by weight,
is usually employed.
Example 55. — Assuming the chemical action in the Edison-
Lalande cell to be represented by the equation k (CuO) +
/ (KHO) + m (Zn) = Cu + (k - i) (CuO) + OH2 + (1-2)
(KHO) + (K2ZnO2) + (m — i)Zn, calculate the amounts of
Fig. 124.
Section through " Inert" Cell.
200 PRACTICAL ELECTRICITY
copper oxide, caustic potash and zinc required per cell to pro-
duce i ampere continuously for 30 days.
Answer. — The quantity of electricity =i x 24 x 30 ampere hours,
=720
and the molecular weights of the substances are in the approxi-
mate ratio, 63-5 + 16 ; 39 + i + 16 ; 65-5, i.e., 79-5 : 56 : 65-5.
Now the electrochemical equivalent of zinc is 0-000339 grammes
per coulomb (approximately) or 1*22
grammes per ampere hour.
Weight of zinc required
= 720 x 1-22 grammes,
= 878
= 1-94 Ibs. approx.
Weight of copper oxide
= i'94 x 79-5.
65-5
= 2-35 Ibs.
Weight of caustic potash
_ 1-94 x 56 x 2*,
65-5
Fig. 125. — Edison-Lalande
Cell. = 3-31 Ibs.
A modified type of this cell, called the " Neoiherm " cell, is now
made, in which the copper oxide, the depolariser, forms a lining
to the containing vessel, which is of iron. When the cell has
become discharged, the deposited copper can be re-oxidised rapid-
ly by heating the iron vessel in an oven. By this means the
cost of re-charging is greatly reduced. A cell weighing 12 Ibs,
will give i ampere for 150 hours ; the E.M.F. is approximately
i volt, and its initial resistance about o-i ohm.
82. Standard Cells, Clark's and Weston's Cells.— The cells
previously described are intended for use as current generators,
and for this purpose high E.M.F. and low internal resistance are
desirable features. Those described in this section are designed
as standards of E.M.F., so that uniformity and constancy are
the principal requirements for this class of cell. Cells which give
E.M.Fs. whose values are accurately known, enable many elec-
trical measurements to be conveniently made (see Chapter IX.),
and much time and trouble have been devoted by many experi-
menters to the production of such cells. The best known and
* From the formula it will be seen that 2 molecules of caustic potash
are used up per i molecule of copper oxide.
CLARK'S STANDARD CELL
201
most useful standards are those devised by Mr. Latimer Clark,
of London, in 1872,* and by Mr. 'Edward Weston, of Newark,
New Jersey, U.S.A., in 1893! respectively. Both types of cell
have been made up in many shapes and forms, but the H-form,
introduced by Lord Rayleigh in 1882, Fig. 126, or some modifica-
tion of it, is now generally used.
In the Clark's cell the materials used are pure mercury,
mercurous sulphate, solution of zinc sulphate, and zinc amal-
gam, the latter being made by dis-
solving pure zinc in pure mercury.
Such an amalgam behaves electrically
like pure zinc. The pure mercury,
M, Fig. 126, previously distilled in
vacuo, is placed in one leg of the
H tube and covered with a layer of
paste, M s, made by mixing mercu-
rous sulphate with a saturated solu-
tion of zinc sulphate. The zinc
amalgam A is placed in the other
leg of the tube and both legs and
the cross tube are nearly filled
with zinc sulphate solution z z, crys-
tals of zinc sulphate being added to H F°rm- About one-half of full size.
ensure that the solution may be
saturated at all ordinary temperatures. Electrical connection
with the mercury and with the amalgam is made by
platinum wires, w w, sealed into the lower ends of the legs.
The upper ends of the vertical tubes are closed, either by corks
and marine glue, or, preferably, by drawing out the tubes in a
blowpipe flame and hermetically sealing them. Cells set up in
this way, using pure materials, have E.M.F.s remarkably equal
in value under specified conditions. Equality within one-tenth
of one per cent, is easily obtained, and cells set up with great care
will not differ in E.M.F. by more than a few parts in one hundred
thousand. This E.M.F. is, for normal cells, given by the expres-
sion
E*=l-4328— 0-0011 (t— 15), international volts, (19)
where t is the temperature in degrees centigrade. From this
formula it will be seen that the E.M.F. of a Clark's cell at
normal temperature, 15° C., is 1-433 (very approximately)
volts, and that a rise of temperature produces a fall of E.M.F.
of i-i millivolts per degree C.
* Proc. Roy. Soc. , vol. xx., p. 444. f The Electrician, vol. xxx.. D. 741.
202 PRACTICAL ELECTRICITY
In using the Clark's cell as a standard of E.M.F. it is desirable
that its temperature be maintained fairly constant, for if the
temperature be altering rapidly the E.M.F. does not change as
quickly as the temperature, so there is a lag of E.M.F., and the
actual E.M.F. may differ appreciably from that given by the for-
mula above. Should the cell be allowed to generate much current
Marine glue*
-Zinc rod.
-Cork
Zirtc sulphate solution,
„ crystals.
•Me rcu rous sulphate
Mercury.
Fig. 127.— Clark's Cell, Board of Trade (1894) foira (full size).
polarisation occurs, and the E.M.F. is temporarily reduced, but
after a period of rest the cell becomes normal. Usually a few
minutes' rest will permit the cell to recover its E.M.F. to within
one part in a thousand, unless the cell has been left on closed
circuit for a long time. Several other forms of Clark's cell are
shown in Figs. 127, 128, and 129.
For transport the form devised by the late Dr. Alex. Muirhead,
and shown in Figs. 129 and 130, where the mercury is replaced
by a well amalgamated platinum wire, has many advantages.
The chemical action which occurs in a Clark's cell may be
represented as follows : —
Before sending a current
A(Hg) + J(Hg2S04) + w(ZnS04) + »(Zn)
After sending a current
(k + 2) (Hg) + (/-i) (Hg2S04) + (m + 1) (ZnS04) + (»- 1) (Zn),
the mercurous sulphate acting as the depolariser.
CLARK AND WESTON CELLS
203
The chief objection to the Clark's cell as a standard of E.M.F.
is its comparatively large temperature co-efficient, which, as
before stated, amounts to 0-08 per cent, (i-i millivolts) per
degree, and to overcome this defect Mr. Weston replaced the
zinc of the Clark's cell by cadmium. The Weston cell, Fig. 131,
Fig. 128.— Kahle's Modification of the Raylei^h H form of Clark Cell (full size)
ZS.s, zinc sulphate solution ; ZS.c, zinc sulphate crystals ; MZ.S, mercurous sulphate and
zinc sulphate paste ; M, mercury ; A , amalgam of zinc and mercury.
therefore contains pure mercury, mercurous sulphate paste, a
saturated solution of cadmium sulphate, and cadmium amalgam ;
crystals of cadmium sulphate being also added to keep the solu-
tion saturated. By this substitution of cadmium for zinc the
temperature co-efficient is reduced to about one-twentieth its
value for the Clark cell, mainly owing to the solubility of cadmium
sulphate changing much less with temperature than that of zinc
sulphate. The substitution also reduces the E.M.F. consider-
ably, but this is no disadvantage ; in fact, for many purposes a
204
PRACTICAL ELECTRICITY
standard cell with an E.M.F. as low as ^ of a volt would be very
convenient.
Of late years numerous measurements have been made of
the E.M.F. of the West on cadmium cell, both in America, Ger-
many, and Great Britain, the most accurate determination
having been carried out by the authors and Mr. F. E. Smith,
O.B.E., F.R.S., at the National Physical Laboratory, Tedding-
ton.*
Marine
Oiue
HBa S04
Paste ~
Fig. 129.— Section of Portable Clark Cell (Muirhead's Form).
The form of cell used in experiments is shown in Fig. 132,
and the value obtained, after reducing to 20° C., and allowing
for the difference between the international ampere and the true
ampere, is
E = 1-0183 volts at 20° C. (20)
This value, at 20° C., was adopted by the International Con-
ference on Electrical Units and Standards which met in London
in October, 1908. The following formula was also agreed to for
calculating the E.M.F. at temperatures between o° C. and 40° C.
E; = E20 - 0-0000406 (t - 20) - 0-00000095 (t - 20)2 +
0-00000001 (t— 20)3. (21)
For many purposes no temperature correction is necessary, for
a change of 10° alters the E.M.F. by less than I part in 2,000, and
for a change of 20° C., the alteration only slightly exceeds I in
1000.
* On a New Current Weigher, and the Determination of the E.M.F. of
the Normal Weston cadmium cell. Phil. Trans. , 1907.
CLARK AND WESTON CELLS
205
The cadmium cell therefore pos-
sesses a marked advantage over the
Clark as regards variation of E.M.F.
with temperature, and on this ac-
count is being adopted internation-
ally as a secondary standard of
electric pressure. A specification
for setting up cells of this type
is given in Appendix I.
In Mr. Smith's form of cell the
constrictions in the sides of the
vertical tubes prevent the crystals
being displaced even if the cell
be turned upside down, and thus
renders the cell much more portable
than it would otherwise be.
Example 56.— What is the E.M.F. of a normal Clark's cell at
I2°.6 C., and i8°.5 C. respectively ?
Answers. — 1'4354 volts and 1-4290 volts.
Fig. 130.— Mail-head's Portable
Clark's Cells (Mounted).
Fig. 131. — Weston's Cadmium CelL
Example 57. — At what temperature will the E.M.F. of a
Clark's cell be 1-434 volts ?
Answer. — 14° C.
Example 58.— Find the E.M.F. of a Weston normal cell at the
206
PRACTICAL ELECTRICITY
following temperatures, 10° C., 15° C., 25° C., and 30° C. to five
significant figures.
Temperatures.
volts.
Answer. —
15° C.
25° C.
30° C.
E.M.F.
1-0186
1-0185
1-0181
1-0178
83. Calculation of the E.M.F. of a Cell from the Energy
Liberated by the Chemical Action. — We have seen that a cell can
cause an electric current to flow round a circuit, and that chemical
changes occur in the cell during the time the current is passing.
If the external circuit consists of a simple wire, there is heat
generated in the wire, and this heat is produced at the expense of
Fig. 132.— Weston Cadmium Cell (F. E. Smith's Form).
A, amalgam; Aft mercury; P, mercurous sulphate paste; C, crystals of cadmium sulphate;
S, solution of cadmium sulphate.
the chemical energy of the constituents of the cell. For example,
in a Daniell's cell zinc is dissolved in the sulphuric acid and copper
deposited on the copper plate. Now when metals are acted on by
acid outside a cell, heat is generated. Experiments made on the
amount of heat generated during solution have shown that 106,000
calories,* approximately, are produced by dissolving 65 grammes
* A calorie is the amount of heat required to raise i gramme of water
CALCULATION OF E.M.F. OF CELL 207
of zinc in dilute sulphuric acid,* and about 56,000 calories by
dissolving 63-5 grammes of copper. In the actual cell the copper
is removed from solution, heat being absorbed in the process,
so the nett amount of heat generated during the time 65 grammes
of zinc are dissolved will be 106,000 — 56,000 calories i.e., 50,000
calories. This, if the law of conservation of energy be true,
must be equivalent to the electric energy or work produced by
the cell, if none is wasted by local action or otherwise.
In Section 48 we have denned P.D. (or E.M.F.) so that the
product of P.D. and quantity of electricity which flows under
T32«.— Weston Cadmium Cell
i". E. Smith's Form, Mounted).
that P.D. shall represent work or energy. Calling the E.M.F.
E and the quantity Q we may write
EQ = Energy
and the energy in h calories is hj, where / is the mechanical
equivalent of heat (42 million ergs per calorie approximately). If,
therefore, we equate the electric energy to the heat energy
we get
or E =-
an expression which gives the maximum possible value for E, as
this assumes no waste.
Now the value of Q can be determined by finding the quantity
* Thomson's Thermo-Chemistry (translated by Burke, 1908), p. 325.
208
PRACTICAL ELECTRICITY
of electricity required to deposit 63-5 grammes of copper or 65
grammes of zinc.
This = 63-5/0-0003286 coulombs (see Section 10),
= 63-5/0-003286 C.G.S. units of quantity.
Fig. 1323.— Tinsley Cell.
Inserting the numbers in the above equation we have
„
E =
,000 x 42,000,000 x 0-003286
C.G.S. units,
63-5
= 1-086 x io8 C.G.S. units,
= 1-086 volts, (since i volt = io8 C.G.S. units).
In this way we deduce from purely mechanical and thermal
experiments and our definitions of E.M.F. and quantity, the
approximate value of the E.M.F. of a galvanic cell, a matter of
great scientific and practical importance, as it shows the intimate
relation that exists between mechanical, thermal, chemical and
electrical quantities. That the E.M.F of a Daniell's cell is about
1-08 volts is now a well known fact. Calculations similar to the
CALCULATION OF E.M.F. OF CELL 209
above were first made by the late Lord Kelvin in 1851 to ascer-
tain the E.M.F. of the Daniell cell in terms of the absolute electro-
magnetic unit of P.D., io8 of which were, several years afterwards,
viz., in 1862 — called I volt. At the present time the E.M.F. of
any cell can be measured directly by a high resistance voltmeter,
but in 1851 no voltmeters or ammeters or resistance coils adjusted
in ohms, or standard cells existed.
NOTE. — Persons who desire further information about primary batteries,
and the cost of electric energy produced by such means, should consult
the 1896 edition of this work.
CHAPTER VI
RESISTANCE ; ITS LAWS AND MEASUREMENT
84. Comparing Resistances: Voltmeter and Ammeter Method — 85.
Ohmmeter: Megger — 86. Simple Substitution Method of Comparing
Resistances — 87. Differential Galvanometer ; A Null Method — •
88. Wheatstone's Bridge: its Principle— 89. Wheatstone's Bridge:
its Use and Simple Method of Constructing — 90. Bridge Key — 91.
Use of a Shunt with the Bridge — 92. Meaning of the Deflection of
a Bridge Galvanometer — 93. Conditions Affecting the Resistance of a
Conductor — 94. Variation of Resistance with Length — 95. Variation
of Resistance with Cross Section — 96. Variation of Resistance with
Material — 97. Resistance of Metals and Alloys per Centimetre
Cube and per Inch Cube. Specific Resistance or Resistivity — 98.
Resistance of Metals and Alloys for a given Length and Weight —
99. Variation of Resistance with Temperature — 100. Conductors of
Large Specific Resistance have Small Temperature Coefficients — 101.
Conductivity and Conductance — 102. Comparison of Electric and
Heat Conductivities — 103. Resistance and Conductance of Several
Conductors in Series or in Parallel — 104. Currents in Parallel
Conductors — 105. Kirchnoff's Rules — 106. Shunts — 107. Multiplying
Power of a Shunt — 108. Usual Method of Constructing a Shunt Box —
109. Increase of the Main Current Produced by Applying a Shunt —
no. Principle of Universal Shunts — in. Method of Constructing a
Universal Shunt Box ; Advantages of Universal Shunts — 112. Stand-
ard Resistance Coils — 113. Ordinary Forms of Wheatstone Bridge —
114. Portable Forms of Wheatstone Bridge — 115. Dial and Bar
Patterns 'of Bridge.
84. Comparing Resistances : Voltmeter and Ammeter Method.
— By the method described in Section 56, and illustrated in Fig.
91, two resistances can be compared if the relative calibration
of a voltmeter only be known. Further, any of the methods
described in Section 62 for calibrating a voltmeter in volts,
which depend on using a conductor whose resistance is known
in ohms, can be used for measuring a resistance in ohms, if the
voltmeter has been previously calibrated in volts. The one
of these methods which is illustrated in Figs. 94 and 940 is par-
ticularly useful when we desire to know the resistance of a con-
ductor which is much heated by the passage of a current through
it — for example, the resistance of the luminous carbon filament
of a glow lamp, or the apparent resistance of the " electric arc."
The name " resistance " here means, as before, the ratio of
210
.
OHMMETERS 211
the P.D. in volts to the current in Amperes, but in these two
instances it is no longer a constant quantity and independent
of the current passing, so that it is only by a sort of extension of
the name " resistance " that it can be used at all in such cases.
Indeed, had the early experience of currents passing through
conductors been always with currents large enough to produce
considerable warmth in the conductor, it is probable that we
should never have acquired the conception we now possess of
a conductor having a definite resistance as it has a definite length
or a definite cross-section.
If in Figs. 94 and 940, the readings of the ammeter and volt-
meter are / amperes and V volts respectively, then the resistance
of the conductor c, will be given by the formula
y
R = — (approximately),
the approximation arising from the current through the voltmeter
in Fig. 94, and the resistance of the ammeter in Fig. 940, being
neglected. Calling the resistance of the voltmeter Rv, and that
of the ammeter Ra the correct expressions for R are : —
y
R = - - , for Fig. 94.
for Fig.
or
85. Ohmmeter : Megger. — Frequently, when we are measuring
the resistance of a conductor traversed by a strong current, as,
for example, the apparent resistance of an electric arc, we desire
to know in addition the current which is flowing. In such a case
the necessity of having to take simultaneous readings of an
ammeter and a voltmeter in order to ascertain the resistance,
is no disadvantage, since two things have to be ascertained,
and, therefore, two measurements must necessarily be made at
the same time. But in other cases, when the resistance alone
has to be ascertained, it may be a disadvantage to have to take
readings of two distinct instruments simultaneously. Hence
an instrument called an " ohmmeter " was devised by Professor
Perry and one of the authors (W. E. A.) to enable the resistance
of any part of a circuit, through which a current is passing, to
be measured by making a single observation.
212 PRACTICAL ELECTRICITY
A simple ohmmeter contains a " current coil " cc (Fig. 133) and
a P.D. or " pressure coil " cc placed usually at right angles to one
another, and both acting on the same magnetic needle. The former
coil has its terminals T T connected with the circuit, the resistance
of some portion of which it is desired to measure, so that c c is
in series with the circuit, while t, t, the terminals of the pressure
coil, are joined with the points
H and j, the ends of that bit
of the circuit whose resistance,
R ohms is wanted, in the
same way as a voltmeter,
would be placed in parallel
with H j.
The resistance of the current
coil is made as low as possible,
while the portion of the ohm-
Fig. las—Diagram of ohmmeter. meter between the terminals t
and / is made relatively very
high, either by the pressure coil c c itself being wound with a very
long fine wire, or by an auxiliary resistance being added to this
coil and included in the instrument between the terminals t, t.
If the needle be short, the force due to the current passing
round either of the two coils will be perpendicular to the plane
of that coil (Figs. 38, 49). Further, if the needle be made of
hard steel so that its magnetism is not altered by the currents in
the coils, these two forces will be directly proportional to the
currents respectively. Hence the needle will be acted on by
two forces at right angles to one another ; one directly propor-
tional to V, the P.D. in volts between the points H and j, the other
directly proportional to 7, the current in amperes passing through
the conductor H j. Consequently, if matters be so arranged that
no other magnetic forces than the two just mentioned act on the
needle, it will place itself so that the tangent of the angle it makes
with the plane of the pressure coil will be directly proportional
to the ratio of V to 7, that is to R, the resistance in ohms of the
conductor HJ (see Section 31).
Further, if all extraneous magnetic action be avoided, then,
whether the needle be short or long, made of soft iron or of hard
steel, it will place itself at right angles to the plane of the current
coil, that is, parallel to the plane of the pressure coil, when t, t are
both connected with the same point H, that is, when the resistance
of the part of the main circuit included between the two terminals
t and t is nought. As the leads to terminals t, t are separated, so as
to make contact with points of the main circuit farther apart, say,
EVERSHED'S OHMMETER 213
with H and K, the P.D. between the terminals of the pressure coil
will increase, and the needle will deflect away from the plane of
the pressure coil.
And, although the tangent of this deflection may not be directly
proportional to the ratio that the P.D. between the points H and
K bears to the current passing through the conductor H j K, the
deflection will be quite constant as long as the terminals t t are
connected with the points H and K respectively, or with any two
other points in the main circuit separated by the same resistance,
whatever may be the current passing through the main circuit.
For if the main current be doubled, the P.D. between the points
Line C K
Generator
Earth
Fig. 134. — Diagram of Connections of Evershed Ohmmeter.
H and K will be also doubled, therefore both the forces acting on
the needle will be increased in the same ratio, the resultant
will be in the same direction, and the deflection will remain as
before. Hence, whatever the shapes and sizes of the two coils
and of the needle, the scale of the ohmmeter can be graduated
to read off resistances directly in ohms, provided that the only
forces acting on the needle be those due to the currents flowing
round the pressure and current coils respectively.
The principle of the ohmmeter has been employed by Mr.
Evershed in constructing a commercial instrument that has been
much used for measuring the resistance to leakage of electric -light
wires and fittings. The connections of the Evershed ohmmeter
are shown in Fig. 134, the lettering being arranged to correspond
with that in Fig. 133. The current necessary to work the instru-
ment is obtained by means of a portable generator G, or by a
battery of a large number of cells. By comparing Fig. 134 with
Fig 940 it will be seen that the connections are the same, and
73
therefore we have R + Ra = j (see Section 84), where Ra is the
resistance of the current coil c c, Fig. 134. For simplicity, the
magnet and pointer have been omitted in the latter figure. As
Ra is constant for any given instrument, the position the pointer
2i4 PRACTICAL ELECTRICITY
takes up when R= o can be marked o, so the resistance of the
current coil can be allowed lor in this way, and the scale graduated
to read off directly the resistance of R.
To obviate errors of the readings which may be caused by
magnetic forces other than those produced by the currents in
the coils c c, and c c, Mr. Evershed has introduced a new form
of instrument called the "Megger" (abbreviation for megohm -
meter)* in which a fixed magnet and moving coils are used,
instead of fixed coils and moving magnets. In fact he has
applied the principle of the moving coil galvanometer to the
ohmmeter, and thereby obtained the comparative immunity from
disturbance by external magnetic fields which is a prominent
feature of moving coil instruments (see Sections 43 and 61). He
also uses the same fixed magnet to form part of the portable
generator employed to produce the necessary currents, and thus
combines in a single instrument the functions of generator and
ohmmeter.
In Fig. 134 an arrangement for increasing the range of resist-
ance which the ohmmeter will measure satisfactorily, is shown.
By moving the switch arm from A to B, thus joining j with B,
the current coil c c is shunted by a resistance K B, so that only
a fraction of the main current flows through it. The field of
the current coil is therefore weakened and a higher reading is
obtained on the instrument. Usually K B is made so that
moving the switch from A to B gives a tenfold reading.
86. Simple Substitution Method of Comparing Resistances. —
If we merely wish to cut off a length of wire which shall have
exactly the same resistance as that of some other conductor , for
example, if we desire to make a resistance exactly equal to that
of a standard ohm, or a standard ten-ohm coil, the following
method may be adopted : — In circuit with the conductor whose
resistance we wish to reproduce, place any convenient current-
generator and a galvanoscope. Neither the resistance nor
the relative calibration, nor the absolute calibration, of this
galvanoscope need be known. Observe the deflection. Next
remove this conductor, and put in its place a piece of the wire
out of which we desire to construct the resistance, of sufficient
length that a smaller deflection of the galvanoscope is obtained
with the same current -generator. Gradually diminish the
length of this wire until the original deflection is obtained, then
the resistance of this wire must be equal to that of the conductor,
if no other change has occurred in the circuit.
* A megohm is one million ohms. The instrument is called a megohm-
meter because it is intended to measure very high resistances.
COMPARING RESISTANCES 215
To detect any possible change in the sensibility of the galvano-
scope, or in the strength of the current -generator during the test
— a change in either of which would, of course, destroy the accuracy
of the reproduction — it is well, after the wire has been shortened
nearly sufficiently, to substitute the original conductor and see
whether the deflection now obtained with it is exactly the same
as it was at first. If it be found to be slightly different, then the
final adjustment of the length of the wire must, of course, be
made with the new deflection of the galvanoscope. Care must
be taken not to shift accidentally the controlling magnet of the
galvanoscope between the interchange of the conductor and the
wire; further, the current -generator should not be allowed to
Unknown
Fig. 135. — Comparing Resistances by Substitution Method.
send a current for so long a time through either the conductor or
the wire that there is any evidence of a falling -off of its power.
In order to connect the galvanoscope and current -generator
quickly, and conveniently, with either the known or the unknown
resistance, a " plug key, or switch " (Fig. 135), may be conveni-
ently employed. It consists of three sectors of brass, each
carrying a terminal, fastened to a slab of ebonite, or hard wood,
and a brass taper plug, P, which fits tightly into either of the
holes, H or h, this plug being provided with an ebonite or a
wooden handle. If, therefore, the plug P is put into the hole h,
the current will pass through the known resistance, while if the
plug be put into the hole H, the current will pass instead through
the unknown.
The current ^generator B, galvanoscope G, and the resistances
R and R' are shown symbolically in the figure, whilst the plug
key is in perspective.
The preceding method of comparing the equality of two
resistances is exactly analogous with Borda's method of double
weighing, by means of which the weight of a body can be
accurately compared with that of known standard weights, no
matter how unequal be the lengths of the two portions of the
beam of the balance, or how unequal be the weights of the scale
pans.
2l6
PRACTICAL ELECTRICITY
If the known resistance Rr consists of a resistance box such as
that shown in Fig. 89, then any unknown resistance within the
range of the box may be measured by first observing the deflection
of the galvanoscope produced when the unknown resistance is in
circuit and then substituting the resistance box and finding by
trial which plugs have to be taken out of the box to reproduce
the deflection.*
87. Differential Galvanometer, A Null Method.— The measure-
ment of resistance by the method just described is not susceptible
of great accuracy, for this depends on the exactness with which
B
G
Fig. 136. — Diagram of Differential Galvanometer Circuit.
the deflections of the galvanoscope can be read and reproduced,
as well as on the constancy of the battery supplying the current.
To get over these disadvantages methods have been devised in
which equality of two resistances is indicated by absence of
deflection of a galvanoscope or galvanometer. Such methods
are called " Null Methods," one of the simplest of these is that
of the differential galvanometer.
If the galvanoscope G in Fig. 136 be wound with two coils c and
c' which exert equal forces on the needle when a given current
passes through either of them, then, if equal currents be sent
through them in opposite directions, there will be no deflection of
the needle. If, further, coil c be put in series with the unknown re-
sistance R, and c' in series with R1 ', and in opposition to c, as in-
dicated in Fig. 136, then, if the resistance of c is equal to that of
c', there will be no deflection of G when Rr = R, for under these
conditions the currents through R and R' , and therefore through
c and c' will be equal. The absence of deflection will thus in*
dicate the equality of R' and R, and if R' be known then R also
is known. By using sufficient battery power the currents through
* In the practical use of resistance boxes and plug keys, it is important
that the plugs and holes be kept quite clean, as well as the ebonite supporting
the blocks. It should also be remembered that each plug acts like a wedge,
and forces the blocks apart to some extent when it is inserted. When a
plug is taken out it allows the blocks to approach each other, and thereby
loosens the plugs in adjacent holes. //, therefore, any plug be withdrawn
from a resistance box those on opposite sides of it should be re-tightened.
DIFFERENTIAL GALVANOMETER ^ 217
the two circuits R c and R' c' would be large enough to cause a
very small percentage difference in the two currents to produce
quite an appreciable deflection of the galvanometer, so the method
can be made very sensitive. Any change in the E.M.F. or resist-
ance of the battery would affect both circuits equally, so this
method of testing does not depend on the constancy of the battery.
A galvanometer with two coils fulfilling the conditions stated
above, viz. equality of magnetic effect and equality of resistance,
is called a differential galvanometer. The two conditions are
realised as follows : — Two reels of silk -covered copper wire are
chosen so that the diameter of the wire on each is as nearly as
possible the same,* and the two wires are wound side by side on
the galvanometer bobbin until it is nearly full ; the wires are
then tested and cut, so that the resistance, but not, of course,
necessarily the length, of each wire is the same. A current is
now sent in opposite directions through the two coils in series,
when it will be found that, although the wires have been wound
on side by side, one of them will have a slightly greater magnetic
effect than the other, partly perhaps because, being a trifle
thicker, it has to be longer than the other, so as to have the same
resistance, or partly because it is, on the whole, nearer the sus-
pended needle than the other. To remedy this, a small portion
of the wire having the greater magnetic effect is unwound, and
without being cut, which would, of course, destroy the equality
of the resistances of the two coils, the portion so unwound is
doubled back on itself and coiled up out of the way in the base of
the instrument. Thus, by unwinding more or less from the coil
that was magnetically the more powerful, a very good balance can
be obtained. In the use of differential galvanometers in which
the needle is suspended by a silk fibre, a final and most delicate
adjustment can be obtained by raising or lowering one of the
levelling screws slightly, so as to tilt the needle nearer to or
farther from one of the coils. And the spirit level attached to
the instrument should then be permanently adjusted so that the
bubble is in the centre of the glass cover of the level, after the
instrument has been tilted in the manner just described.
When a differential galvanometer is in adjustment no deflection
will be produced if a current be passed through the two coils in
parallel opposing, or in series opposing.
A differential galvanometer can be used not only to indicate
the equality of two resistances, but also to show when one resist-
ance is any multiple or submultiple of another. For example, if
* The wire on the two reels may, with advantage, have been cut from
the same long length of wire.
2i8 PRACTICAL ELECTRICITY
the terminals of the coil c (Fig. 136) be connected by a wire whose
resistance is equal to c but which is arranged to exert no magnetic
force on the needle of the galvanometer, then to produce balance
R' must be equal to 2R ; for the coil c will only carry half the
current passing through R (the other half passing through the
wire in parallel with it) so that to give balance the current in R
must be twice that in R'. This condition will be satisfied when
R' — 2R, for then the resistance of the path R' and c' will be double
that of R and c with c " shunted " by a resistance equal to itself,
and as the two paths are subjected to the same P.D., viz., the P.D.
between p and Q, the currents in them will, by Ohm's law, be
inversely as the resistances of the two paths ; the current in R
will therefore be twice that in R'. Similarly if we put a second
" shunt " on the coil c of resistance equal to the coil itself, balance
would result when R' = $R. A single " shunt " of resistance equal
to half that of either would produce exactly the same result as the
two together. From the foregoing we can formulate a rule
relating to shunted differential galvanometers, viz., if one oj
its coils be shunted by a resistance -ih of its own resistance then
n J
balance will be produced when the resistance in series with the
unshunted coil is n + i times that in series with the shunted coil,
88. Wheatstone's Bridge : its Principle.— The differential gal-
vanometer is a very convenient apparatus for ascertaining
whether one resistance is a certain definite multiple of another ;
but for accurately and rapidly comparing any two resistances,
whether equal to one another or whatever may be their ratio,
the " Wheatstone's bridge," or " Wheatstone's balance," as it is
sometimes called, is more convenient.
As the late Sir Charles Wheatstone explained, when he first
gave a public description of the balance method of comparing
resistances, the credit
of its conception was
due to Mr. Christie.
The name of the better
i K ^-^jz^^_^> — /, #j known man, however,
Fig. 137. has been universally
attached to the ar-
rangement, which is shown symbolically in Fig. 137.
Two conducting branches, P s Q, P T Q, are joined in parallel,
and a current sent through the arrangement, as indicated by the
arrows, then in passing from p to Q, either along the conductor
p s Q, or along the conductor P x Q, there are points having all
potentials between the potential of P and that of Q ; therefore it
WHEATSTONE'S BRIDGE
219
follows that for every point in the conductor P s Q, there must be
a point in the conductor P T Q having the same potential. Let
s and T be two such points ; then, if they were joined with the
terminals of an electrostatic, or of a current voltmeter, or indeed
with the terminals of any galvanometer, there would be no
deflection. Given one point s, the corresponding point T can,
therefore, be experimentally
found by joining one ter-
minal of an electrostatic
voltmeter, or of any gal-
vanometer, to s and touching
the other conductor P T Q at
different points with a wire
attached to the other ter-
minal of the voltmeter or
galvanometer, until a point
T is found for which there
is no deflection. In practice
a galvanometer is generally
employed, since a galvano-
meter can be constructed so
as to be a much more sensitive detector of a P.D. than an
electrostatic voltmeter.
Let Ia be the current flowing along P s, then Ia must be the
current flowing along s Q also, since no current passes through a
galvanometer connecting the points s and T (Fig. 138). Let /&
be the current flowing along P T Q, and let Ra, R^, Rc, Rj, be the
resistances respectively of P s, s Q, P T, T Q* ; then, since the
potential difference between P and s is the same as the potential
difference P and T,
I a Ra — h RC-
Similarly, since the potential difference between s and Q is the
same as the potential difference between x and Q,
Fig. 138.— Simple Diagram of Wheatstone's
Bridge.
Therefore, combining these two equations, we have
Ra - Rc / x
~FT ~ 7T \22}
Kf) t\d
which is the law connecting together the resistances of the four
" arms " of the Wheatstone's bridge when balance exists, f
* p s, s Q, P T and T Q, are called the " arms " of the bridge,
t If no current passes through the galvanometer, when current flows
through the arms, the bridge is said to be " ^ ««'••* "
balanced.
220 PRACTICAL ELECTRICITY
This law may also be written in the form,
RaRd-RbRc (23)
or in words, the products of the resistances of opposite arms in a
balanced bridge are equal.
This law may also be proved graphically, thus : — Let o, A, B, c
(Fig. 139) be points in a conductor through which a steady current
is flowing and let o A, A B, B c be drawn so that the lengths of
the lines represent, on some
convenient scale, the resist-
ances of the parts of the con-
ductor between the points o
and A, A and B, and B and c
respectively, then if lines o P,
A Q, B R, c s be drawn per-
pendicular to the straight line o A B c and of such lengths that
they represent the potentials at the points o, A, B and c respec-
tively, it follows from our fundamental definition of resistance
that the points P, Q, R and s all lie in one straight line, and that
the tangent of the angle this straight line makes with o A B c
measures the current. The trigonometrical tangent will, however,
only measure the current in amperes if the length of the hori-
zontal line that represents an ohm is the same as the length of
the vertical line that represents a volt.
Suppose now that P P'
(Fig. 140) represents the
P.D. between the points
p and Q in Fig. 138, and
suppose that P s repre-
sents the resistance Ra,
SQ represents Rb, PX
represents Rc, and TQ
represents Rd, then, if the
points P' and Q in both
the figures be joined by
Fig* 140< straight lines, and per-
pendiculars s s', x x' be erected, it follows these perpendiculars
represent the P.Ds. between the points s and Q and x and Q re-
spectively of Fig. 138, on the same scale that P p' represents the
P.D. between the points P and Q. But the points s and x are by
hypothesis selected such that no current flows through a galvano-
meter used to join them, therefore s s' equals x x'.
Further, from the properties of similar triangles, we know
that-
WHEATSTONE'S BRIDGE 221
therefore, since s s' equals T x', we have
Rb Rd
Ra+Rb Rc+Rd'
; ; . or #f=^j
the same relationship as was previously arrived at as the law of
the Wheatstone's bridge.
The last equation may also be written in the form
_Ra Rp_
RC " Rd'
and this is the equation that we should have obtained for no
current through the galvanometer, had its terminals joined P
and Q, and the current generator been placed between s and T.
Hence, when balance is obtained with a Wheatstone's bridge, the
balance will nut be disturbed by interchanging the galvanometer
and battery.
89. Wheatstone's Bridge : its Use and Simple Method of
Constructing. — Any one of the four resistances, Ra, R^, Rc, Rd
can be expressed in terms of one of the other resistances multiplied
by the ratio of the two remaining resistances to one another. For
example,
or d = oX-,
Ka
etc. If then the bridge be " balanced," that is, if two points s
and x have been found of the same potential, and we know the
resistance of one of the arms, say R^, in ohms, and the ratio of the
resistance of two of the other arms, say Rc to Rd, but not neces-
sarily the values of either Rc or Rd in ohms, we can, from the
first equation given above, find at once the value of the resistance
of the fourth arm, Ra, in ohms. Similarly, if we know Rc in ohms
and the ratio of R^ to Ra, but without necessarily knowing either
Rb or Ra, we can at once find the value of R^ in ohms, from the
222
PRACTICAL ELECTRICITY
third equation, etc. Hence, one mode of using the bridge to
measure the resistance of Ra is to keep the ratio of Rc to R# con-
stant, and simply vary the resistance of Rb until no current
passes through the galvanometer. Another method consists in
keeping R^ constant and varying the ratio of Rc to Rj. For ex-
ample, the resistances Rc and R& may be the resistances of different
lengths of the same kind of wire, in which case we know that Rc will
be to Rd simply as the
ratio of these lengths
whatever be the abso-
lute resistance in ohms
of the two parts (see
Section 94). In both the
above cases Rc and R&
are called the " ratio
arms," or the " pro-
portional arms."
A form of Wheat-
stone's bridge in which
P T Q, of Fig. 137, was one piece of stretched wire, and the ratio of
the " proportional arms " Rcto Rg, varied by moving the connec-
tion of the wire leading to one terminal of the galvanometer, was
originally employed by the Electrical Committee of the British
Association, and is, for this reason, sometimes called the " British
Association bridge " ; at other times, the " metre bridge" from
the stretched wire being often a metre long. The wire may be
made of platinum, or better still, of platinum-iridium which re-
sists wear.* In Fig 141, P Q represents the stretched wire and K
a sliding key which can make contact with it at any point.
Fig. 141. — Diagram of Metre Bridge.
Fig. 1410;. — Commercial Form of Metre Bridge.
To protect the platinum-iridium wire from being accidentally
knocked or damaged, it may conveniently be placed in a groove
cut in the edge of an ebonite or slate disc, D (Fig. 142), and contact
made with any point of it by means of the spring key K carried
at the end of the movable radial arm A, and shown in detail in
Fig. I42«. The small pin under the knob K is to prevent the
knob being pressed down so much as to damage the platinum-
* On account of their comparative cheapness german silver, platinoid or
manganin are frequently used for bridge wires.
METRE BRIDGE
223
indium wire. The circuit of the battery B (Fig. 142) is closed by
a separate key K'.
The scale round the edge of the disc in Fig. 142 is divided into
centimetres and millimetres, but for rapid work it is more con-
venient to have this scale divided into ratios, as indicated for a few
points in the following table, where the top line of numbers
gives the length of the bridge wire measured from the left hand,
the second line of figures the ratio of the length on the left
Figs. 142 and 1433. — Circular Metre Bridge.
to the length on the right, and the third line the ratio of the length
on the right to the length on the left : —
o 10 20 30 40 50 60 70 80 90 zoo
o o-in 0-250 0-429 0-667 l I-5°0 2-333 4 9 oo
oo 9 4 2-333 1-500 i 0-667 0-429 0-250 o-iii o
A form of metre bridge of greater range is shown in Fig. 143.
It has three stretched wires w w, each a metre in length, and so
arranged that either one of them alone, or two of them in series,
or all three in series, can be made use of to form the two sides Rc
and Rj of the Wheatstone's bridge (Fig. 141). When the plug E
is, as in the figure, placed in the hole H, the current simply passes
through the stretched wire which is nearest to the observer. If,
on the other hand, the plug E be put in the hole h, then, since
the brass plate P is permanently connected with the plate p'
by a thick copper strip under the base of the instrument, the
224
PRACTICAL ELECTRICITY
stretched wire nearest to the observer is short-circuited and the
middle wire is in series with the one farthest from him. Lastly,
if the plug be removed altogether, the three wires are in series.
The object of thus lengthening the wire is to increase the
accuracy of the test when desired (provided the galvanometer is
sufficiently sensitive ), and a still further increase in the accuracy
can be effected by removing the short-circuiting pieces slt S2,
and inserting coils of known resistance in place of them. For
example, suppose that the ratio of the unknown to the known
resistance be f , then the slider K must be placed so as to divide
the stretched wire into two parts having this ratio. Hence,
if one of the three wires only be used, the lengths of the two parts
which will give exact balance will be 60 and 40 centimetres, and
BRIDGE KEYS 225
an error of I centimetre in the position of the slider will correspond
with an error in the determination in the ratio of
6i_6o
30 40
X 100 per cent., or 4-3 per cent.
If, on the other hand, the three wires in series be employed, then
the lengths into which the three metres of wire must be divided
to obtain exact balance will be 180 and 120 centimetres, and an
error of one centimetre in the position of the slider will correspond
with an error in the determination of the ratio of
181 180
no 120
— r - X 100 per cent., or 1-4 per cent.
If now two coils, each having a resistance equal to, say, 500
centimetres of the stretched wire, be inserted in place of the short
circuit pieces sx and S2, an error of a centimetre in the position of
the slider will only correspond with an error of
781 780
510 520
- x 100 per cent., or 0-32 per cent.
Contact between the platinum-tipped knife-edge k and one
or other of the stretched wires, is produced by depressing the
knob K, which causes the lever, L L, to which this knife-edge is
attached, to turn on an axis A A. On removing the pressure,
the lever is pressed up by a spring underneath it. The slider
should never be moved with the knife-edge k depressed, as this
would scrape the stretched wire and alter its cross-section. To
prevent the wire being cut by the knife edge k, if K be pressed
down with great force, it is desirable that k be carried on a spring
so that the force between the knife-edge and the wire is limited.
In order to enable k to make contact with either the first,
second, or third wire, the knob K is not fastened rigidly to the
lever, but can slide along it in a slot, and be so placed that the
near end of the spring s rests in either one of three grooves on the
top of the lever, L L, corresponding with the three positions
of k when it is in contact with the three stretched wires respec-
tively.
90. Bridge Key. — In using a Wheatstone's bridge it is desirable
to send the current through the four arms of the bridge Ra, Rfr,
Rc, Rj, before it is allowed to pass through the galvanometer,
and this is especially important when testing the resistance of
P
226 PRACTICAL ELECTRICITY
the copper conductor of a long submarine cable, since the current
in such a case takes an appreciable time to reach its final
value and become steady, due to the cable acting as a "condenser."
Hence, if the galvanometer circuit were completed when the
battery was attached to the bridge, an instantaneous swing of the
galvanometer would be produced, even if the ratio of Ra to R^ be
the same as the ratio of Rc to Rj. And although, since the ratio
of resistances having been effected, the current through the galva-
nometer would become nought as soon as the currents in the four
branches of the
bridge became
steady, great delay
in the testing
would be caused
by this first swing
of the needle. A
similar difficulty
would occur in
measuring the re-
sistance of an
electro -magnet or
even of any coil
without an iron
core, if it were not
specially wound ,
because whenever
Fig. 144-Bridge Key. a Coil ls SO WOUnd
that a current
passing through it produces magnetic action, a short interval
of time has to elapse, after putting on the battery, before
the current reaches its maximum, or steady, value, arising from
what is called the " self-induction "* of the coil.
A key for sending the current through the four arms of the bridge
before it is allowed to pass through the galvanometer is shown
at K (Fig. 144) , and is a modification of the one originally em-
ployed by the Electrical Committee of the British Association.
On pressing down the button, contact is first made between the
flexible piece of brass A and the flexible piece of brass B. This
completes the battery circuit, and causes the current to flow
through the four arms of the bridge shown symbolically in Fig.
144 by the spiral lines. On the button being still further pressed
down, B is brought into contact with a little knob of ebonite E
on the top of the flexible piece of brass c. This does not complete
* Defined in Section 195.
BRIDGE GALVANOMETER SHUNT 227
any other electric current ; but on the button being still further
depressed c is brought into contact with D, and the galvanometer
circuit is completed.
This form of key is to be preferred to the ordinary bridge key,
because all the connections are above the base of the key and in
sight, whereas when the connections are made under the base, it
occasionally happens that, without its being noticed, the pieces
of gutta-percha covered wire used to make the connections are
either badly insulated, or are loosely connected at their ends with
the terminals of the key, and so introduce unnecessary and
unsteady resistance.
91. Use of a Shunt with the Bridge. — It is desirable to employ
also another key k (Fig. 144), which may be quite simply made of
a twisted bit of hard brass wire, bent so as to press up against
a sort of bridge of wire. When this key is not depressed, a
portion of the current is shunted past the galvanometer
through any convenient shunt 5. the resistance of which need
not be known, as it does not enter into the calculations. The
object of this shunt is merely to diminish the sensibility of the
galvanometer when the first approximation to balance is being
made. As soon as this has been done the key k should
be depressed, and all the current in the galvanometer circuit
arising from want of perfect balance allowed to pass through the
galvanometer itself, and the resistances adjusted until perfect
balance is obtained. Another device to expedite the testing,
and also to prevent powerful currents being sent through the
galvanometer, consists in not holding the key K down when the
first rough approximation is being made, but merely giving it a
tap, which has the effect, when the balance is far from perfect,
of giving the needle of the galvanometer a slight impulse to one
side or the other, according as the ratio of Ra to R^ is larger or
smaller than that of Rc to R#, instead of causing the needle to
violently swing against the stops on one side or the other as it
would do if the key K were held down before the balance was
approximately arrived at. '
92. Meaning of the Deflection on a Bridge Galvanometer. —
A considerable amount of time will be saved in testing if the
meaning of a deflection of the galvanometer needle, say to the
right, be once for all definitely ascertained, and a note be made
whether it means that the ratio of Ra to R^ is too large or too
small. The simplest way of recording this, if we assume, for
example, Ra to be the unknown resistance, is to put the words
" increase R^ " and " diminish R^ " one on each side of the gal-
vanometer, these being the directions to be followed according
228 PRACTICAL ELECTRICITY
as the needle deflects towards one or other of them.* The position
of these two directions must, of course, be reversed if the terminals
of the galvanometer, or of the testing battery, be reversed.!
Example 59. — In measuring a resistance on the Wheatstone's
bridge the resistances of the arms p T and T Q (Fig. 138) are 1000
and 100 ohms respectively. The unknown resistance is placed
in the arm P s, and the resistance in s Q is adjusted until balance
is as nearly as possible obtained. It is found that when the
variable resistance s Q is 546 ohms the galvanometer deflection
is 15 divisions to the left, while if s Q is made 547 ohms the deflec-
tion is 27 divisions to the right. Find the value of the unknown
resistance, assuming proportionality of deflection for small
changes in the resistance s Q.
A change of i ohm in s Q produces a change of 42, i.e. (15 + 27)
divisions in the deflection, hence a change of — , or 0-36 ohm in
s Q would cause a change of 15 divisions in the deflection. Con-
sequently, if s Q were 546-36 ohms the galvanometer deflection
IOOO
would be zero, therefore the resistance tested is x 546-36,
or 5463-6 ohms.J
93. Conditions Affecting the Resistance of a Conductor.— The
resistance of a conductor depends on four distinct conditions :—
(1) Its length.
(2) Its cross-section.
(3) The material of which it is composed, Hie purity of the
material, and the hardness and density.
(4) The temperature.
It is therefore important that the student should ascertain
by experiment how much change is produced in the resistance
by varying each of these four conditions separately. And gener-
ally, in experimenting, it is to be remembered that when it is
possible to change several of the conditions under which the experi-
ment can be madet it is of the utmost importance that only one of
the conditions should be varied at one time. The effect produced
by the variation of one condition should be fully inquired into
before any one of the other conditions is in any way altered,
* The words " unplug " or " plug " are also commonly used when plug
resistance boxes are employed in the adjustable arm.
t When the coils of a bridge do not enable exact balance to be obtained
the method of " proportional parts " explained in Example 59 may be used.
J This method of " proportional parts " can also be used with the
differential galvanometer, in cases where the resistance coils available
do not permit of exact balance being attained.
RESISTANCE :: LENGTH
229
otherwise it will often be quite impossible to gather from the
results, what portion of the variation of the effect was produced
by a particular change in the conditions.
94. Variation of Resistance with Length. — In Section 49 we
saw that when a steady current passed througn a uniform
conductor the P.D. between any two points was proportional to
the length of the conductor between the points Combining
Fig. 145. — Apparatus for Proving that Resistance is Proportional to Length.
this fact with the fundamental definition of resistance (Section 51),
page 142), it follows at once that the resistance of a uniform
conductor is proportional to its length.
This law may be proved independently by using a high -resist-
ance galvanometer as a voltmeter. For it is to be remembered
that, although it would not be justifiable to prove that Ohm's
law were true by using a cwm^-voltmeter, seeing that the
possibility of employing a galvanometer as a voltmeter depends
on the fact that Ohm's law is true, galvanometers could be used
as accurate voltmeters when once Ohm's law has been proved
to be true, even if the distribution of potential along a uniform
wire conveying a steady current followed some law other than
it actually does.
Fig. 145 shows a simple arrangement for testing the distribu-
tion of potential in such a case. A current is sent through a
uniform wire, say of German silver, stretched along a graduated
230
PRACTICAL ELECTRICITY
bar between the points w w'. A tangent galvanometer, whose
coil has a high resistance compared with that of the straight wire
w w', has one of its terminals, B, connected with one end of this
wire, w, while its other terminal, B', can be connected with any
point of the stretched wire by means of the loose flexible wire
and the sliding key s'.
Then experiment shows,
if the sensibility of the gal-
vanometer is kept unchanged
by the adjusting magnet
not being moved during the
experiment, and if the cur-
rent flowing through the
wire w w' be kept quite con-
stant, that the tangent of
the deflection of the galvano-
meter, and therefore the
P.D. between its terminals,
is directly proportional to
the length of the wire w s'.
In using this apparatus the
contact must be loosened be-
fore the slider is moved along, otherwise the wire will be scraped
and its cross-section no longer remain perfectly uniform.
If it be desired to try this experiment with a longer wire than
can be conveniently used in a straight form, we may employ the
frame (Fig. 146), consisting of six or more wooden cylinders
having a screw groove cut on each. Lengths of say 5, 10, 20,
30, 40 and 50 feet of wire of the same material and having exactly
the same thickness throughout, say o-oi of an inch, may be wound
in the grooves on the respective cylinders, and by connecting
the binding screws together in pairs the whole of the wire may
be joined up in series. If a current be sent through the whole
of the wire joined up in series from left to right through the wire
on the first cylinder, right to left through that on the second,
etc., and if the current be maintained constant, it will be found
that the P.D. between the terminals at the ends of any one
of the cylinders is proportional to the length of wire on that
cylinder, thus proving that resistance is proportional to length.
95. Variation of Resistance with Cross-Section. — For ascer-
taining the law of variation of the resistance of a conductor with
its cross-section, the spiral grooves in another set of six cylinders
(similar to those in Fig. 146 except that all screw grooves are of the
same pitch) have wound in them wires all composed of the same
RESISTANCE OF METALS 23*
material and of exactly the same length (say, twenty feet), but
having diameters respectively of, say, 0-0076, 0-0092, 0-0108,
0-0136, 0-0164, 0-02 of an inch.* The resistances of these wires
may be tested by any of the methods described in the Sections 84
to 89, or 94, in terms of some one resistance taken as a standard ;
and when this is done, it is found that the resistances of the different
conductors of the same material are inversely as the squares of their
diameters — that is, inversely as their sectional areas.
96. Variation of Resistance with Material.— The cylinders
in this case have wound on them wires of exactly the same length
(say, twenty feet) and having exactly the same diameter (say,
o-oi of an inch), but made of the following materials respec-
tively— copper, brass, platinum, iron, lead, and German silver ;
and when the resistances are tested by any of the methods
described in Sections 84 to 89, it is found that the metals, as
given in this list, are arranged in increasing order of resistance,
this being, roughly, as the numbers I, 4, 5}, 6, 12, 13.
97. Resistance of Metals and Alloys per Centimetre Cube
and per Inch Cube. — The " specific resistance " or " resistivity " of
a material is usually expressed as the resistance in " microhms "
or millionths of an ohm, at o° C. of a centimetre cube,| or of an
inch cube — that is, the resistance from one face to the opposite
face across the cube. It has been customary hitherto in books
to give a table of the specific resistances of a number of pure
materials and alloys expressed to four significant figures as deter-
mined by Dr. Matthiessen nearly fifty-five years ago ; and such
a table will be found in the early editions of " Practical Elec-
tricity." But during recent years a number of investigations
have been carried out on the resistance of copper — the material
generally employed for electric conductors — and it has been
found that a diminution of from 3 to 4 'per cent, can be produced
in the resistance of copper by compressing it, without any change
being made in its chemical composition. Electrolytic refining of
copper has led to the production of the metal on a large scale
of a high degree of purity, so it is now quite common to find
" commercial copper " of greater purity arid smaller specific
resistance than the " pure copper " tested by Matthiessen.
Difference in density of a material alters the specific resistance
as also does the mechanical treatment or annealing to which it
* These sizes correspond with No. 36, 34, 32, 29, 27 and 25 Standard
Wire Gauge.
f We may point out that the expression " centimetre cube," as here
used, is merely an abbreviation for " a conductor one centimetre long and
one square centimetre cross-section." A similar meaning applies to
" inch cube."
232 PRACTICAL ELECTRICITY
has been subjected, so the numbers given in the following table
(No. VI.) must be regarded as being only approximately correct,
and they are, in most cases, stated only to three significant
figures. The substances are arranged in order of increasing
specific resistance, and the unit employed is the international
microhm.
From the table on page 233 we see that of the various pure
metals, annealed silver is the one having the least, and bismuth
the one having the greatest, resistance for a given length and
sectional area.
The numbers given in the table can be used to ascertain
the resistance of a wire or rod of any length and of any cross -
section composed of any one of the materials at o°C. For
example, if p be the specific resistance per centimetre cube, in
microhms, / the length, and d the diameter of the wire in centi-
metres, the resistance is
— - ^ microhms. (24)
or more generally we may write,
length
Resistance = — : — X specific resistance, (25)
cross-section
Example 590. — Find the resistance at o° C. of an annealed
copper wire ^th of an inch in diameter and 100 yards long.
Taking the specific resistance as 0-61 microhm per inch cube
and using the formula —
R .= — -^ we have
4 x 100 x 36 x 0-61
R
3-1416 x — x io6
100
= 0*279 ohm, approximately.
Example 60. — What length of hard drawn copper wire, No. 16
gauge (diameter 0-064 inch) will have a resistance of I ohm at
o°C. (assume p =0-640) ?
Answer. — 5,020 inches.
140 yards, approx.
Example 61. — What must be the diameter of a platinum silver
wire so that it may have a resistance of one ohm per metre at
o°C.?
Answer. — 0-0555 cm.
0-555 mm.
SPECIFIC RESISTANCES
233
TABLE VI:
PURIFIED SUBSTANCES ARRANGED IN ORDER OF INCREASING RESISTANCE
FOR THE SAME LENGTH AND SECTIONAL AREA.
Name of MetaL
Resistance in International
Microhms at 0° Centigrade.
Relative
Resistance.
Centimetre
Cube.
Inch Cube.
Silver, annealed ...
•48*
0-583
( from
Copper, annealed j tQ
'55
•6l
o-6io
0-633
°4
•09
,, ,, (International, 1913)!
•588
0-6250
•07
,, (Matthiessen)
•594
0-6277
•077
Silver, hard drawn
•58
0-622
•07
Copper, hard drawn ... < tQ
'59
1-64
0*626
0-646
•07
•II
,, (Matthiessen) ...
1-630
0-6418
•io
Gold, annealed ...
2-05
0-807
1-38
Gold, hard drawn
2-089
0-822
1-41
Aluminium, annealed
2'43
0-96
1-64
Silicium Bronze ... (about)
2'5
0-98
1-69
Zinc, pressed
5-01
2-21
379
Tungsten ... ...
6-4!
2'S
4' 3
Nickel, annealed ...
T- +
6 '94
J
2 73
T" j
4-69
Phosphor bronze ... (about)
7-8
3-07
5-27
Platinum, annealed
9-04
3'55
6-09
Iron, annealed
9'7
3-82
6-56
Gold-silver alloy (2 oz. gold,
i oz. silver), hard or annealed
10-8
4-27
7-33
Tin, pressed ...
13-2
5'«9
8-9
Lead, pressed
19-6
7-71
13-2
German silver { f ™m
19-0
42-O
7-48
n-8
12-8
2O'2
Platinum-iridium alloy, Density
21-32
22'2
8-73
I5-0
Platinum-silver alloy (i oz.
platinum, 2 oz. silver), hard
or annealed
24 '3
9-58
16-4
Platinoid (about)
34
i3H
23
Antimony, pressed
35'4
I3-9
23-8
Manganin (about)
42
167
28-7
Nickelin .
43
17
2Q
Eureka
T" J
47
18-5
~\7
•32
Constantan
48
j
19
J6
33
Ta Ta
e;i
20
34*e
KruDcin
Jx
8«;
33
OH- J
CT
Nichrome
D
89
3*5
~/ /
60
Mercury
94-08
j j
37'°4
63-6
Bismuth, pressed ...
1 08
42-5
73
Carbon ... ... (about)
4,000 to
i, 600 to
2,700 to
10,000
4,000
6,700
* Profs. Dewar and Fleming give 1-468, and Mr. Fitzpatrick 1-481.
t The International Standard for Annealed Copper at 2o°C is : — i metre
length of i square millimetre cross section has a resistance of ^ ohm. (0-017241) .
| Varies between 5-0 and 6-6,
234 PRACTICAL ELECTRICITY
Example 62. — Determine the cross- section of a column of
mercury at o° C. whose resistance will be o-i ohm per metre ?
Answer. — 0-094 sq. cm.
9-4 sq. mm.
Example 63. — Which has the greater resistance, a copper wire
20 feet long, 0-015 inch in diameter, or a platinum -silver wire 10
feet long, 0-037 mcn m diameter at o° C. ?
The resistance of the copper wire will be to that of the platinum -
20 X 1-55 . 10 X 24-3
silver as ~ is to - — , or as 0-79 to I.
0-0152 0-0372
Hence, the copper wire has rather more than three-quarters
of the resistance of the platinum -silver wire.
98. Resistance of Metals and Alloys for a Given Length and
Weight. — As metals are usually sold by weight it is frequently
convenient to know, not the resistance of a given volume of a
material of specified length, but the resistance of a given length
having a given weight. In the following table (No. VII.)
will be found the resistances in international ohms at o° C. of
wires one foot long weighing one grain, and one metre long
weighing one gramme, the substances being arranged in increasing
order of resistance for a given length and weight, this order being
different from that employed in Table VI., where the sub-
stances were arranged in increasing order for the same length
and cross -section.
From Table VII. we see that of the metals aluminium has
the least resistance for a given length and weight ; whereas we
saw from Table VI., that for a given length and cross-section it
was annealed silver that had the least resistance.
Since weight = volume X density,
= length X cross-section X density, we have
cross-section =- : — ^3 r— , and by substituting in formula
length X density
(25) we get as the relation between resistance, length, and weight,
w
where A is the density in grammes per cubic centimetre and
w the weight in grammes.
Example 64. — What will be the weight of an iron wire 100
yards long, having a resistance of i ohm at o° C. ?
An iron wire I foot long, weighing I grain, has I -08 ohms
RESISTANCE, LENGTH AND WEIGHT 235
TABLE VII.
PURIFIED SUBSTANCES ARRANGED IN ORDER OF INCREASING RESISTANCE
FOR THE SAME LENGTH AND WEIGHT.
Name of Metal.
Resistance in International
Ohms at o« Centigrade of
a wire
Relative
Resistance.
i foot long
weighing igrn.
i metre long
weighing i grm.
Aluminium, annealed
0*090
0-063
I
Copper, annealed* { f™m
0-199
O-2O9
0-139
0-143
2'2I
2-27
„ ,, (International 1913)
O-2026
0-1413
225
„ „ (Matthiessen) ...
0-2037
0-1421
2-26
Copper, hard drawn* { £r°™
0-208
0-218
0-142
0-146
2-26
(Matthiessen)...
0-2078
0-1449
2-3I
Silver, annealed
0-218
0-I52
2-42
Silver, hard drawn
0-238
0-166
2-64
Zinc, pressed
0-575
0-401
6-4
Gold, annealed
0-402
6-4
Gold, hard drawn ...
0-587
0-409
6-5
Nickel, annealed .„
0-84
°'59
9 '4
Phosphor bronze ... (about)
i-o
0-70
in
Iron, annealed
I -08
075
12-0
Tin, pressed ...
1-38
O'Q6
15-3
Tungsten
* J
1*64
\S VJW
18-2
Gold-silver alloy (2 oz. gold,
• WT
i oz. silver), hard or annealed
2-36
1-65
262
German silver { rom
2-37
1-66
26-4
I to
2 "87
2"OI
32-0
Platinum, annealed
2-74
i '93
307
Lead, pressed
3' 19
2-22
35-3
Antimony, pressed
2*38
37-8
Platinum-silver (i oz. platinum
2 oz. silver), hard or annealed...
4-19
2-92
46-5
Platinoid (about)
4-40
3-03
48
Manganin (about)
57
Nickelin
_ ,g
VQO
59
Eureka
6-1
j yw
4'2CJ
64
Constantan ...
6-2
*t •** j
4-33
66
" la fa"
4«cc
69
Platinum-iridium (Density 21-32)
6-76
JJ
4-73
V/^J
75
Kruppin
Q'9
6-9
1 08
Nichrome . ... ...
-7 y
10*6
7*4
118
Bismuth, pressed
15-2
T-
io-6
168
Mercury ...
18-36
12 '80
204
* The standard adopted by the International Electro-technical Commission
in 1913 for a length of one metre weighing i gramme, is as follows : —
Annealed copper, 0-15328 ohm at 2o°C.
For commercial purposes a temperature coefficient (constant mass) of
0-00428 per degree C. from o° C. is adopted and 0-00393 when 20° C. is taken
as standard temperature. This gives —
Annealed copper, 0-1413 ohm at o° C,
Hard drawn copper, 0-1443 „ ,,
236
PRACTICAL ELECTRICITY
resistance at o° C. Hence, an iron wire x feet long, weighing x
grains, has % x 1-08 ohms at o° C. If the weight is y grains, the
resistance is - x x 1-08. Now x is here 300, and the resistance is
r ohm
300'
therefore, x 1-08=1, or y
y
: 97,200 grains.
Answer. — 13-9 Ibs.
Example 65. — What is the resistance of a mile of hard drawn
copper wire weighing 20 Ibs. ? (Assume I footgrain 0-210 ohm).
Answer. — 41-8 ohms.
Example 66. — The weight of wire to be used for a 10 ohm
platinum silver coil is not to exceed 5 grammes ; find (a) , the
length and (b), the diameter of the wire required ?
Answers. — Length =4- 13 metres.
Diameter =0-358 mm.
99. Variation of Resistance with Temperature. — To ascertain
the way in which the resistance of metals and alloys varies with
the temperature, small coils of silk-covered wire composed of the
different materials may be conveniently wound on paper cylinders
and inserted in narrow glass test-tubes, GV G2, G3, and G4 (Fig.
147), the test-tubes being supported from a wooden disc. One
end of each of the coils may be soldered
to a common terminal, T, while the
other ends of the coils are soldered to
the terminals TV T2, T3 and T4. The
test-tubes are inserted in the water-
bath w (Fig. 148) , which can be warmed
with the Bunsen burner B, standing
on a sheet of asbestos, A A, to a tem-
perature which is indicated by the ther-
mometer 1 1, enclosed in a brass tube
to prevent mechanical injury ; and the
resistances of the different coils of wire
can be measured with a Wheatstone's
bridge, differential galvanometer, or
other suitable arrangement, the measur-
ing apparatus being protected from the
heat of the burner by means of the
double screen s.
In carrying out experiments of this
kind, it must be borne in mind that, as the glass bulb
of a thermometer is very thin, and as mercury is a substance
Fig. 147. — Coils of Wire used in the
Apparatus for Measuring the
Variation of Resistance with
Temperature.
RESISTANCE AND TEMPERATURE 237
having a very small " specific heat," * a thermometer rapidly
acquires the temperature of the liquid in contact with it ;
whereas a mass of metal inserted in the same liquid may
have a very different temperature from the liquid which
immediately surrounds it, especially if the temperature be
Fig. 148. — Calorimeter for Measuring the Coils of Wire shown in Fig. 147.
rapidly rising or falling. Further, a liquid, being a bad conductor
of heat, the temperature in different parts of it will be different,
unless it be kept constantly in motion ; therefore a stirrer, s s,
is provided with the heating vessel seen in Figs. 147 and 148.
Lastly, the water-bath w is made in two separate parts in order
that the current of hot water which rises by " convection " from
the heated bottom of the water-bath may not come directly
into contact with the glass tubes. By using petroleum in the
inner vessel and perforating the tubes G, Fig. 147, the tem-
perature of the wires can be altered more rapidly and determined
with greater accuracy. This arrangement, however, is more
i
* The specific heat of a substance is the ratio of the amounts of heat
required to raise equal masses of the substance and of water through i°.
238 PRACTICAL ELECTRICITY
dangerous than the water bath, and for elementary work is not
necessary. $ -;>:
Before taking a measurement of the resistances of the coils of
wire at any particular temperature, it is well to adjust the flame
of the Bunsen burner so as to maintain the temperature of the
thermometer constant for some minutes, the liquid being con-
stantly agitated with the stirrer s s during the time. For the
longer the time during which the temperature of the water in
the bath is all kept at a uniform and constant temperature, the
greater is the probability that the coils of wire have acquired the
temperature indicated by the thermometer /.
The exact law connecting the variation of the resistance of a
metal with the temperature depends not only on its chemical
constitution, but on its molecular condition, such as its hardness,
density, etc. To a first approximation the law is a linear one,
i.e., the resistance increases uniformly with the temperature,
the rate of increase for common pure metals being about 0-38 per
cent, per i° C. Nickel and iron have larger " coefficients " as
will be seen from Table VIII., whilst mercury and alloys have
smaller ones. Copper is the material of most importance to
electricians, and for this metal the approximate simple rule, the
resistance of copper increases about 0-4 per cent, per i° C., should
be remembered. This simple rule is practically exact if 15° C. in-
stead of o° C. be taken as the temperature of reference.
More accurately the relation between resistance and tempera-
ture may be represented by the formula,
Rt=R0(i + at + btz), (27)
where Rf is the resistance at temperature t and R0 its resistance
at o° C., and a and b are coefficients depending on the material.
For copper, platinum and mercury the approximate values are : —
TABLE VIII.
Material.
a.
b.
Copper, hard drawn . .
„ annealed
Platinum
>» • • • •
Mercury
+ 0-00408*
+0-00427
+0-00345
+0-0036
+ 0-000888
+ 0-000,001, 12f
— 0-000,000,53 J
— 0-000,000,11)
to U
— 0-000,000,64 )
+0-000,00103
It should be understood that the values of the coefficients
differ somewhat for every specimen, and if it be necessary to
* Swan and Rhodin. f Clark, Forde and Taylor.
J Callendar. § Ayrton and Kilgour.
RESISTANCE THERMOMETERS 239
make use of the relation between resistance and temperature for
accurate work, the law of variation for* the actual piece employed
should be determined experimentally. One of the principal uses
to which the relation between resistance and temperature has
been put is the measurement of temperatures electrically. When
once the law connecting resistance and temperature has been
determined for a given specimen, a measurement of its resistance
is in effect a measurement of its temperature, and by this means
temperatures of ovens, flues, furnaces, etc., can be readily found.
Platinum, is the material generally used for these purposes, and
"platinum thermometers " form at the present day, one of the most
accurate means of measuring temperatures. Copper is sometimes
employed for temperature measurements, and in electrical
machinery the temperature of the copper coils is frequently
deduced from the resistances of the coils themselves ; the resist-
ance at some known temperature having been previously deter-
mined.
100. Conductors of Large Specific Resistance have Small
Temperature Coefficients. — On comparing Table VI. with Table
IX. it will be observed that, if the metals and alloys be arranged
in increasing order of specific resistance, they are arranged
roughly in decreasing order of temperature variation, or, in
other words, the poorer the conductor the smaller its variation of
resistance with temperature. And not only does the tempera-
ture variation become less and less as the specific resistance
of the material increases, but it passes to the other side of zero
and is negative in the case of a bad conductor like carbon, which,
in the form used in electric arc -lamps, has a specific resistance
of about o-oi ohm per centimetre cube — a value, roughly, 6,000
times as great as that of copper. That is to say, the resistance
of carbon diminishes with increase of temperature ; for example,
the resistance of the carbon filament of a glow-lamp, when glowing
at its normal brilliancy, is only about three-quarters of the resist-
ance it possesses when cold. This property of carbon has been
utilised by making a resistance of a metallic wire in series with
a carbon filament, so arranged that the increase of the resistance
of the wire caused by rise of temperature was practically balanced
by the simultaneous diminution in the resistance of the carbon
filament.
When we came to still poorer conductors, such as gutta-percha
or indiarubber, which are, therefore, usually termed insulators,
the temperature coefficient is not only negative, but is numerically
much larger than it is for any metal. For example, the gutta-
percha which is usually employed in the manufacture of
240
PRACTICAL ELECTRICITY
submarine cables sometimes has a specific resistance of about
350 x io12 ohms per centimetre cube at 24° C., or about 200
million million million times the specific resistance of the copper
conductor ; but this high resistance is diminished to one-ninth
by an increase of temperature of only 15° C.
TABLE IX.
SPECIFIC RESISTANCE AND PERCENTAGE TEMPERATURE VARIATION OF
MATERIALS USED FOR RESISTANCES.
Material.
Approximate s
Resistance in Inter-
national Microhms per
Centimetre Cube at
o°C.
Percentage Variation
of Resistance per
i°C.
Tungsten
6*4
6*9
7*8
9-0
97
io'8
19 to 30
22*2
24 '3
34 to 36
39 to 42
43
47
48
Si
85
89
94
4,500 to 10,000
0-3 to 0-5
o'6
0*08
°'3S
o'S
0-065
0*04 to 0*028
0-13
0*027 to 0*03
0'02 tO O'OlS
0*0017 to o'ooo
o "0024
0-0048
0*0014 to
— 0*0029
— 0-0029 to
— 0*0076
0*074
0*024
0*089
—0-03
Nickel
Phosphor bronze ...
Platinum ...
Iron
Gold-silver alloy (2 oz. gold, i oz.
silver)
German silver
Platinum-indium (Density 21*32) ...
Platinum-silver alloy (i oz. plati-
num, 2 oz. silver)
Platinoid ...
Manganin (85 oz. copper, 12 oz.
manganese, 3 oz. nickel)
Nickelin
Eureka
Constantan
"Ja-/a "wire
Kruppin . .
Nichrome
Mercury ...
Carbon
The connection between high specific resistance of a metallic
alloy and low temperature coefficient has led people to seek for
metallic alloys of higher and higher specific resistances. About
1888 Martino found that adding a trace of tungsten to German
silver raised its specific resistance from about 20 microhms to
34 microhms per centimetre cube, and lowered its temperature
coefficient from about 0-044 to °'°2 Per cent* Per l0 C. The
substance thus produced is called " platinoid," and has been
much used in the construction of resistances.
Going still farther, Mr. Weston, by adding manganese to
copper instead of, or in addition to, nickel, succeeded in pre-
paring alloys whose resistance does not vary appreciably for
ordinary changes of temperature, or, like the resistance of
carbon, actually diminishes with rise of temperature.
TEMPERATURE COEFFICIENTS 241
These manganese alloys have been very fully investigated
at the Physikalische Technische Reichsanstalt, the German
Government physical laboratory at Berlin, and meltings, with
as much as 30 per cent, of manganese and a specific resistance
over 100 times that of copper, have been prepared. The par-
ticular alloy, however, which the work of this Institute has
shown to be the best for ordinary purposes is one containing
85 per cent, of copper, 12 per cent, of manganese, and 3 per
cent, of nickel by weight, and is called " manganin." Manganin,
which has a specific resistance of about 42 microhms per centi-
metre cube, or about 28 times that of copper, is now manu-
factured commercially, and, excepting when the most minute
accuracy is desired, the variation of the resistance of com-
mercial manganin may be regarded as zero for ordinary changes
of temperature.
To protect the wire of standard resistance coils from oxidation,
it is customary to coat them with wax or shellac. Manganin
coils protected by shellac have been found to vary very slightly
in resistance, according as the atmosphere is moist or dry, and
to prevent these changes such coils must be hermetically sealed.
With an improved form of standard coil devised by Dr. C. V.
Drysdale, this sealing is unnecessary. The wire is of constantan*
having a negative temperature coefficient, and is electroplated
with nickel of a thickness sufficient to make the temperature
coefficient of the plated wire practically nil.
Example 67. — To find the resistance of a wire 52 metres long,
i square millimetre in section at 22° C., made of pure copper,
hard drawn, specific resistance 1-63.
Resistance required in ohms.
1-63 52 x 100 ,
= - <r X - — - — - X (l + 0-00408 X 22).
Too
Answer. — 0-923 ohm.
Example 68. — To find the resistance of a wire no feet long,
of an inch in diameter, at 46° C., made of pure annealed platinum.
Taking as a mean correcting factor for platinum (i + 0-0036^
000,000,38^), the resistance in ohms equals
- — ~ x ^ — x (i + 0*0036 x 46 — 0-000,000,38 x 46 2).
i X 202
Answer. — 278 ohms.
Example 69. — At what temperature, approximately, would a
German silver coil, which had i B.A. unit of resistance at 16° C.,
have the resistance of i international ohm ?
* A nickel-copper alloy.
Q
242 PRACTICAL ELECTRICITY
One international ohm equals 1-0136 B.A. unit, therefore the
temperature must be raised sufficiently to increase the resistance
of the coil by 1-36 per cent. ; German silver increases in resistance
by about 0-04 per cent, per i°C. (Table IX.), therefore if t be
the increase of temperature above 16°,
0-04 X t = 1-36,
or t = 34°6 C. approximately.
Answer. — The B.A. coil will have a resistance of I international
ohm at about 50° C.
Example 70. — At what temperature would a metre of mercury
i square millimetre in section have I international ohm resist-
ance ?
Answer. — 65°-5 C.
Example 71. — A set of resistance coils made of platinum -
silver are correct at 14° C. Between what limits of temperature
approximately may they be used without correcting the results,
if the temperature error is not to exceed J- per cent. ?
The resistance of platinum -silver increases about 0-03 per cent,
per i° C., as stated in the last table ; therefore, if t be the number
of degrees above or below 14° C., within which the coils may be
used without the error exceeding J per cent.,
0-03 X t = 0-25,
.'. t= 8°.
Answer. — The limits of temperature are, therefore, approxi-
mately 6° and 22° C.
Example 72. — If the greatest change of temperature at some
particular , place between summer and winter is from — 8° to
25° C. in the shade, what is the greatest percentage variation in
the resistance of a set of German silver coils ?
Answer. — 1-3 per cent, approximately.
161. Conductivity and Conductance. — For many years the word
" conductivity " was used to mean the reciprocal of resistance,
and the name mho was suggested by Lord Kelvin as the name
for the unit of conductivity so as to indicate its relation to the
ohm as the unit of resistance. The word " conductance'' however,
is now employed to signify reciprocal of resistance, and conduc-
tivity is denned as the conductance of unit cube. Conductivity
is therefore synonymous with specific conductance in the same way
as resistivity is synonymous with specific resistance. Conductivity
and resistm/y denote specific properties of materials per unit
cube, whereas conduct a nee and resistance are general terms.
The specific resistance or resistivity of silver being 1-50
CONDUCTIVITY 243
microhms (the centimetre being taken as the unit of length), the
conductivity of silver is , or 666,000 mhos.
0-000,001,50
It is common to speak of specimens of copper as possessing
various percentage conductivities, such as 95, 98, or 101 per cent.
What is meant in such cases is that the specific resistance of such
copper bears to Matthiessen's standard, or to the International
standard for annealed copper, the ratio of 100 to the number
mentioned. Thus hard-drawn copper of 98-5 per cent, conduc-
tivity (Matthiessen's standard), has a resistance per centi-
TOO
metre cube at o°C. of —=— x 1*630 microhms, or 1-655 microhms.
Example 73 — If the resistance of a sample of commercial metal
is 97-5 ohms, whereas the resistance of the same piece of metal,
if quite pure, would be 94-3 ohms at the same temperature, what
is its percentage conductivity in terms of that of the pure metal ?
The conductance of the sample of commercial metal = .
97-5
The conductance of the same, if pure, would = • — — ;
.'. if x be the percentage conductivity,
i x i
X
97-5 100 94-3
.'. x = 9672.
Answer. — 96-72 per cent, conductivity.
Example 74. — What will be the resistance at 37° C. of a copper
wire 20 metres long, weighing 12 grammes, and having 92 per
cent, of the conductivity of pure annealed copper according to
the International standard ? (Take a = 0-0038.)
Answer. — 5-84 ohms.
102. Comparison of Electric and Heat Conductivities. — The
reciprocals of the numbers given in column 4 of Table VI.
will express the relative electric conductivities of the metals
for the same length and sectional area. These numbers are
given in column 2 of Table X., the electric conductivity of
silver being called 100. On comparing these with the conduc-
tivities of the metals for heat for the same length and sectional
area as given in column 3 of Table X., we observe that the
metals arrange themselves approximately, but not absolutely,
in the same order for the two conductivities.
244
PRACTICAL ELECTRICITY
TABLE X.
APPROXIMATE RELATIVE CONDUCTIVITIES PER CUBIC UNIT.
Name of Metal.
Electric.
Heat.
Silver
100
100
Copper
96
94
Gold*
73
55
Aluminium
57
51
Zinc
26-2
27-4
Brass
24-6
25-8
Cadmium
21-8
22'2
Platinum*
16-6
9*4
Tin
15-4
16-3
Nickel
12-8
143
Iron
11-7
15-0
Steel
9-0
n-8
Lead
7.9
8-5
Platinoid
4'5
5'9
German silve
__
3'9
5'7
Manganin
3-5
5-1
Bismuth*
i-i
1-8
If, however, we experiment with worse and worse conductors
we find that the electric conductivity diminishes much more
rapidly than the heat conductivity. For example, the electric
conductivity of copper is about lo20 times the conductivity of
vulcanised indiarubber, whereas the heat conductivity of copper
is only about io4 times that of vulcanised indiarubber. Hence,
while we can obtain insulators for electricity, or bodies which,
relatively to the metals, do not practically conduct electricity at all,
solid or liquid insulators for heat are unknown.
103. Resistance and Conductance of Several Conductors in
Series or in Parallel. — In Section 53 we have seen that when
two equal conductors are connected in series the total or com-
bined resistance of the whole is equal to twice the resistance
of each, and when two such conductors are connected in parallel
the combined resistance is half the resistance of either. An
extension of the reasoning there given enables us to show that
the resistance of any number of conductors in series is equal to the
sum of the resistances of the conductors, and the conductance of any
number of conductors in parallel is equal to the sum of the several
conductances.
Suppose several conductors of resistances Ra, Rb, Rc, Rd ohms
to be connected in series, Fig. 149, and let the P.D. between their
terminals be Va, V^, Vc, V^ volts respectively when a current
* Wiedemann and Franz's results ; the others are deduced from Prof..
C. H. Lee's numbers. Phil. Trans., 1908.
COMBINED RESISTANCE 245
/ amperes is passing through them. The P.D. between the be-
ginning of Ra and the end of Rd will be given by
Fi.?. 149. — Four Conductors in Series.
Dividing through by 7 we get
Z -- V«+Vb , VG Vd
T T T T T-
y
But - is, by definition, the resistance R of the combination, whilst
-j^> -~> ~> -jT> are respectively the resistances of the several
parts ; hence, we can write
R=Ra + Rb + Rc-\-Rd (28)
The same may be proved for any number of conductors.
If the conductors be connected in parallel as shown in Fig.
150, then the P.D. between the ends of all the conductors is
Fig. 150. — Branch Circuits in Parallel.
the same. Calling this potential difference V, the current
V
through the conductor of resistance Ra will be given by Ia = -
R*
Similarly, the currents /&, Ic and Id, through the conductors of
V V V
resistance, R^, Rc, and Rd, will be — , — and - respectively; whilst
Kb Kc Kd
the main or total current / will be equal to the sum of the several
246 PRACTICAL ELECTRICITY
currents (see Section 7). Hence we have
JL L Jl Jl
^#fl + fy+l^^ Rd'
J i i i i
r=i^+^+^+^-
Now —represents a current divided by a potential difference,
and is therefore the reciprocal of a resistance, or a conductance,
whilst — - •— — and — — are the conductances of the several
RO, Kb KG ««•
branches, hence we see that the conductance of a number of
conductors in parallel is equal to the sum of the conductances of the
several conductors.
A single conductor which would allow a current / to flow
through it when a P.D. equal to V existed between its ends, would
have the same conductance as the several conductors in parallel,
and consequently the same resistance as these conductors so
arranged. Such a single conductor is said to be " equivalent "
to the several conductors and its resistance is equal to the
" combined resistance " of these conductors. Calling the com-
bined resistance R, the above equation becomes
and en taking the reciprocals of both sides
'
.
(30)
From this we deduce the rule : the combined resistance of
several conductors (or resistances) in parallel is equal to the re-
ciprocal of the sum of the reciprocals of the several resistances.
The particular case of two conductors in parallel occurs so 'fre-
quently that it is convenient to remember the relation in another
form.
From the above we have
T
R =
i i
PARALLEL RESISTANCE 247
or. in words, the combined resistance of two conductors in parallel
is equal to the product of the resistances divided by their sum.
The truth of the rule for the resistance of conductors in parallel
can be proved experimentally. A set of four coils on one
bobbin, designed for this purpose, is shown in Fig. 151, one end
of each coil is connected with the binding screw sx and the
other ends respectively with the small brass mercury cups clt C2,
C3, C4, as indicated symbolically by the zig-zag lines on the face
of the bobbin. A brass bar R R carries another terminal S2, and
has four holes in it opposite c1, C2, C3, cd respectively, which
R
Fig. 151. — Set of four Coils used for Testing the Resistance of Conductors in Parallel.
contain mercury. By using one of the wire bridge pieces BJ,
B2, etc., any of the four coils can be connected between s2 and slt
and its resistance measured. As shown in the figure the coils
Cj and C2 are in parallel, and the resistance between sx and S2
will give the combined resistance of Cj and C2, when in parallel.
By suitably placing the bridge pieces any combination of the
four coils can be arranged, and measured, and the formulae
given above experimentally verified.
Example 75. — Resistances of 25, 32, 17, and 40 ohms are put
in parallel with one another. What is the combined resistance ?
i i i I i - V
— = A .'. x = 6-4 ohms.
x 25 32 17 40
Answer. — 6-4 ohms.
248 PRACTICAL ELECTRICITY
Example 76. — A coil of wire has 1,125 ohms resistance. What
resistance placed in parallel with it will make the combined
resistance 1000 ohms? Answer.-g.ooo ohms.
Example 77. — The wire of a resistance coil has 10,000 ohms
resistance, but the surface of the ebonite between the terminals,
having been imperfectly cleaned, has a resistance of only 870,000
ohms. What is the combined parallel resistance between the
terminals ? Answer.— 9,886 ohms.
104. Currents in Parallel Conductors. — If Ia, Ib, Ic, 1$, etc.,
be the currents in the various branch circuits (Fig. 150), and
/ be the current in the main, we have
_
ft *•'
d
&c.;
la
ia + ib + ic + id + &c. -i- + -L + .L + -1.+ &c.
*« *& *c Kd
i
^0 *o~
or
Similarly, ~^ — —
I J- + JL + — -j- _£_ -j- &c.
^a ^6 ^c ^rf
Example 78. — Resistances of 12, 7, 2, and 30 ohms are placed
in parallel with one another, and a current of 10 amperes, as
measured by an ammeter in the main circuit, passes through the
combination. What are the currents in the respective branches ?
Answer. — 1-097, 1-881, 6-583, and 0-439 amperes respectively.
105. Kirchhoff 's Rules. — In the foregoing section we have made
use of the fact demonstrated in Section 7 that when a current
divides into two or more parts, the whole is equal to the sum
KIRCHHOFFS RULES
249
of the parts. The statement of this fact is usually known as
Kirchhoff's First. Law, or Kirchhoff's iFirst Rule, and is given in
two forms, viz. : — (i) The sum of the currents flowing to any point
is equal to the sum of the currents flowing away from that point,
or (2) the algebraical sum of all the currents meeting at a point is
zero. Symbolically, the rule may be written,
2/=o. (32)
It is of fundamental importance and of great use in electrical
calculations.
Another useful rule formu-
lated by Kirchhoff is known
as Kirchhoff's Second Rule,
which says that in every
closed circuit the algebraical
sum of the products of the
currents into the resistances
equals the algebraical sum of
the EM.Fs. in the circuit
w
(33)
B
I'K
E
W
This rule is really a de- Fig. 152.
duction from Ohm's Law.
For a simple circuit its truth is self-evident, for the expression
may in this case be written : —
or
?-*
which is merely the algebraic representation of Ohm's Law.
Consider next the closed circuit w G w' w, in Fig. 152. In
this circuit there is no electromotive force, therefore
= o.
Let the letters Ib, Ig, and Ir in this figure represent currents,
and Rfj, Rg and R resistances, let the P.D. between w and w' be
V, and the E.M.F. of the battery, E.
By Ohm's Law we have
V
or
V=IgRg,
= JL, or V = IrR,
Hence 7^ = IrR.
250
PRACTICAL ELECTRICITY
If we trace out the closed circuit, w G w' w, in a clockwise
direction, the direction of the current Ig must be considered
positive, whilst that of Ir is negative.
? (IR) thus becomes IgRg + ( - IrR), or
IgRg —
and as V = IgRg = IrR,
we have IgRg — IrR = o,
In the closed circuit B w R w' B, the E.M.F. is E, and
is IR + IR
Fig. 153-
where 7?6 is the resistance of the battery branch between w and w'.
Applying the rule we therefore get
IbRb + IrR = E,
but IrR = V, by Ohm's Law,
/. E =V+IbRb.
or, in words, the E.M.F. of the battery is equal to the potential
difference between the terminals, plus the product of the current
passing and the resistance of the battery, a result previously
arrived at in Section 55.
Again, for the closed circuit B w G P w' B (Fig. 153),
?(IR) is IbRb + IgRg>
where Rb is the resistance of the path P w' B w, and Rg that of
WGP,
2E = £-£'.
Hence, according to KirchhofFs second rule we have
'IbRb + IgRg = E-E'.
SHUNTS 251
This may be written'
E-IbRb = E' + IgRg.
Now E — Ib Rb gives the P.D. between the points w and p, and
E' + IgRg, also represents the same P.D., for if the current Ig
passes in the direction of w G p, the potential difference V must
exceed the electromotive force E', which acts in the opposite
direction ; and the excess of V over E' must be such as will
cause a current Ig to pass through the resistance Rg, viz., IgRg-
Hence V - E' = IgRg,
or V = E' + IgRg)
But V=E-IbRb,
or IbRb + IgRg = E-E'.
or 2 (IR) = 2E.
From these examples we see that KirchhofFs Second Rule is
consistent with Ohm's Law. This rule enables us to write
down simultaneous equations which represent the relations be-
tween currents, resistances, and E.M.F.s in circuits constituting
more or less complicated networks, and from these equations the
unknown quantities can be expressed in terms of the known ones.
106. Shunts. — One of the commonest instances of parallel
circuits occurs when a galvanometer is " shunted." We have
already seen (Section 19), when calibrating a galvanometer
by comparing it with a standard galvanometer, and again when
using a Wheatstone's bridge (Fig. 144), that it is sometimes
convenient to employ a bypath, or shunt, to convey a portion
of the current, so that the current
passing through the galvanometer
is less than the current in the main
wires connected with it. We will
now consider what must be the
relative resistances of the shunt
and galvanometer to allow any
particular fraction of the whole
current to pass through the galvano- Fig. 154.
meter.
Let Rg be the resistance of a galvanometer, Rs that of the wire
shunting it, and let V be the P.D. between the terminals of the
shunted galvanometer which is joined to the mains Mx and M2
(Fig. 154). Then if Ig and Is be the currents that pass respectively
through the galvanometer and shunt,
252 PRACTICAL ELECTRICITY
or the currents in the galvanometer and shunt bear to one another
the inverse ratio of the resistances.
This relation may be deduced by Kirchhoffs second rule, as
follows : —
^ (IR) = ^E = Q (in this case).
or IgRg — ISRS = o,
la Rs
whence -f- = — .
ls Kg
Also, by a well-known rule of proportion, it follows that
R>
_
Ig + IS Rg + Rs
Is
but Ig + Is, the sum of the currents flowing through galvano-
meter and the shunt respectively, is equal to the current / in
the mains MJ or M2, hence
I* K.
Rg + R
(34)
and T = pV (35)
/ Kg + Ks
or the current in either branch bears to the whole current, the ratio
of the resistance of the other branch, to the sum of the resistances
of the two branches.
107. Multiplying Power of a Shunt. — Since
_ Rg + R5 j
Rs *
7? -4-7?
the fraction —g— — - is frequently called the " multiplying power
KS
of the shunt "^that is, the number by which the current flowing
through the galvanometer must be multiplied to obtain the
total current, or current in the main.
MULTIPLYING POWER OF SHUNTS 253
As an example of the last equation, let us suppose that we
desire that Ig shall be one-tenth of /; then
or generally, if we desire that -th of the whole current shall pass
through the galvanometer,
Rg + Rs
or Rs = -I— Rg. (36)
n — i
Example 79. — A galvanometer of 2,572 ohms resistance is
shunted with a resistance of 285-8 ohms. What fraction of the
main current passes through the galvanometer ?
Answer h _ Rs . 285'8 _ _!_
f -Rg + Rs ~ 2857-8 ~ 10*
Example So. — A galvanometer has 5,461 ohms resistance,
what must be the resistance of the shunt in order that yjo^h
of the main current may pass through the galvanometer ?
7? T
Answer. — — ,. 5 • = , therefore R& = 55-16 ohms.
5461 + Rs 100
Example Si. — A galvanometer and its shunt are both wound
with copper wire. The multiplying power of the shunt is 100
when the temperatures of the galvanometer and of the shunt coils
are the same. What is the multiplying power when the tem-
perature of the galvanometer coils is 5° C. above that of the
shunt ?
Answer. — 102.
108. Usual Method of Constructing a Shunt Box. — Three coils,
having respectively the Jth, ^th, and ¥ Jgth of the resistance of
the galvanometer, are usually inserted in a small box b (Figs.
155-6), which generally accompanies a galvanometer. The ter-
minals of the galvanometer, as well as the two wires which connect
the galvanometer with the rest of the circuit, are joined to the bind-
ing screws, s, s on the shunt box, and each of the first three shunt
coils has one of its ends connected with the brass piece c, while
the 0ther ends are connected respectively with the brass pieces D,
254
PRACTICAL ELECTRICITY
Fig. 155. — High Insulation Shunt Box.
E, and F, as indicated
symbolically in Fig. 156.
If, then, the brass plug p'
be inserted in the hole
between the brass bar A B
and the brass piece c, all
the current will pass from
A B to c, through the plug,
and practically none will
pass through the galvano-
meter, since the resistance
of the path from A B to c
through the plug is ex-
tremely small compared
with that through the gal-
vanometer. If, on the
other hand, the plug be
inserted in the hole be-
tween A B and D, as in
Fig. 155, current will pass
from AB to D through
the plug, and from D to c
through the coil in the shunt box, which connects with c. And
as this coil has jjth of the resistance of the galvanometer, r\jth
of the total current will pass through the galvanometer. Simi-
larly, if the plug be inserted in the hole between A B and E or
in the hole between A B and F, ^ooth or ^1 o* ^e whole
current will pass through the galvanometer.
Instead of employing three coils whose
resistances respectively are Jth, ^gth, and
5J^th of that of the galvanometer, and
joining one end of each of these coils to
the brass piece c, the coils may be joined
up in series between the brass pieces c and
D, D and E, E and F respectively, like the
coils of an ordinary resistance box (Fig.
157). In this case the coils must have
resistances ^Rgt (fo--fffa)Rg, and
( J — ¥V ) Rg and the bl°ck marked F will
correspond with the Jth shunt, while that marked D will corre-
spond with the g^-gth shunt, as indicated symbolically in Fig. 157.
In order to obtain very good " surface insulation " the
brass pieces, A B, c, D, E, and F are, in the particular shunt
box shown in Fig. 155, mounted on ebonite pillars P, P, P,
Fig. 156. — Top of Shunt Box,
showing Parallel Arrange-
ment of Shunts.
SHUNT BOXES
255
p , and, to avoid the insertion of the, plug into one or other of
the holes pushing these pillars outwards and so preventing the
plug making firm contact with the pieces of brass on each side
of it, there is a spring cap c c, sliding on
the plug, which passes over the two
vertical pins on each side of the hole,
and so holds the brass pieces together
against the wedging action which tends
to force them asunder when the plug is
pressed in. The plug has a long ebonite
handle i, which should be held by the
flat part at the end to minimise the
leakage taking place along the surface
of the handle and through the body of
the experimenter to the ground.
109. Increase of the Main Current Produced by Applying
a Shunt. — Although the current passing through an unshunted
galvanometer is the same as the current in the main, and although
the current passing through a shunted galvanometer is always
•
times the current in the main, it must not be assumed
Fig. 157.— Top of Shunt Box,
showing Series Arrange-
ment of Shunts.
K
that the application of a shunt to
the current passing through it in the ratio of unity to
a galvanometer diminishes
RS
Rg+ R<
For the application of the shunt diminishes the resistance in the
7? 7?
circuit by the difference between R and -^— ^-^— , and this
diminution of the resistance of the cir-
cuit increases the current in the main,
more or less, depending on the arrange-
ment of the circuit ; so that the current
in the main after the application of the
shunt is greater than the current in the
main before the shunt was applied by an
amount that may be very small or may
be very large.
Let the circuit consist of a resistance
Rm in series with a galvanometer of re-
sistance Rg, and let a fixed P.D. of V volts be maintained between
the terminals of this circuit (Fig. 158), then Igl, the current
passing through the main or through the galvanometer, equals
V
l£
256 PRACTICAL ELECTRICITY
Next let the galvanometer be shunted with a shunt or resistance
Rs (Fig. 159), and let the P.D. of V volts be still maintained be-
tween the outer terminals of the circuit shown in Fig. 159, then
the current now passing along the main equals
V
and Ig2, the current now flowing through the galvanometer equals
_ _ RSV _ ,
-~ ' U7)
+
T Z? / Z? i U \ i ZP ~D
2gl Km (Kg + Kg) + KgKs
Now the value of this ratio depends
on the value of Rm as well as on Rg and
for example, if Rm be very large compared with Rg,
^2 Rs
whereas if Rm be very small compared with Rg and Rs,
-f^- = unity (approx.).
2gi
That is to say, if the resistance external to the galvanometer lie very
large, the galvanometer current after the application of the shunt,
bears to the galvanometer current before its application the ratio
ofRs to Rg + Rs ; while, on the other hand, if the resistance external
to the galvanometer be very small, shunting the galvanometer
produces very little effect on the current passing through it. And
this arises from the fact that on applying the shunt in the first
case the main current is not appreciably changed, while in the
second it is increased by an amount almost exactly equal to the
current that is shunted past the galvanometer.
For example, let Rg be 1000 ohms and Rs Jth of Rg ;
(i) let Rm be 100,000 ohms, then the true ratio of Ig2 to Igl is
in-i x 101,000
— r -- . or 0-1009 about,
100,000 x 1,111-1 + iii,m"
T-)
whereas the value of — — ?— is o-i, which differs by about i per
KS + K
CIRCUITS IN PARALLEL 257
cent, from the true ratio, so that the current through the galvano-
meter is reduced practically to one-tenth of its previous value ;
(2) let Rm be 10 ohms, then the true ratio of IS2 to ISI is
in-i X 1,010
, or o«Q2 about,
IO X I,III'I + 111,111
whereas the approximate value of the ratio is unity, which differs
by about 8 per cent, from the true ratio, so that the current
Fig. 1 60.— Part of the Plan of an Electrically Lighted House.
s, Street Mains ; H, Mains to House ; M, Supply Meter ; D, Distribution Board ;
/, Leads to the Rooms ; /', Branch Leads ; L, Glow Lamps.
through the galvanometer remains nearly unchanged by the
application of the shunt.
An important example of this independence of currents in
parallel circuits that can be produced by making the value of R^
in Fig. 159 very small, occurs in the wiring of a house for electric
lighting. -The glow lamps are all connected in parallel with
the house mains as indicated in Fig. 160, which represents a
portion of the plan of the ground floor of a house, and shows
the way in which the electric lighting mains and branch mains
are run. At the place where the house mains, H, are connected
with the street mains, s, a constant, or nearly constant, P.D.
is maintained by the Electric Supply Company, the value of this
258 PRACTICAL ELECTRICITY
nearly constant P.D. being frequently 100 volts. Each lamp,
L, or each group of lamps, is provided with a switch so that the
current can be turned on to, or off from, each lamp, or group of
lamps, independently ; and it is obviously important that the
turning on, or off, of a switch in one part of a house shall not
sensibly affect the light given by the glow lamps in some other
part of the house. Now a glow lamp is a very sensitive in-
dicator of any variation of the current passing through it, for
the light given out by a glow lamp, when glowing at about its
normal brilliancy, varies about four per cent, for each one per
cent, variation of the current passing through it. Hence it is
extremely important to arrange matters so that the current
passing through each lamp shall be practically independent of
the current passing through any other lamp, and this result is
attained by making the resistance of the house wires H, /, /'
small compared with the resistance of the filaments of the
lamps, in accordance with the principle discussed in this section for
a galvanometer and shunt.
Example 82. — A galvanometer of 8,100 ohms' resistance is
in a circuit having 500,000 ohms' resistance external to the gal-
vanometer. What is the percentage change in the main current
made by shunting the galvanometer with a |th shunt ?
Answer. — 1-46 per cent.
Example 83. — If a galvanometer have 1,980 ohms' resistance,
and a shunt be attached so that the current passing through the
galvanometer is only ^th °f *ne total current, what will be
the resistance of the shunt, and by how many ohms will the resist-
ance of the circuit be diminished by employing the shunt ?
Answer. — Resistance of shunt =20 ohms.
Diminution of resistance=ig6o-2 ohms.
Example 84. — A pair of " leads " or branch conductors runs
from the street mains, where a P.D. of 100 volts is maintained,
to a hall where 150 glow lamps are in use. Each of the lamps
would take 0-5 ampere at 100 volts. What must be the resist-
ance of the leads in order that, when all the lamps are burning in
parallel, the P.D. between their terminals is 98 volts ?
Answer. — The resistance of each lamp is — • = 200 ohms.
o-5
08
Hence the current taken by each lamp at 98 volts is -^— , or 0-49
ampere, and the total current through 150 lamps in parallel is
150 X 0*49, or 73-5 amperes. The resistance of the leads must
UNIVERSAL SHUNTS
259
be such that there is a " drop " of pressure of 2 volts when the
2
current is 73-5 amperes : hence the resistance is -- , or 0-0272
/ o o
ohm.
In some cases it is desirable that shunting a galvanometer
should not alter the current passing in the main circuit. This
necessitates the insertion of an additional resistance to compen-
sate for the diminution
produced by shunting the
instrument. Shunt boxes
arranged to effect this
compensation are called
" constant total current
shunts " ; one form is
shown diagrammatically
in Fig. 161, and in plan
in Fig. 162.*
no. Principle of Uni-
versal Shunts. — When
using a shunt to compare
the relative strengths of
two currents, it is un-
necessary to know what
is the exact fraction of
the main current that /
passes through the gal- /
vanometer, for all that
iJ
s»
H
3jj^
fatal size
Figs l6lA.l6a._ Constant Total Current Shunt Box.
has to be known is the
way in which this fraction is varied when the shunt is altered.
Carrying out this idea, the authors have devised a method of
applying shunts to a galvanometer in which the resistances of
the coils of the shunt box need have no special relation to the
resistance of the galvanometer itself. Hence the same shunt box
can be used with any galvanometer.
For example, let a galvanometer of any resistance Rg ohms be
permanently shunted with any resistance Rs ohms (Fig. 163),
and when a current of / amperes conies along the main M2 and
leaves the main MX, let the deflection of the shunted galvano-
meter be a. Next, let the main M2 be moved from the point d
to the dotted position at the point c, the fraction of Rs between
the points b and c being -. Now when a stronger current of /'
* A useful exercise for the student is to work out the values of Rsl, Rsz, Rs3,
Rlt R2, and R3, say, for a galvanometer of 1000 ohms resistance.
260
PRACTICAL ELECTRICITY
amperes comes along the dotted main M2 at the point c and
leaves by the main Mlf let the deflection be a' ; then, if the
deflections of the galvanometer are directly proportional to the
currents passing through it,
/' a9
— = n — -
/ a
whatever be the values of Rg and of Rs.
Fig. 163. — Principle of Ayrton and Mather's Universal Shunt.
For let the galvanometer current in the first case be Ig, and in
the second I'g , then
D
7g= P ,5p I [formula (34)],
and Pg =
R
£
• £=L L
"la n ' I '
and when — = -/
a Ig
I' a'
n
n
For example, if n be 10 or 100, tne ratio of the currents /'
to / will be exactly 10 times or 100 times the ratio which the de-
flection a' bears to a, independently of the values of Rg and of Rs.
in. Method of Constructing a Universal Shunt c Box, and
its Advantages. — A " universal shunt box " constructed on this
principle is seen in Fig. 164. The terminals A and B of the shunt
UNIVERSAL SHUNT BOX
261
box are permanently connected respectively with the terminals
of the galvanometer, while the terminals B and c of the box are
connected with the two main wires which lead the current up to
and away from the galvanometer and shunt. The ends of a coil
of any resistance Rs ohms are permanently connected as shown,
7? 7? 7?
and at points in this coil corresponding with — ^ —
100 1000
ohms,
Fig. 164. — Plan of Ayrton and Mather's Universal Shunt Box.
permanent connections are made with the several blocks of the
shunt box as illustrated.
Then, whatever be the resistance of this coil Rs compared with
the resistance of the galvanometer Rg (either, or both, of which
may, therefore, be unknown), it is easy to show that if Ig amperes
be the current flowing round the galvanometer when a plug is
h
amperes
placed in the hole marked d, it will be — £ — -, — -
10 100 1000
respectively when the plug is put instead into the holes marked
c, b and a respectively, if there be the same current in the main
circuit.
262 PRACTICAL ELECTRICITY
This method of altering the shunting of a galvanometer by
using a fixed resistance Rs and varying the position of the mains,
instead of keeping the mams fixed and varying the resistance of
the shunt, as in Fig. 155, has several important advantages, viz. : — •
(1) The same shunt box can be used with any galvanometer,
etc.
(2) Variations of the temperature of the room produce no
error, for if all the coils be of the same material, change of
temperature will not alter the ratios of the resistances.
(3) The coils of the universal box can, by a proper choice of Rs,
be made integral numbers of ohms, and therefore more
easily and cheaply adjusted than fractional values such as
are necessitated by J, ^, and ^^ of Rg.
(4) Lastly, whatever be the value of Rg, n, and Rs (Fig. 164),
provided that Rs is less than RK (n + ^n2 + n), the use of the
universal shunt produces less change in the total resistance of
the circuit than would be caused by an ordinary shunt
of equal multiplying power.*
The one disadvantage of the
universal shunt, which, how-
ever, is usually of little im-
portance, is that the application
of the shunt to the galvano-
meter reduces the sensitiveness
of the instrument. For the
very few cases in which full
sensibility is necessary, pro-
_ vision is made for unshunting
the galvanometer. Thus in
Fig. 165.— Universal Shunt Box. ,-,.
-big. 104, removing the plug
from the box and connecting the right hand terminal of the
galvanometer to c instead of to A, will cause all the current to
pass through the instrument.
A more recent form of Universal Shunt Box is shown in
Fig. 165, having multiplying powers of i, 3, 10, 30, 100, 300 and
1000. Instead of plugs, a switch arm touching metal studs is
used to alter the point of contact of the main wire with the shunt.
Fig. 166 shows a universal shunt intended for carrying large
currents (up to 100 amperes), with multiplying powers of i, 2, 5,
10, 20, and 5o.f A great advantage of this shunt is that the
sensibility can be changed by merely turning the switch handle.
* A proof of this is given in the 1896 edition of this book, on page 308
t A diagram of connections of this shunt is given in Fig. 2560
LARGE CURRENT SHUNT
263
The main circuit is not in-
terrupted by this movement,
and the ammeter remains
connected to the same two
points of the shunt, which-
ever multiplying power is
used.
Example 85. — A Universal
Shunt, 7,000 ohms in resist-
ance, is employed with a
galvanometer having a resist-
ance of 1,270 ohms. What
fractions of the main current
pass through the galvano-
meter if the part of the
shunt included between the ,|
mains is 10 ohms, 70 ohms, J
700 ohms, and 7,000 ohms|
successively ?
Answer. — The ratio of the
galvanometer Current tO the Fig. 166.— Ayr ton and Mather Shunt for Strong
main current is Currents' designed by Mr' DuddeU'
IO
70
700 , 7000
and
8270' 8270' 8270 ' 8270
respectively, or the fractions are in the ratio ^, ^~, ~ to I.
Example, 86. — Taking the galvanometer and shunt referred
to in the preceding question, find the percentage difference
in maximum sensibility between the galvanometer used with
the universal shunt and used in the ordinary way.
Answer. — If the universal shunt is used, the maximum sensibility
is obtained when the mains are across the galvanometer terminals,
and the galvanometer takes
7000
8270
= 0-846 of the main current.
If the ordinary method is employed, the galvanometer takes the
whole of the main current for maximum sensibility. Hence,
the universal shunt gives 15-4 per cent, less maximum sensibility.
Example 87. — If a galvanometer of 1,270 ohms resistance
be employed, and if the resistance of the circuit external to the
galvanometer be 200,000 ohms, calculate the percentage variation
that will be made in the main current when the sensibility of the
galvanometer is diminished from its maximum to one-hundredth
of the maximum, first by using a shunt specially constructed
264
PRACTICAL ELECTRICITY
for the particular galvanometer, secondly a universal shunt
of 7,000 ohms in resistance.
Answer. — The percentage change in the main current will be
0-62 when using the ordinary shunt, and 0-50 when the universal
shunt is employed.
112. Standard Resistance Coils. — A resistance coil, when used
as an accurate standard, is wound inside a brass box, B (Fig. 167),
which is inserted in a
vessel of water or oil, v v,
and the temperature of
the liquid is accurately
measured by means of
the thermometer t. The
hollow cylindrical brass
box B, which holds the
coil, is made of large
diameter outside and in-
side, so as to expose as
much surface as possible
to the liquid in order
that the coil inside may
acquire the temperature
of the bath as quickly as
possible. It is desirable
to provide a stirrer for
agitating the liquid and
bringing it all to one tem-
perature ; and the vessel
v v may with advantage
have double sides, with
an air - space between
them, as seen in the
figure, to check transference of heat between the water and
the outside space.
The tubes T, T are to prevent the coil being short-circuited
by water getting into the holes through which the rods w, w
attached to the ends of the coil, are brought out. These tubes
are made of brass, but they are lined with tubes of ebonite to
prevent electric contact between the brass tubes and the rods
w, w. Electric connection with these rods is made by dipping
their ends E E into little cups containing clean mercury.
Within recent years the form of standard coil shown in
Fig. 168 has come into extended use. It is known as the " Reich-
zanstalt " form, and is arranged so that the ends of the copper
E
Fig. 167. — Standard Resistance Coil.
STANDARD RESISTANCES 265
rods which dip into the mercury cups may be about level with the
top of the bath in which the coils are immersed when in use.
Two terminals, known as " potential terminals" are provided
on the rising parts of the rods, and the coil is adjusted so that
the resistance between these points is of the value marked on
the coil. Its advantages are (i) the length of copper terminal
rod (which has a high temperature coefficient), especially the
part included between the
measuring points, is much
smaller than in the form shown
in Fig. 167 ; (2) this part of
the copper may be immersed
in the oil bath, and conse-
quently its temperature can be
controlled and measured more
readily than if in air ; and (3)
errors which might be produced
by defective contacts in the
mercury cups are eliminated.
Coils of this form are particu- Fig. 768.— standard ohm o>a
. . „ •. i f tt ^ (Reichsanstalt Form).
larly well suited for potentio-
meter measurements " described in Chapter IX.
As it is frequently necessary to know the resistance of standard
coils to a very high degree of accuracy, say, one part in one
hundred thousand, it is evidently an advantage to make such
coils of metal whose variation of resistance with temperature is
very small. For this reason it is customary to construct them of
German silver, platinum -silver, platinoid, eureka, constantan,
or manganin wire. The coils in resistance boxes are also made
of low temperature coefficient alloys.
113. Ordinary Forms of Wheats tone Bridge. — In Section 89
two forms of Wire Bridge are described. By such instruments
it is theoretically possible to measure any resistance, how-
ever large or small, by comparison with a unit coil, but
practically there is a limit to the range of measurements, for if
the resistance be either very great or very small the slider will
have to be moved very near one end of the wire to obtain balance,
and it will be impossible to read off the short length of wire
between the end and the slider with accuracy. For example,
in comparing a resistance of about 100 ohms with a unit coil,
the shortest segment of the metre wire would be about I centi-
metre, and this length could not be read off from an ordinary
scale nearer than a tenth of a millimetre, which would mean a
possible inaccuracy of one per cent. Further, the ends of a bridge
266 PRACTICAL ELECTRICITY
wire usually differ somewhat in resistance for a given length,
from the middle portion, because of their having been heated in
soldering to the copper bars, so that the true ratio of the resist-
ances of the two arms may differ appreciably from the ratio of
their lengths. Also, a bridge arranged unsymmetrically is not
so sensitive as one with nearly equal arms, i.e., it requires a
greater change made in the ratio of Rc to Rd (Fig. 141) to pro-
duce an appreciable deflection of the galvanometer.
Fig. 169.— Top of a Commercial Wheatstone's Bridge.
For these reasons it is customary to have several coils of
values, say, i, 10, 100, 1000, as the known arm of a metre bridge,
and use one or other to suit the resistance to be measured. A
resistance box is sometimes employed and arranged so that
the point of balance comes somewhere near the middle of the
stretched wire.
We have already seen that a simple bridge cannot be used to
measure a resistance whose ratio to the known arm is 100 or ^jo>
to within i per cent., so with a unit coil the range of the bridge
(measuring to about i per cent.) is from o-oi ohm to 100 ohms.
By the aid of coils 10, 100, and 1000, this range can be extended
to 100,000 ohms. If, however, it is desired to measure within
£o of one per cent., the total range will be approximately o-i ohm
to 10,000 ohms. In practice it is necessary to measure resistances
much smaller than o-i ohm and far greater than 10,000 to a
higher accuracy than YQ%, and for these purposes other forms
of bridge are employed. One of the commonest forms is the
" coil bridge/' such as is shown in Fig. 169. It consists essentially
of a resistance box containing coils from i to 10,000 ohms
COIL BRIDGES 267
(total), and two sets of coils 10, 100 and 1000 as shown. The
latter are called the " proportional arms " or " ratio coils " of the
bridge, and the former the " adjustable arm," whilst the resistance
Rx to be measured, is spoken of as the " ' unknown ' arm." A
copper link seen at the top left-hand corner of the box, connects
the left-hand proportional arm with the adjustable arm. The
range of a bridge of this kind may be taken as TJn to 1,000,000
ohms ; the accuracy attainable depending chiefly on the sensi-
tiveness of the galvanometer and on the battery used.
In the wire bridge balance is obtained by changing the ratio
of the arms of the bridge by varying both of them, whereas in
using a coil bridge it is customary to obtain balance by varying
one arm only, the adjustable arm. This makes it possible to
read off the value of a previously unknown resistance directly ;
for if we make the proportional arms equal to each other, either
10 and 10, or 100 and 100, or 1000 and 1000, balance will be
obtained when the adjustable arm is made equal to the unknown
arm ; the resistance of the latter can therefore be read off
Fig. 170.— Post Office Wheatstone's Bridge.
directly. For measuring resistances not exceeding 100 ohms,
we may make the proportional arm adjacent to the adjustable
arm greater than the arm opposite, say, 1000 and 10 respectively,
in this case balance will exist when the adjustable arm is 100
times the unknown. By this means the value can be read off
directly to two decimal places, i.e., to o-oi ohm. For resistances
268 PRACTICAL ELECTRICITY
GALV. &
LINE
INFlN
NFIN
300
GALV.
LOGO IOO IO 10 100 1,000
20O ' IOO 410 30 20
Fig. i7oa.— Plan of Post Office Bridge.
Fig. 1706. — Improved " Post Office Bridge."
COIL BRIDGES 269
between 100 and 1000 ratio of 1000 to 100 would be convenient,
and for coils between 1000 and 10,000 a ratio of 1000 to
1000. When the unknown resistance exceeds 10,000, it is
necessary, with such a bridge, to make the adjacent arm less
than the opposite one, and to multiply the value of the adjustable
arm which gives balance, by 10 or 100, according to the ratio
used, to obtain the resistance of the " unknown " arm.*
I
Fig. 171.— Portable Wheatstone's Bridge with Battery and Galvanometer combined.
The proportional coils of a bridge can be adjusted to equality
or to ratios of 10 to i, or 100 to i, with a very high degree of
accuracy (about one part in 100,000), so the chief objection to
using high ratios in a wire bridge does not exist in a coil bridge,
and for this reason coil bridges are more generally useful.
In all forms of bridge intended for very accurate measurement
provision is made for measuring the temperature of the coils,
for in such cases variation of resistance with temperature must
always be taken into account.
114. Portable Forms of Wheatstone Bridge. — Another form of
coil bridge is the one adopted by the Post Office Telegraphs Depart-
ment, and known as the " Post Office Bridge," or " Post Office
Box." It is shown in Fig. 170, and Fig. ijoa gives a diagram
of connections of the instrument. From these it will be seen
that keys for the battery and galvanometer circuits respectively
are placed at the front of the box, the proportional arms at the
back, and the terminals to which the ends of the resistance to be
measured are to be joined, are marked LINE. Sometimes they
* When measuring a resistance on a given bridge, with a given gal-
vanometer and battery, the most sensitive arrangement is generally
obtained by making the four arms as nearly equal as possible, and joining
the galvanometer or battery, whichever has the greater resistance, between
the junction of the two highest arms and the junction of the two lowest arms,
270
PRACTICAL ELECTRICITY
are marked LINE and EARTH respectively, the term " earth "
being used because the currents sent through most telegraph
lines return through the earth, and one terminal of the bridge is
Fig. iyia. — Diagram of Connections of Portable Bridge.
connected to earth in most of the resistance measurements
made in the service. An infinity plug (INFIN.), i.e., a plug which
breaks the circuit of the adjustable arm when removed from its
Fig. 172. — Dial Pattern of Bridge (Silvertown Co.).
hole, is provided. It is useful in testing whether a line is broken
or disconnected, and for other purposes.
Fig. 1706 shows in plan an improved form of Post Office Bridge
in which the units, tens, hundreds and thousands are arranged
PORTABLE BRIDGES
271
in separate columns. This facilitates reading off, and also makes
the box-easier to clean. The values of the coils are I, 2, 3, 4,
and decimal multiples of these numbers, a system which is
gradually replacing the i, 2, 2, 5, arrangement formerly in
common use, and in each proportional arm there are four coils,
i, 10, 100, and 1000 ohms.
Fig. 173. — Bar Pattern of Bridge (Gambrell Bros.).
A portable bridge, complete with double key, for galvanometer
and battery, is illustrated in Fig. 171. The battery is contained
in the space below the galvanometer, access to which is provided
for by the door D at the near end of the box. There is space for
four small dry cells. The plugs on the box belong to the adjust-
able arm, whilst the ratio coils are fixed inside the box and
arranged like a universal shunt ; the small switch seen on the top
of the box connects the galvanometer to either one of three points
on the shunt, so that the bridge reads direct, or multiplies or
divides by 10, as indicated in the diagram, Fig. 1710. With 14 -
coils in the adjustable arm, o-oi to 20 ohms, the bridge can be
used from o-ooi to 400 ohms, and forms a very handy instrument
in an electrical engineering laboratory.
115. Dial and Bar Patterns of Bridge. — These forms are shown
in Figs. 172 and 173 respectively. In both of them the ad-
justable arm consists of sets of nine equal coils, units, tens,
hundreds and thousands arranged in a ring or alongside a bar.
Only one plug is required for each dial, but the number of coils
necessary to obtain a given resistance is more than doubled.*
* The system i, 2, 4, 8, 16, &c., advancing by powers of 2, is the one
which requires the smallest number of separate coils, but it is incon-
venient for use with the common scale of notation.
272
PRACTICAL ELECTRICITY
The bar pattern has an advantage over the dial form because
of the greater ease in cleaning the insulation.
Fig. 174. — Portable Bridge,~with Switch Contacts.
For industrial purposes dial bridges with switch contacts, Fig.
174, instead of plugs, are frequently employed. They are easy
to use, and there are no plugs to get lost, but the resistance
of a switch contact is not so small or constant as that of a well-
fitting clean plug. The bridge shown in Fig. 174 is of the
latest design, having been made during the autumn of 1910 by
Messrs. Gambrell Bros., for the City Guilds Engineering College.
It is fitted with a single pivot moving coil galvanometer, battery,
and keys, and has a Mather's ratio-coil switch, as indicated
diagrammatically in Fig. 1710.
CHAPTER VII
ELECTRIC ENERGY AND POWER
116. Work done by a Current — 117. Electric Unit of Energy: Joule —
118. Heat Produced by a Current — 119. Measuring the Heat Equi-
valent of Electric Energy — 120. Power — 121. Electric Unit of
Power: Watt — 122. Joule's Law — 123. Instruments for Measuring
Power : Wattmeter — 124. Commercial Forms of Wattmeters — 125.
Joule, or Energy Meter : Clock Form — 126. Board of Trade Unit
of Energy — 127. Energy Meter : Motor Form — 128. Quantity or
Ampere-hour Meters — 129. Electric Transmission of Energy — 130.
Power Developed by a Current Generator — 131. Connection between
the E.M.F. of a Battery, the P.D., between its Terminals, the
Resistance and the Current — 132. Electromotive Force of any
Current Generator — 133. Power Absorbed in the Circuit Exterior
to the Generator: Back E.M.F. — 134. Distribution of Power in an
Electric Circuit — 135. External Circuit that Receives Maximum Power
from a Given Current Generator — 136. Arrangement of n Cells to give
Maximum Power to an External Circuit of Fixed Resistance — 137.
Minimum Number of Cells required to give a Fixed Amount of Power
to a given External Circuit — 138. Importance of Low Resistance and
High E.M.F. for Large Powers — 139. Modifications introduced into
the Previous Results by Limitation of the Maximum Current a Cell
may Produce — 140. Efficiency — 141. Efficiency of Electric Trans-
mission of Energy — 142. Connection between Electrical Efficiency of
Transmission and the Ratio of Power Received to the Maximum Power
Receivable — 143. Economy in Electrical Transmission of Energy:
Kelvin's Law.
116. Work Done by a Current. — Whenever an electric current
flows through a circuit work is done, just as whenever a water
current flows through a pipe or along a river bed the flowing
water does work on the obstacles that obstruct its passage. When
a water stream of Q cubic feet per second falls down a height
of / feet, the work done in t seconds equals
62-43 Q / t foot pounds very approximately,
62*43 being approximately the weight of a cubic foot of water in
pounds. So when an electric current of / amperes flows from a
point L to a point M through any circuit, the potential at M being
V volts lower than the potential at L, the work done on the part
L M of the circuit by the electric current in t seconds equals
0-7372 I V t foot pounds, very approximately, or
44-23 IV t' foot pounds, very approximately, in t' minutes.
s 273
274 PRACTICAL ELECTRICITY
The constant 0-7372 is derived from the fundamental definition
of the ampere and volt (see Sections 8, 48, and 54), and the
known relation between the erg and the foot pound given in
Appendix II.
In Section 48 we explained that the unit of potential difference
was chosen so that the product of the P.D. between two points
and the quantity of electricity passed from one to the other should
be equal to the work done by the electric current between those
points. Now the expression
IV t can be written V (It).
Here (It) represents the quantity of electricity in coulombs,
and to fulfil the above condition the product of V and (//)
must represent work or energy in joules (see Section 52).
Now i joule = io7 ergs, by definition, and
i foot Ib. = 30-48 x 453-6 x 981 ergs (Appendix II.)
= 1-356 x io7 ergs,
= 1-356 joules ;
/. 77* joules - — ^ 7/Hootlbs.,
1-356
= 0-7372 77*foot Ibs.
Neither the current of water nor the current of electricity
mentioned at the beginning of this section is changed, but the
current of water in falling from one level to a lower level, and the
current of electricity in falling from one potential to a lower
potential, gives up energy, provided that there is no apparatus
in the part of the circuit in question which gives energy to
the current instead of receiving* energy from it.
When the stream of water is a steady one, and when it flows
through a uniform tube such as / 1 (Fig. 81), all the energy lost
by the water between any two points PJ and P3 is converted
directly into heat, and is employed in slightly warming the
water and the tube ; so, in the same way, when a steady electric
current flows through a wire, the wire and the surrounding bodies
being at rest relatively to one another, the energy lost by the
current is turned directly into heat and the wire is warmed. If,
however, the obstruction to the passage of the water be produced
not merely by objects at rest but by the buckets of a water-wheel
which can be moved by the falling water, then a portion of the
energy lost by the water appears as mechanical energy given to
the water-wheel ; so, in the same way, when there is a magnet
or a piece of iron near the wire conveying the steady electric
WORK DONE BY A CURRENT 275
current, and when the relative positions of the wire and the magnet
or iron can be changed by electromagnetic attraction, then a
portion of the energy given up by the current is employed in
doing work on the movable system. For example, when a current
is sent through a galvanometer with a pivoted needle, or through
a coil of wire suspended in a magnetic field, or through the coil of
an electromagnet with a movable armature, or, generally, through
any " electromotor," the current not only does work in heating
the wire through which it flows, but it also does work in producing
mechanical motion against the controlling or resisting force.
As soon as the galvanometer needle or the suspended coil has
been deflected to such a position that the force due to the
current is balanced by the controlling force, or when the armature
of the electro -magnet has been pulled down against some stop,
or the electromotor has been brought to rest by some opposing
force becoming greater than the electromotor can overcome, no
more mechanical work is done by the current, and all the energy
it subsequently loses is directly turned into heat and goes to
warm the wire through which the current flows.
The expression 44-23 IV V foot pounds may be divided into
two parts, one part representing the energy which is lost by the
current and turned directly into heat, and the other the energy
lost by the current which is converted into some form of energy
other than heat. If an electromotor be driven by the current
and be employed to grind corn or to turn a grindstone, this
second portion of the energy will also be turned into heat ; but
this heat will not be produced by a direct conversion of electric
energy into heat, but by a conversion first of electric energy into
mechanical energy, and secondly of mechanical energy into heat.
If, on the other hand, the electromotor be used to raise blocks of
stone to the top of a scaffolding for building purposes, then this
second part of the energy will not be turned into heat at all.
If the circuit through which the current flows contains an
electrolytic cell, then, although no mechanical work will be
done by the current in this cell, chemical change will be effected,
and when, as a consequence, chemical energy is added to the elec-
trolytic cell, the work done by the current in producing this
chemical energy is analogous with the work done in producing
mechanical energy, and must be added to the work done by the
current in directly heating the conductor to obtain the equivalent
of the expression 44-23 IV t' foot pounds.
If, on the contrary, chemical energy disappears from the cell
on the passage of the current, this energy is transformed into
electric energy, and the electrolytic cell, therefore, acts as a
276 PRACTICAL ELECTRICITY
current generator and introduces electric energy into the
circuit.
In this case the amount of electric energy thus introduced
into the circuit must be subtracted from the amount spent in
heating the portion of the circuit considered, to obtain the energy
transferred to that portion from the remainder of the circuit. The
net amount of energy may, therefore, be either positive or negative
according as the energy introduced by the cell is less or greater
than the energy spent in heating the conductor.* When the
current flows from L to M and the potential of the point L is
higher than that of M, the current flows in the direction of the
P.D. and the energy transferred to LM is positive, but when
L is at a lower potential than M and the current flows from L
to M, i.e., in opposition to the P.D., in that part of the circuit,
the energy transferred to L M is negative. This means that the
portion L M of the circuit generates more energy than it dissi-
pates, and therefore it causes energy to be transferred from L M
to the remainder of the circuit. In all cases, however, we may
say that if a P.D. of V volts be maintained between any two points
L and M in a circuit, the amount of electric energy transferred in
t' minutes between the portion of the circuit L M and the res,t of
the circuit by a current of I amperes, equals in all cases 44-23
IVt' footpounds.
In certain exceptional cases the electrolytic cell may act
simply as a resistance and be merely warmed by the passage of
the current, but for that to be the case the work done in producing
chemical action at one plate of the cell must be exactly balanced
by the work given out in the same time by the chemical action
at the other plate.
Example 88. — An arc lamp takes 12 amperes at 50 volts
pressure. How many foot pounds of energy does it receive per
minute ?
Answer. — 26,538.
Example 89. — A resistance coil of 1,500 ohms has a P.D. of
12 volts maintained between its terminals. How many foot
pounds of energy does it receive per minute ?
Answer. — 4-25.
Example 90. — What current at 100 volts' pressure will supply
1000 foot pounds per second to a given circuit ?
* From this it will be seen that a current generator may, if its internal
resistance be high enough, abstract energy from a circuit even when its
E.M.F, helps the current.
THE JOULE 277
Foot pounds per second = - --— x 77,
,. , 1000 x 60
therefore I = - - amperes.
44-23 x 100
Answer. — 13*56 amperes.
117. Electric Unit of Energy : Joule. — In the previous section
we have shown that the work done in t seconds by a current of
/ amperes flowing in a path L M between the ends of which there
is a P.D. of V volts is given by the expression,
0-7372 IV t foot pounds (approximately),
but as one foot pound equals 1-356 x io7 ergs, we may write,
work done in t seconds = IV t x io7 ergs,
or =IVt joules.
By choosing io7 ergs as the practical unit of electrical work or
energy, no numerical coefficient other than unity is required in
the expression for electrical work done. This is a distinct advan-
tage, for we can now write,
W=IVt, (38)
where W is the number of joules produced in t seconds by a current
of I amperes at a P.D. of V volts. If I and V and / be all unity,
then W = i, from which we see that the work done in one second
by a current of one ampere flowing through a circuit between the
terminals of which a P.D. of one volt is maintained, is one joule.
The joule is therefore the practical unit of electrical energy
corresponding with the ampere, and volt, and second, and is
consequently of great importance. Its relation to the foot
pound is expressed by the equations
1 joule = 0-7372 ft. Ibs., very approximately,
and 1 foot pound = 1-356 joules, very approximately.
Example 91. — A pressure of no volts is maintained between
the electric -light mains of a house, and twenty glow lamps in
parallel, each taking a current of 0-3 ampere, are turned on for
five hours nightly for thirty nights. How much energy in joules
does the house receive ?
Answer. — 20 X 0-3 x no X 5 X 3600 x 30, or 356-4 million
joules.
118. Heat Produced by a Current. — When a circuit acts simply
like a resistance, so that the whole of the energy given up by a
current flowing through it is converted directly into heat, Ohm's
law holds in its simple form. Hence, if R be the resistance in
278 PRACTICAL ELECTRICITY
ohms of the circuit, / the current flowing through it in amperes,
and V be the P.D. between its terminals in volts,
V = IR
or the work in joules done by a current of / amperes in t seconds
in heating a circuit of R ohms equals I2Rt. But we know from
the investigations carried out by Joule— which have been repeated
subsequently, with even greater accuracy, by Prof. Rowland,
Prof. Reynolds, and others — that the heat required to raise the
temperature of one pound of water by i° C. when the water is at
15° C. is the equivalent of 1,400 foot pounds of work.* Therefore,
if we take this as our unit of heat, it follows, since one joule equals
07372 foot pound very approximately, that h, the number of
these heat units generated in t seconds in the circuit, is given by
h = 0-000,526,6 I2Rt, very approximately ;
or if t' be the time in minutes,
h = 0*031,60 I2Rt' ', very approximately.
Lastly, if a " calorie " be denned as the heat required to raise
the temperature of i gramme of water by i° C. when the water
is at 15° C., then c., the number of calories generated in t seconds
by a current of / amperes in a resistance of R ohms, is given by
c — 0-2390 I2Rt, very approximately, (39)
or the number of calories generated in t* minutes is given by
c = 14*34 I2Rt' , very approximately,
and i calorie = 4-184 joules.
119. Measuring the Heat Equivalent of Electric Energy. — The
formulae given in the last section may be verified by sending a
known current for a certain time through a coil of wire of known
resistance immersed in a measured mass of water, and by ob-
serving the rise of temperature with a delicate thermometer. As,
however, a portion of the current passes through the water, the
resistance in the circuit is a little smaller than that of the coil of
wire ; also the resistance may vary by warming during the course
of the experiment. Hence greater accuracy will be obtained if,
instead of attempting to measure the resistance of the circuit
directly, we observe from time to time the current that flows,
say / amperes, and the P.D. between the terminals of the coil,
say V volts ; then, if I' and V be the mean values of the current
* The "British Thermal Unit" is the amount of heat required to raise
i Ib. of water from 60° F. to 6ip F., and is the equivalent of 778 ft. Ibs. of
work, very approximately.
HEAT EQUIVALENT OF ENERGY 279
and the pressure during a period of ^seconds, the electric energy
that has been given to the coil and water during that time is
I'V't joules, which must therefore be proportional to the amount
of heat produced in that time.
If the product I'V be small, electric energy will be given to
the circuit slowly; therefore the heat will be produced in it
slowly, and it will not be possible to accurately ascertain the
amount of heat generated in a given time, without allowing for
the heat that is lost by
radiation, convection, and
conduction during the experi-
ment. If, however, the pro-
duct TV be made fairly
large, and the quantity of
water employed in the ex-
periment be not too great,
the time taken for a rise of
temperature to be produced
that can be accurately read
on a sensitive thermometer
need not be long enough for
any serious loss of heat to
occur. Further, if the vessel
containing the water be made
of very thin glass, the heat ^
absorbed in raising the tern- -
perature of the vessel may be ":
neglected unless very great
accuracy is desired ; also, if Fi«
the wire be composed of a
substance of high specific resistance and small temperature co-
efficient, not only will the change of resistance of the coil through
warming become negligible, but its mass may be small and still
a considerable amount of power may be given to it. Hence the
heat absorbed by the coil to raise its own temperature may be so
small compared with the heat absorbed by the water that the
former may be neglected, unless great accuracy be desired.
The problem of properly proportioning the parts, and of gener-
ally arranging the apparatus so that a beginner may obtain con-
siderable accuracy by using it, without its being necessary to make
any corrections for the loss of heat by radiation, convection, and
conduction, was worked out by Mr. Haycraft, formerly one of the
staff at the Central Technical College, and one of the authors ;
and they found that with the apparatus illustrated in Figs. 175
280
PRACTICAL ELECTRICITY
and 176, which fulfils the conditions they have theoretically
arrived at, students can easily obtain results not differing by as
much as one per cent, from the truth.
A strip of manganin about J inch wide, 0-03 inch thick, and
about 10 feet long, is wound so as to form the top and bottom of
a sort of cylindrical box, M M, about 5 inches across and 3 inches
high (Figs. 175, 176), the convolutions of the strip being kept
Fig. 176. — Apparatus for Measuring the Heat Equivalent of Electric Energy.
from touching one another by being screwed to a light frame-
work composed of two horizontal strips of vulcanised fibre, F, F,
joined by three thin vertical rods of ebonite, E, E, E. The two
ends of the strip are soldered to two stiff vertical copper wires,
c, c, about J inch thick and 6 inches long, the soldered joints
being covered over with varnish to prevent galvanic action
taking place at the joint (see Section 68), and the strip M M,
and the upper wires c, c are also varnished to prevent electrolysis
being produced by the current leaking through the water. The
whole is immersed in about 122 cubic inches or 2 litres of water
contained in a thin glass beaker, GG (Figs, 175, 176), which is
just wide enough to take the framework of manganin strip, and,
to diminish the risk of this beaker being broken, a piece of felt N
is placed between it and the base board o o.
Electric connection is made with the stiff wires c c by means
of two insulated very flexible leads, L L, each composed of a
strand of about 210 thin copper wires, the copper wires being each
about o-oii inch thick. The current is measured with an accur-
HEAT EQUIVALENT OF ENERGY 2Si
ate ly -calibrated ammeter, A, and the P.D. set up between the
upper ends of the stiff copper wires by means of an accurately-
calibrated voltmeter, v (Fig. 176).
The object of using a flat conducting strip and forming it
into the box shape seen in the figures is to enable the conductor
itself to act as an efficient stirrer when it is moved up and down
in the water, the flexible leads L, L, which are fastened to a wooden
rod p p fixed to the base board o o, as shown in Fig. 176, serving
as a handle to the box M M. The heat generated in the strip is,
therefore, given off fairly uniformly to the water, and the mean
temperature can be read with considerable accuracy on a single
stationary thermometer, t.
With an apparatus constructed as above described, and used
with a current of 30 amperes, the temperature of the water rises
at approximately the same rate as that of the leads L L and wires
c c, so that the conduction of heat to or from the water by the
copper is practically nil, and error from this cause is eliminated.
Below we give a set of results actually obtained by students of
the Central Technical College, using 2 litres of water. In calcu-
lating the last column allowance has been made for the water
equivalent of the glass vessel and stirrer, which amounted to
47 grammes.
Time in
Temperature C°.
Current in
Mean P.D.
Calories
Seconds.
Initial.
Final.
Rise.
Amperes.
in Volts.
per Joule.
120
18-40
22-O2
3-62
30
8-634
0-2383
1 80
I3-25
18-70
5'45
30
8-634
0-2390
1 80
13-60
19-00
5'4°
30
8-648
0-2367
I2O
12-97
16-58
3-6i
3°
8-656
0-2375
I2O
12-64
16-26
3-62
30
8-698
0-2365
I2O
12-89
16-49
3-60
30
8-662
0-2364
120
12-11
15-72
3-6i
3<>
8-666
0-2368
120
I2-IO
15-74
3-64
30
8-642
0-2395
I2O
I3-I3
16-75
3-62
30
8-692
0-2367
Mean 0-2375.
Average deviation from the mean = o-ooi = 0-42 per cent.
Now we saw in Section 118 that the true number of calories
per joule was about 0-2390, hence only two of the preceding
results obtained by the students differs by more than I per cent,
from the truth, while the mean of the nine observations gives a
result which has an error of only about one half per cent. Conse-
quently the result aimed at in designing this apparatus has been
achieved.
In carry out the investigation we may vary either —
(i) The time during which the current is allowed to flow ;
282 PRACTICAL ELECTRICITY
(2) The current made to flow through the strip ;
(3) The resistance of the conductor, by using similar stirrers
made of somewhat thicker or thinner manganin strip ;
and when a series of experiments is made varying each of these
three conditions, one at a time, it is found that the rise of tempera-
ture of the water, and therefore the amount of heat produced, is
proportional to the time, proportional to the square of the cur-
rent, and proportional to the ratio of V to / — that is, to the
resistance of the arrangement. Further, if we take as the calorie
the heat required to raise the temperature of i gramme of water
by i° C. when the water is at a temperature of about 15°, we find
that the relationship between the number of calories, the current
in amperes, the resistance in ohms, and the time, is practically
that given in Section 118.
Example 92. — A current of 30 amperes is passed through a
coil of wire immersed in water for five minutes, a voltmeter
reading 10-3 volts at its terminals. The volume of water is
2,000 cubic centimetres, and the temperature rises from 15-7°
to 26-66° C. What result does the experiment give for the heat
equivalent of one joule in calories ?
Answer. — 0-2364, a result about one per cent, too low, no
corrections having been made for cooling during the experiment.
Example 93. — A temporary resistance is made by putting a
coil of wire of 4 ohms resistance into a wooden bucket containing
37 pounds of water. If a current of 40 amperes be sent through
the coil, what about will be the rise of temperature of the water
in the first three minutes ? Answer. — 16° C.
120. Power. — " Power " is the name given to the rate of
doing work — that is, the rate of transformation of one form of energy
into another — and it must be carefully distinguished from the
amount of work done, there being the same sort of difference be-
tween power and work that there is between a velocity and a
distance. The word power was, however, used in the older books
on dynamics to stand for the applied force, and that is the
meaning of the word power in such expressions as " the mechani-
cal advantage of a machine is the ratio of the weight to the power."
Again, the word power is sometimes wrongly used for energy, as
in the expression the " storage of power/' Beginners must,
therefore, be on their guard against being misled by such loose
expressions, and they should never employ the name power, or
" activity," as suggested by Lord Kelvin, in any other meaning
than the rate of doing work. In that sense, of course, power
cannot be stored, for while a certain quantity of water in a reser-
POWER 283
voir at the top of a hill represents a certain store of energy, the
power that this water can exert at any time when flowing out of
the reservoir will depend on the rate at which it is allowed to
flow.
When work is being done at a constant rate, the power is
constant, and it is measured by dividing the number which
expresses the work done in any time by the number expressing
the time. If, however, the rate of doing work at one moment
is greater than at another — for example, when a person runs
upstairs quickly at first and then more slowly — we do not mean
by the power expended at any moment, the actual work done
in a minute, or even in a second, for the rate of doing work may
be changing very rapidly. In such a case the power at any time
is the limiting value of a ratio obtained thus : — Measure the work
done in a very short time, a portion of which precedes, and the
remainder of which follows, the instant at which we wish to
measure the power ; divide the work done in the very short time by
that time, then this ratio more and more nearly represents the
power being expended at the moment in question, as we make
the very short time shorter and shorter.
When, however, electric energy is being transformed into some
other form of energy, the power may be very easily ascertained,
whether the rate of doing work is constant or not, without its
being necessary to measure a small time. For the work done
in t seconds by a constant current of I amperes flowing
through a circuit under a constant P.D. of V volts equals
IVt joules,
provided that there is no apparatus in the circuit that gives
energy to the current instead of receiving energy from it ;
therefore the rate of doing work in joules per second equals simply
IV.
Hence, if at any moment we measure the current and the P.D.
simultaneously, the product of the two measurements gives us
the instantaneous value of the power being expended at that
moment, and no measurement of time need be made. Conse-
quently the rate of transformation of electric into some other
form of energy may be varying, but as long as it is not varying
so rapidly as to prevent accurate readings of an ammeter and
voltmeter being taken, the instantaneous value of the power
can be ascertained at any moment.
121. Electric Unit of Power : Watt. — When work is being
done at the rate of one joule per second the power exerted is
284
PRACTICAL ELECTRICITY
called a " watt " ; therefore the power of one watt is developed
when work is done at the rate of
107 ergs per second,
or 1 joule per second,
or 0-7372 foot pound per second, very approximately,
or 44-23 foot pounds per minute, very approximately ;
and since when work is being done at the rate of 550 foot pounds per
second, or 33,000 foot pounds per minute, one " horse-power "
is said to be exerted,
33 000
1 horse-power = TJ- watts, very approximately, '
1 horse-power = 746 watts, very approximately,
1 watt = 1 /746th of a horse-power, very approximately,
1 kilowatt = 1000 /746th, or 1-340 horse-power, very ap-
proximately.
.'. 1 kilowatt = 1J horse-power, roughly.
Further, if P be the power in watts expended in a circuit between
the ends of which a P.D. of V volts is maintained and through
which a steady current of I amperes is flowing,
P = IV (40)
provided that the circuit contains no apparatus that gives energy
to the current, instead of receiving energy from it.
Example 94. — What power in watts is expended in the arc
lamp and in the resistance coil referred to in Examples 88 and
89. Answer. — 600 watts, and 0-096 watt respectively.
Example 95. — What power in kilowatts is expended in the coil
in Example 92 ? Answer. — 0-309 kilowatts.
Example 96. — The adjoining figure shows the " load diagram "
LOAD DIAGRAM. DECEMBER.
1000
of a central station for December — i.e., the curve giving the out-
put of the station in amperes throughout the twenty-four hours.
JOULE'S LAW 285
If the station pressure is 440 volts, what is the output in H.P.
(horse-power) at 7 a.m., 12 noon, 6.30 p.m., and 10 p.m. ?
Answer— jj H.P., 23 H.P., 612 H.P., and 286 H.P.
respectively.
Example 97. — Two glow lamps, each giving 16 -candle power,
take 1-75 and 1-25 watts per candle respectively. How many
lamps can be supplied per horse-power expended in the two cases,
and how many candles per horse-power will they give ?
Answer. — Twenty-seven and thirty-seven lamps respectively ;
426 and 597 candles.
Example 98. — How many candles per horse -power are given
by an arc lamp taking n amperes and 50 volts, and giving a
mean candle-power of 1,750 in all directions ?
Answer. — 2,374 candles per horse-power.
122. Joule's Law. — From the above it follows that if P be
the power in watts expended in heating a circuit of resistance
R ohms through which a current of / amperes is flowing, then
P - I*Rt
for P=IV,
and V = IR, by Ohm's Law,
/. P = I*R; (41)
or, the rate at which heat is generated in a resistance through which
a current is flowing, is proportional to the product of the square
of the current and the resistance ; this is known as Joule's Law.
y
Since / = - the above expression may be written
K
a form which is useful in many cases.
Further, the energy transformed into heat in t seconds is by
Section 117, given by the equation —
W = IVt joules,
and this may be written
W = I2Rt joules, or
F2
W=~.t joules
R
according as / or V is given.
Expressed in calories we have
c = 0-2390 I2Rt, (43)
V'2
and 0 = 0-2390—*, (44)
256 PRACTICAL ELECTRICITY
as the forms most convenient to use when the current is given,
or the P.D. is given, respectively.
Example 99. — Calculate the power in watts expended in a con-
ductor whose resistance is 2-5 ohms when a current of 20 amperes
is flowing through it. Determine also the energy used in one
hour. Answers (i) 1000 watts, or i kilowatt
„ (2) 3,600,000 joules, or i kilowatt hour;
Example 100. — One of the lamps of an electric radiator con-
sumes 250 watts at 210 volts ; what current passes through it,
and what is its resistance ? Answer (i) 1-19 amperes approx.
,, (2) 176-4 ohms approx.
Example 101. — An ammeter graduated to 150 amperes has a
resistance of 200 microhms ; find the power expended in the
instrument at maximum reading. Answer. — 4-5 watts.
Example 102. — What power is used in a voltmeter of 8,000
ohms resistance when a P D. of 220 volts exists between its
terminals ? Answer. — 6-05 watts.
123. Instruments for Measuring Power : Wattmeters. — The
electric power used in a circuit can be determined, as shown in
the preceding sections, by finding the current and P.D., and multi-
plying them together ; when the current is steady this method is
comparatively simple, but in many cases it is more convenient
to use an instrument, the reading of which gives the power direct -
ly. Instruments for this purpose (called wattmeters) were first
made in England by Professor Perry and one of the authors in
1881. One of the simplest forms of wattmeter resembles the
electrodynamometer described in Section 39, in having a fixed
coil and a moving coil, and a torsion head whereby the moving
coil can be brought into a definite relative position to the
fixed coil. In the electrodynamometer both coils carry the
same current, but in the wattmeter one of the coils carries the
main current of the circuit in which the power is to be measured,
whilst through the other coil a current proportional to the P.D.
between the terminals of the circuit passes. They are called the
current coil and the pressure coil respectively.
The current coil c c, Fig. 177, which is made of a few turns of
thick wire, is inserted in the main circuit ; while the other coil,
c c, consisting either of many turns of fine wire, or, better, of a
few turns of fine wire in series with a stationary high resistance,
w, is connected as a shunt to that portion of the circuit L M the
power given to which we desire to measure. The current passing
through c c is therefore proportional to the P.D. between the
WATTMETERS 287
ends of L M, while the current passing through c c is the sum of
the currents flowing through L M and' through c c. If, however,
the resistance of the fine-wire circuit of the wattmeter is very
large, the current passing through it will be very small compared
with the current flowing through L M, so that the current passing
through c c will be practically that flowing through L M. Hence
the part of the wattmeter between the terminals Tlt T2 acts as
an ammeter, while that between the terminals tlt t2 serves
as a voltmeter. Consequently the product of the currents in
c c and c c is proportional to the power given to L M. But this
Fig. 177. — Diagram of Wattmeter.
product is directly proportional to the couple exerted between
these two coils if the coils be always brought into the same
position relatively to one another. Hence the power to be measured
is proportional to the torque that must be exerted on the movable
coil of the wattmeter to keep it in a fixed position relatively to the
stationary coil.
The torque required to be exerted on the suspended coil c c
in order to maintain it in a fixed position relatively to the station-
ary coil c c, may be conveniently produced by turning the head
H and the pointer p attached to it. This twists the thin vertical
wire supporting the movable coil, as the upper end of this wire is
rigidly fastened to the head H. And, since the angle through
which one end of a wire is twisted relatively to the other end is
directly proportional to the torque exerted, the power given
electrically to the portion of the circuit L M will be directly pro-
portional to the angle through which the pointer p has been turned
to keep the coil ccm the position it occupied when no current was
passing through the coils.
288
PRACTICAL ELECTRICITY
Another way of joining up a wattmeter is to connect tv t2,
the terminals of the fine-wire circuit, to L and T2 respectively, so
that the fine-wire circuit is a shunt to both L M and the thick wire
coil c c of the wattmeter. In that case the current passing
through c c will be accurately the current that flows through
L M, but the current passing through c c will now be proportional
to the P.D. between the points L and T2, and not between the
points L and M. The difference between these two P.Ds. will,
however, be very small if the
power spent in sending the cur-
rent through c c is very small
compared with the power spent
in sending it through L M, and
this result can be practically
attained by making the resist-
ance of the coil c c as small as
possible.
124. Commercial Forms of
Wattmeters. — Commercial watt-
meters based on the principle
described in the last section have
been constructed by several
people. A compact form, de-
signed by Mr. Swinburne, is
,seen in Fig. 178, which shows
'the instrument with the outer
cylindrical cover removed so that
the interior may be visible. The
stationary coil c is made in two
sections, the front one having been removed in the figure so that
the suspended coil c can be better seen. The position of this
suspended coil is sighted by means of a small pointer which is
rigidly attached to the bottom of the vertical rod hanging down
from the small moving coil c, and when a measurement is made,
the milled head H is turned until the small pointer is exactly
over a black line marked on a silvered plate which is fixed
to the base of the instrument just under the little pointer.
Parallax is avoided by the pointer and this line being looked at
through a small window w w in the dial plate at, the top of the
wattmeter.
Instead of measuring the torque that has to be exerted to keep
the suspended coil in its initial position, as in using the zero watt-
meters shown in Figs. 177 and 178, we may observe the angle
through which the moving coil is turned against the action of a
Fig. 178. — Swinburne Wattmeter, with
cover removed.
WATTMETERS
289
spring or gravity. A deflectional wattmeter with spring control
is shown in Fig. 179. The case has been removed to show the
working parts more clearly. From the figure it will be seen that
the moving coil to which the pointer is attached, is outside the
fixed or thick wire coil — c c being the ends of the thick wire coil,
and c c those of the thin coil. Near the bottom of the figure are
four magnets between the poles of which an aluminium disc fixed
to the moving coil passes. They serve to " damp " the move-
Fig. 179. — Elliott's Deflectional Wattmeter.
ment, and bring the pointer quickly to rest, but otherwise do not
affect the indications of the instrument. The power to be
measured will not be directly proportional to the angle through
which the pointer attached to the movable coil turns, but the
scale seen in Fig. 179 has been constructed by sending various
known currents through the two coils respectively, and making
a direct reading scale by a process similar to that described in
Section 22.
Example 103. — In a certain wattmeter it is found that the head
must be turned through 125° to bring the pointer to zero, when
the current in the main coil is 20 amperes and the P.D. between
the ends of the shunt coil is 120 volts. How much must the
head be turned to bring the pointer to zero if the wattmeter is
measuring the power taken by a resistance of 7-3 ohms through
which a current of 30 amperes is passing ?
The wattmeter reading is proportional to the product of the
currents in the two coils, and the current in the shunt coil is
proportional to the P.D. between its terminals, which is 7-3 x 30,
PRACTICAL ELECTRICITY
or 219 volts in the second case. Hence, if 0 is the angle through
which the head must be turned,
_0_ _ 30 x 219
125 ~~ 20 X 120*
Answer. — The head must be turned through 342°.
Example 104. — If the resistance of the shunt coil in the above
wattmeter is 6,542 ohms, what additional resistance in the shunt
circuit will make the constant of the instrument 20 watts per
degree ? Answer. — 272-6 ohms.
Example 105. — If the current passing through the circuit, the
power given to which we desire to measure, is 20 amperes, while
the P.D. maintained between its terminals is 30 volts, and if the
resistances of the thick -wire coil and of the fine-wire circuit of
a wattmeter are o-oi and 1000 ohms respectively, calculate the
error that will be made by using the wattmeter when joined up
in the two ways described in Section 123.
Answer. — When the wattmeter is joined up as shown in Fig.
177 the current passing through the thick -wire coil cc will be
30
20 H amperes instead of 20 amperes — that is, will be 0-15
per cent, too large ; therefore the power measured by the watt-
meter will be 0-15 per cent, greater than the power given to the
circuit L M. If, on the other hand, the wattmeter be joined up as
described at the end of Section 123, the current passing through
the fine-wire circuit of the wattmeter will be produced by a P.D.
of 30 + 20 X o-oi, or 30-2 volts instead of 30 volts, the 0-2
volt being the P.D. expended in sending the current through
c c. Hence the current through the fine-wire circuit, and
therefore the power measured by the wattmeter, will be 0-67
per cent, too large. Consequently the former method of joining
up the wattmeter would give the more accurate result in this
particular case. It should here be noted that in both cases the
wattmeter reading is greater than the true power given to L M.
125. Joule-or Energy Meter: Clock Form. — As shown by
Professor Perry and one of the authors in 1882, any pendulum
clock can be easily converted into a " joulemeter ; " that is, into
an instrument which records the energy given to an electric
circuit in any definite time, and Fig. 180 illustrates the first
electric energy meter, called originally an " ergmeter," that was
constructed in this way.
The ordinary pendulum bob is replaced by a bobbin B, on which
is wound a coil of fine wire, the coil being wound on in two
ENERGY METER
291
parts, c, c, for convenience of attachment of the bobbin to the
pendulum rod. These two halves of the fine-wire coil are joined
in series with one another, and the terminals of this coil, tlf t2, are
connected as a shunt with that portion of the circuit LM, the
energy given to which we desire to record. Fixed to the clock
case in the position shown is a coil consisting of a few turns of
thick wire ; this coil being also constructed in two parts, c c, so
that the pendulum coil may swing symmetrically between them.
These two halves of
the thick -wire coil
are joined in series
with one another,
and . the terminals
of this coD, TJ, T2,
are connected up
as shown, so that
the main current
through L M passes
through the station-
ary coil c c. Cur-
rents, then, flow
both round the
moving coil cc and
the stationary coil
c c, and produce,
therefore, an at-
traction or a repul-
sion between these
coils, depending on
whether the coils are joined up so that the currents circulate
round them in the same direction or in opposite directions. The
force exerted between the coils will vary with their relative posi-
tions, but its mean value will be proportional to the product of
the currents flowing in the two coils ; that is, it will be propor-
tional to the power given to the part of the circuit L M.
The action of this force on the swinging pendulum will be
approximately the same as if the action of gravity had been
increased or diminished ; hence, if the coil be joined up so that
there is an attraction, the clock will be caused to gain, whereas
if the ends of the fine-wire coil be interchanged, the clock will lose.
And if this force is small compared with the weight of the pendu-
lum bob it may be shown in the following way, that the total gain
or loss of the clock in any period is directly proportional to the
energy given electrically to the circuit in that period.
Fig. 180. — Ayrton and Perry's Original Gaining
Clock Joulemeter.
292 PRACTICAL ELECTRICITY
From the binomial theorem we know that when b is small
compared with a, then (a ± b)n = an ± n an~l b, approximately,
so that the difference between (a + b)n and an is proportional to
b, when a and n are constants. Now it follows from Section
27, that the number of vibrations a pendulum makes in t
seconds is proportional to the square root of the controlling
force, and therefore n equals J in this case, and if the con-
trolling force be increased or decreased by a small amount, the
difference in the number of vibrations in a given time caused
by this change will be proportional to the change. We thus
see that the gain or loss of the clock produced by the magnetic
force between the fixed and moving coils will, if this force be
small compared with the
weight of the pendulum,
, be proportional to the
power given to the circuit
L M, and to the time-. In
other words the gain, or
loss, is proportional to
'K' the energy expended in
the circuit. Further, if
instead of observing the
gain or the loss of the
. l..-er,npaeann8 o e roD meter ^ ^ j^^j by
comparing its indication
at the beginning and end of the interval with a good clock or watch,
we place two clocks inside the " supply meter," one clock being
an ordinary one, and the other a clock having for its pendulum
a pressure coil swinging near a stationary coil, through which
passes the main current flowing through the house, the energy
given to the house in any time will then be directly proportional
to the difference between the number of vibrations that the two
clocks have made in the interval. This difference can be read
off on a counting mechanism like that used on a gas meter if the
staff F F (Fig. 181), driving this counting mechanism be connected
by means of " differential gearing" with the two clocks.
The staff F F (Fig. 181), is rigidly connected with the balanced
arm, A A, which carries at one end a planet wheel P P. This
gears into two crown wheels K K, K' K', turning loosely on the
staff F F. These crown wheels have also teeth cut on their circum-
ferences like ordinary toothed-wheels, and are geared with the
two clocks respectively, one wheel being rotated by one of the
clocks right-handedly and the other left-handedly. When no
electric energy is being supplied to the circuit LM (Fig. 180),
ARON ENERGY METER
293
the crown wheels are driven by the clocks at equal rates in oppo-
site directions, the wheel p p therefore is simply turned round on
the arm A A as an axis, but the arm itself is not moved. But when
energy is supplied to
the circuit the clock
with the magnetic pen-
dulum goes faster, the
crown wheel driven by
it, therefore, also ro-
tates faster than the
other crown wheel, and
the pinion p p not only
is rotated on the arm
A A, but the arm itself,
and the staff F F at-
tached to it, are driven
round, and move on the
dial hands at a rate
depending on that at
which electric energy is
supplied to the circuit.
In the modern form
of clock energy meter
designed by Dr. Aron,
the two clocks are
driven by the same
mainspring, each pen-
dulum carries a coil c, c,
whose axis is vertical
when in the mid-posi-
tion, and these swing
over and near to fixed
coils c, c, as seen in
Figs. 182 and 1820.
The coils are so con-
nected that one of the
pendulums gains and
the other loses, when
current passes through
the meter, and the difference in the number of vibrations
in a given time is doubled by this device, and this difference
is registered on the meter dials by differential gearing as
described above. Two difficulties experienced with the original
form of energy meter, viz., winding the clocks, and the necessity
Fig. 182. — Aron Energy Meter (cover removed).
PRACTICAL ELECTRICITY
of the clock keeping correct time when no current was passing
through the current coils, have been overcome by (a) fitting an
automatic winding device operated electrically, and (b) reversing
the current through the pendulums at equal intervals of time,
and at the same instant reversing the connection of the differential
gearing with the registering dials. Both the latter operations are
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Fig. 1820. — Connections of Aron Energy Meter.
performed automatically by electrically driven mechanism, and
thus the effect of any want of synchronism in the two pendulums
when no current is passing is eliminated. By properly choosing
the toothed gearing between the differential gear and the regis-
tering dials, and by adjusting the position of the stationary
coils relative to the pendulums, the dials can be arranged to
read the energy in joules, or any convenient multiple thereof.
126. Board of Trade Unit of Energy. — The units of energy we
have hitherto considered are the " erg," the " foot pound," and the
BOARD OF TRADE UNIT 295
" joule/' the former being the unit in the "C.G.S." system, and
the latter the unit in the " practical " system of electrical units.
Although the joule (or the " watt second ") is ten million ergs, the
joule is found inconveniently small for commercial purposes,
and when electric energy began to be supplied to the public by
electric lighting companies in the early eighties, the Board of Trade
adopted a much larger unit, viz., the " kilowatt hour " or 1000
watt hours as the commercial unit. The word " kelvin " has been
suggested as a name for this unit, but is not yet generally adopted.
" The Board of Trade Unit," the words " of energy " being
generally omitted, is the name given to the work done in a
circuit when the power exerted in watts multiplied by the time
during which it is exerted in hours equals 1000, or
1 Board of Trade unit = 1000 watt hours,
„ = 3,600,000 joules,
„ = 36 x 1012 ergs,
„ „ „ = 2,653,800 foot pounds, very approxi-
mately,
„ „ „ = 1-340 horse-power hour, very approxi-
mately.
„ „ „ = 1£ horse-power hour, roughly.
A Board of Trade unit is, therefore, a thing that can be bought
and sold at a specified price, like a ton of iron, and this price can
be regulated by agreement or by law, just as the price of gas per
1000 cubic feet, or as cab fares are regulated. The average price
per Board of Trade unit supplied to large consumers in London
for lighting purpose is about sixpence, whilst for heating, cook-
ing and electric-driving of machinery, i|d. per unit is a common
price.
Example 106. — If electrical energy is supplied at 4d. per
Board of Trade unit, determine whether it is more economical
to use 16 -candle power carbon lamps taking 2-5 watts per candle
and lasting 500 hours, or 16 -candle power lamps taking 3-5
watts per candle and lasting 900 hours, the cost of a new lamp
being in each case lod.
Answer. — Using 2-5 watt lamps : —
2-5
Cost for energy per candle hour = X 4
IOOO
= o-oio penny.
Cost for lamp renewals per candle hour . = —
ID X 5OO
= 0-0012 penny.
Total cost per candle hour = 0-0112 penny.
296 PRACTICAL ELECTRICITY
Using 3-5 watt lamps : —
Cost for energy per candle hour = 5- x 4
Cost for lamp renewals per candle hour
1000
= 0-014 penny.
10
16 x 900
= 0-0007 penny.
Total cost per candle hour = 0-0147 penny.
Therefore, in this particular case, it is more economical to use
the lamp having a shorter life but taking less power per candle. *
Example 107. — Is the same conclusion true if the lamps are
8 -candle power, all other things remaining the same ?
Answer. — The total costs per candle hour become 0-0125 and
0-0154 penny, so that the shorter-life lamp is still the cheaper.
Example 108. — Compare the cost for equally lighting the same
area with gas at 2s. 6d. per 1000 cubic feet (the mantles used
giving 60 candles for 5 cubic feet per hour) with metal filament
incandescent lamps using electric energy at 4d. per unit (the lamps
taking 1-25 watts per candle), and with flame arc lamps supplied
with electric energy also at 4d. per unit (the lamps taking J watt
per candle).
The cost of renewal for broken mantles and glow lamps, and
the carbons for the arc lamps, not to be included.
Answer. — Relative costs : — Gas, 2-5 ; electric glow lamps,
5-0 ; arc lamps, i.
Example 109. — What is the reduction in a consumer's bill
of £80 per annum for electric energy supplied, (a), if the price of
a unit is reduced from 6d. to 5d. ; (b), if lamps taking 1-2 watts
per candle are used instead of lamps taking 3-5 ?
Answer. — (a) A reduction of £13 6s. 8d. ; (b) a reduction of
£52 us. 5d.
Example no. — How many Board of Trade units are consumed
by a loo-volt 20-candle power lamp taking 28 watts burning
continuously for one year ? What is the cost at id. per unit ?
Answer. — 245 units ; £i os. 5d.
127. — Energy Meter: Motor Form. — In the last section was
described the method of recording the sum of the products of the
power into the time ; that is, the total amount of electric energy
given to a circuit, by using the attraction between the current and
pressure coils of a wattmeter to alter the rate of going of a clock.
* The costs in examples 106-110 are based on pre-war prices.
MOTOR ENERGY METERS 297
But, as pointed out by Professor Perry and one of the authors in
1882, in the same patent specification, this attraction may, in-
stead, be employed to drive the counting mechanism, and give a
direct record of the energy supplied to any circuit if the current and
pressure coils be made to form the stationary and moving parts
respectively of an electromotor without iron, and if the rotation
of the motor be resisted by a torque proportional to the velocity
of rotation. This principle has been used by Professor Elihu
Thomson in the construction of a very large number of joulemeters.
For some reason this instrument as constructed by Professor
Elihu Thomson has been called a " recording wattmeter " ; this
name is, however, a misnomer, since it is the total amount of
energy in Board of Trade units, and not the variations of the
power in watts, which the instrument records.
It is impossible to obtain continuous motion by the mutual
action of the currents in two coils unless the current in one of the
coils, at any rate, be periodically reversed. For, suppose currents
flow round two coils in such directions that the coils attract one
another, the coils, if one or both of them be free to move, wilJ
approach one another, the force of attraction will rapidly increase,
causing them to finally rush together, when they will press against
one another, and any further motion will be clearly impossible.
On the other hand, if the directions of the currents be such that
the coils tend to repel one another, either it will happen that one
of the coils will turn round, when they will approach as before,
or, if neither of the coils be free to turn, they will recede from one
another until the distance separating them becomes so great
that the force of repulsion is too small to overcome any fri^tional
resistance that may oppose the motion.
To keep up a continuous motion, then, of one coil relatively to
another, there must be employed some form of " commutator " or
arrangement for reversing the current through one of the coils ;
further, if we wish that the force producing the motion shall
remain fairly constant, either the moving or stationary part of
the motor must consist of a number of coils so arranged that, as
the rotation of the motor changes the position of one coil in the
magnetic field, its place in the field is taken by the next coil.
This part of the motor is called the " armature" while the other
part is called the " field," and if the armature has a sufficient
number of coils on it the torque exerted between the field and
the armature remains practically constant, in spite of the motion
of the one relatively to the other.
The armature of the Elihu Thomson energy meter is the rotating
portion, and it consists of eight coils, cv c2 . . . cg, wound on
298
PRACTICAL ELECTRICITY
Fig. 183.— Rotating
Armature of the
Elihu Thomson
type of Motor
Energy Meter.
this shunt current
is led into and
out of the commu-
tator by two station-
ary " brushes," B, B,
the current dividing
into two parts at
each brush and fol-
lowing the paths
shown by the arrows
(Fig. i83«).*
* In the modern
form of House Service
Meter illustrated in
Fig. 184, the armature
is made spherical in
shape, and the brushes
B, B, Fig. 183, are made
of small silver tubes
supported on fine brass
wires.
a light framework, as seen in
Fig. 183, which shows the
armature detached from the
complete meter in order that
the construction of the
former may be clearly seen.
The end of each coil is
electrically connected with
the beginning of the next,
and is also connected by
means of one of the wires
ze>i, ze>2 . . . . WQ (Fig. 183)
with one piece of the eight-
part commutator klt kz . . . .
k8. The armature, which is
in series with a stationary
resistance, is joined as a
shunt to the portion of the
circuit the energy given to
which it is desired to record,
like the pressure coil of the
wattmeter, Fig. 177, and
Fig. i83«. — Diagram showing the Directions of the Currents in
the Armature of the Elihu Thomson type of Energy Meter.
THOMSON ENERGY METER
299
This figure is a diagrammatic sketch of the armature, commuta-
tor, and brushes at the moment when the two pieces k7 and ka
of the commutator are touching the brushes, and to avoid con-
fusion in this sketch only these two pieces of the commutator are
shown connected with the coils. But in reality each of the eight
commutator pieces ^ .... k8 is joined respectively with the
end of one coil and the beginning of the next, and, since the
brushes are stationary
while the armature
and commutator re-
volve, the direction of
the currents in the coils
would appear exactly
the same whether the
pair of commutator
pieces touching the
brushes were kL k5,
The result is, that
although the armature
rotates, the current
flowing round it pro-
duces a magnetic field,
in a nearly fixed posi-
tion, indicated by the
dotted line N s.
The stationary field
coils c c, seen in perspective in Figs. 1836 and 184, are placed
in series with the portion of the circuit the power given to
which we desire to measure, so that the main current passes
through these field coils and produces another stationary magnetic
field, which is almost at right angles to that produced by the
armature, and the action of the one field on the other causes a
continuous rotation of the armature.
As these two fields have always the same relative position,
the torque exerted will be directly proportional to the product of
the strengths of the fields, and, as no iron is used in either the
armature or the field coils, the magnetic fields will be directly
proportional to the currents producing them ; hence the torque
producing the rotation will be directly proportional to I V,
the power in watts given to the portion of the circuit under
consideration.
The motion of the armature c (Fig. 184) is resisted by the
horizontal aluminium disc D. which is rigidly attached to the
Fig. 183$. — Interior of Thomson Energy Meter (Old Form).
300
PRACTICAL ELECTRICITY
armature spindle E, being rotated in the magnetic field produced
by four stationary permanent magnets M, M, M, M. The south
pole of each of these magnets (Fig. 184) is above, and the north
pole below, the disc, so that the lines of force produced by these
permanent magnets are vertical and at right angles to the plane
of rotation of the disc. This rotation between the poles of
Fig. 184.— Thomson House Service Meter (Modern Form).
the magnets causes currents, called " Foucault currents " or
" eddy currents," to be induced in the disc, and the attraction
between these currents and the stationary magnets impedes
the turning of the armature. Now, the strength of these induced
currents is proportional to the angular velocity a, so that the
torque which resists the motion is proportional to a.
We have, therefore, as shown by Professor Perry and one of
the authors, a driving torque proportional to IV and a retarding
LAW OF ENERGY METER 301
torque proportional to a ; hence, if the frictional resistance to
motion introduced by the bearings of the armature, the rubbing
of the commutator against the brushes B, B (Fig. 183), and the
train of wheels in the counting mechanism driven by the screw
or worm s (Fig. 183), be very small, the armature must rotate
at such a speed that the electromagnetic driving torque, which
is proportional to IV, is exactly equal to the electromagnetic
retarding torque, which is proportional to a, or
IV oc a.
If, now, during any time t seconds, the power supplied to the
circuit be constant, IV will be constant for that time, and so
also will a ; therefore
IV t a at,
but IV t is the energy in joules and a t is the angle turned
through by the armature in that time. Consequently for each
period of time during which the energy is supplied at a constant
rate, the angle turned through by the armature, and therefore
the advance of the counting mechanism, is directly proportional
to the energy supplied in that time. Therefore, adding together
all the amounts of advance of the counting mechanism and ail
the amounts of energy supplied for each of the periods during
which energy is supplied at various constant rates, we may
conclude that the total advance of the counting mechanism
in any interval will be directly proportional to the total amount
of energy supplied in that interval, whether the energy has been
supplied at a uniform or at a variable rate.
The friction at the bearings of the armature may be rendered
small by using a very light armature, and by forming the ends
of the armature spindle of hard metal, carefully pointed, and by
supporting them in jewels, as is done in good watches. The
friction and inertia of the counting mechanism can be overcome
by making the parts small and light ; and the friction of the
commutator &t . . . . k8 against the brushes B, B, Professor
Elihu Thomson has found, can be reduced to a workable limit by
constructing the commutator of silver, as well as the parts of the
brushes that rub against it, and by making the diameter of the
commutator very small.
The clock type of meter has the great advantage over the
motor form that, no matter how small be the rate at which
electric energy is supplied to a circuit, the clock meter actually
records the total amount of energy supplied, whereas, in conse-
quence of friction, a motor meter will not start until the currents
passing round its coils reach a certain value. Hence, if the
302
PRACTICAL ELECTRICITY
electric power that a circuit receives be always very srr.all, the
armature of a motor meter may never move, and so the meter
will record no energy received, even though the period during
which this very small amount of power has been supplied has
been so long that the total amount of energy that ought to be
recorded is considerable. To overcome this defect it is now
customary to put a
"starting coil" j, Fig.
184, near the main
coils, consisting of a
moderate number of
turns of fine wire in
series with the arma-
ture. These turns
produce a magnetic
field in the space oc-
cupied by the arma-
ture and give a torque
nearly equal to the
friction torque. The
meter is therefore on
the point of starting
when no current is
being used in the
lamps, and switching
on a single small lamp
causes the armature
to turn.
128. Quantity or
Ampere-hour Meters.
— In practically all
cases where electric
energy is supplied to
consumers the supply
authorities are re-
Fig. 185'. — Bastian Meter (front cover removed).
quired to maintain a constant P.D. between the mains which
convey the currents, and as the energy is Vlt the quantity
(It) is proportional to the energy supplied when V is constant.
If, therefore, we can measure It, the whole energy can be
determined by multiplying by an appropriate constant. For
example, if the supply pressure be 200 volts a current of one am-
n • r x 200 x i x i
pere flowing for one hour will mean a consumption of
1000
i.e., 0*20 Board of Trade units.
AMPERE-HOUR METERS
303
The simplest form of quantity meter is the voltameter (Sections
7, 8, n, 12, and 13), and both copper and acid (or alkali) volta-
meters are used for this purpose. In one form of the latter, the
Bastian Meter (Fig. 185'), a solution of caustic soda is electrolysed,
Fig. 185. — Chamberlain and Hookham Metet.
the resulting gases being allowed to escape, and the diminution of
volume of the liquid measures the quantity of electricity passed
through it. A layer of oil about half an inch deep is placed on
the surface of the electrolyte to prevent evaporation, and the
current is led into and out of the liquid by rods R', R termina-
ting in cylindrical nickel electrodes near the bottom of the
glass vessel. The scale of the instrument is graduated to read
304
PRACTICAL ELECTRICITY
Board of Trade units at a particular pressure. Most ampere-
hour meters are, however, of the motor form. Two of these,
the " Chamberlain and Hookham," and the " Ferranti " meters
are shown in Figs. 185 and 186. A section of the former meter
is given in Fig. 1850, and a part plan in Fig. 1856. A metal
disc D, supported by the spindle F, is situated in a circular cavity
formed between two blocks of ebonite, E E, and containing mer-
cury, and on the opposite faces of this cavity are conical poles,
B B, made magnetic by the large permanent magnet A. Wires
-((<§
Fig. 1856. — Chamberlain and Hook
Meter (Plan of Disc).
Fig. 1 8 5 a. — Chamberlain and Hookham
Meter (Section).
conveying the current project slightly into the cavity at j and
K, so that the direction of current flow is mainly radial, and be-
tween the poles B B. Here we have conductors, the mercury
and disc, conveying a current in a magnetic field ; a force will
therefore exist causing the mercury and the disc to revolve, and
this force will be proportional to the strength of the current flow-
ing. The speed of the disc will become constant when the retard-
ing torque due to Foucault currents induced in the disc by the
poles B is equal to the driving torque, and, as seen in the last
section, the retarding torque due to eddy currents is proportional
to the velocity. The velocity of rotation will, therefore, be pro-
portional to the current strength, and the number of revolutions
in a given time proportional to the product of the current and
AMPERE-HOUR METERS
305
the time, or to the quantity passed. By a suitable choice of
gear, depending, amongst other things, on the pressure of supply,
the revolution of the spindle F is transmitted to the registering
Fig. 186.— Ferranti Meter (Front View).
dials shown in Fig. 185, which read off direct in Board of Trade
units. A coil i on an' iron core, seen to the right of Fig. 1850,
is used to correct the slight error arising from the fluid friction of
the mercury. Fluid friction increases more rapidly than the
velocity, so that unless this was compensated in some way the
3o6
PRACTICAL ELECTRICITY
velocity of rotation would not be quite doubled by doubling
the current. The coil i, however, is arranged to cause more
lines of force to pass through its core as the current increases,
and thereby weaken the magnetic field between the poles B,
thus reducing the retarding torque due to the eddy currents in
C,
Fig. i86a.— Ferranti Meter. (Section through Disc.)
the disc. At first sight one might imagine that weakening the
field between the poles would decrease the driving torque just
as much as it decreases the retarding torque, but this is not the
case, for the retarding torque varies as the square* of the strength
* That the retarding torque is proportional to the square of the strength
of field may be seen by remembering that the force exerted on a conductor
carrying a current in a magnetic field is proportional to the product of the
strength of current and the strength of the field, and that the strength of
the induced eddy current is proportional to the strength of the field ; the
product, therefore, varies as the square of the strength of the field, all
other things remaining constant.
AMPERE-HOUR METERS 307
of field, and the driving torque as the first power ; reducing the
field strength by one per cent, would therefore lessen the driving
torque by one per cent, and the retarding torque by two per
cent, approximately.
The Ferranti continuous current meter is illustrated in Figs.
186 and i860. In principle it resembles the Chamberlain and
Hookham instrument, and differs mainly in having an additional
magnet with poles s B, Fig. i860, which acts as a brake only,
whilst the magnet with poles s D, s D, produce a magnetic
field which acts both as driver and brake. Compensation for
fluid friction is effected by a coil, c, c, which strengthens the
poles s D, and weakens s B. The driving force is, therefore, in-
creased more rapidly than the current as the current increases,
whilst the retarding torque remains practically unchanged owing
to one pair of poles being weakened as much as the other is
strengthened. The path of current through the, meter is from
the positive terminal Cj, through the copper disc c D, and
mercury in a radial direction to C2, and by way of the coil c c
to the negative terminal. A worm and wheel, seen at the top
of the vertical spindle supporting the disc, are used for recording
the number of revolutions, and therefore the quantity passed
through the meter.
It will, of course, be understood that meters of the motor type
cannot be exactly correct under all circumstances, and to secure
reasonable accuracy without necessitating exceptional perfection
in construction with its attendant high cost, errors not exceeding
-for — 2 per cent, are tolerated.* This is the permissible error
between full load and ^ of full load. For currents between ^ and
1*0 full load, the percentage error must not exceed 2-5 per cent,
and at ^o fc^ l°ad 4*5 per cent. On the other hand, all meters,
to comply with the specification, must start running when a cur-
rent equal to i per cent, of full load current passes through the
main circuit, provided this current be not less than -£•§ of an am-
pere; if this current be less than ^o of an ampere then the meter
must start and run steadily with ^o ampere.
Example in. — A current of 68 amperes at no volts is passed
through a meter for twenty minutes, and the change of reading
produced is 2-54 Board of Trade units. Calculate the percentage
error of the meter. Answer. — 2 per cent. fast.
Example 112. — The gearing between the revolving disc of a
meter, and the recording dials is such that the former makes
48,000 revolutions whilst the " Units " dial makes one complete
* " Btsh. Eng. Stand. Specification for Consumer's Electric Supply Meters. "
3o8 PRACTICAL ELECTRICITY
revolution (10 units). On being tested by passing 5 amperes
at 200 volts through the meter, the disc makes 60 revolutions in
46 seconds. Find the error of the meter.
Answer. — Meter reads 1-3 per cent. slow.
Example 113. — A quantity meter correctly adjusted for a
pressure of 220 volts is used temporarily on a 100 volt circuit ;
what multiplier must be employed to convert its readings into
true Board of Trade units ?
100
Answer. , or 0-4545.
129. Electric Transmission or Energy. — In order to maintain a
steady electric current we must have a closed electric circuit such
as K L M N K (Fig. 187) , and any complete circuit usually consists
of two essentially distinct parts.
In the one part K L M N the current
flows in the direction in which the
potential diminishes — that is to
say, the potential at K is greater
than that at L, the potential at L
greater than that at M, and so on
— and at every point throughout
this portion of the circuit electric
energy is being turned into heat, or into heat and also into some
other form of energy, such as chemical or mechanical energy. This
part of the circuit corresponds with the overhead telegraph wires
and the telegraph instruments which are placed at the ends of
the wires used to receive the telegraph messages, or it corresponds
with the electric-light insulated copper mains under the streets,
the wires, the glow and arc lamps in the houses, and the electro-
motors used to do work in houses and factories which are supplied
with current from the street mains. And in nearly all the calcu-
lations which have been made in this book hitherto regarding
current, P.D., energy, and power, it is this part of the circuit
K L M N only that we have been dealing with. So in the same
way we might have been studying the flow of water in the water
mains under the streets or in the water pipes in our houses, or
the flow of water along a river where the water moves under the
action of gravity.
The water which produces the stream may be obtained from
a reservoir or an elevated cistern, or from some pond at the top
of a hill ; but, unless there be some contrivance for keeping the
reservoir filled by raising the water from a low level to a high
level against the action of gravity, the reservoir will run dry
TRANSMISSION OF ENERGY 309
and the water stream will cease. Hence to maintain a continuous
stream, the water must continuously, or at any rate from time
to time, be carried up in buckets, or be raised by some form of
pump, or by the evaporating power of some hot body like the
sun ; in fact, as much work must be done on the water in raising
it as it does in its descent through the pipes or along the river bed.
So in the same way in some part N K (Fig. 187) of any complete
electric circuit there must be some apparatus for raising the
electricity from a low to a high potential, and the energy which this
apparatus thus puts into the electric circuit must be withdrawn by
the apparatus from some outside store of energy. In the sending
of currents to produce telegraphic signals, or to ring an electric
bell, the battery forms the pump which raises the electricity from
a low to a higher potential as the current passes through it,
and the chemicals placed in the battery constitute the store
of energy on which the battery draws ; while in the sending of a
current to produce the electric light or to work electromotors
in a town, the dynamo at the "Electric Light Station" is the
pump, and the coal in the bunkers at the " Generating Station,"
which is used to generate steam, is the store of energy on which
the dynamo indirectly draws, through the medium of the steam
engine and boiler.
A complete electric circuit which includes an electromotor
is something like one of the pipes of the London Hydraulic
Company under the streets with a pump at one end and a water-
motor at the other. The pump takes energy from some outside
source and gives it to the water, this energy is partly wasted in
heating the running water and the pipes, in consequence of friction,
but the greater part of the energy is given out by the water to
the water-motor at the other end of the pipe. Here the pipe
corresponds with the electrical conductor, the pump with the bat-
tery or the dynamo, and the water-motor with the electromotor.
There is, no doubt, an important difference in the two cases, the
water which flows out through the water-motor at one end of
the pipe need not be immediately returned to the pump at the
other end, indeed it may not be the same water at all which is
pumped up again by the pump to maintain the water stream,
whereas in the electric circuit the same electricity is regarded as
flowing round and round the circuit. But there is this important
similarity, that, just as the water is not the energy, so electricity
is not energy, in spite of erroneous statements that have been
sometimes made to the contrary. Pressure is given to the water
by the pump at one end of the pipe, and pressure is given out
by the water to the motor at the other end, so potential is given
3io PRACTICAL ELECTRICITY
to the electricity by the battery at one end of the wire, and
potential is lost by the electricity at the other end of the wire,
where the electricity flows through the electromotor.
Probably the closest analogy with the electric transmission of
energy is the driving of one pulley by another by means of an
endless belt (Fig. 188). Energy is put into the belt as it passes
round the driving pulley plt energy is given out by the belt as
it passes round the driven pulley P2. The running belt corre-
sponds with the electric current, the driving pulley PJ with the
battery or dynamo, and the driven pulley P2, with the electro-
motor. The model (Fig. 189) shows in a rough symbolical way
what takes place in
the transmission of
energy with pressure-
water, compressed air,
an endless belt, or elec-
tricity. The working
stuff, water, air, belt,
Fig. 188.— Transmission of Power with an Endless Belt. °r electricity, IS first
raised in pressure, and
has energy given to it symbolised by the ball, B, being raised
in the carrier c through the height N K against the action of gravity;
the ball then gradually loses pressure (or height) as it proceeds
along the tube or wire K L, which conveys it to the other end of
the system, shown by the ball falling as it proceeds from K to L,
and the energy thus lost is spent in heating the tube or wire. At
the other end there is a great drop of pressure as the ball falls.,
in the carrier c', through L M, corresponding with a transference
of energy to the motor m which drives a little air-fan, and finally
the ball comes back along the return pipe or wire M N, losing, as
it returns, all that remains of the energy given to it initially in
the pump, or elevator at N K. The ball has, in fact, come back
to its original level.
If the circuit external to the battery is simply a resistance
containing no electromotor nor electrolytic cell, then the circuit
is analogous with the model seen in Fig. 190, the balls B, B falling
by gravity along the rails K L M N, and being raised against the
action of gravity through the height N K. The balls are lifted by
their being picked up by the hooks attached to the endless belt
b b, the right-hand side of which is made to rise continuously by
the handle H being turned.
There is another way of transmitting energy through a pipe
which is wholly different from the methods previously consideried,
and that is by means of coal gas, but in this case the quality of
MECHANICAL ANALOGIES
Ihe material sent through the pipe and not its pressure is the
important consideration. The energy contained in coal gas is not
pressure-energy, but chemical energy ; therefore, as long as the
pressure of the gas is sufficient to make it come out of the pipe at
a suitable rate, it does not matter, as far as the amount of energy
contained in a given weight of gas is concerned, whether the
pressure be small or large. But the chemical constitution of the
coal gas is of great
importance. On the
other hand, when
energy is trans-
mitted by water, or
by air or electricity,
the pressure is as
important a factor
in estimating the
amount of energy
delivered as the
quantity of the
working substance.
Apart from the
action of the tide,
water at sea -level is
quite useless for
working machinery,
no matter how
much water be available, so also, electricity at zero potential
is useless for working an electromotor or producing an electric
light. It is, therefore, all important to the user of the
water supplied by the London Hydraulic Company whether its
pressure is 750 pounds per square inch or 500 pounds per square
inch, but it is of no importance to him whether the water be
ordinary river water or be chemically pure.
Hence, while practically no restriction is imposed by law
on the pressure that the Gas Companies must maintain in the
gas as supplied to a house, the public Electric Light Companies
are prohibited by law from allowing the P.D. between the electric
light mains, where they join the house mains, from varying more
than 4 per cent, from the standard pressure.
This fundamental difference between the transmission of energy
by coal gas and by electricity must be fully grasped, for it is
probably a want of appreciation of this important difference
that has led people to make such erroneous statements as that
electricity is a form of energy.
Fig. 189.— Mechanical Model illustrating the Transmission of
Energy from a Generator, N K, to a Motor, L M.
312 PRACTICAL ELECTRICITY
130. Power Developed by a Current Generator. — If I be the
current in amperes flowing round the circuit K L M N K (Fig.
187), and if V be the P.D. in volts between the points K and N,
the work done per second on the part of the circuit K L M N
equals / V joules. In addition, if the resistance of the portion
of the circuit between N and K be Rb ohms, the current will do
work in heating this resistance at the rate of IzRb joules per
second. Hence, the total power developed by the current equals
(/7+/2#&)watts. (45)
Now, from the conservation of energy it follows that the work
done per second by the current on the circuit must equal the work
IL
Fig. 190. — Mechanical Model illustrating an Electric Circuit composed of a Current
•Generator and an External Resistance.
done per second on the current by the apparatus between N
and K, which converts some form of energy into electric energy.
Therefore, whatever be the construction of this apparatus, the
rate at which the transformation of energy takes place in it, the
rate, in fact, at which it introduces electric energy into the
circuit, must equal
There are three distinct classes of apparatus that may be
employed for introducing electric energy into a circuit, viz. : —
(1) A battery, which transforms chemical energy into electric
energy ;
(2) A " the* "mo-pile," which transforms heat into electric
energy ;
(3) A dynamo, which transforms mechanical energy into
electric energy ;
and in all cases, whether the current generator be of the battery,
thermo-pile, or dynamo type, the rate at which the current
generator withdraws energy from some outside source and
introduces it into the electric circuit equals
(IV+ I2Rb) watts, or / (V + IRb) ;
E.M.F. AND P.O. OF BATTERY 313
where the quantity (V + IRb) is called the E.M.F. of the
generator.
131. Connection between the E.M.F. of a Battery, the P.D.
between its Terminals, the Resistance, and the Current. — If E
be the E.M.F. of a battery in volts, Rb its resistance in ohms, V
the P.D. between its terminals in volts, R the external resistance
in ohms, and / the current in amperes produced by the battery,
we have from the last section
E=V+IRb; (46)
therefore, since V equals 7 R, we have
or = + b.
These equations can be most conveniently written in the
following forms : —
V = E - IRb.
V- R E
iT~I E>~ •
K+ Kb
V
and as /= — ,
K
From the last equation it follows, since E is a constant for a
given battery, that when R is very large I is very small, and
from the first equation we see that when / is very small, V is
equal to E. Hence to find the E.M.F. of a battery we must measure
the P.D. between the terminals when the battery is sending no current
at all, or but an extremely small one. A voltmeter whose resistance
is very high compared with that of the battery must, therefore,
be used in measuring E, and the only current that the battery is
allowed to send must be that passing through the voltmeter.
If a current / amperes be sent through a battery of resistance
Rb ohms, in the direction opposed to that in which the battery
would itself send a current, then the P.D. of V volts maintained
between the battery terminals has to send this current against
the battery resistance Rb, as well as to overcome the E.M.F. of
the battery, say E volts. Hence in this case
V = E+IRb. (460)
Example 114. — A battery having an E.M.F. of 15 volts, and
an internal resistance of 25 ohms, is sending a current through
314 PRACTICAL ELECTRICITY
an external resistance of 5 ohms ; what is the P.D. between the
battery terminals ? Answer. — 2\ volts.
Example 115. — What current must the battery in the last
question send so that its terminal P.D. may be 7-5 volts ?
Answer. — 0-3 ampere.
Example 116. — The P.D. between the terminals of a battery
is 15 volts when the battery is sending a current of 2 amperes,
and 12 volts when the current is 3 amperes ; what is its internal
resistance ?
If E be the unknown E.M.F. of the battery, and R^ its resist-
ance,
we have 15 = E — zR^,
also 12 = E-^Rb,
or Rb — 3 ohms. Answer. — 3 ohms.
Example 117. — A battery having an E.M.F. of 55 volts, and
an internal resistance of 0-25 ohm, is sending a current of -20
amperes through an external resistance. How many watts are
spent in the external resistance, and in the battery itself ?
Answer. — The total watts developed are 20 x 55, or 1,100.
The watts taken by the battery itself, due to its resistance, are
2O2 x 0*25, or 100.
Hence, the watts spent in the external circuit are 1,000.
Example 118. — A battery having an E.M.F. of 2-2 volts, and
a resistance of 0-18 ohm, is opposing a current sent through it
by a more powerful battery. If the current passing through it
is 15 amperes, what is the P.D. between its terminals ?
Since, generally, V = E + IR^,
we have V = 2-2 + 15 x 0-18 ;
.'. V = 4-g.
Answer. — 4-9 volts.
132. Electromotive Force of any Current Generator. — If Rb
be the internal resistance of any current generator, and V be
the P.D. in volts between its terminals, when the current that
the generator is producing, or is helping to produce, is / amperes,
it is customary to call the expression
(V+ I Revolts,
the E.M.F. of the generator, even when the expression is not
independent of the value of I. In such a case the E.M.F. of the
generator is not a constant, as it is very approximately in the
case of a battery, but varies with the current passing, and it must
then be regarded merely as a name for the value that (V + I Rb)
E.M.F. OF ANY GENERATOR 315
may happen to have. A dynamo is an example of a very impor-
tant type of current generator, the E.M.F. of which often varies
greatly with the current passing, and the name E.M.F. which,
like the name resistance, originally came into existence to desig-
nate a constant property which was not altered by varying the
current, is now used in an extended sense in connection with a
dynamo, as is the name " resistance " when speaking of the
apparent resistance of the electric arc (see Section 84).
When the E.M.F. of a current generator varies with the current,
we cannot find its value, as we did in the case of a battery, by
stopping the current and measuring the P.D. between the
terminals of the generator, since the stoppage of the current
would alter the value of the thing to be measured. The values
of V and / can, however, be measured at any moment by means
of a voltmeter and an ammeter, and if the generator be, for
example, a dynamo, whose resistance is practically independent
of its E.M.F. and of the current passing (except in so far as the
current warms the coils of the machine), we can stop the rotation
of the armature, which reduces the E.M.F. to nought, and then
measure R^, the resistance of the dynamo, by means of a Wheat-
stone's bridge or ammeter and voltmeter, as we would measure
the resistance of any other coil of wire.
In Section 130 it was shown that the power developed by a
current generator equalled
(IV+I2Rb) watts,
or I (V + IRb) watts,
therefore, if we decide in all cases to call the expression (V + IR^)
the E.M.F. of the generator, whether it be constant and inde-
pendent of the current or not, it follows that the electric power
developed by any current generator equals the product of the current,
that is flowing, into the E.M.F. of the generator at the time. Hence,
we may define the E.M.F. of any current generator in volts as
the ratio which the electric power developed by the generator, in watts,
bears to the current flowing through it, in amperes, this ratio being
a constant in the case of a good battery, but varying greatly
with the current in the case of other types of current generators,
such as dynamos. We then have
E-P
T
or P = IE.
133. Power Absorbed in the Circuit Exterior to the Generator ;
Back E.M.F. — When power is given by a current of / amperes
316 PRACTICAL ELECTRICITY
to a circuit between the ends of which a P.D. of V volts is main-
tained, the power so given equals IV watts. Of this a portion,
I2R watts, will be spent in heating the circuit where R is its
resistance in ohms, and if
the circuit acts as a simple resistance, the whole of the electric
energy given to it being converted directly into heat.
If, however, no thermo-pile be in circuit, and if
there must be some apparatus in the circuit which transforms
electric energy into some form of energy other than heat, and the
rate at which this transformation takes place equals
IV -I2R.
Two classes of apparatus may be employed for removing
electric energy from a circuit without directly converting it into
heat, viz. :—
(1) An electromotor ; *
(2) A cell, or battery, placed in the circuit so that its E.M.F.
opposes the current,
and we know from formula 460 that, if the E.M.F. of an opposing
cell has a constant value of E volts, then
E=V-IR; (47)
so that (IV — I2R), which represents the rate at which electric
energy is withdrawn from the circuit by the cell and not converted
into heat, equals
IE,
where E is the " back E.M.F/' in the circuit, or the E.M.F. of
the cell opposing the current.
When there is an electromotor, or thermo-pile in the circuit,
the expression (V — IR) will not usually be a constant and
independent of the current, as it is in the case of a good cell,
but we are led by analogy to call the expression (V — IR) in all
cases the back E.M.F. in the circuit, whether it be constant and
independent of the current or not. So that in all cases, apart
from the heating due to resistance, the rate of conversion of
electric energy in a circuit into some other form of energy, equals
the product of the current into the back E.M.F. in the circuit at
the time ; or we may define the back E.M.F. of any apparatus,
in volts, as the ratio which the rate of conversion, in watts, of electric
* The type of electromotor dealt with throughout in this chapter is
the "series," or single circuit electromotor, having its armature and
the field magnet in series with the main circuit.
BACK E.M.F. 3i7
energy into some other form of energy bears to the current, in amperes,
flowing through the apparatus.
If the back E.M.F. is independent of the current, when, for
example, it is produced by a battery which is inserted in the
circuit so as to oppose the current, we can find its value by stop-
ping the current and measuring the P.D. between the ends of
the circuit containing the back E.M.F. When, however, the back
E.M.F. varies with the current while the resistance of the appara-
tus producing it does not, as, for example, in the case of a motor,
the value of the back E.M.F. can be ascertained at any moment
by taking simultaneous observations of a voltmeter and ammeter,
to determine the values of V and I, then, having stopped the
rotation of the armature of the motor to reduce the back E.M.F.
to nought, the resistance of the motor, R, can be measured with
a Wheatstone's bridge, or in any other convenient way.
When there is a back E.M.F. of E volts in a circuit of resistance
R ohms, and between the ends of which a P.D. of V volts is
maintained, the current
T V~E
1= — 5 — amperes.
K
If, now, R and V be kept constant, and E be increased, / will
diminish ; when E becomes equal to V the current will be
nought ; when E is made larger than V the current becomes
negative, the change of sign meaning that the current begins
to flow in the opposite direction ; and the apparatus that
previously had a back E.M.F., and was withdrawing electric
energy from the circuit and transforming it into some other form
of energy, begins to act as a generator, exerting a forward E.M.F.
and introduces electric energy into the circuit.
Example 119. — A current generator having a resistance of
0-3 ohm, maintains a P.D. of 100 volts between its terminals
when producing a current of 45 amperes. What is its E.M.F. ?
Answer. — 113^ volt.
Example 120. — A current generator has an E.M.F. of 67 volts,
and maintains a P.D. of 63 volts between its terminals when it
is producing a current of 12 amperes. What will be the current
when the E.M.F. is 105 volts and the terminal P.D. 98 volts ?
Answer. — If the resistance of the generator is constant the
difference between the E.M.F. and the terminal P.D. is propor-
tional to the current, therefore the required current is J X 12 or
21 amperes.
Example 121. — A battery of 3 cells in series, each having 1-08
volts E.M.F., is joined up in circuit with two lead plates immersed
3i8 PRACTICAL ELECTRICITY
in dilute sulphuric acid. The resistance of the whole circuit,
including the battery and the lead cell, is 2-7 ohms, and the cur-
rent is found to be 0-385 ampere. What is the back E.M.F.
of the lead cell ? Answer. — 2-2 volts.
Example 122. — A battery sends current through a cell consist-
ing of two lead plates in dilute sulphuric acid, the cell having a
back E.M.F. of 2 volts. What is the resistance of the cell if the
P.D. between the terminals is 5 volts and the current 1-5
amperes ? Answer. — 2 ohms.
Example 123. — The resistance of a motor is 0-24 ohm, and when
a P.D. of 83 volts is maintained between its terminals a current
of 25 amperes passes. What is the back E.M.F. of the motor ?
Answer. — 77 volts.
Example 124. — If the resistance of a motor is 1-2 ohms, and
when a P.D. of 100 volts is maintained between its terminals it
runs at such a speed that its back E.M.F. is 91 volts, what is the
current flowing through the motor ? Answer. — 7^ amperes.
Example 125. — A current of 30 amperes is flowing through a
motor of J ohm resistance, and it is running at such a speed that
its back E.M.F. is 76 volts. What is the P.D. that is maintained
between the motor terminals ? Answer. — 91 volts.
134. Distribution of Power in an Electric Circuit. — When a
current generator sends a current of / amperes through a simple
circuit consisting of several resistances in series, the power spent
in any part of the circuits is given by I2 R, where R is the resist-
ance of the part considered. We therefore see that in such a cir-
cuit the power is distributed amongst the several parts, in propor-
tion to the resistances of those parts. Calling the resistance of the
generator R^ and that of the remaining parts Rv R2, Rs, etc., the
electric power spent in heating the generator is I2Rb, and that in
the parts Rlf R2, R3, etc., is I*Rlt I2R2, I2R3, etc., respectively.
If the total resistance of the circuit be R', the total power is
I2R', and the fraction of the total power used in a part of the
circuit, say, R2, is given by
This may be also written
- -2
TR'OTJT'
CIRCUIT TAKING MAXIMUM POWER 319
where F2 is the P.D. between the terminals of the part of the
circuit of resistance R2, and E is the £.M.F. of the generator.
When the circuit contains an electromotor or battery of back
E.M.F., E', the effective E.M.F., in the circuit is
E-E'y
•p _ -pi
and the current I — — — — (48)
K
The rate at which electric energy is transformed into some
other kind of energy, mechanical or chemical, will be
IE',
or^'.£'. (49)
135. External Circuit that Receives Maximum Po'wer from a
Given Current Generator. — Let E be the E.M.F. of the current
generator in volts, and R^ its resistance in ohms, then, if / is the
current in amperes produced when the terminals of the generator
are connected to the ends of some external circuit, and if P is the
power in watts given to this external circuit, Pl the electric
power in watts produced in the generator, and P2 the power
in watts wasted in heating the generator,
P, = IE,
(50)
in all cases.
The change, however, produced in the value of P by varying
the external circuit so as to alter the value of /, will depend on
whether the values of E and R^ are constant and independent of
the value of /, or whether one or both vary with /. If the
generator be one having a fixed E.M.F. and resistance, an examin-
ation of the change of the value of P with a variation in the cur-
rent, /, is quite simple. For when the external circuit is so selected
that / is very small, then P is obviously very small ; if, now, the
circuit be gradually altered so as to make / increase, then P
increases ; on the other hand, when / has nearly its maximum
77
value, viz. -— , which is, of course, attained on the generator being
Rb
short-circuited, P is very small again. As I is continuously
increased there must, therefore, be some value of / at which P
ceases to increase and begins to diminish, or, in other words, there
must be some value of / which makes the expression just given
for P a maximum.
320
PRACTICAL ELECTRICITY
0-36
that
/
s
To ascertain this value of I we may employ various methods ;
for example, we may give arbitrary values to E and R^, plot a
curve connecting P and 7, and find out by inspection the approx-
imate value of 7 for which P is a maximum. Such a curve is seen
in Fig. 191, the values of 2 volts and 3 ohms having been arbi-
trarily given to E and R^ respectively in calculating the values of
the expression for P. From this curve we see that P is a maxi-
mum for some value
of 7 between 0-32
ampere and
ampere, and
.this value of
somewhat nearer
0-32 than 0-36 am-
pere. If the curve
be drawn on a much
larger scale, so that
the value of 7 that
makes P a maxi-
mum can be read
off with still greater
accuracy, it is found
that this value of 7
is 0-33 or J ampere.
Now J ampere is
half | ampere,which
is the current that
the generator would
produce if short-circuited, and the same result would be
arrived at whatever values were given to E and Rb ; there-
fore, generally, we may conclude that the external circuit
which receives maximum power from a current generator, of
fixed E.M.F. and resistance, is the circuit which makes the
current half as great as it would be if the generator were short-
circuited.
The following is another way of obtaining the same result : —
Since P = 7 (E - IRb), formula (50),
or
Fig. 191. — Curve showing the Value of the Current that gives
;the Maximum Power to an External Circuit.
IE\.
E*
therefore, subtracting and adding we have
CONDITIONS FOR MAXIMUM POWER 321
£= (l*
- ~y
Rb
--('-
2RbJ
CE \ 2
/ - — =r— J is a square it can never be negative,
2 J\b j
p
therefore — - will be a maximum when
The above relations may also be written
2 IRb = E,
or IRb=(E-IRb).
Now 7.R& is the voltage required to send a current / through
the generator, and (E — Rb) the P.D. on the external circuit, so
we see that when maximum power is given to the external circuit,
half the E.M.F. of the generator is used in the generator and half
in the external circuit.
£
Since I = „ " _, we see that
Kfr + jR.
when I =
R = Rb, (52)
or the resistance of the external circuit which receives maximum
power from a given generator of fixed E.M.F. and resistance, is
equal to the internal resistance. Fig. 192 shows a curve between
P and R plotted for the case E = j volts and Rb= 2-5 ohms,
from which it will be seen that P is maximum when R = 2-5.
Further, when
E
Now the maximum power that the generator can produce is
— . E, for — - is the maximum current, or short-circuit current.
Hence, the maximum power the generator can develop is — - ,
v
322
PRACTICAL ELECTRICITY
from which we conclude that the greatest power a generator of
fixed E.M.F. and resistance can give to an external circuit is one
quarter the power which the generator would develop if short-cir-
cuited.
Should the external circuit include another generator of E.M.F.,
Ef, the current will be given by
, E+E'
where R' is the resistance of the external circuit, and the power
Fig. 192. —Curve connecting the Power received by an External Circuit and
the Resistance of that Circuit,
given to the external circuit by the primary generator will be
I (E— IRb) as before. This, as shown previously, has a maxi-
£
mum value when 7 = — ^—, so the condition for maximum power
2Kb
in the external circuit is
Rb + R' 2Rbf
or E'
2Kb
or
(54)
(55)
CONDITIONS FOR MAXIMUM POWER 323
Similarly, if an electromotor or battery of back E.M.F., E' be
in the external circuit, the value of E' which permits of maxi-
mum power being given to the external circuit of resistance R ', is
E'=^.E. (56)
and when both E and E' are fixed and R' alone is variable we have
Rf = E~EE> ' Rb' (57)
It is to be observed that the preceding results are all generally
true whatever be the nature of that portion of the external
circuit which we desire shall receive maximum power. For
example, the reasoning would be exactly the same whether the
portion of the external circuit under consideration were composed
of a variable resistance, or whether it contained in addition a
forward E.M.F. produced by some current generator that could
be altered, or a back E.M.F. produced by some electrolytic cell,
or by a running electromotor, the E.M.F. of which could be
adjusted to bring the current to the required value.
From what precedes, then, we may conclude : —
(1) If an external circuit be a simple resistance of R' ohms, then
in order that it may receive maximum power from a generator
having a fixed E.M.F. of E volts and a fixed resistance of Rb ohms
R' must equal Rb.
(2) If the external circuit contain in addition a forward E.M.F.
of E' volts,
E+E' E
— must equal
2Rb'
_. , R — Rh ,->
or E must equal - x E.
2 Rb
(3) If it contain instead a back E.M.F. of E' volts,
E' must equal -^—. — — x E.
2Rb
If we desire that a current generator of fixed E.M.F. and re-
sistance shall give maximum power to a portion of an external
circuit — for example, if the generator be connected by long leads
of fixed resistance RI ohms to a motor or to lamps at a distance,
and we desire to arrange the motor or the lamps so that they
shall receive the maximum power — then the fixed resistance of
the leads must be added to the fixed resistance of the generator ;
hence for Rb in what precedes we must substitute Rb + RI.
324 PRACTICAL ELECTRICITY
Example 126. — What is the maximum horse-power that can
be given to any external circuit by a battery of 50 cells in series,
each having a resistance of 0-05 ohm and an E.M.F. of 2-2 volts ?
Answer. — The maximum power will be one-quarter of the
power which the battery would develop if short-circuited, on
the assumption that short-circuiting the battery did not affect
its E.M.F. or resistance. Therefore the maximum power that
can be given to any external circuit equals
I 50 X 2-2
- X - - X 50 X 2-2, or 1,210 watts,
4 50 x 0-05
which equals 1-622 horse-power.
Example 127. — How many glow lamps, each requiring a current
of J ampere and a P.D. of 100 volts between its terminals to
make it glow properly, can be used with the above battery of
cells, and how should the lamps be arranged ?
Answer. — In order that the battery may give maximum
power to the lamps, the lamps, as they contain no forward or back
E.M.F., must be grouped so that the resistance of the group equals
the resistance of the battery. The latter is 2-5 ohms, while that
TOO
of one lamp is — , or 300 ohms ; hence the lamps must be placed
in parallel, and, if p be the number of lamps arranged in parallel,
the lamps will receive maximum power when
.
that is, when p = 120.
It does not follow, however, that 120 lamps can be used in
parallel and each receive J ampere ; indeed, all the preceding
shows us is that arranging lamps in parallel up to the number
of 120 is the method for causing the group of lamps to receive the
maximum power from the battery of 50 cells. To find the actual
number of lamps, p, that can be employed in parallel, each lamp
receiving a current of J ampere, we have
p 50 X 2-2
.*. p = 12,
or twelve is the greatest number of lamps that can be used, and
they must be arranged in parallel.
Example 128. — A current generator having a fixed E.M.F. of
80 volts and a resistance of 0-7 ohm is used to drive an electro-
EXAMPLES 325
motor having a resistance of J ohm, the electromotor being con-
nected with the generator by mains having a resistance of 2
ohms. What should be the back E.M.F. of the motor so that
it will receive the maximum power ?
Answer. ^~ -— x 80, or 32-6 volts.
2 (07 + 2)
Example 129. — What should be the back E.M.F. of the motor
in the last question so that it may develop the greatest mechanical
power ? Answer. — 40 volts.
Example 130. — If a battery of 50 cells in series, each having
an E.M.F. of 2-0 volts and a resistance of 0-05 ohm, be giving the
maximum power to an external circuit, what is the current that
flows, and by how much per cent, will the power given to the
outside circuit be reduced if the circuit be altered so that the
current flowing is diminished by 20 per cent. ?
Answer. — 20 amperes ;
By 4 per cent.
Example 131. — If the external circuit in the last question consist
of a simple resistance, what is the value of this resistance when it
receives maximum power, and by how much per cent, will the
power given to the external circuit be reduced if its resistance is
(a) 50 per cent, smaller, (b) 40 per cent, larger, than that which
corresponds with maximum power ?
Answer. — 2-5 ohms;
By 1 1 -i per cent.,
By 2-8 per cent.
Example 132. — A generator having a fixed E.M.F. of 220 volts
drives a motor. What should be the back E.M.F. of the motor
so that it may develop the greatest mechanical power, and by
how much will the power it develops be reduced if the back
E.M.F. be increased by f rds above this value ?
Answer. — no volts ;
By|ths.
136. Arrangement of n Cells to give Maximum Power to an
External Circuit of Fixed Resistance. — Since the power expended
in a resistance R is equal to I2R, the problem resolves itself into
finding the arrangements of the
cells which will produce maxi-
mum current through the circuit.
A number of cells may be
grouped in various ways ; they
may be put all in Series Or all Fig. i93.-Four Cells joined in SerieT
3*6
PRACTICAL ELECTRICITY
Fig. 194. — Four Ceils joined in
Parallel.
in parallel with each other, as shown in
Figs. 193 and 194 respectively, or they
may be arranged partly in series and part
in parallel as shown in Fig. 195. Fig. 196
indicates another of the many possible
groupings of cells.* We will, however,
confine our attention for the present to
" regular " groupings in which all the
cells have equal currents through them. Suppose the cells n
in number are all alike, and each have an E.M.F. E volts and
resistance Rb ohms, and let them be arranged s in series and
p in parallel. Only certain values of s and p are possible,
for to satisfy the assumed condition s p must equal n. The
internal resistance of s cells in series is sRb, and of p cells in
7~>
parallel — , consequently, for a
P
battery with s cells in series and
p in parallel, the internal resist-
. sRh
ance is - -, or
', and the
Fig. 195. — Six Cells joined Three in Series
and Two in Parallel.
E.M.F. is sE.
The current through an external resistance R will, by Ohm's
Law, be
sE
I = -
or =
(58)
,
n s
As E is constant, 7 will be a maximum when the denominator is
a minimum. This occurs when the two terms are equal to ead
other, for
n * s/ * n
* As a general rule, cells having different E.M.Fs. should not be con-
nected in parallel.
GROUPINGS OF CELLS
327
R
Fig. 196. — Mixed Grouping of Cells.
and the right hand side is a minimum when
i.e., when
or
-p-
But — - is the internal resistance of the battery when arranged s
P
46 8 10 12
Values of 5
Fig. 197. — Curve connecting the Current and the Number of Cells in Series when the
total number of Cells and the External Resistance are fixed.
cells in series and p in parallel, hence the current through the
circuit is a maximum when the cells are grouped in such a way
328 PRACTICAL ELECTRICITY
(if possible) that the internal resistance of the battery is equal to
the resistance of the external circuit.
It may happen that no regular grouping of the cells may make
the internal resistance equal to the external resistance, in which
case the two possible arrangements which give internal resist-
ances nearest to R should be tried to determine which of the two
gives the largest current. Usually if the two arrangements give
values of — - equally near to R the one with the largest number
of cells in series gives the largest current.*
Fig. 197 is a curve showing the relation between s, the number
of cells in series, and the current in amperes, produced when
n = 12, E = i, Rb = i, and R = 0-5, from which it will be seen
that s = 2 gives the best practical arrangement.
Example 133. — What arrangement of 24 cells, each having
a resistance of 0-47 ohm, will send the maximum current
through an external resistance of 1-2 ohms ?
We have s p = 24
and Rb = 0-47,
also, when the cells are arranged to send the greatest current
through the external circuit,
$Rb -1-2
— - — L Z,
P
hence s = 7-83,
and p = 3*06.
Answer. — 8 cells should be placed in series and 3 in parallel.
Example 134. — What is the maximum current that can be
sent by 100 cells, each of 1-4 volt E.M.F. and 3 ohms resist-
ance, through an external resistance of 20 ohms ?
Answer. — 0-904 ampere.
Example 135. — What is the maximum current that can be
sent by 80 such cells through the same resistance ?
Answer. — 0-800 ampere.
Example 136. — Would it be possible to arrange 48 Grove's
cells, each having an E.M.F. of 1-87 volts, and a resistance of
0-14 ohm, so as to develop J a horse-power in an external
resistance of 0*1 ohm ?
We have s p = 48,
and Rfr = 0-14 ;
* Problems of this nature are treated in detail in a small book on " The
Grouping of Electric Cells," by W. F. Dunton.
GROUPINGS OF CELLS 329
also, when the cells are arranged to, give the greatest power to
the external circuit,
P
s = 6
and p = 8.
With this arrangement of cells the current will be 54-7
amperes ; consequently, the power developed in the external
circuit will be about 299 watts, which is about 0-4 of a horse-
power.
Answer. — It is not possible to develop \ a horse-power in the
external circuit in question with any arrangement of the
particular cells ; but if they be placed 6 in series and 8 in
parallel, the power given to the external circuit will be about
0-4 of a horse-power.
Another way of treating the problem is to find the maximum
E2
power that one cell can give to an external circuit, viz., — — , and
4^b
multiply this by the number of cells. We then have
Total external power = — - - — watts.
4 x 0-14
= 300 watts,
= 0-40 horse-power.
This shows that half a horse-power cannot be obtained from the
battery.
137. Minimum Number of Cells required to give a Fixed
Amount of Power to a given External Circuit. — This problem may
be approached in two ways. In the first place we may find the
maximum power P one cell can give to an external circuit, and
divide the required power P' by P. This will give n, the mini-
mum number of cells. From formula (53), Section 135,
«P=P',
•-. -^ (59)
Secondly, since the external circuit is given, its resistance R is
known, and the power spent in the circuit is
or
330 PRACTICAL ELECTRICITY
The problem, therefore, resolves itself into finding the minimum
number of cells required to send a current of / amperes through
a resistance of R ohms. When the number of cells is a minimum,
then each one cell must be giving out as much power as possible,
and its P.D. must therefore be half its E.M.F. If we have s cells
in series, the total E.M.F. is sE, and the P.D. = — .
2
Hence IR = — ,
2V
E'
The conditions also necessitate the internal resistance being equal
to the external resistance :
P
or
/. n,orsp= — ,
2lR
\ E J R
4P'Rb
= - , as before.
The second method of treatment enables us to find the arrange
ment of cells required,
2lR
for s =
E
/P7 R
= 2V P -P'
7^ £
= 2
E
(60)
EXAMPLES ON CELLS 33i
Example 137. — It is required, by .means of cells each having
an E.M.F. of 1-8 volts and a resistance of 0-3 ohm, to maintain a
terminal P.D. of 12 volts when a current of 8 amperes is flowing.
What is the least number of cells that must be used, and how
should they be arranged ?
2 X 12
and p = = 2.6
£2
8
Take, therefore, p equal to 3, and recalculate s from the equation
which gives s = 12.
Answer. — 36 cells, 12 in series and 3 in parallel.
Example 138. — What is the least number of Daniell's cells,
each having an E.M.F. of i-i volts and a resistance of 0-5 ohm,
that will send a current of 4 amperes through a resistance of
i ohm ?
Answer. — 27 cells, arranged 9 in series and 3 in parallel, will
send a current of 3-96 amperes through the external circuit ;
while 28 cells, arranged 7 in series and 4 in parallel, will send a
current of 4-1 amperes through the external circuit.
Example 139. — A circuit is to receive 250 watts at a pressure
of 20 volts from cells having an E.M.F. of 1-5 volts each, and a
resistance of o-i ohm. What is the least number of cells required,
and how should they be arranged ?
The power developed by one cell, if short-circuited, would be
— or 22-^ watts. Hence, when the least number of cells is
o-i '
22"^
used, each cell will give — -, or 5-625, watts to the external
4
2^0
circuit ; and, therefore, at least — |— , or 44-44 cells, are necessary.
2 X 2O
The number that must be placed in series equals - .
or 26-7 ; practically, then, 23 cells in series and 2 in parallel is
what is necessary.
Answer. — 46 cells, 23 in series and 2 in parallel.
Example 140. — 18 glow lamps, each requiring 5 volts and I
ampere to glow properly, are to be used with cells each having
332
PRACTICAL ELECTRICITY
an E.M.F. of 2 volts and a resistance of 0-2 ohm. Calculate the
minimum number of cells required, and give the arrangements of
lamps and cells that may be employed with about that number
of cells.
Answer. — We have R^ = 0-2, P' = 5 x i x 18 = 90, E = 9, ;
hence
_
E*
= n,
and
4xo-2x9O_ ~
~^~ '
or 1 8 is the smallest number of cells necessary.
To give maximum external power, each cell must supply a
current of amperes, i.e., 5 lamps, and have a P.D. of —
2 iv£ 2,
i.e., i volt. As each lamp requires i ampere at a P.D. of 5
volts, the lamps should be arranged not less than 5 in parallel,
and the cells not less than 5 in series. From this it is clear that
the practicable arrangements of lamps and cells requiring a
number of cells not differing much from the minimum are —
Number of Lamps.
Number of Cells.
Total
In Parallel.
In Series.
In Parallel.
In Series.
Cells.
18
I
4
5
20
9
2
2
9
18
6
3
I
J9
T9
Example 141. — It is desired to expend 100 watts for heating pur-
poses in a coil of wire, the current being supplied by cells having
each an E.M.F. of 17 volts and a resistance of 0-3 ohm. What
is the least number of cells that must be employed, and what are
the various resistances that can be given to the coil so that the
required amount of power can be developed in it with the least
number of cells ?
Answer. — The minimum number of cells equals
£2*
,that
4 X 0-3 X IPO
""
or 41-5 ; so that practically 42 cells must be
is,
- /
used. These 42 cells may be arranged either 42, 21, 14, 7, 6,
3, 2, or i in parallel, and the corresponding resistances of the coil
A. ^* O*Q^ "V TOO
must equal - — — , where p has the values just given.
Hence we have —
CELLS FOR LARGE POWERS 333
Number of Cells Resistance of Coil
in Parallel. in Ohms.
42 . . . . . . . . 0-007065
21 . . . . . . . . 0-02826
14 .. .. .. 0-06357
7 •• 0-2543
6 . . . . . . . . 0-3460
3 •• i'385
2 .. .. .. 3-II5
I . . . . . . . . 12-46
138. Importance of Low Resistance and High E.M.F. for
Large Powers. — On examining the equation n = 4 P'R^/'E2 we
observe that n is proportional to Rj, and inversely proportional to
E2. Hence the smaller the internal resistance of the type of cell
used, and the larger its E.M.F., the smaller the number of cells
required. Halving the internal resistance would halve the num-
ber of cells required, whilst doubling the E.M.F. of such would
reduce the number to one quarter. High E.M.F. and low
internal resistance are, therefore, factors of great importance where
cells are required for producing large powers*
139. Modifications Introduced into the Previous Results by
Limitation of the Maximum Current a Cell may Produce. — With
some types of cell, particularly secondary cells or accumulators,
the internal resistance of which is usually very low, the maximum
safe value of the current is not limited by the E.M.F. and resist-
ance of the cell, but by the fact that the plates are liable to disin-
tegrate, if currents exceeding certain values per square foot of
plate surface are permitted to flow. Primary cells polarise
rapidly, and therefore vary in effective E.M.F., if currents approxi-
mating to — — are taken from them, unless R^ is large. A limita-
?«6
tion of the permissible current is consequently necessary in many
cases, and this leads to a modification in the solution of the
problem considered in Sections 135, 136, and 137, when the
j£
maximum current allowable is less than — — -.
zRb
For example the arrangement of a given number of cells n,
to produce the maximum current through a given external resist-
ance, must in such a case be limited by the condition that / must
not exceed p I' , where p is the number of cells in parallel, and
/' the maximum current which may pass through one cell.
* A modified form of two-fluid Bichromate Cell (the Bleeck-Love Cell),
having the very high E.M.F. of 27 volts, was put on the market by the
Silvertown Co. about 1910.
334 PRACTICAL ELECTRICITY
Now in Section 136, we have
n „
T *E P
JL = ~
sRb R nRb | R
= p I' as a maximum ;
gives the smallest permissible number of cells in parallel.
To determine the minimum number of cells required to produce
a current of / amperes and a given terminal P.D. of V volts
^
when /' is less than we have,
2 K
S-r/'Jfc'
F7
/. n=sp=-— — — — -. (63)
1 (b - I Kb)
The above expression gives also the solution to the problem
of the number and arrangement of cells required to supply
a power of IV ( = P) watts to an external circuit. It may also
be written „
and as (pP) is the total current, we have
/' (64)
Example 142. — What arrangement of 20 cells, each having
an E.M.F. of i-i volts, and a resistance of 0-5 ohm, will send the
largest current through an external resistance of 4 ohms, if no
cell is to produce a larger current than I ampere ? What is the
value of this maximum current ?
/ 20 /i-i \ / —
Answer.— p = \J ( 0-5 1 = y 3 ;
therefore, the cells must be arranged 2 in parallel and 10 in series.
The current will be 1-69 amperes.
EFFICIENCY 335
Example 143. — With the cells referred to in the last question,
and with the same condition as to the maximum current a cell
may produce, what is the least number of cells that will maintain
a P.D. of 10 volts between the terminals of an external circuit
when sending a current of 3 amperes through it ?
Answer. — p = 3
10 ,-..
s = - = 16-6 ;
i-i — 0-5
therefore 48 or 51 cells must be employed, the former maintaining
a P.D. of rather less than 10 volts, and the latter a P.D. of more
than 10 volts, when producing a current of 3 amperes.
Example 144. — It is desired to give a power of 125 watts to an
external circuit by means of storage cells, each having an E.M.F.
of 1*9 volts and a resistance of o-oi ohm, on the condition, how-
ever, that a cell may not produce a larger current than 10 amperes.
What is the least number of cells required, how should they
be arranged, and what should be the resistance of the outside
circuit ? I2-
Answer. — sp = - - = 6^94 ;
10 x i '9 — 100 x o'oi
therefore, 7 cells must be used, and as 7 is a prime number the
cells must, for " regular " grouping, be placed all in series or all
in parallel. There will consequently be two values of R, viz., the
highest values given by the equations
- I25> and +K* = I2S
Hence, the external circuit must have a resistance of 1-271 ohms
when the cells are in series, and 0-0259 ohm when the cells are
in parallel.
140. Efficiency. — When, by means of any machine, or contriv-
ance, one form of energy is converted into another form, some
heat is produced ; hence, if heat energy is not the form in which
the energy is required after the conversion, some portion of the
energy which has been used up has been converted into a useless
form as far as the object in question is concerned, and may,
therefore, be regarded as wasted. Consequently, in all cases the
amount of useful energy produced is less than the amount of
energy used up. For example, when a machine is used to do
work there is a waste of energy in the heating of the bearings of
the machine ; if falling water is employed to turn a water wheel
there is in addition waste of energy in the eddies set up in the
water, in the splash of the water against the blades of the wheel
as well as in the friction of the water stream against the sides of
336 PRACTICAL ELECTRICITY
the channel which guides it to the wheel. When oil, wax, gas,
etc., are consumed as illuminants only a very small fraction oi
the available energy is converted into the special form of energy,
called light, which affects the retina of the eye, and the greater
part is wasted in heat, whose action on the eye does not differ
from its action on other parts of the body. Again, in a battery
a certain amount of chemical energy is wasted in the heat pro-
duced by " local action " (see Section 68), which goes on even
when the battery is producing no useful current ; further,
on the battery being used to send a current through some exter-
nal circuit a portion of the chemical energy that is converted
into the electric energy is always wasted in heating the battery
in consequence of its resistance.
The value of any machine or contrivance for effecting a conver-
sion of one form of energy into another depends first on the rate
at which energy in a useful form is evolved by it — that is, the
useful power the machine develops, and which is sometimes
called its " useful activity " — secondly, the value of the contrivance
depends on the ratio of the amount of useful energy produced to
the amount used up in the process, and this ratio is called the
" effioiency."
Efficiency, then, is expressed by a number, less than unity,
such as J, 0-63, 75 per cent., 84 per cent. Sometimes, how-
ever, it is found convenient to employ different units of energy,
or of power, in the numerator and denominator of the fraction
which represents the efficiency. For example, while the true
efficiency of a carbon glow lamp does not generally exceed o-oi
— that is, not more than one -hundredth of the electric energy
supplied to it is converted into light — the efficiency of such a glow
lamp is sometimes spoken of as J candle per watt, meaning that
an electric power of 4 watts must be supplied to the lamp to
produce as much light as is given out by i standard candle.*
When any current generator developing an E.M.F. of E volts
and having a resistance of R^ ohms is sending a current of I am-
peres round any circuit, the ratio which the power in watts given
to the external circuit bears to the electric power in watts de-
veloped in the generator is I (E — IR^)
IE
or •£ (65)
where V is the P.D.
Therefore, this fraction is the efficiency of the generator.
* It is customary to speak of the "efficiency" of a lamp as so many
" watts per candle " ; the word " inefficiency " would be more correct.
EFFICIENCY 337
The name " electrical efficiency " is sometimes given to the
preceding expression to distinguish ' it from the " commercial
efficiency " of the generator, which means the ratio of the useful
electric power it produces to the total power it consumes. Now,
the total power consumed is always greater than the total electric
power the generator produces. For example, if the generator
be a battery some of the chemicals will often be wasted in local
action, or if it be a dynamo there will be power wasted in friction
at the bearings of the machine, etc. Hence the commercial
efficiency of a generator is always less than
I(E-IRb) V
IE E'
Similarly, if E' be the back E.M.F. of a motor in volts, Rm its
resistance in ohms, and / be the current in amperes flowing, the
electrical efficiency of the motor is
IE'
E'
or v>
while its commercial efficiency, or the ratio of the useful mechani-
cal power it produces to the electric power it receives from the
mains, will be less than this, since some of the mechanical power
which the motor produces will be wasted in the friction at its
bearings, as well as in the friction between the rotating commu-
tator and the brushes, etc. The commercial efficiency, however,
of very large well-made dynamos and motors is as high as 96
per cent.
Another useful definition of commercial efficiency is
Output
Input '
Where the word Output means the useful power which the appara-
tus gives out and Input is the power put in. The difference
between the input and the output is converted into some form
of energy (usually heat), other than the form desired, and is there-
fore spoken of as " loss." From the law cf Conservation of
Energy we know that
Output -f loss = Input.
.•. the above expression may be written,
„ . Output
Efficiency == •F— — — ,
Output + loss
Input — loss
or =— ^ .
Input
w
368 PRACTICAL ELECTRICITY
Example 145. — What must be the resistance of a current gener-
ator so that 95 per cent, of the power produced by it shall be given
to the outside circuit, consisting of a simple conductor having a
resistance of 35 ohms ?
We have 35 n = -25-,
35 + Rb 100
if Rb be the resistance of the generator ;
.*. Rb = 1-842.
Answer. — 1*842 ohms.
Example 146. — loj horse-power is spent in driving a dynamo
which maintains a P.D. of 100 volts between its terminals when
it is generating a current of 65 amperes. What is the commercial
efficiency of the machine ?
Answer. — loj horse-power equals loj X 746, or 7646-5 watts,
while a current of 65 amperes at a P.D. of 100 volts equals 6,500
watts ; therefore, the commercial efficiency is 85 per cent.
Example 147. — A motor having i£ ohms resistance develops
a mechanical power of J a horse when a P.D. of 60 volts is main-
tained between its terminals and a current of 9 amperes is sent
through it. What are the electrical and the commercial efficiencies
of the motor ?
Answer. — The power wasted on account of the resistance of the
motor is 121-5 watts, while the power received is 540 watts ;
therefore, the electrical efficiency is — =, or 77-5 per cent.
540
The mechanical power developed is 373 watts ; therefore, the
o/^o
commercial efficiency is 2L2. or 6q-i per cent.
540
Example 148. — An electromotor is required to work a pump
raising water through a height of 120 feet. If 15,000 gallons are
to be raised per day of ten hours, what current will the motor
take at 200 volts' pressure, supposing the " combined efficiency "
of motor and pump to be 60 per cent. ?
A gallon of water weighs 10 Ibs. ; hence the work to be done
in ten hours equals 15,000 X 10 X 120 ft. Ibs., so that the power
exerted in foot pounds per minute equals 30,000. But as 40 per
cent, of the power given to the motor is wasted in the machinery,
the motor must receive — X 30,000 or 50,000 ft. Ibs. per minute,
o
Hence 50,000 = 44-23 X I X 200,
.-. / = 5-65.
Answer. — 5-65 amperes.
EFFICIENCY OF TRANSMISSION 339
Example 149. — If electric energy is supplied by public mains
to a factory at id. per Board of Trade unit, and an electromotor
works with an efficiency of 80 per cent., how much does the energy
used to drive the machinery in the factory cost per horse -power
hour ?
Answer. — One Board of Trade unit equals 1-340 horse-power
hour, and of this 80 per cent, is delivered by the motor to the
machinery ; therefore, 1-072 horse-power hour costs id., or
one horse-power hour costs o-93d.
Example 150. — If a glow lamp gives light equal to 16
candles when a current of 0-21 ampere passes and a P.D. of
100 volts is maintained between its terminals, how many watts
are required per candle ? Answer. — 1-31.
141. Efficiency of Electric Transmission of Energy. — If a
stream of water be used to work a turbine that drives a dynamo
producing a current which- flows through long leads and works an
electromotor at the other end of the leads, the commercial effi-
ciency of the whole arrangement is the ratio of the useful mechani-
cal power developed by the motor at the one end of the system to
the power of the falling water supplied to the turbine at the other.
The whole power given by the falling water to the turbine would
not, however, be available for driving the machinery in a factory
even if the factory were built close to the falling water, for some
of the power will be wasted in the turbine itself ; hence the
" commercial efficiency of transmission " from one end of the sys-
tem to the other may be taken as the ratio that the useful me-
chanical power given out by the distant electromotor bears to
the mechanical power given by the turbine to the dynamo at the
near end. The " electrical efficiency of transmission " in such a
case is the ratio that the electric power which is converted
into mechanical power in the motor bears to the electric power
which is produced in the dynamo, or the electrical efficiency of
transmission equals
JF' F'
if OI E ' (66)
where E is the E.M.F. of the dynamo and E' that of the motor
If Rd, RI, and Rm be the resistances in ohms of the dynamo,
the leads, and the motor respectively,
. E -E' E-E'
when R equals the total resistance of the circuit, therefore,
eliminating E' , the electrical efficiency of transmission equals
340 PRACTICAL ELECTRICITY
(67)
Now, whether E, R$, and Rm are constant and independent of
the current, or whether they change their values with the current,
the preceding expression varies from zero when the external
p
circuit is such that / equals ~ (which will happen when the
R
motor is held at rest so that it has no back E.M.F. and acts simply
like a resistance) to unity when the external circuit is such that
/ is zero, provided, of course, that neither R^ nor Rm becomes
extremely large when / becomes very small.
The electrical efficiency of transmission is, therefore, the
greater the smaller is the current. Diminishing the current, how-
ever, diminishes the power developed by the generator unless
its E.M.F. be increased. Similarly, diminishing the current
diminishes the power that can be received by the distant motor
unless its back E.M.F. is increased. Hence, to electrically
transmit a large amount of mechanical 'power over a long distance
with high efficiency we must employ a dynamo producing a large
E.M.F. at the one end and a motor producing a large back E.M.F.
at the other, and the current that flows must be kept small.
For precisely similar reasons, if we desire to employ water to
transmit a large amount of power through a long pipe with high
efficiency, the water must be at a high pressure and the stream
must be small. Hence the London Hydraulic Company use
water at 750 pounds per square inch pressure in their pipes, and
boiler makers employ a pressure of as much as 1,400 pounds'
pressure per square inch with portable tools for riveting, etc.,
by hydraulic pressure.
It is interesting to consider how the E.M.F. of the generator
must increase with the amount of power to be transmitted and
with the resistance of the circuit, in order that the loss of power
due to the resistance of the circuit may not exceed a certain
percentage of the power to be transmitted.
The electric power Pl developed in the „ _ £,
generator - x E watts.
the electric power P/ converted into E —E'
mechanical power by the distant motor = — — — x E' watts,
R
therefore the power PI lost on account of _ £,. 2
the resistance of the circuit = — -J- watts ;
K
EFFICIENCY OF TRANSMISSION 341
hence pl = (Y x R, (68)
so that — - — , the percentage of the power developed in the
generator which is lost on account of the resistance of the circuit,
equals
"»fl«- (69)
Consequently, if this percentage loss is to be a constant, E2 must
increase proportionately to the product of P1 into R.
For example, if we desire to transmit 10,000 watts along a
circuit having a resistance of 2 ohms, and to keep the loss of power
down to 4 per cent.,
/IOO X IO,OOO X 2
E= \/ -
4
= 707 volts,
or the generator must have an E.M.F. of 707 volts.
If in addition to, or instead of, the motor at the other end of
the leads there be some apparatus of resistance R' ohms in which
we wish to develop heat or light, then this resistance R' must not
be included in the preceding expressions, for the heat developed
in this resistance is what we desire shah1 be produced, and there-
fore must not be regarded as energy wasted in heat. For example,
if the arrangement receiving energy at the other end of the leads
be simply a group of glow lamps, having any resistance of R'
ohms, it follows, from what precedes, that the percentage of
the power developed by the generator, which is lost on account
of the resistance of the circuit, equals
ioo-|§ (Rb + RI).
Although the transmission of signals by electricity through
wires many hundreds of miles in length has been successfully
carried on for more than half a century, the history of the electric
transmission of considerable amounts of power is all comprised
within the past forty years. In the following table are given
the results of attempts to accomplish this object, and it is seen
how the employment of higher and higher P.Ds. has enabled
larger and larger amounts of power to be transmitted over longer
and longer distances with increasing efficiencies. During 1919 a
scheme for transmitting 500,000 kilowatts 570 miles in California
was designed, the transmitting pressure being 220,000 volts.
CO
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ELECTRICAL EFFICIENCY 343
142. Connection between Electrical Efficiency of Transmission
and Ratio of the Power Received to the Maximum Power Receiv-
able. — When the current generator has a fixed E.M.F. and
resistance like a battery, we have seen in Section 135, that
whether we desire the whole of the external circuit, or a portion
of the external circuit, to receive maximum power, or whether
we desire that the transformation of electric energy into non-
heat energy shall proceed most rapidly in an electromotor or
electrolytic cell possessing a back E.M.F., the electric power
usefully employed must be equal to the electric power wasted
in heating the circuit, so that the electrical efficiency of trans-
mission must be one-half. If, however, the part of the circuit
under consideration be arranged so that it receives less than the
maximum power receivable from the given current generator,
the electrical efficiency of transmission may be greater than
one-half, and the following calculation gives the connection
between the electrical efficiency of transmission and the ratio of
the power received to the maximum power that could be received.
If E is the fixed E.M.F. of the current generator in volts and R^
its resistance in ohms, Rx the resistance in ohms of the part of the
circuit under consideration, and RI the resistance in ohms of the
leads connecting it with the generator, we have
D
electrical efficiency = ~, where R is the total resistance
K
of the circuit, and
_ power rece
maximum power receivable
~ R
power received _ _ \^ R
4 («-*,)
= 4 efficiency (i — efficiency). (70)
Now this is a quadratic equation connecting the efficiency
with the ratio of the power received to the maximum power
receivable, which ratio we will call r for brevity, therefore each
value of r will be given by two different values of the electrical
efficiency of transmission, the sum of the two values being equal
to unity ; for example, whether the efficiency is J, or f , the
344
PRACTICAL ELECTRICITY
8
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r Electrical efficiency )
: transmission . . j
r Ratio of power]
ceived to maximum I
>wer receivable . . j
0 0
IH P*
wlw
K
preceding equation gives r equal to -J£ .
In the following table, however, which
rv T~>
gives corresponding values of — , or -=£'
h K
the electrical efficiency, and of r, only
the larger value of the efficiency is given
corresponding with any particular value
of r.
From this table we see that when Rx, the
resistance of the part of the circuit under
consideration, is increased until the electri-
cal efficiency of transmission is 75 per cent.,
the power this part of the circuit receives
is | of the possible maximum, also that Rx
may be increased until the electrical
efficiency of transmission is over 85 per
cent, without the power received being
less than J of the possible maximum.
The figures given in the first two lines
of the preceding table are equally true
whether the E.M.F.- and resistance of the
generator are constant, or whether they
vary with the current, but the figures in
the third line, for the ratio of the power
received to the maximum power receivable,
are only true when both the E.M.F. and
the resistance of the generator are con-
stant ; indeed, it may be shown that an
external circuit receives maximum power
from a dynamo when the external resist-
ance is smaller than that of the dynamo,
and when the electrical efficiency is there-
fore less than J.
If the external circuit, instead of being
a simple resistance, contains an apparatus
of resistance Rm ohms and back E.M.F. of
E' volts, and if it receives power through
leads of fixed resistance RI ohms from a
generator having a fixed E.M.F. of E
volts and resistance of R^ ohms, we know
that the ratio which the rate of trans-
formation of electric energy due to the back
E.M.F. in this apparatus bears to the maxi-
mum rate of such transformation, equals
ELECTRICAL EFFICIENCY 345
E-E'
xE
, where R = Rb + RI + Rm,
is*
E-E'
F_/ _&\
E'
and as the electrical efficiency of transmission = — ,
therefore the ratio which the rate of transformation of electrical
energy due to the back E.M.F. bears to the maximum rate of such
transformation, equals
4 x efficiency (i — efficiency).
This is exactly the same equation as was obtained in the
previous case, see (70) above, and therefore must lead to the
same numerical connection between the values of the electrical
efficiency of transmission and the ratio which the rate of trans-
formation of electrical eneigy due to back E.M.F. bears to the
maximum rate of such transformation.
Example 151. — A battery having an E.M.F. of 30 volts and a
resistance of 4 ohms is sending a current through an outside
circuit consisting of leading wires having a resistance of i ohm
and 4 glow lamps arranged in parallel. A P.D. of 12 volts is
maintained between the lamp terminals, and each lamp produces
3j candles. Calculate the number of candles that is produced per
watt, and the percentages of the power generated in the battery
that are given to the lamps and wasted in the battery and leading
wires.
Answer. — The current = — — , or 3-6 amperes.
4 + 1
The power given to the 4 lamps
= 3-6 X 12, or 43-2, watts.
Therefore, as the total illumination is 4 X 3|, or 14, candles,
0-324 candle is produced per watt.
Also the power generated by the battery
— 3*6 X 30, or 108, watts.
The power wasted in the battery
= (3:6)2 X 4, or 51-84, watts.
The power wasted in the leading wires
= (3*6)2 X i, or 12-96, watts.
346 PRACTICAL ELECTRICITY
Therefore, of the 108 watts produced by the battery 43-2
watts, or 40 per cent., is given to the lamps, and 64-8 watts, or
bo per cent., is wasted in heating the battery and the leading
wires.
Example 152. — A dynamo of 0-2 ohm resistance is supplying
current to a group of glow lamps in parallel placed at the ends of
leads having 1-8 ohms resistance. The lamps take 75 amperes,
and a P.D. of 100 volts has to be maintained between their
terminals. If 32 horse- power is spent in driving the dynamo,
what are the electrical and commercial efficiencies of the trans-
mission, and what are the electrical and commercial efficiencies
of the dynamo alone ?
Answer. — 40, 31-4, 94 and 73-8 per cent.
Example 153. — A dynamo haying a resistance of 2j ohms, and
an E.M.F. of 1000 volts, develops 40 horse-power. What may
be the resistance of the leads so that 60 per cent, of the power
developed by the dynamo is delivered to some apparatus at the
other end of the leads ? Answer. — 10-9 ohms.
143. Economy in Electrical Transmission of Energy. — Kelvin's
Law. — The question of efficiency has been dealt with in the pre-
ceding paragraphs, from which it will be seen that the higher
the P.D. and the lower the resistance of the lines the greater
the efficiency of transmission of a given amount of electrical
energy will be. It might therefore be surmised that by using
the highest possible P.D., and lines of very small resistance, maxi-
mum economy would be secured. This, however, is not the case,
for although such an arrangement would give a " high efficiency,"
the previous reasoning takes no account of the cost of the
apparatus and transmission line, or of the serious leakage of
current that takes place from wires when very high P.Ds. are used.
The cost of machines per kilowatt output increases considerably
when the pressure is very large, and the cost of the line would be
greatly increased if wires of large cross-section were used, with a
view to making their resistance small. Very high pressures
enable smaller, and therefore qheaper, conductors to be used
without excessive I2 R loss in the conductors, whilst the cost
of apparatus and loss by leakage are increased. For very long
lines the cost of the line becomes of prime importance, whilst for
short distances the cost of the machinery is the principal item.
It will, therefore, be evident that there will be some particular
pressure and some particular size of wire for which, in a given case,
the total yearly cost, made up of interest on capital, upkeep, and
cost of the energy wasted in the circuit, will be a minimum.
KELVIN'S LAW 347
Transmission at medium pressure is most economical for short
distances, whilst for long distances/ such as those mentioned in
columns 8 — n of the table on page 342, very high pressures are
essential to economy.
Considering the transmission line alone we may notice that
small wires would be cheaper than large ones, both to buy and to
erect, but the energy wasted in heating would be increased by
their use. As energy has a money value, the total yearly cost
of a line (interest on capital, and cost of energy wasted in heating)
will not be a minimum if either very small or very large conductors
be used. In fact, as was shown by the late Lord Kelvin, in 1881,
the total cost will be. least when the interest on capital spent in
erecting the line is equal to the cost of the energy wasted in heating
the line. This is " Kelvin's Law," and is of great commercial
importance.
CHAPTER VIII
QUANTITY AND CAPACITY.
144. Electric Quantity and its Measurement — 145. Ballistic Galvanometer
— 146. Measurement of Quantity by Ballistic Galvanometer — 147.
Correction of Ballistic Galvanometer for Damping — 148. Determina-
tion of Decrement and Logarithmic Decrement — 149. Constant
of a Ballistic Galvanometer — 150. Comparison of Quantities — 151.
Capacity — 152. Condensers : Mechanical Analogies — 153. Units ot
Capacity : Farad : Microfarad — 154. Variation of the Capacity of a
Condenser with the Area of its Coatings and the Distance between
them — 155. Relation between Electrostatic Unit of Capacity and
the Farad — 156. Capacity of Spherical and Plate Air Condensers
in Farads — 157. Capacity of Cylindrical Condensers — 158. Specific
Inductive Capacity — 159. Dielectric Strength of Insulators — 160.
Resistivity of Insulators — 161. Construction of Condenser of Large
Capacity — 162. Condensers for Large P.Ds. ; Leyden Jars— 163.
Comparison of Condensers — 164. Potential Divider — 165. Combined
Capacity of Several Condensers— 166. Charged Condensers are
Stores of Electric Energy, not of Electricity — 167. Energy Wasted
in Charging a Condenser from a Source of Constant P.D. — 168.
Absolute Measurement of Capacity — 169. Measurement of Specific
Inductive Capacity and Resistivity of Insulators — 170. Standard
Air Condensers — 171. Ratio of Electromagnetic and Electrostatic
Units of Quantity — 172. Use of Condensers for Comparing E.M.Fs.
of Cells, or other Current Generators — 173. Condenser Method of
Measuring the Resistance of a Cell.
144. Electric Quantity and its Measurement. — We have already
defined (Section 10) the coulomb or unit of quantity, as the
quantity that flows in one second through a conductor conveying
a current of one ampere, or the " ampere second." When the cur-
rent is a steady one it is quite easy to find the quantity which
passes in a long interval, by observing the current in amperes
and the time in seconds, and multiplying them together, just as
one can calculate the quantity of water that passes a given point
in a channel of known cross-section in a given time by observing
the velocity of the stream and the time, supposing, of course,
the stream to be a steady one. If the electric current be varying
then the quantity may be determined by summation (or integra-
tion). The whole time is supposed to be divided into a very large
number of very small intervals, during any one of which the
current will be practically constant (say / amperes), and the pro-
348
Fig. ig8a.
Needle and Coils
of Simple
Ballistic
Galvanometer.
MEASUREMENT OF QUANTITY 349
duct of the current during that interval and the length of the
interval (81 seconds, say) is the quantity that passes during
the interval. The whole quantity is then expressed by
Q = 2 / 8t coulombs.
To measure such a quantity practically, one way is to pass
it through a voltameter and observe the amount of chemical
decomposition produced, from which the number of coulombs
could be calculated (see Sections 9 and 10). Another way
is to pass the current through a quantity meter, such as
described in Section 128. These methods can, however, only
be used when the quantities are fairly large (say many coulombs),
for the decomposi-
tion produced by
a single coulomb
in the most sen-
sitive voltameter
known is very
small. For ex-
ample, in a silver
voltameter one
coulomb deposits
0-001118 gramme
of silver, and in a
copper voltameter
0-0003286 gramme
Of COpper, Whilst Fig. I98.— Simple Ballistic Galvanometer.
in an acid volta-
meter one coulomb liberates 0-1734 cubic centimetre of mixed
gases at normal temperature and pressure.
145. Ballistic Galvanometer. — To measure small fractions of
a coulomb, especially if the quantity passes in a very short time,
it is evident from the above that some other method must be
used ; and for this purpose galvanometers may be employed.
Instruments intended for this purpose are generally called
" ballistic galvanometers," and should fulfil the following con-
ditions : — (a) the periodic time of vibration should be great
compared with the duration of the current, so that the whole
quantity passes before the needle has moved appreciably from its
zero position ;* (b) the frictional (or damping) forces tending to
bring the needle to rest after displacement, should be small, i.e.,
the needle should make many oscillations before coming to rest.
* A current of short duration is called a " transient current," and the
quantity conveyed by such a current is spoken of as a " charge " or a " dis-
charge."
350 PRACTICAL ELECTRICITY
In other words, the amplitude of one swing should not differ much
from the next.*
An instrument satisfying these conditions fairly well is shown
in Figs. 198, 1980. The needle is formed of a " bell-shaped
magnet " (see Fig. 1980) and a fine glass pointer p serves for
reading the deflection. The magnet is suspended by a single
fibre of silk, so that it hangs centrally in the axis of the two
Fig. 199. — Reflecting Ballistic Galvanometer.
circular coils of wire which form the winding. The coils are
drawn open in Fig. 1980 to show the magnet.
Another ballistic galvanometer intended for measuring very
small quantities is illustrated in Fig. 199. Here the coils
are separated to show the magnets forming the needle. This
needle consists of three bell magnets fixed to a vertical wire,
the magnets being placed so that the upper and lower ones
have their north poles pointing in a direction opposite to that
of the middle magnet. When the magnetic moment of the
one magnet is nearly equal to that of the other two, the system
* The ratio of the amplitude of one swing to the next is called the
" decrement," and the Napierian logarithm of this ratio, the " logarithmic
decrement"
LAMPSTAND AND SCALE 351
Fig. 200.— Lampstand and Scale for Reflecting Galvanometer (Front View).
Fig. 2ooa- Lampstand and Scale (Back View).
An electric lamp is contained within the conical shade.
352
PRACTICAL ELECTRICITY
has little directive force in a uniform magnetic field, and this
lengthens the period of vibration, and increases the sensitiveness.
Such a system of magnets is called an " astatic needle," and has
the great advantage of being little affected by stray magnetic
fields. By placing one of the magnets inside the coil and the
others outside, the forces exerted on the magnets by a current
in the coil all tend to deflect the needle in the same direction.
To magnify the
movement of the
needle caused by the
quantity passing
through the coils, a
mirror is attached to
a wire stem support-
ing the suspended
magnets, and a spot
of light from a lamp
is reflected from this
mirror to a finely
graduated scale
placed several feet
away. By this means
a very small deflec-
tion of the needle can
be measured. A con-
venient form of
lampstand and scale
Fig. zooft.— Reading Telescope and Scale. ]s shown in FigS. 200
and 2000.
Another method of observing such deflection is by viewing
in the mirror the reflection of a scale by the aid of a telescope,
such as indicated in Fig. 2006.
Moving coil galvanometers can often be used with advantage
for ballistic work, and one of the narrow-coil form, devised by the
authors, is illustrated in Fig. 201. The coil M is shown separately
with its mirror L attached, and the mirror is also seen between
F and F in the complete instrument. The coil hangs inside
a brass tube in a narrow gap between the poles of the
magnet A. Tubes containing suspended coils of different resist-
ances are supplied for use with the same magnet ; one of these
is shown at E D H, near the base of the galvanometer. Where
fine adjustment of the zero of the instrument is required, a tan-
gent screw is fitted to the torsion head, as seen in the upper
right hand corner of the figure.
BALLISTIC GALVANOMETER
353
146. Measurement of Quantity by Ballistic Galvanometer. —
Suppose a quantity Q is passed through a tangent galvanometer
in correct adjustment, and that at time t the current has a value
of / amperes, we shall then have
81 being a very short interval of time.
Fig. 201. — Ayrton and Mather's Reflecting Galvanometer.
If the coil has n convolutions of mean radius r centimetres
the strength of field at the centre will be given by
2 TC / n
F =
10 r
(see Sections 30, 35) and this will produce a turning moment on
a small magnet at the centre of FM dyne-centimetres, where
M is the magnetic moment of the needle (Section 26). The
forces exerted on the needle will, at any instant, be proportional
x
354 PRACTICAL ELECTRICITY
to the value of the current at that instant, and if the whole time
during which the current flows is very short, the result is that the
needle receives a sudden twist, or impulse, which produces an
angular momentum Kco (where K is the moment of inertia of the
needle and &> is the angular velocity produced by the impulse)
equal in value to the sum 2 FM 8t*
Hence K co = 2 F MM,
lor
10 r
2 nn M
Q, coulombs,
10 r
= gMQ, (71)
TT n
where £ = 2 —
10 r'
the " constant " of the galvanometer coil, or the strength of field
produced at the centre of the coil by a current of i ampere.
From this we see that the angular velocity o> is proportional
to the quantity Q discharged through the instrument, and if we
can measure this velocity we have then a measure of Q.
When there is no frictional resistance offered to the movement
of the needle, ay can be measured by observing the angle through
which the needle swings before it begins to return towards the
zero position. This is called the "first swing," the " throw " or
the " kick " of the needle. The kinetic energy stored up in the
needle when its angular velocity is w is given by the expression
— , and as the displacement increases, this energy is trans-
formed into potential energy of position. At the instant of
maximum elongation &> = o, and all the kinetic energy has been
converted into the potential form. Frictionless motion being
postulated, no waste of energy occurs, so the potential energy
must equal - — , and the work done against magnetic forces
as the needle moves from the zero position through an angle st
where s is the "first swing " of the needle, is - — .
Now the work done in turning a magnet of moment M through
* The case is analogous with the momentum generated by a force /',
acting as a mass m for a short time 5t, for m v = 2j'5t.
BALLISTIC GALVANOMETER 355
an angle s from its position of equilibrium in a uniform magnetic
field of strength H is
ME (i - cos s),
or 2 MR sin2 -,
2
Hence - = 2MHsm*-,
2
, IMH , s
and co = 2——sm-
I
j
# o> = 2 V MHK sin -,
2
but Kcu= gMQ, from (71)
gMQ - 2 y/MHK sin -,
«=w
HK . s
sin — ,
gM M 2
(72)
As H, g, K, and Af , are constants in a given case, we learn that
the quantity discharged through a galvanometer is proportional
to the sine of half the first swing which it produces, the needle being
controlled by a uniform magnetic field.
Now in Section 27 we have shown that when a magnet of
magnetic movement M, and moment of inertia K oscillates in a
magnetic field of strength H, its time of small vibrations is
so the above equation may be written
H T s
<?=--sm2,* (73)
where T is the complete periodic time of vibration, i.e., the
interval between two successive passes through the zero position
in the same direction.
This expression gives the value of Q in coulombs when H, g, T
and s are known. The two latter, T and s, are easily measured,
* This formula is true, not only for a tangent galvanometer, but for any
galvanometer in which the needle is controlled by a uniform magnetic
field, and the direction of the deflecting force is at right angles to the
controlling force. It can, therefore, be used when a sensitive galvanometer
with the wire fairly near the needle, is employed,
356 PRACTICAL ELECTRICITY
and the ratio of H to g can be readily found by observing the
steady deflection d of the galvanometer produced by passing a
current of known strength* through the instrument. If a
current 7 gives a steady deflection d, we have from the law
of the tangent galvanometer (formula 12 and formula 71),
I = — tan d,
g
H J^
~g := tanl'
and the value of Q becomes
sin -
(74)
a comparatively simple formula for the measurement of quantity.
When a reflecting instrument is used the angles s and d are
/y o
usually very small, and — - is approximately equal to —•
so in this case we have
T
Q= -- 7 --, approximately.! (75)
2 TT a
147. Correction of Ballistic Galvanometer for Damping —
When the needle of a galvanometer comes to rest after a com-
paratively small number of swings, i.e., the instrument is
appreciably damped, the first swing of the needle due to the
passage of a given quantity of electricity through the galvano-
meter will be less than if no damping existed, for during the
first swing the frictional forces, such as air friction, etc., will have
dissipated part of the energy initially stored in the needle, and,
in consequence, the potential energy stored at the end of the
swing will be less than — - . In other words, s will be lessened
by the presence of damping. The expression for Q in the last
paragraph will give a value smaller than the proper one, if
the motion of the needle is damped, so to correct it a multiplying
* A convenient means of obtaining a small current of known strength.
is to put a cell of known E.M.F. in circuit with the galvanometer and
a large known resistance.
t Observe that when T = 2 IT, Q'= I, if s = d. In other words, when
the periodic time is 2?r seconds, the swing per microcoulomb is equal to
the steady deflection per microampere.
LOGARITHMIC DECREMENT 357
factor greater than unity must be employed. This factor will
depend on the " decrement," the number expressing the ratio of
one swing to the next ; the larger the decrement the larger
the correcting factor will be. Writing the value of the decrement
as (i-f-y), for the decrement always exceeds unity, an approxi-
(y\
i -f '- ) when y is small, and a more
accurate factor is ( i + - j where X is the logarithmic decrement,
viz., log.e (i+y).
We may, therefore, write the expression for Q as
T Sin2/' y\
+— ), approximately, (76)
.
rp sin — / ^\
or Q = —f - - ( H --- ), very approximately. (77)
TT tan d >
These corrections can only be used with accuracy when the
amount of damping is small, and for cases where the decrement
is large a more complicated correcting factor is necessary.*
When the angles of swing and steady deflection are small,
as is usually the case with reflecting instruments, the formulae
reduce to
Q = - ~1SI+' aPProximately, (78)
and Q = - —s(i-\ — j, approximately, respectively. (79)
148. Determination of Decrement and Logarithmic Decrement.
— The motion of a swinging needle may be represented by a
curve as shown in Fig. 202, where vertical distance represents
" displacement " and horizontal distance " time."
This figure corresponds with a moderately damped oscilla-
tion, the amplitude being reduced to half its initial value in about
* This correcting factor, the calculation of which is too elaborate for
an elementary work, is
TT *
where e is the base of Napierian logarithms. Two approximations
are (i + 0-5 y — 0-277 y* + 0-130 y»),
and (i + 0-5 A — 0-027 A-a — °'°54 * 3)> respectively.
358
PRACTICAL ELECTRICITY
fourteen swings. An undamped vibration is represented in Fig.
203, and a well damped one in Fig. 204, the motion being
nearly destroyed in six swings.
1
Time — *•
Fig. 202. — Diagram representing a moderately damped vibration.
The motion of a vibrating needle is very nearly isochronous,
i.e., the time of a complete vibration is the same whether the
swings are fairly small or very small, and the ratio of one swing
to the succeeding one is found to be constant. In the above
figure the decrement
(i -f- y) = — = — = — = — etc., etc., and
also
= etc.
Hence
and
Time — ^
Fig. 203.— Diagram representing an undamped vibration.
so we see that the decrement can be found by observing the
magnitude of one swing and the nth swing reckoning from the
one first observed, and finding the (n — i)th root of the ratio.
Time
Fig. 204. — Diagram representing a well-damped vibration.
thus
BALLISTIC GALVANOMETER CONSTANT 359
and x=— 5— log.A*
n— i €sn
or x = 3°3 log-io ~ » (approximately).
H — I 5||
for log.€ 10 = 2-303 approximately.
149. Constant of a Ballistic Galvanometer. — In Sections 146 and
147, formulae are given connecting quantity and the swing
produced by discharging that quantity through the coil of a
galvanometer. The expressions include T, IT, - • > or — » and y
tan d d
or \, all of which are constant for a particular instrument used
under definite conditions. We may therefore write the equations
(74) and (75) of Section 146 as
and Q = kr s' , respectively,
where k and A/ are constants, and have the iollowing meanings : —
viz., k is the quantity which will produce a swing of 180°, kf the
quantity which will produce a swing of i division of the scale of
the reflecting instrument, and s' the first swing expressed in
scale divisions.
When there is damping, the correcting factor fi-f--jor(i-f--j
being constant under given conditions, may be included in " the
constant " of the instrument, so that whether there is damping or
not the formula
0=£sin|. (80)
expresses the quantity passed through a ballistic galvanometer,
when the swing produced is so large that the sine differs appreci-
ably from the angle, while
Q=k'J. (Si)
expresses the quantity approximately when the angles of swing
are small.
The constants k and k' differ considerably. In fact, for the
same instrument, k is usually a much larger number than k't
but the ratio of the two depends on the angle corresponding with
i division of the scale in the second type of formula.
* The percentage accuracy to which A is determined by this method
spends on the value of — , a value approxin
Sn
Napierian logarithms, 2-718 gives the best result.
depends on the value of — , a value approximating to e, the base of
360 PRACTICAL ELECTRICITY
Example 154. — A ballistic galvanometer gives a swing of 60°
when o-ooi coulomb is discharged through it. Find the constant
k of the instrument.
Answer. — k — 0-002 coulomb, i.e., a quantity of 0-002 coulomb
would produce a swing of 180°.
Example 155. — If the galvanometer mentioned in the last
question were fitted with a mirror and used with a millimetre
circular scale placed at i metre away from the mirror find the
constant k' of the instrument.
Answer. — Here we must remember that when the mirror
moves through any angle 0, the ray of light reflected from it
turns through double the angle, viz., 26. A quantity of -^ oi
o-ooi coulomb would produce a swing of 0° where
6 60° sin 30°
sin - = sm — ~ 100 —
2 2 100
. 6 i
/. sin - = — = 0-005
2 200
and 0 = 34'
Now an angle of i° will be subtended by
27T X IOOO .... ,,
millimetres on the scale,
360
.'. i° = 17-4 divisions,
and iiV = «jZ divisions,
= 19-8.
Hence TJ0 of o-ooi coulomb produces a swing of 19-8 divisions,
and the quantity per division is
o-ooooi f
— _ = 0-000000506,
or k' = 0-506 microcoulombs.
Example 156. — The periodic time of an undamped reflecting
galvanometer is 10 seconds and a current of ^ milliampere gives
a steady deflection of 200 scale divisions. Find the quantity of
electricity which produces a swing of 100 divisions.
T I
Answer. — Using the formula Q = - - ~ s , (Section 146),
2 71" w
10 i
we have Q = •7— „ X 100
6-28 10,000 x 200
= 80-5 microcoulombs.
COMPARISON OF QUANTITIES 361
Example 157. — Calculate the quantity in the last example if
the instrument had been damped so that its decrement was 1-06.
Answer. — 80-5 X 1-03 approximately,
= 83 micrccoulombs.
Example 158. — The following readings of the swings of a
galvanometer were taken to find the decrement and logarithmic
decrement. Determine their values. (Odd suffixes denote swings
to the right of zero and even ones swings to the left.)
Values of \ Sl S* Sa $* S* SG S? S* Sj> SIQ SH
( 312 281 253 227*5 205 184 166 149 134 121 io8;5
Answer. — Decrement (i + y) = i-in.
Log e dec. X = 0-105.
150. Comparison of Quantities. — Quantities of the same order
of magnitude may be compared by noting the swings produced
when they are discharged successively through the same ballistic
galvanometer, the adjustment of which remains unchanged.
If the quantities be Ql and Q2 and the swings produced be st
and sa then,
. s
or = -i, approximately,
S2
when the angles are small. To make such comparisons it is
not necessary to know H, g, T, y or X, nor the deflection for a
known current.
If one of the quantities be much greater than the other, and
the galvanometer be sufficiently sensitive to give a reasonably
large swing with the smaller quantity, the instrument must be
shunted when the larger quantity is discharged. Calling the
multiplying power of the shunt m^ and assuming Ql to be the
greater of the two quantities, we have,
JO = m1 — , approximately,
Qz sz
a relation which is strictly true when a Universal Shunt is em-
ployed, for in this case the damping of the instrument is not
altered by changing the multiplying power of the shunt. With
an ordinary shunt, however, changing the shunt alters the re-
sistance between the galvanometer terminals, and the currents
induced (see Section 187) by the motion of the magnets near the
362
PRACTICAL ELECTRICITY
coils, causes the decrement to increase as the multiplying power
of the shunt is increased. Allowance for change of damping
must consequently be made by finding the decrements under the
two conditions, when an accurate comparison is required. It
is sometimes necessary to use a shunt of multiplying power m2
say, when the smaller quantity is discharged ; then,
0+ m^s*
— , approximately.
Q2 ™2S2
151. Capacity. — On page 82, we have already stated that the
name " capacity " has been given to the ratio,
Quantity
Potential Difference
The equation C = ^therefore defines " capacity " C just as
the expression of Ohm's Law,
R = _ defines " resistance " R.
Ohm's Law relates to conductors at constant temperature,
through which electric currents are passed, and says in effect,
that the ratio of potential difference to current is constant for a
given conductor. Experi-
ment also shows that when
two conductors are in a
fixed relative position, and
far removed from other
bodies (or when one is com-
pletely surrounded by the
other), the ratio of Quantity
to Potential Difference is
constant. The value of this
constant depends on the
sizes, shapes, and proximity
of the conductors. This in-
fluence of proximity can be
readily shown by the ar-
rangement of gold leaf
electroscope and plate M
sketched in Fig. 205. The
electroscope may be charged with a definite quantity of
electricity by placing the plate M very near p, (but not
actually touching), and* charging the arrangement by connect-
ing one pole of a battery to p and the other to M, and then
Fig. 205.— Condensing Gold-leaf Electroscope.
CONDENSERS 363
disconnecting it. On moving M away from p the leaves of
the electroscope will diverge, showing that the potential of
the leaves rises. The quantity of electricity on the electro-
scope, however, is not altered by the movement of M, for the
metal system attached to p is highly insulated. As Q is con-
stant and V increases, -,- decreases, i.e., the capacity decreases.
When one conductor is
completely surrounded by
another, the capacity of
the inner one is the number
of coulombs required to be
given to the inner one to
Fig. 206. produce a P.D. of I volt
between the two. For ex-
ample, the capacity of A, Fig. 206, is the number of coulombs
on A when there is a P.D. of i volt between A and B.
If a metal plate A, Fig. 207, be surrounded with a flat metallic
box B, the top and bottom of the box being parallel to and very
near A, then the capacity of A will be very large, since it will
require a large charge of electricity to be given to A in order to
raise the P.D. between A and B to i volt.
152. Condensers : Mechanical Analogies. — An arrangement
of conductors such as are shown in Fig. 207, is called a " conden-
ser," so a condenser may be denned as two conductors separated
by an insulator and so placed relatively to one another that the
capacity of the arrange-
ment is large compared
with the size of the con-
ductors. The conductors
are called the " coatings " Fig. 207.
of the condenser.
Condensers behave like mechanical springs. When a spring
is subjected to pressure (or tension) the spring is strained, and
work is done, this work being stored up as potential energy of
deformation of the spring. The energy remains in the spring
so long as the pressure is maintained, and on relieving the pressure
work is done by the spring. Similarly when a condenser is
subjected to electric pressure, electric energy is stored in the
condenser, and is given out again when the pressure is removed.
Other ways in which springs and condensers behave similarly
are (a), the deformation of the spring is proportional to the force
applied, and the electric displacement in a condenser is propor-
tional to the electric pressure used ; (i), the energy stored in a spring
364
PRACTICAL ELECTRICITY
is proportional to the square of the force, and that in a condenser
proportional to the square of the potential difference ; and (c),
if the force to which the spring is subjected is excessive the
spring will break, whilst excessive electric pressure on a condenser
will cause the dielectric* to break down.
Another analogy exists in gas cylinders used for storing and
for transporting compressed gases. The quantity of gas in a
cylinder is proportional to the pressure, and the energy due to
the compression proportional to the square of the pressure.
Excessive pressure would cause the cylinder to burst.
r,^-r----~_^T-"j\c
Y
Fig. 208. — Hydraulic Analogue of Condenser.
A hydraulic analogue to the action of a condenser is represented
in Fig. 208. Let P be a close fitting piston attached by a spring
s, to a fixed point A in a smooth horizontal pipe A B, and the
whole pipe A B c D filled with water. When the paddle wheel w
is stationary, the pressure of water on the two sides of the piston,
will be equal, and the piston will be in equilibrium, and the spring
s is unstretched. If now w be rotated clockwise, the pressure
on the A side of P will be greater than that on the B side and the
piston will move until the difference of pressure is balanced by
the tension of the spring. For a definite speed of w, p will take up
a definite position in the pipe, and a definite quantity of water
will have passed a given cross-section of the pipe, say x Y. Increas-
ing the speed of w will increase the difference of pressure, increase
the displacement of p and also of the water past x Y, whilst a
decrease of speed will result in a diminution of the displacement
and cause a quantity of water to move in the direction B A. So
long as the speed of w remains constant and no leakage past the
piston occurs, there will be no movement (or current) of water
* The insulating medium between the conductors is called the
dielectric.
UNIT OF CAPACITY; FARAD 365
in A B, but any change of speed will cause a transient current to
pass in one direction or the other, according as the speed rises
or falls. In the same way an electric condenser allows transient
currents to pass (i.e. permits quantities of electricity to be displaced)
when the P.D. between its coatings changes, but allows no current
to pass when the P.D. is constant. Increasing the P.D. will
cause a transient current in the direction of the P.D., whilst a
decrease of P.D. will result in a transient current in the opposite
direction. If the pipe A B, Fig. 208, be of small cross-section,
the quantity of water displaced for a given change of pressure
will be small, whilst if the cross-section be large the displace-
ment produced by the same change of pressure will be great.
A pipe of small cross section therefore corresponds with a con-
denser of small capacity, and large cross-section of pipe with a
condenser of large capacit}^.
153. Units of Capacity ; Farad ; Microfarad. — A condenser
having a large capacity does not mean one that would hold a
large quantity (or charge) without its insulation breaking down,
but one that will hold a large charge relatively to the P.D. between
its coatings ; i.e., — is large.
If A, Fig. 207, be charged with positive electricity, there will
be a charge of negative electricity on B, whereas, if A'S charge be
negative, that on B will be positive. Experiment also shows that
the quantity on A is exactly equal in amount and opposite in sign
to that on the inside of B, for if the outer conductor of a charged
condenser be connected momentarily to earth and then insulated,
the condenser will show no external signs of electrification,
although the condenser remains charged.
To make a condenser of large capacity we may either use very
large plates, or make the distance of the plates apart very small.
There are obviously prac-
tical difficulties in making
the distance separating
the plates extremely
small, for the plates might
tOUCh, Or a Spark might pig. 209.— Diagrammatic Representation of a Condenser.
pass across the interven-
ing space, if a moderate P.D. was set up between the plates,
and so discharge them.*
On the other hand if we make the plate A, Fig. 207, and the
* To reduce the risk of this occurring, it is customary to place thin sheets
of solid insulating material, such as mica or paraffined paper, between the
plates.
366 PRACTICAL ELECTRICITY
box B very large, the apparatus becomes cumbersome. To over-
come this difficulty two series of plates, A and B, connected as
shown in section in Fig. 209, are employed, and a condenser is
often represented symbolically by such a figure. A simpler
diagram, representing a condenser, is given in Fig. 2090.
A condenser which holds
i coulomb when the P.D.
between its coatings is I
Fig. 209*.— Simple diagram of Condenser. Volt is Said to have U
capacity of one "farad."
The "farad " is therefore the unit of capacity. For practical
purposes a capacity of one farad is far larger than is convenient,
so a millionth of a farad, i microfarad, is adopted as the com-
mercial unit.
As i volt is io8 C.G.S. electromagnetic units of E.M.F. or P.D.,
and i coulomb equals ~ of a C.G.S. unit of quantity, it follows
that i farad is io~* C.G.S. units of capacity, and i microfarad
=icT15 C.G.S. units.
154. Variation of the Capacity of a Condenser with the Area
of its Coatings and the Distance between them. — That the capacity
of a condenser is directly proportional to the effective area* of either
coating hardly needs proof, but an experimental proof can
readily be obtained by charging two or more similar condensers
to the same P.D. and discharging them first separately, and then
putting them in parallel and discharging them all together,
through a ballistic galvanometer. It will be found that the
quantity in the latter case is equal to the sum of the several
quantities in the former, thus showing that the combined capacity
of several condensers connected in parallel is equal to the sum
of their individual capacities. Now a large condenser is usually
formed of a number of smaller parts all connected in parallel,
so the experiment described proves the statement in italics above.
From the experiment with the condensing electroscope (Fig. 205)
described in the previous section we see that increasing the distance
between the coatings diminishes the capacity of the condenser, but
the law between capacity and distance could not be determined
satisfactorily by this apparatus. Plates of large area would be
required to obtain quantities measurable by a ballistic galvano-
meter unless very high potential differences were employed. We
* The effective area of the coating of a plate condenser is usually rather
greater than the actual area, for near the edges of the coatings the lines
of electric force passing from one to the other spread beyond the edges.
An approximate correction may be made by assuming the smaller plate
is increased all round by a strip of width 0-4 times the distance between
the coatings.
CAPACITY AND DIMENSIONS 367
can, however, show theoretically in a simple way that the capacity
oj a condenser with plane parallel plates is inversely proportional
to the distance between the coatings, and this conclusion is verified
by experiment.
In Chapter II., page 82, we have shown that the capacity
of a condenser formed of concentric spheres is
in electrostatic units, where r± and r2 are the radii in centimetres
of the inner and outer spherical surfaces respectively.
If (r2 — rj be called t, the thickness of the insulator, we have,
capacity = ^ ( -.
t
and the capacity per unit area of inner sphere is
which may be written --
47T*
If we now suppose the sphere to become infinitely large the
opposing surfaces will become plane, and rL= cc . The term -
47T7-J
will become zero and we get the capacity per unit area of two
plane parallel plates is - electrostatic units ; i.e., the capacity
47T/
is inversely as the distance between the coatings.
Combining the two conclusions we may say that the capacity
of a plate condenser is directly proportional to the effective area
of the plates and inversely proportional to their distance apart.
155. Relation between the Electrostatic Unit of Capacity
and the Farad. — On page 81, we have stated that the electro-
magnetic C.G.S. unit of quantity is approximately 3 X io10
electrostatic units of quantity, and a method of proving this
experimentally will be found in Section 171. In both the c.G.s.
electromagnetic system of units and the c.G.s. electrostatic
system the dyne and the erg are the units of force and energy
respectively, and in both systems potential difference is defined
so that the work done when a quantity of electricity passes
* The area of a sphere of radius ?'j = 4 IT r\ -.
368 PRACTICAL ELECTRICITY
from one point to another is equal to the product of the quantity
and the potential difference between these points (see Section
48). As the work done when one c.G.s. electromagnetic unit of
quantity passes between two points whose P.D. is one C.G.S.
electromagnetic unit of P.D. is one erg, and the work done when
one c.G.s. electrostatic unit of quantity passes between two
points whose P.D. is one C.G.S. electrostatic unit of P.D. is also
one erg, we see that the magnitudes of the units of P.D. in the
two systems must be inversely as the magnitudes of the units
of quantity. Accordingly the magnitude of the electrostatic
unit of P.D. must be 3 x io10 times as large as the c.G.s. electro-
magnetic unit of P.D., and as I volt is io8 C.G.S. units (see Section
55a), one electrostatic unit of P.D. must be 3 x io10 -*- io8
volts, i.e. 300 volts, approximately.
If a condenser of capacity i farad had a P.D. of 300 volts (i
electrostatic unit of P.D.) between its coatings, the quantity on
each coating would be 300 coulombs, and as i coulomb is 3 x io9
electrostatic units, this equals
300 x 3 X io9 electrostatic units of quantity, approximately,
i.e. 9 X io11 electrostatic units of quantity, approximately.
The quantity on each coating of a condenser whose capacity is
i electrostatic unit, would only be i electrostatic unit of quantity
under the same P.D., so that I jar ad is 9 X io11 electrostatic units
oj capacity, and i electrostatic unit of capacity = i-in micro-
microfarads.
156. Capacity of Spherical and Plate Air Condensers in Farads.
—The foregoing numerical relation enables us to express the
capacity of spherical and plate condensers in electromagnetic
units of capacity (farads) as follows : —
Capacity of isolated sphere of radius r centimetres equals,
— - — - farads; (82)
9 X io11
Capacity of concentric spheres of radii r± and r2 centimetres equals,
farads ; (83)
9 x io11 (r2 -
Capacity of parallel plate condenser of effective area A square
centimetres and distance apart of t centimetres equals,
== farads,
9 x io11 X 4 TT t
^
or : — — — farads; (84)
1-131 X io13/
CALCULATION OF CAPACITY 369
Expressed in microfarads (Cm) the formulae become,
r
(85)
1^2
r6('a-'i) (86)
and
1-131 x io7* (87)
respectively, and if the measurements be taken in inches instead
of centimetres we get,
For an isolated sphere,
_ 2-822 r"
For concentric spheres,
2822 r\ r£
and for plate condensers,
(90)
the two dashes (") above the letters signifying inches, and square
inches in the case of A.
157. Capacity of Cylindrical Condensers. — Another form of
condenser of great practical importance consists of two long
concentric cylinders, for the insulated wires and cables used in
the distribution of electrical energy, and in submarine telegraphy,
approximate to this shape. The capacity of such a condenser
whose axial length / centimetres is very great compared with the
diameter, and whose dielectric is air, can be shown to be
— — electrostatic units, (91)
2l°g6J
where D and d are the inner diameter of the outer cylinder and
the outer diameter of the inner cylinder respectively.*
If common logarithms be used, the formula becomes
— n ° 4^43 , JV electrostatic units, (92)
2 (log D — log d}
and converting to farads and microfarads we get,
~~~io™ (log D - log d) '
* D and d should both be measured in term? gf the sa.me unit, but the.
unit employed is immaterial,
PRACTICAL ELECTRICITY
and Cm = - (log p log gf (94)
/ being in centimetres.
When / is measured in inches the latter expression becomes,
6-128 I"
Cm = io7 (log D - log d)'
Example 159. — Express in electrostatic units of capacity and
in microfarads (a), the capacity of an isolated spherical conductor
of i metre diameter, (b) , that of the earth considered as an isolated
sphere whose diameter is 12,756 kilometres.
Answers. — (a) 50 electrostatic units, -^ microfarads.
(b) 6-378 X io8 electrostatic units, 709 microfarads.
Example 160. — Find the capacity of a spherical conductor i foot
in diameter placed concentric with a hollow sphere of I2j inches
inside diameter, air being the dielectric.
Answer. — 747 electrostatic units, or 0-830 milli-micro farad.
Example 161. — Determine the capacity of an air condenser,
having parallel plates of effective area 2,000 square inches
spaced ~ of an inch apart.
Answer. — 4,043 electrostatic units, 4-492 milli-microfarads.
158. Specific Inductive Capacity. — In Section 157 we have con-
sidered the coatings of condensers to be separated by layers of air.
If instead of air, solid or liquid insulators, such as glass, gutta-
percha, indiarubber, oil, etc., be used, we find that the capacity
is increased in definite proportions, depending on the nature of
the insulator employed. For example, if an air condenser be
submerged in paraffin oil, so that the air between the coatings is
replaced by the liquid, the capacity is found to be about 2-1 times
greater than before, whilst if gutta percha be used as the dielectric
instead of air, the capacity will be about 4 times as great. The
ratio in which the capacity of a condenser is altered by substi-
tuting some other material for air between its coatings is called
the " specific inductive capacity " of the substance. In the cases
just mentioned we may say that the specific inductive capacity
of paraffin oil is 2-1, and the specific inductive capacity of gutta
percha, 4.
The following table gives a list of the " specific inductive
capacities " of many important dielectrics as determined by
various experimenters using different specimens of material : —
TABLE XI
APPROXIMATE " SPECIFIC INDUCTIVE CAPACITIES," OR
" DIELECTRIC CONSTANTS " OF SUBSTANCES
Substance.
Specific Inductive Capacity.
Air at 760 mm. pressure
„ „ 5 »'
Carbon dioxide at 760 mm.
Hydrogen ,, ,,
Sulphur dioxide „ „
Alcohol
Oil, Castor
,, Linseed
„ Olive
„ Paraffin (Light)
(Heavy White) . .
,, Resin
,, Sperm
,, of Turpentine
Water
Amber
Balata ..
Chatterton's Compound
Ebonite
Glass, Plate .. ;;•• ..
„ Flint (Very Light)
„ (Dense)
„ „ (Double extra dense)
Gutta Percha
India Rubber, Pure
,, ,, Vulcanized
Jute .. ..x> ..
Marble
Mica
Paper
„ Impregnated with oil
,, Dry cellulose
Pitch
Porcelain
Quartz, Fused
,, Crystalline
Resin
Shellac
Sulphur
Wax, Paraffin
Sealing
i-o (Taken as Standard.)
0-9985 to 0-9994
1-00069,, I'QOoS
0-9997 ,. 0-9998
1-0037
26 (about)
4-62 to 4-67
3'35
3'i6
2-04
2'55
37
3'i
2-2 tO 2-43
80 (about)
2-8
2-4 to 3-6
4-0
2-56 to 3-15
6-1
6-57 '
7'4
io-i
3-6 to 4-43
2-1
27
3
6-1
5-0
1-8
2-8
6-7
1-8
4-4
378
4-27 „ 4'6
2-55- 3'i
2'5 » 37
2-58,, 4-03
1-92 , 2-47
4'5 » 5'2
„ 2'3
» 5'5
» 4
,, 6-6
„ 2-2
„ 3-8
6-8
372 PRACTICAL ELECTRICITY
From what has been said above it will be evident that the
capacities of condensers with dielectrics other than air can be
obtained from the formula for air condensers of the same dimen-
sions, by multiplying by the specific inductive capacity of the
dielectric used.
Example 162. — Find the capacity of i mile of gutta percha
covered wire, 2 millimetres diameter, covered to 6 millimetres,
assuming the specific inductive capacity of the material to be 4-2.
6-128 X 1760 X 3 X 12 , ,
Answer. -- — — '-? - ^ — -— X 4-2= 0-342 microfarads.
io7 (log. 6 — log. 2)
Example 163. — What must be the area of each coating of a
condenser whose capacity is to be one microfarad, and the dielec-
tric mica u\y of a millimetre thick (Spec. Ind. Cap. 5) ?
Answer. — 1-13 X io4 sq. cms., or 1-13 sq. metres.
159. Dielectric Strength of Insulators. — Not only is the
capacity of a condenser increased by using (say) glass, mica, or
wax, instead of air, as the " dielectric " or insulating material
between its coatings, but the resistance to the loss of charge by
sparking from one coating to the other is greatly increased by
the change. With a glass condenser far greater P.Ds. can be used
than is possible with an air condenser of the same size. The
resistance to sparking does not depend on the insulating quality
of the substance, but on its rigidity and the resistance it in consequence
opposes to rupture.-
The property of resisting rupture by electric pressure is spoken
of as " dielectric strength " or electric strength, and is usually ex-
pressed as the potential gradient* in volts per centimetre, or volts
per millimetre, at which breakdown occurs. For example, if a
P.D. of V volts exists between two faces of a plate of thickness / the
V V
potential gradient is — , and the value of — which causes rupture
t t
of a substance is called the dielectric strength, or more shortly the
electric strength of that substance. Electric strength is influenced
by many conditions, such as temperature, time of application of
of the P.D., etc., so it is difficult to make precise measurements.
In the case of gases the electric strength increases as the pressure
increases, and nearly in direct proportion. For air at normal
pressure and temperature, the electric strength is approximately
3,800 volts per millimetre. Approximate values for various
substances are given in Table XII.
* Potential gradient means '
DIELECTRIC STRENGTH & RESISTIVITY 373
TABLE XII
APPROXIMATE DIELECTRIC STRENGTHS OF SUBSTANCES
IN KlLOVOLTS PER MILLIMETRE
Air 3-8
Ebonite . . . . . . . . . . 53
Glass (ordinary) . . . . . . . . . . 16
Mica . . . . . . 50 to 60
Micanite . . . . . . . . . . . . 18 to 40
Paraffined paper . . . . . . . . . . 34
Porcelain (hard) . . . . . . . . . . 18
Press-spahn . . . . . . . . % . . . . 9 to 22
Rubber (pure) . . . . . . . . . . 47
Rubber covered Cable . . . . . . . . 10 to 25
160. Resistivity of Insulators. — In making condensers another
property of insulators which has to be considered is their resistivity.
A perfect condenser is one that allows no current whatever to
pass through, when a steady P.D. is maintained between its
coatings, and the greater the resistivity of the insulator used in
the condenser the nearer this perfection is attained. It is, there-
fore, important to use very high resistance materials for the
purpose. In the case of metals and other good conductors the
resistivity is constant (if the temperature remains constant),
however long the current is passed. With insulators this is not
so, for in almost all cases where the P.D. used is much below
that required to produce rupture of the material, the resistance
increases with the time of application of the P.D., but increases
more slowly as time goes on. This phenomenon is called " electri-
fication," and to obtain more consistent results in measuring
the resistances of insulators it is usual to make the necessary read-
ings after a constant P.D. has been applied for definite intervals
of time. For electric light wires and telegraph cables one minute
is now adopted as the standard time of electrification.
Temperature has a very large influence on the resistivity of
bad conductors (or insulators), their resistance decreasing as the
temperature rises. In the case of gutta percha the resistivity is
halved by raising the temperature about 5° C., and for indiarubber
a rise of about 15° C. halves the resistance. Approximate data
relating to the resistivity of insulating materials in common use
at about normal temperature will be found in Table XIII.
374 PRACTICAL ELECTRICITY
TABLE XIII
APPROXIMATE RESISTIVITY OF INSULATORS
( Substance.
Ohms per Centimetre Cube.
Amber
155 x lo12
Canada Balsam
280 x io12
Cellulose (Dry)
i, 600 x io12
Ebonite
450 x io12 to
30,006 x io12
Glass . . . . . . , *
50 Xio12 „
300 x io12
„ Flint (Density 4-1)
250 x io12
„ . „ ( „ 3'3)
9,900 Xio12 ,,
2O,000 X IO12
Gutta Percha
25 xio12 ,,
5,000 x io12
India Rubber
1,500 xio12 ,,
l8,OOO X IO12
Jute (Impregnated)
3,000 x io12
Marble
500 x io12
Mica
4 xio12 „
8,800 X IO12
Micanite
2,500 x io12
Paper
0-5 x io12
Porcelain
2,100 x io12
Press-spahn . ^V* .
o-oi x io12
Quartz (Fused) . .
i, 600 x io12
„ (Parallel to Axis)
153 x io12
(Perpendicular to
Axis)
20,000 X IO12
Resin
7,000 x io12
Resin Oil
0-2 XIO12
Shellac
1,500 XIO12 ,,
9,000 x io12
Slate
0-08 xio12 „
10 X IO12
Sulphur
4,000 xio12 ,,
8,200 XIO12
Wax (Paraffin)
49,000 XIO12 ,,
294,000 xio12
For some of the substances in the previous tables, two values
are given representing the variation of resistivity of different
specimens of the material. These indicate to some extent the
great variations that exist between different samples, and show
the necessity of testing in all cases, where it is important to know,
even roughly, the actual resistivity of a particular specimen.
161. Construction of Condenser of Large Capacity. — When
a very large capacity is required the dielectric employed
consists usually of sheets of paper or of mica, which have been
soaked in melted paraffin wax or in a solution of shellac in
alcohol.
CONSTRUCTION OF CONDENSERS 375
The sheets of tinfoil are shaped as, shown in a (Fig. 210), one
corner being cut off, and the sheets of insulating material b are
made about two inches wider and two inches longer, and have
two corners cut off. On a sheet of insulating material there is
first laid a sheet of tinfoil, as in c, then a sheet of insulating
material is laid on the top, then a second sheet of tinfoil with
its uncut corner turning the other way, and so on, so that finally
there are a number of alternate sheets of tinfoil with their corners
projecting over the sheets of insulating material to the right, and
the other set of alternate sheets of tinfoil, with their uncut
corners projecting over to the left. Each of the exposed sets
of corners is soldered together, and forms an electrode or terminal
of the condenser.
When paraffined paper is employed as the insulating material,
the paper is first very carefully examined by holding it up to the
Fig. 210.
light, sheet by sheet, so that the existence of any small holes
may be detected, and any sheet possessing such holes discarded.
The good sheets are then placed in a bath of melted paraffin wax
warmed by steam to about 110° C., or a little above the boiling
point, so that all water may be driven off. On a horizontal slab
of cast iron, also warmed by steam to about the same tempera-
ture, the sheets of paraffined paper and tinfoil are' laid in the
way just described, the sheets being carefully smoothed with a
flat strip of wood as they are laid on. Two sheets of paper are
placed between each pair of sheets of tinfoil to avoid the possi-
bility of a hole in the paper causing leakage, it being, of course,
most improbable, even if there were a minute hole in each sheet,
that the holes would come exactly opposite one another. After
the condenser has been built up in this way it is placed between
two warm metal plates, and pressed with a heavy weight while it
is cooling, in order that the surplus paraffin wax may be squeezed
out and the whole consolidated.
It is not desirable to use the paraffin wax in the baths more
than once, since even when the temperature is not raised to more
376 PRACTICAL ELECTRICITY
than about 110° C. or 120° C, slight decomposition of the wax
may occur, which diminishes its high specific resistance.
Within recent years " rolled " condensers and "foiled paper "
condensers* have come largely into use for telephone and other
purposes, for which the precise constancy of capacity is not of
prime importance. " Rolled " condensers are formed by taking
two long strips of tinfoil (like wide ribbons), separated and
covered by lengths of paper somewhat wider than the foil,
and rolling them up together. The resulting roll is then flattened
under a press. The two strips of foil are insulated from each other
by the separating papers, and form the two coatings of the
condenser.
Foiled paper resembles the paper frequently used for wrap-
ping up packets of tea, and is made by coating long strips
of paper with a layer of finely powdered tin mixed with an
adhesive ; after being dried, the coated paper is passed between
rollers and burnished. To make a condenser from foiled paper,
two strips of it are interleaved with plain paper, and rolled up
together, the length of paper rolled depending on the capacity of
the condenser required. Condensers of one microfarad can be
made from strips about n feet long and about 7 inches wide.
After rolling, the papers are dried, waxed, and pressed. Such a
condenser may occupy a volume as small as 4 cubic inches and
weigh about 4 ounces.
162. Condensers for Large P.Ds., Leyden Jars. — The charge,
or quantity, in a condenser of capacity F is given by
Q = CV coulombs,
and this charge can be made great by making V very large, even
if C be of moderate magnitude. Condensers for large P.Ds.,
such as are produced by frictional electrical machines, must be
constructed so that they will not break down under the high
pressures, and to fulfil this condition it is desirable that the
insulating material should have large dielectric strength and be
of sufficient thickness to reduce the potential gradient to a safe
working value. The material most used for such condensers
is glass, either in the form of sheets, tubes, or jars.
A very common form of condenser for large P.Ds. is shown
in Fig. 211, which represents a " Leyden Jar." The name is
derived from the town of Leyden, at which the property of electric
capacity was accidentally discovered in 1746, by Musschenbroek,
and his pupil, Cunens. Desiring to collect the supposed electric
fluid, the}'- used a bottle partly filled with water, into which
* Devised by Mr. G. F. Mansbridge, of the Postal Telegraphs Department.
LEYDEN JARS
377
Fig. an. — Leyden Jar.
dipped a nail, passing through the cqrk, to carry the supposed
fluid from the electric machine to the water, and on Cunens
touching the nail with one hand, the bottle being held in the
other, he received a shock.
In the ordinary Leyden jar the
coatings are sheets of tinfoil, one
pasted inside and the other out-
side. Electric connection is made
with the inside either by a metal
rod or foot resting on the bottom,
or more commonly by a chain or
flexible wire, hanging from a brass
rod supported by a wooden cover,
resting on the top edge of the
jar. The use of a wooden cover
supporting the rod or chain is
objectionable from the fact that it
facilitates surface leakage of electricity between the two coatings
by short-circuiting the inner surface of the glass between the top
edge of the jar and the upper edge of the tinfoil inside the jar.
An improved form of jar is shown in Fig. 212, in which the
outer tinfoil is not carried so high up the jar, and the inner foil
is replaced by strong sulphuric acid s s.
A lead foot and stem L supports a
metal rod I from the bottom of the jar,
and serves to make contact with the
acid. The rod I passes through a large
hole in the wooden cover w w, which
hole may be closed by the cork c,
sliding on i when the jar is not in use.
When the cork is raised, as in the figure,
no electricity can pass to the cover
from the rod, and the surface leakage
path from inside to outside is up the
inside of the jar, over the edge, and
down the outside. The inner surface
of the jar being kept very dry by the
' presence of the strong acid, is, if
Fig. 212.— An improved form of properly cleaned, an excellent insulator,
and enables such a jar to retain its
charge for many hours without much loss.*
* We may here remark that, with high P.Ds., leakage over surfaces is
often far more serious than the passage of electricity from one coating to
the other through the dielectric.
378 PRACTICAL ELECTRICITY
In Fig. 212, a cone with spherical end is shown supported on
the rod i. When the jar is charged, the cone will be charged also,
and by means of a " proof plane" the density (quantity of
electricity per unit area) at different points of the conductor may
be investigated.
Fig. 213. — Three Ley den Jars in Parallel.
Ley den jars are sometimes used for wireless telegraphy,
which are charged and discharged many thousands of times a
second. Under these conditions tinfoil coatings are not very
satisfactory, for want of intimate contact with the glass causes
local heating to occur. This can be avoided to a great extent
by silvering the glass by chemical deposition, and the coatings
on the best jars are formed of deposited silver.
Fig. 214. — Three Leyden Jars in Series.
Where it is necessary to have condensers of capacity greater
than that of a single jar, of the largest size obtainable, several
jars are connected in parallel, forming a battery of Leyden jars,
as illustrated in Fig. 213, and if the P.D. is too great for one jar
to withstand, several jars may be connected in series, care, of
course, being taken to insulate them in a suitable manner. (See
Fig. 214, where the letters I.S. indicate insulating stands.)
COMPARISON OF CONDENSERS
379
163. Comparison
of Condensers. —
The simplest way
of comparing two
condensers of
about the same
capacity is to
charge them to the
same P.D. by a
suitable battery,
and observe the
swings produced
on discharging
them in succession
through a ballistic galvanometer. The ratio of the swings pro-
duced (or the sines of half the swings when the angles are large)
gives the ratio of the capacities,
Fig. 215. — Charge and Discharge Key.
£'
c.
sin
_± = _if or =
sin-
any damping that may exist in the instrument, being the same
in the two cases, cancels out.
A form of key, called a charge and discharge key, suitable for the
comparison of condensers, is shown in Fig. 215, and a scheme of
connections is indicated diagrammatically in the same figure.
Fig. 216. — Condenser Circuit in which
the Charge only is Measured.
Fig. 217. — Condenser Circuit in which both
Charge and Discharge are Measured,
L is a brass spring, supported on a corrugated ebonite pillar, and
normally is in contact with the platinum tipped screw s2. When
the left-hand end of L is depressed by touching the ebonite push
p the coatings of the condenser c are charged to the P.D. of the
batten^ B, and on allowing P to rise, the quantity on the coatingr
38o PRACTICAL ELECTRICITY
of c is discharged through the galvanometer G, thus giving a
throw, which is a measure of the capacity. The object in sup-
porting the several terminals of the key on ebonite pillars is to
obtain very good insulation, and the pillars are grooved to
increase the length of surface from terminals to base and thereby
lessen surface leakage.
Arrangements of condenser circuits, including charge and dis-
charge keys, are shown in Figs. 216 and 217 ; in the former the
galvanometer is in series with the battery and measures the
charge that passes through the condenser when the key is pressed,
but the discharge does not pass through the instrument. Fig.
215 illustrates connections whereby the discharge only is measured,
whilst in Fig. 217 both charge and discharge pass through the
galvanometer. It is usually advisable to measure discharge only
by the arrangement in Fig. 215, for less error is introduced if the
condenser be imperfect, and there is no risk of damaging the
galvanometer if the condenser be short-circuited or very leaky.
When the condensers to be compared differ greatly in capacity
they may be charged to the same P.D., and shunts, preferably
of the " Universal " type, used, as already described in Section
150, to compare the resulting quantities ; or the two condensers
may be charged to different P. D.'s so as to make the quantities
discharged of the same order of magnitude. Calling the P.D.
used on the larger condenser Ft and that on the smaller one V^
we have
c
2
In cases of extreme inequality it may be necessary to adopt a
combination of shunts and different P.Ds., the formula in such
cases being
sin —
or — r = -r^ — — , approximately, (98)
C2 v\ m2 s2
mt and m2 being the multiplying powers of the shunts used.
164. Potential Divider. — From equations (96) — (98) it will be
noticed that the ratio of P.Ds. VJV^ must be known, but the
absolute value of either V^ or V^ in volts is not required. A simple
POTENTIAL DIVIDER
means of getting two P.Ds. in known ratio for condenser work
is to close the circuit of a battery through a set of resistance coils
whose relative values are known. (See Fig. 218.) The P.D.
between any two points in the circuit will then be proportional
to the resistance between
them. For example, if — Illlllll • jl
the resistance between A
and c in Fig. 218 be R1
ohms, and that between
A and D, R2 ohms, then A^^AA/V^JA/VV\AAAAAAAA^-JE
R
-v,
D
Fig-
°btaining tw°
The arrangement in Fig.
218 may be called a
" Potential Divider." Any ordinary resistance box can be used
for a potential divider, provided the wire with which the coils are
wound is sufficiently thick to carry safely the current which the
battery will send through them. A device more convenient than
an ordinary box is shown in Fig. 219, where a number of equal
Fig. 219. — Simple Dial Potential Divider.
coils arc connected in series, and their junctions joined to
studs arranged in a circle so that a switch arm s, pivoted at P,
can make contact with any of them. If there be m coils between
A and c and n coils between A and E, the ratio of the P.D. between
A and T to that between A and E will
m
be—.
With two sets ot
382 PRACTICAL ELECTRICITY
coils, one (say) of 10 coils of unit resistance and the other of 9 coils,
each 10 units (Fig. 220), a P.D. can be divided into hundredths ;
for when s is at 4 and S1 at 7, the P.D. between T and ^ will be
£ga of that between A and E'. Increasing the number of dials or
the number of coils per dial enables finer subdivisions to be
obtained ; for example, with two dials, one having 100 one unit
coils and the other 99 coils, each 100 units, a subdivision to i part
in 10,000 is possible.
Fig. 220. — Two-Dial Potential Divider.
165. Combined Capacity of Several Condensers. — Condensers
may be connected together either in parallel or series, as shown
for Leyden jars in Figs. 213 and 214.
When several are joined in parallel (Fig. 221), and a P.D.
applied to the terminals D E, each condenser has the same
P.D between its coating, viz. V, so the quantities of electricity
on Cj, C2, C3, etc., will be
etc., etc.,
and if the battery be removed and D and E connected together
by a wire, each of the condensers will be discharged and the total
quantity, Q, that passes between D and E will be the sum of
Qi, Q*> Qz> etc.,
- CjV + C2V + C3V + etc.
= (d + C2 + C3 + etc.) V
GOiMBINATIONS OF CONDENSERS 383
Hence -^ = Cl + C2 + C3 + etc.
But p. is the capacity of the combination,
.*. combined capacity C = C1 + C2 + C3 + etc.
D
(99)
A_L
f
F,
*
F,
Fig. 221.— Three Condensers in Parallel.
from which we learn that the combined capacity of a number of
condensers joined in parallel is equal to the sum of their several
capacities.
Condensers connected in series, as represented in Fig. 222, have
a combined capacity less than the capacity of either, which may
be calculated from the formula
^-+ ^,— + etc.
(100)
i i i i
^r -~ r T; r ;r
t»j ^2 Oj,
vSuppose we take three condensers as shown in Fig. 222, and let
them all be completely discharged to begin with. When the P.D.
D
1
due to the battery is applied,
the quantity Q1 on condenser
C]. is the same as that on C2,
for the lower plate of Cl and
the upper plate of C2, with the
connecting wire, form an in-
sulated system whose total
charge is zero ; there must,
therefore, be as much posi-
tive electricity on the upper
plate of C2 as there is nega-
tive on the lower plate of Clt
and as the quantity of nega-
tive on Cl is the same as the amount of positive on the other
coating of Cv we see that the quantity on the coating of C1 is
the same as that on C2, and also the same as on C3.
Fig. 222. — Three Condensers in Series.
the quantity that will pass on discharge.
< - CF < = CF,
But
384 PRACTICAL ELECTRICITY
and the total P.D., F, equals the sum of the P.Ds. on the several
condensers.
i.e. V = Fx + F2 + F3,
^+ + *t by definition of
C
capacity, Section 151.
C2
Q Q Q
— _£ __ L. JC __ L ^ •
~ c c c '
1 2 3
. I-i+J 1
£ C1+C2+C3>
The same may be proved for any number of condensers. It is
interesting to notice that capacities in parallel have the same
law of combination as resistances in series, and capacities in series
the same law as resistances in parallel.
Example 164. — Two condensers of capacities 2-3 and 4-2
microfarads are connected first in series and second in parallel.
Find the capacities of the two combinations;
Answers. — 1-486 microfarads.
6-5 microfarads.
Example 165. — What capacity must be put in series with the
two condensers of Example 164, when coupled in parallel so that
the capacity of the whole combination may be 1-5 microfarads ?
Answer. — 1-95 microfarads.
Example 166. — A submarine telegraph cable 2,300 nauts*
long has a capacity of 0-345 microfarads per naut. Find the
quantity of electricity required to charge the copper conductor
to a potential of 40 volts.
Answer. — 0-0317 coulombs.
Example 167. — Three condensers of 20, 10 and 5 microfarads
respectively are available, how could they be combined so as to
make up a capacity of 12 microfarads approximately ?
Answer. — The 20 and 10 in series, and these in parallel with
the 5. Actual capacity, 11-67 m-^s-
166. Charged Condensers are Stores of Electric Energy, not
of Electricity. — If a suitable galvanometer be inserted in each
of the wires connecting the two coatings of the condenser c
with the two ends of the battery B (Fig. 223), it will be found
on completing the circuit by closing a key at K, that the first
swings on the two galvanometers are such as indicate equal
* A naut is a nautical or geographical mile, =6,087 feet approximately.
ENERGY STORED IN CONDENSERS 385
quantities of electricity passing through them. And if when the
condenser is charged the battery be removed, and the condenser
be discharged by connecting together the wires P and Q coming
from the galvanometers, then the first swings of the galvanometer
needles will again be such as to indicate that equal quantities of
electricity pass through them, but in this case in the opposite
direction to that in which the electricity passed during the charge.
Hence, both on charging
and on discharging A con-
denser, as much electricity
passes into one coating as
passes out of the other,
and there is
or accumulating, of elcc- " Fig. 223.
tricity. In fact, so far
as the galvanometer deflections during the charge show, we
could not say whether there was a condenser at c or a resistance,
the value of which was, from some cause, rapidly increased, to
practically infinity, on completing the circuit. The sudden
deflections, however, produced on the galvanometer when the
wires P and Q are joined together after removing the battery,
could not be produced if c were a resistance, since no alteration
of the value of a resistance can, by itself, and without any current
generator, produce a current. When the condenser has a large
capacity and when the P.D. employed in charging it is large, the
current obtained on discharging it may produce very powerful
effects. Hence, we are led to conclude that, although a charged
condenser contains no store of electricity, it contains a store of
electric energy, and it can be shown that, if the capacity of the
condenser be C farads, and if it be charged with a P.D. of V
volts, the store of electric energy, or the work this store can do
when the condenser is discharged, equals
C x V*
footlbs.
2712
For the unit of P.D. is chosen (Section 48) so that the work
done when a quantity of electricity passes between two points
whose P.D. is V, is equal to the product of Q and V. When
Q is expressed in coulombs and V in volts, QV will be in joules.
Now a condenser of capacity C farads charged to a P.D. of
V volts contains a quantity equal to CV . coulombs on each
coating, and if this quantity were discharged at a constant
P.D. of V volts the work done would be CV x V, i.e. CV2.
But the P.D. falls as the discharge proceeds and eventually
z
386 PRACTICAL ELECTRICITY
becomes zero, the average value being half the initial value,
y
viz. — . The energy of discharge is therefore
CV2
or - - joules, (102)
and as I joule equals 07372 foot-pounds (see Section 117), the
energy is
0-3686 CV2 ft. Ibs. approximately.
CV2
167. Energy wasted in charging a Condenser from a Source
of Constant P.D. — In this case the whole quantity CV passes
under a pressure V, so the work done by the source is CV2 ; but,
as proved above, the energy stored in the condenser is only
CV2
— , so that half the total energy is wasted in the process, and
appears as heat in the circuit. It is possible, however, to charge
a condenser without appreciable loss if this be done gradually
from a source whose P.D. rises steadily from zero to the maximum
value, as can be done by means of a dynamo, or a potential divider.
Example 168. — How many times per second would a con-
denser of 10 microfarads have to be charged with 86 volts and
discharged, so that it would give out about Yoob °^ a horse-
power ? Answer. — About 20.
Example 169. — If a battery having an E.M.F. equal to 200
volts be used to charge a condenser of 2oXio~8 farads, how
many foot Ibs. of work are wasted in the charging ?
Answer. — 0-295.
Example 170. — Find the energy stored in (a)t a Leyden jar
of capacity 3-5^ of a microfarad charged to a P.D. of 10,000 volts,
and (b), the 1894 Atlantic cable, whose capacity is 775 microfarads
when charged to a P.D. of 50 volts.
Answers. — (a) ~ joule, or 0-0123 ft.-lb.
(b) 0-97 joule, or 0-71 ft.-lb.
Example 171. — If an air condenser be formed of two parallel
metallic plates, each two square feet in area, placed ^th of an
inch apart, and charged with a P.D. of 250 volts, what amount of
work must be done in separating the plates, so that the distance
between them is increased to th of an inch, if the wires used
ABSOLUTE MEASUREMENT OF CAPACITY 387
in charging the condenser be removed before the plates are
separated, so that the charge in the condenser remains unaltered
during the separation ?
Answer. — As the distance is made three times as great the
capacity will be reduced to J, and the P.D. raised to 250 X 3 =
750 volts. If we calculate the energy stored in the condenser
before and after the plates are separated, the difference will give
the amount of work done. This equals 8-94 Xio~5 ft.-lbs.
Another way of considering the problem is to notice that as
the energy depends on the square of the P.D. and the first power
of the capacity, the energy is tripled by the separation, and
therefore the work done in the separation is, in this particular
case, twice that originally stored.
168. Absolute Measurement of Capacity.— If a constant source
of P.D. whose value is known in volts, such as may be secured by
means of standard cells (Section 82) be used, the capacity of a
condenser in absolute measure can be found by measuring the
quantity of electricity which passes into or out of the condenser
on charge or discharge by the method described in Section 146 ;
for if the E.M.F. of the cells battery used be E, then
Q = CE,
and as C = — ,
E
.'. C = 5 ( i -f- )/ E, approximately, (104)
2 7T d \ 2'
(formula 79), when the swing and damping are small.
The current / may be measured in terms of E if a good resist-
ance box is available, for if R1 be the resistance through which
the E.M.F. E will produce a current / amperes, giving a steady
deflection d on the galvanometer, we have
£
/ = — -, and the above formula becomes
Ri
C =
(i + -j approximately. (105)
27U Kld\ 2-
This shows that it is not necessary to know the value of E when
the value of R1 is known in C.G.S. measure, and the above re-
lation between C and Rl is one of great interest, for it enables
us to measure a capacity in terms of resistance and time, or a
resistance in terms of capacity and time.*
* If an air condenser be constructed so that its capacity can be cal-
culated (see Sections 156, 157), the above relation enables a resistance Kv
to be determined in absolute measure if v (the ratio of the electro-
magnetic to the electrostatic unit of quantity) be known. See Section 171.
388
PRACTICAL ELECTRICITY
Fig. 224.
In making the above measurement it is seldom possible to get
a resistance large enough to prevent the deflection d being too
great to measure, when the whole E.M.F. necessary to produce
a swing s that can be accur-
ately read, is used to send
the steady current, so it is
usual to employ a potential
divider, to obtain a known
fraction of the whole E.M.F.
(or P.D.). An arrangement
for this purpose is shown in
Fig. 224, where the left-
hand side shows the con-
nections for obtaining s, and
the right-hand side for
determining d. The value of Rl can be found from the
TD
equation Rl = R — Rr + — Rg* approximately, where R and R'
are the resistances indicated in Fig. 224, Rg the resistance of
the galvanometer circuit from N to o, and the resistance of the
battery is small compared with R and Rg. •
169. Measurement of Specific Inductive Capacity, and Re-
sistivity of Insulators. — As denned in Section 158 the specific in-
ductive capacity of an insulating material (or dielectric) is the
ratio in which the capacity of an air condenser is altered when the
air between its coating is replaced by the material. If, therefore,
we can construct a condenser in which the change from air to
another dielectric can be made without varying anything else,
and measure the capacities in the two conditions, the ratio of the
capacities will give the value required. The specific inductive
capacity of gases and liquids can be
measured in this way, but for solid
materials difficulty arises in excluding air
from between the surface of the metal
coating and the dielectric to be tested,
and also in obtaining a sheet of the
material of exactly the same thickness as
the air between the coatings of the air
condenser.
To avoid these difficulties we may make a condenser of the
material by pasting sheets of tinloil of known area opposite each
other on the two sides of a sheet of the material, the thickness of
* Students should deduce this as an exercise.
Fl's- 225<
Plate
SPECIFIC INDUCTIVE CAPACITY 389
which has been carefully measured. , Fig. 225 shows a condenser
made in this way in which x is one coating and D the dielectric
sheet supported on an insulating block B to keep the coating
away from the table. From the dimensions of the condenser
the capacity of an air condenser of the same size can be calculated
by formula (84) in section 156, and the capacity of the actual
condenser can be measured either absolutely (Section 168) or by
Fig. 226. — Circular Plate Condenser with Guard Ring.
comparison with one whose capacity is known (Section 163), from
which we can find e, the specific inductive capacity or " in-
ductivity " of the material, for
__ capacity of actual condenser
capacity of air condenser of the same size
Standard air condensers which may conveniently be used foi
the comparison are described in Section 170.
To determine the resistivity of an insulating material, which
can be obtained or made into sheet form, a condenser like the one
shown in Fig. 225 may be constructed, and, by employing a very
sensitive galvanometer, the current which a high P.D. will
cause to pass from one coating to the other may be measured,
after the P.D. has been applied for i minute, and the resistance
calculated from
«-£
390
PRACTICAL ELECTRICITY
The resistivity or specific resistance of the material is given by
area of coatings
P =R X tfrckness of sheet aPProx'mately (*« Sections 93-95).
In practice the values of V and / are not determined, but the
galvanometer is " standardised " by observing the deflection
Fig. 227. — High Resistance Galvanometer with Highly Insulated Coils.
produced by a known small fraction of the P.D. through a very
high resistance of known value, usually one megohm. Let
the deflection produced after one minute electrification of the
dielectric be dt when the whole P.D. is used, and suppose a
deflection d2 given by -th of the P.D. through a megohm in series
n
with the galvanometer, then
R = n — - (1,000,000 + Rg) ohms, approximately,
ai
when the deflection of the galvanometer is proportional to the
current strength, and Rg is the resistance of the galvanometer.
RESISTIVITY OF INSULATORS 391
The arrangements of circuits shown on the left and right of
Fig. 224 are suitable for observing dl and d2 respectively.
Results of experiments on the resistance of insulating materials
are liable to be seriously vitiated by surface leakage, unless great
precautions are taken. For example, leakage may occur from the
tinfoil T, Fig. 225, over the uncoated surface of D to the foil on
the other side, unless this surface be carefully cleaned and dried,
Fig. 228.— Standard Air Condenser,
and any such leakage would cause the resistance measured to be
smaller than the true resistance of the material which it is the
object of the experiment to measure. One method of avoiding
such errors is by use of a guard wire or guard ring, suggested
in 1895, by Mr. W. A. Price, and since then considerably
developed by the authors. This is illustrated diagrammaticaUy
in Fig. 226, as applied to testing a circular sheet of insulating
material, D. Here T represents a tinfoil sheet and T' an annulus
of tinfoil or other conductor, in contact with the surface of the
dielectric and joined to one galvanometer terminal as shown.
With this arrangement there will be no tendency for current to
leak from T to T', for they are practically at the same potential,
on account of the current through the galvanometer being so very
small, and any leakage from T' to the lower electrode is not
392
PRACTICAL ELECTRICITY
measured by the galvanometer, so that error from this cause is
eliminated.
To determine the resistivity of good insulators in this way
necessitates the use of very sensitive galvanometers having a
very large number 0} convolutions of wire wound as near as possible
to the needles, so that the force exerted on the needle by the
Fig. 229. — Improved Form of Plate Air Condenser.
extremely small current which passes through the insulator may
be as large as possible. Fig. 227 shows a reflecting galvanometer
constructed for the authors for testing insulators, which has a
total length of wire on the four coils of about thirteen miles,
and a resistance 360,000 ohms. To insulate the coils from earth
they are supported from corrugated ebonite rods, which hang
from a brass ring, R, carried on three corrugated ebonite pillars
from the slate base, and these rods are artificially dried by
strong sulphuric acid contained in the vessel v.
170. Standard Air Condensers. — Fig. 228 shows a form of air
condenser which can be made quite easily and whose dimensions
may be measured with moderate precision. Sheets of plate glass
about 12 inches square, are used to support tinfoil coatings, but
do not act at the dielectric of the condenser. The top sheet,
STANDARD AIR CONDENSERS
393
Fig. 229«.
T T, in the figure, which is removed
from the condenser to show the
second one, is covered all over with
tinfoil, as is also every alternate
sheet in the pile of plates. The
intermediate ones, i.e. the even
numbers counting from the top, are
only partially covered on both
sides, as is seen at p p, the sheets
of foil being ten inches square.
Small pieces, F F F, of " patent
plate " glass about ~ of an inch
thick serve to keep the plates apart,
and thus determine the distance between adjacent plates, allowance
being made, of course, for the thickness of the foil. The two
sheets of foil on opposite faces of the even numbered plates are
connected together and all joined to the terminal B, whilst the
sheets on the odd numbered plates are all joined to terminal A.
The smaller sheets, P p, etc., therefore, form the inside coating of
the condenser, and x T, etc., the outside coating. As there are
thirteen glass plates altogether, seven form outer coatings and
six inner coatings, and this gives an
approximate area of the inner
coating, 6 X 2 X 100, i.e. 1,200
square inches, and the distance
apart being ^ of an inch approxi-
mately, the capacity will be
[formula (90) ],
2-246 X 1,200 . , ,
— ^ microfarads, approx.
io7 X^
i.e. = 2-7 milli-microfarads, approx.
An improved way of
making a plate air-
condenser is to silver
the glass plates all
over * by the ordinary
process, and cut a
narrow circular groove
in the deposits on both
* Or platinise them by
covering them with " plat-
inizing liquid " and then
Jbig. 2296. Fig. 229c. applying heat.
394
PRACTICAL ELECTRICITY
faces of even numbered plates, leaving a tongue projecting over
the same edge, as indicated in Fig. 2290. The deposit will thus
be divided into two parts insulated from each other by the
groove, the inner parts on the two faces forming circular discs
united by a narrow strip of conductor, as shown developed in
Figs. 2296 and 229$, and acting as the inner coatings of the con-
denser when the plates are assembled as in Fig. 229. The outer
portion of the deposit
on the even numbered
plates is used as a
" guard ring " like T',
Fig. 226, whilst the
deposit on the odd
numbered plates acts
as the outer coating
of the condenser as
described above. Con-
nections between ter-
minals and electrodes
are made by means
of metal springs, s s s,
pressing against the
edges of the plates
at suitable places, as
Fig. 230. — Diagram of Connections for Testing Guard
Ring Condenser.
shown in Fig. 229. The use of a guard ring practically eliminates
leakage error in testing, and at the same time enables the capacity
to be calculated with greater accuracy.
For determining the capacity of such a condenser in electro-
magnetic measure, or for standardising a ballistic galvanometer,
connections as shown in the diagram, Fig. 230, may be used,
the inner coating, guard ring, and outer coatings being designated
by T, T' T', and i" respectively.
Another form of standard air condenser is illustrated in Fig.
231, which is employed at the National Physical Laboratory,
Teddington. As will be seen from the vertical section the con-
denser is made up of. many concentric cylinders, alternate ones
being connected together to form one coating, and the remainder
forming the other coating. Its capacity is about 20 milli-
microfarads. Fig. 232 shows a condenser formed by two con-
centric spheres used by Dr. Rosa, at the Bureau of Standards,
Washington, in determining " v." (See Section 171.)
171. Ratio of Electromagnetic and Electrostatic Units of
Quantity. — This ratio is of fundamental importance in many
branches of electrical work, such as the calculation of capacities
RATIO OF UNITS
395
of transmission lines and cables, and in telegraphy and telephony,
both ordinary and " wireless." Its value has already been
stated as 3 x io10 approximately, and used in Sections 155 and
156 ; the ratio is generally designated as " v."
Fig. 231. — Cylindrical Standard Air Condenser.
Although the accurate determination of "»" requires very
delicate instruments and great experimental skill, it is possible
to give a simple explanation of one of the best methods of making
the experiment.
In Section 154 and Chapter II., page 82, we have shown
how the capacities of air condensers of certain simple forms
396
PRACTICAL ELECTRICITY
can be calculated in elec-
trostatic units, and in
Section 168, a method
of measuring the capacity
of a condenser in farads
is given. Since one
farad equals io"9 C.G.S.
(electro-magnetic units of
capacity), the result of
the measurement divided
by io9 gives the capacity
in terms of this unit.
Let c be the calculated
capacity of an air con-
denser in C.G.S. electro-
static units, and C its
measured capacity in
C.G.S. electro - magnetic
units, whilst v' and V
represent in electrostatic
and electro-magnetic
units respectively the P.D. to which the condenser is charged in
the experiment. If q and Q denote the quantity discharged
expressed in these two systems of units, then
Fig. 232. — Standard Spherical Condenser.
^ - = v (definition of v)
and
also
Q = cv.
Hence £ = — —.
Now in both systems of units the unit of P.D. is chosen so
that one erg of work is done when unit quantity passes between
two points in a circuit, between which unit P.D. exists.
qv' =
or
Consequently, the above equation may be written
COMPARING E.M.F.s BY CONDENSER 397
or
c
C'
C
(106)
When c has been calculated and C measured as described in
Section 168, the value of v is determined.
Experiments carried out by Prof. Perry and one of the authors
in 1878 gave v = 2-98 x io10, whilst the value obtained by Rosa
and Dorsey in 1907 at the American Bureau of Standards is
2-9963 x io10. This number differs very little from 2-9986 x io10,
the velocity of light in centimetres per second. For ordinary
purposes, 3 x io10, the number employed in Section 155, is
sufficiently exact.
172. Use of Condensers for comparing E.M.Fs. of Cells or
other Current Generators. — A diagram of connections suitable
for the above purpose is given in Fig. 233. A and B indicate
the generators whose @
E.M.Fs. are to be
compared, c the con-
denser, G a galvano-
meter, K a charge and
discharge key (Fig
215), and P a three-
way plug key (Fig.
135), whereby either
A or B may be con-
nected to D. When
A is connected to D,
and the key K pressed
and released, the con-
denser discharges
through the galvano-
meter a quantity pro-
portional to the E.M.F. of A, producing a swing slf and when B
is joined to D the swing s2 on discharge is proportional to the
E.M.F. of B ; if G is a reflecting instrument we have
E s
•— = — approximately (107)
^B S2
This method can be used satisfactorily with cells that polarise
rapidly when on closed circuit, and also with cells of high internal
C
Fig. 233.— Comparison of E.M.Fs. by Condenser Method.
398
PRACTICAL ELECTRICITY
resistance, such as standard
Clark and Weston cells. By
using a universal shunt on
the galvanometer, generators
whose E.M.Fs. are of different
orders of magnitude can be
satisfactorily compared.
173. Condenser Method of
Measuring the Resistance of
a Cell. — In Section 131, we
have seen that when a genera-
tor of constant E.M.F. E
and resistance R^ is on closed
circuit through an external
resistance R, the P.D. between the terminals is
I'ig. a34.— Arrangement of Key and Condenser
for Testing Resistance of Battery.
V=
E
'
c/
and when R is infinite, V = E.
If, therefore, we charge a condenser Cj C2 (Fig. 234) from the
generator B on open circuit, and obtain a swing slf on discharging
it through the galvanometer, and a swing s2, when the circuit
of the generator is closed through a resistance R sl will be pro-
portional to E and s2 to V. We can therefore write the above
relation
R
(108)
K
from which we deduce
With cells that
polarise quickly, the
circuit should not be
closed longer than
necessary ; Fig. 235
shows an arrange-
ment of circuits in
which the circuit is
opened by the act of
releasing the key K'.
The expression for
Rb contains (sl — s2),
and it is interest-
ing to notice that
this difference may Fig< 23S._ Finding Resistance of Battery by Condenser Method.
EXAMPLES 399
be measured directly by observing the^swing produced on break-
ing or making the circuit of R whilst the key K is pressed. This
may be conveniently done by lifting or depressing the key K' in
Fig. 235.
Example 172. — Find the resistance of a cell which produces
a swing of 250 divisions when an open circuit and one of 200
divisions when its circuit is closed through 5 ohms.
Answer. — 1-25 ohms.
Example 173. — What swing would be produced on the ballistic
galvanometer in the previous question, by changing the external
resistance from five to infinity ? Answer. — 50 divisions.
Example 174. — A storage battery on open circuit causes a
swing of 340 divisions on a ballistic galvanometer shunted with
a j^ shunt, and one of 282 divisions on the unshunted instru-
ment, when the circuit of the battery is closed through 0*05
ohm, the key K, Fig. 235, being previously pressed. Find the
resistance of the battery. Answer. — 0*0045 ohm.
CHAPTER IX
POTENTIOMETER MEASUREMENTS
174. Poggendorff's Method of Comparing the E.M.Fs. of Cells or Batteries
— 175. Principle of the Potentiometer — 176. Calibration of Potentio-
meter Wire — 177. Industrial Forms of Potentiometer — 178. Modern
Form of Crompton Potentiometer — 179. Dial Potentiometer —
1 80. Calibration of Voltmeters. Volt (or Ratio) Boxes — 181. Stand-
ard Resistances for Current Measurements — 182. Calibration of
Ammeters — 183. Comparison of Resistances by Potentiometer —
184. Measurement of Power — 185. Advantages and Disadvantages
of Potentiometer Measurements.
174. Poggendorff's Method of Comparing the E.M.Fs. of
Cells or Batteries. — A way of measuring the E.M.F. of cells by
means of a voltmeter has been described in Section 131. This
method, although convenient and moderately accurate for cells
having small internal resistance, and which do not polarise on
sending a current, cannot be used for comparing E.M.Fs. of
" standard cells," the internal resistance of which is usually very
high, unless a sensitive electrostatic voltmeter or electrometer be
available.
The condenser method, Section 172, removes to a great extent
objection arising from resistance and polarisation of the cells
tested, but if either condenser or electrometer be used, the accur-
acy of the measurements would be limited by the exactness
with which the deflection of the instrument could be read, just in
the same way as the accuracy of measuring a resistance by the
substitution method of Section 86 depends on the unavoidable
errors in observing the deflection of the galvanoscope employed.
Now in measuring resistances we saw (Section 87) that by using
a " null method " much greater precision could be obtained,
and in the comparisons of E.M.Fs., the introduction of null
methods by Poggendorff in 1841 contributed greatly to the
accuracy of such measurements.
The principle of the method is to balance an E.M.F. against
the P.D. produced between two points in a circuit through
which a current flows from an independent source. Suppose B
is a constant battery sending a current through the long stretched
400
POGGENDORFFS METHOD 401
wire D o, Fig. 236 ; there will be a certain P.D. between D and o,
D being at a higher potential than o, because the current flows
from D to o, and there is no source of E.M.F. between them.
I I B
ill
o1
c
Fig. 236.— Simple Circuit through Stretched Wire.
The P.D. between the point o and a point c between o and D,
will be less than that between o and D, and if the point c be
supposed to move gradually from o to D, the P.D. between c and
•m-B
1
E
Fig. 237. — Balancing an E.M.F. against the P.D. between two points in a wire,
o will gradually increase from zero to V, where V is the P.D.
between D and o. If, therefore, we have a cell of E.M.F., E,
not greater than V, and connect its negative terminal to o, as
x 1
0
E
A
C,
E
Fig. 238. — Poggendorffs Method of Comparing E.M. Fs.
shown in Fig. 237, it will be possible to find a point c between
o and D, such that the P.D. between c and o is equal to E. The
potential of this point c will then be the same as the potential of
c', a conductor connected to the positive pole of the cell, and if
2 A
402 PRACTICAL ELECTRICITY
c' be brought into contact with c, there will be no current through
the galvanoscope G, because E is balanced by the P.D. between
c and o. Another cell or battery, of E.M.F. Elt Fig. 238, may
similarly have its E.M.F. balanced by the P.D. between c^ and o,
and if both E and El are balanced simultaneously, we have
E = IR,
and El = IRlt
where 7 is the current passing through the wire D o, and R
and RI, the resistances of the wire between c and o, and Cj and o
respectively.
Hence E _ R
£1 «i'
I
= /?
if the wire o D be uniform, and / and ^ are the lengths o c and
OC
G
E,
Fig. 239.— PoggendorfPs Method, using only one Galvanometer.
When the current I is quite constant, there is no need to use
two galvanometers, or even two sliding contacts c' and c/, for
by using a two-way key or switch, K, as shown in Fig. 239, the
balance points corresponding with E and Elt may be found in
succession. After obtaining the second balance, the first one
should be tested again, and if any change has occurred, the
balancings should be repeated.
The accuracy attainable in the above tests depends, of course,
on the precision with which / and ^ can be measured, and on the
sensibility of the galvanometer. Usually the galvanometer
can be made to give a large deflection for a small alteration of /
(especially if a reflecting one be used), so the measurement of the
lengths of wire is the controlling factor as regards accuracy
of the test. It is, therefore, when great accuracy is required,
desirable to make the lengths to be measured as large as con-
venient, in order that a given error of reading the lengths, say a
PRINCIPLE OF POTENTIOMETER 403
fraction of a millimetre, may introduce a very small error in the
ratio of E to Er A long straight wire would occupy much bench
room, and be awkward to use, so to avoid this inconvenience,
the wire may be arranged in zigzag fashion, or several lengths
connected in series, as shown in Fig. 240, where five metres
of wire are placed on a board about no centimetres long. The
slider s can be moved along the graduated scale, and the contact
B
Fig. 240. — Five Wire Potentiometer.
piece c brought over any one of the five wires by moving it along
the slot L in the spring -key part of the slider.
To utilise the wire to the best advantage, the battery B should
be chosen so that the P.D. between the ends of the wire is only
slightly greater than the largest of the two E.M.Fs. to be com-
pared. A variable resistance may be inserted at Rx Figs. 239
and 240, for reducing the P.D. on the wire when necessary.
We may here point out that in all balance measurements of the
kind above referred to, the question of " polarity " is of great im-
portance, for unless the two voltages oppose each other, as regards
the galvanometer, no balance can possibly be obtained.
175. Principle of the Potentiometer. — From the previous
section it will be understood that any E.M.F. or P.D., not greater
in value than the P.D, between D and o, can be balanced by the
P.D. between two points on the wire, through which a current is
flowing, and if the drop of potential per unit length of the wire,
or the resistance and current, be known, the balanced P.D. or
E.M.F. may be measured in this way. This is the principle
of potentiometer .measurements.
The drop of potential per unit length of wire may be found
404 PRACTICAL ELECTRICITY
in several ways. The usual one is to employ a standard cell,
either Clark or Cadmium, whose E.M.F. is known, and to find the
length of wire / (o c, Fig. 239, say), the P.D. between the ends of
which balances E, the known E.M.F. ; then the drop of potential
£
per unit length is — . This may give an inconvenient number,
so it is customary to adjust the strength of current flowing through
Fig. 241.— Knife Edges for Calibrating Wire.
the wire by means of a variable resistance Rx in series with the
battery B, until the P.D. per unit length is a round number, say,
foo' food" or Toooo °f a V0^- For example, if we use a cadmium
cell for which E = 1-0184 volts at 17° C., the P.D. drop per cm.
of the wire in Fig. 240 may be made equal to one-hundredth of
a volt by placing the slider s and contact piece c so that it
touches the wire at a point 101-84 centimetres from o, and
adjusting Rx until no deflection of the galvanometer occurs on
pressing the key c against the wire. By balancing the cadmium
cell at a point 2x101-84 (203-68) cm. from o, a drop of ~o vo^
per centimetre can be obtained. If a Clark cell be used, for
which E = 1-433 volts at 15° C., the balance points must be
143-3 cms. and 286-6 cms. from o, to get P.D. drops of —^ and
a£o of a volt per centimetre respectively.
176. Calibration of Potentiometer Wire. — The potentiometer
measurements above described depend for their accuracy on the
uniformity of resistance of the stretched wire, as also do the
measurements of resistance made by a metre bridge. It is,
therefore, of importance to have some convenient means of
testing the uniformity. This may be done roughly by measuring
the P.Ds. on equal lengths of the wire, say, by a high resistance
reflecting galvanometer, when a constant current is passing
through the wire. The deflection will then be approximately
proportional to the resistances of the parts of the wire tested.
A convenient appliance for making this test is a rectangular
bar of wood w, say 10 centimetres long, with a metal plate p P
CALIBRATING POTENTIOMETER WIRE 405
having a knife-edged notch in it, fixed to each end, as shown in
Fig. 241. Wires from the terminals T, T lead to the galvanometer G,
Greater accuracy can be obtained by employing a high resist-
ance differential galvanometer arranged as in Fig. 242, where R
is a fixed resistance, approximately equal to that of the wire
between the knife edges of the bar w. The difference between
i\\
D.G.
Fig. 242.— Calibrating Wire by Differential Galvanometer.
R and the resistance of the length of wire under test may be found
by observing the deflection of the instrument, the sensitiveness
of the galvanometer being determined by observing the change
of deflection caused by shunting R with a known resistance.
Supposing R to be ~ of an ohm, shunting it with 100 ohms
would produce a change of resistance of 10Q00 of an ohm
Fig. 243. — Calibrating Wire by Differential Galvanometer.
approximately, so that the deflection produced by the act of
shunting would correspond approximately with a ten -thousandth
of an ohm.
Another method of calibrating the wire is to measure the P.D.
on successive equal lengths by a second potentiometer, the
initial test being repeated at intervals, to ascertain whether any
change is taking place in either of the circuits.
Instead of the wood bar w, with contacts at a fixed distance
apart, we may use, either with differential galvanometer or
potentiometer, two independent sliders, s, s', Fig. 243, similar
to the one shown in Fig. 240, to subdivide the wire into parts
406 PRACTICAL ELECTRICITY
of equal resistance. Putting s at the zero point of the wire, we
may move s' to a point such as will give true balance on the
differential galvanometer, the resistance between s and s' will
then be equal to R. Now move s to the position of s' and again
adjust s' to give balance, then the wire between the new
positions of s and s' will have a resistance Rt*
Proceeding in this way the whole length of wire may be
tested, and a relative calibration curve drawn between length
of wire and resistance, taking R as the unit, which unit need not
be known in ohms, so far as the relative calibration of the wire
is concerned.
To lessen the necessity of using a calibration curve for poten-
tiometer and bridge wires, great care is taken in drawing wires
intended for these purposes, in order that they may be very
nearly uniform. It is also important that the wire be not easily
oxidised, hard enough to resist wear and indentation by
contact with the slider, that it be of material having a small
temperature coefficient for resistance, and have small thermo-
electric force,f with respect to copper and brass. German
silver, manganin, platinum -silver, and platinum -iridium are
frequently employed for these purposes.
177. Industrial Form of Potentiometer. — Instead of using
a very long wire, some forms of potentiometer are provided
with short wires 100 to 105 units J long, and a number of coils
in series with the wire, each of which has a resistance equal to
that of 100 units of length of the wire. Such an arrangement
is represented diagrammatically in Fig. 244. Each coil is then
equivalent to 100 divisions of the wire, and 14 such coils (as shown
in the figure) have a resistance equal to 1,400 units of the wire ;
the coils and the wire have therefore a combined resistance
equal to 1,500 — 1,505 units. The choice of fourteen coils was made
* In order that this test may be correct, the index marks on the two
sliders s and s" should be placed so that when they are put successively
at a given point on the scale, their contact makers touch the wire at
exactly the same point along its length. The index error, if any, may be
found by interchanging the positions of s and s' and reversing the wires
to the galvanometer. Index error may be avoided by leaving s' in the
first position, which gave balance, and moving s to s" (Fig. 243)
and adjusting it so that balance is produced when the connections
between R and the differencial galvanometer are reversed, as shown in
dotted lines.
t When a circuit includes different metals and the junctions are not
all at the same temperature, an electric current usually flows round the
circuit. Bismuth and antimony give comparatively large currents, and for
this reason are used in " thermopiles," or " thermo- junctions," instruments for
detecting radiant heat or indicating differences of temperature.
J Some makers take the centimetre as the unit of length for the potentio-
meter wire, whilst others use £ of an inch as the unit.
SIMPLE POTENTIOMETER 407
to permit of the Clark cells (E.M.F. 1-433 at 15° C) being conveni-
ently used for adjusting the P.D. drop per unit length of wire
to Y^th of one volt. For by connecting the negative terminal
of the cell to 14 and the positive one through the galvano-
meter to the slider placed at 33 on the wire, and varying the
current through the wire by resistance Rx until balance exists,
the required adjustment is made. When so adjusted the potentio-
meter can be used to measure any E.M.F. between zero and 1-5
50 100
Fig. 244. — Diagram of Simple Potentiometer.
volts, for there is a P.D. drop of one-tenth of a volt on 100
divisions of the wire, and on each of the fourteen coils. P.Ds. not
exceeding o-i volt can be balanced on the wire itself, those
between o-i and 0-2 by the aid of the first coil and the wire.
P.Ds. between 0-2 and 0-3 require the use of the first two
coils and the wire, and so on.
Now that the cadmium cell is much used as a standard
of E.M.F. (1-0184 volts at 17° C.) ten coils would be
sufficient, but the limit of P.Ds. measurable would then be
reduced to about i-i. As the addition of a few coils is not
very costly, it is undesirable to reduce the range of the
instrument by omitting the four coils n to 14 ; in fact, the
addition of four coils, making 18 in all, is to be recommended,
for the range of the instrument is then increased to 1-9 volts.
A number of coils greater than 18 would necessitate the use of
more than one storage cell for producing the constant current
through the circuit,* because the E.M.F. of such a cell, in dis-
charging, falls below 2 volts, but if allowed to fall below 1-9
becomes unsteady. Some potentiometers have been constructed
with 25 to 35 coils for special purposes, and two cells employed
for supplying the constant current.
The range of an ordinary potentiometer may, however, be
doubled by using two storage cells and balancing the standard
cell at half value, e.g. 0-7165 (seven coils and 16-5 divisions* of the
wire) for the Clark cell at 15° C., and 0-5092 for the cadmium cell
* Storage cells have been found to be by far the most convenient source
of current for this purpose.
f Here the word " division " means a main division of the scale ; these
are often subdivided into ten parts.
408
PRACTICAL ELECTRICITY
at 17° C. Under these conditions the drop of P.D. per division
of the wire is TQ2oo °f a vo^ (2 millivolts), and the potentio-
meter readings have to be doubled.
In practice the equal coils of the potentiometer are arranged
(between contact studs) round a circular dial, and a switch arm
makes contact with any one of them desired, as shown at Q in
Fig. 245. The variable resistance Rx, Fig. 244, is generally made
in two parts, Rxl and Rxz, Fig. 245, one for rough and the other
oFo
Fig. 245. — Crompton Potentiometer Diagram.
for fine adjustment, each section of Rxl being nearly equal to
the whole of RX2 ; the latter is usually a circular slide wire con-
tinuously adjustable.
Another device fitted to potentiometers for measuring a number
of P.Ds. in quick succession is a multiple double pole switch
M, Fig. 245, sometimes called a selector switch, and several
pairs of terminals (usually six pairs) connected with this switch,
whereby any pair of these terminals may be joined to the
measuring points, Q and s. The pairs of terminals are marked
A, B, c, D, E, F, in the figure, but to avoid confusion the con-
nections of only one pair (c) are shown.
178. Modern Form of Crompton Potentiometer. — The scheme
of connections described in the last paragraph is substantially
that adopted in the Crompton Potentiometer, of which large
numbers are in actual use. The latest form, however, differs
from this in several details, chief amongst which is that Q, Fig.
246, carries another contact arm insulated from Q and joined to
the lower of the two terminals marked POTENTIOMETER COILS, in
Fig. 246, which gives an outside view of the instrument.
Q is joined to the upper of these two terminals, and by
using them the resistance of any of the coils can be readily
tested. To keep the contacts free from dirt they are all placed
under glass, and the slider s is moved by a handle H, outside
the case. A triple successive contact spring key is placed
INDUSTRIAL POTENTIOMETER
409
in the galvanometer circuit. Pressing the key lightly com-
pletes the circuit through a high resistance ; greater pressure
brings the second contact into operation and short circuits
most of the resistance, whilst still greater pressure cuts all of
it out of circuit, thereby permitting the full sensibility of the
galvanometer as a voltmeter to be utilised.
Example 175.— A Clark cell at 18° C. is balanced by the P.D.
on 103-5 centimetres of the potentiometer wire in Fig. 240 ;
Fig. 246.— Crompton Potentiometer. (General View.)
find the P.D. drop between the extreme ends of the stretched
wire. Answer. — 6*907 volts.
Example 176. — What current must be passed through a poten-
tiometer wire having a resistance of 0-436 ohm per metre, so that
the P.D. drop per division (quarter of an inch) may be one milli-
volt ? Answer. — 0-3613 ampere.
Example 177. — The wire of a potentiometer has a resistance of
thirteen milliohms per centimetre, and is 106 centimetres long ;
find (a) , the resistance of each of the fourteen coils in series with
the wire, (b), the resistance required external to the wire and
coils when the drop per centimetre of wire is o-ooi volt, and the
P.D. of the storage cell used is 2-05 volts.
Answers.— (a) 1-3 ohms,
(b) 7-07 ohms.
Example 178. — By how much must the external resistance in
the case above be reduced when the P.D. of the cell falls to 1-95
volts? Answer. — 1-3 ohms.
179. Dial Potentiometer.— In this form of instrument the
slide wire is replaced by a series of coils arranged around dials
as shown in Fig. 247. Here the dial on the right takes the place
of one-tenth the slide wire in Figs. 244 and 245, and enables steps
of P.D. of yi^ of the P.D. between adjacent studs on the left-hand
dial to be obtained.
The connection between the dials is exactly the same as
that of the two-dial potential divider, shown in Fig. 220, the
4io
PRACTICAL ELECTRICITY
only difference being that the left-hand dial has 150 coils instead
of 9, and the right-hand dial 100 coils instead of 10 ; the whole
100 coils of the right-hand dial have a resistance equal to that of
Fig. 2-17.— N.C.S. Dial Potentiometer.
one coil in the left-hand dial. Variable resistances Rxl, RX2, Fig.
2470, corresponding with RX1 and RX2 of Fig. 245, are provided
for adjusting the current through the coils to give a drop of
Fig. 247*. — N.C.S. Potentiometer. (Diagram.)
P.D. of o-oi volt per coil in the left-hand dial, and there-
fore of o-oooi volt per coil in the right-hand dial. This
adjustment is carried out by putting the arms of the main
dials at readings corresponding with the E.M.F. of the stan-
VOLTMETER CALIBRATION
411
dard cell used, and varying Rxl and R%2 until balance is obtained.
For a Clark cell (E = 1-433) at 15° C. the arm on the left-hand
dial would be set at 143, and that on the right-hand one at 30,
whilst for a Weston cadmium cell (E = 1-0183 at 20° C.) the
corresponding position would be 101 and 83 respectively.
The N.C.S.* potentiometer has three pairs of measuring termi-
nals A, B and v, Fig. 2470;, joined to a selector switch M. To
the pair A -f , A — , the standard cell is usually attached ; B + and
Fig. 248. — Calibration of Low Reading Voltmeter by Potentiometer.
B — can be used for any P.D. within the range of the dials
(1-51 volts), whilst the pair v + and v — are connected with a sub-
divided resistance N, Fig. 2470, by means of which any voltage
up to 300 times the range of the dials may be measured. This
subdivided resistance is a single dial potential divider, with un-
equal coils giving ratios of I, 3, 10, 30, 100, and 300.
180. Calibration of Voltmeter by Potentiometer: Volt (or
Ratio) Boxes.— In nearly all cases a separate source of current is
used to produce the necessary P.D. between the voltmeter ter-
minals, and this P.D. is measured in one of two ways, depending on
whether the maximum reading of the instrument is below or
above the range of the potentiometer. For low reading volt-
meters (say below 1-5) the terminals of the instrument may be con-
nected with one of the pairs of measuring terminals on the potentio-
meter, and the P.D. which produces any given deflection of the
voltmeter measured directly. Different scale readings on the
voltmeter may be obtained by altering the rheostat Rx, in series
with the generator B2, Fig. 248, which gives a scheme of con-
* The letters N.C.S. are the initials of the partners of the firm of
Nalder Bros., Ltd. (Nalder, Crawley & Soames), the original makers of the
instrument.
412
PRACTICAL ELECTRICITY
nections suitable for the purpose, when the resistance of the volt-
meter is not very high. For high resistance instruments a rough
potential divider, indicated in Fig. 249, a form used at the Gity
Guilds College, would be more convenient. By aid of it any
desired reading can be produced on the voltmeter.
Switch
To potentiometer
Fig. 249. — Potential Divider for Voltmeter Calibration.
The coils in the rough potential divider are made of open
spirals of bare platinoid wire, No. 20 S.W.G., and will carry
currents up to three amperes without excessive heating. It can
therefore be used on any voltage not exceeding 750, but the
power used is considerable at high voltages.
To voltmeter /To voltmeter
Fig. 250. — Diagram of " Volt-Box " Connections.
To calibrate a voltmeter reading higher than 1-5 on a Crompton
potentiometer it is necessary to have some means of obtaining a
known fraction of the P.D. between the voltmeter terminals.*
This is generally done by means of a high resistance with tappings
* The N.C.S. instrument is, as already explained, provided with a
subdivided resistance by which P.Ds. up to 450 volts (300 times 1-5) can be
measured.
VOLT- OR RATIO-BOXES
along its length at points giving convenient ratios. Thus, in
Fig. 250, if R be the total resistance between p and Q, and con-
nections be made at points R, s, and T such that the resistance
1-5 4-5 15 45 ISO 450VOlt5
To poCenbiomefcer
Fig. 251. — Ratio-Box Connections.
PR, PS, and PT are |, J, and ^ of R respectively, the P.Ds.
on these portions, when a current is passing through the whole
resistance, will be, half, a fifth, or a tenth that between p and Q.
If, therefore, the whole resistance be connected across the terminals
of the voltmeter v, Fig. 249, the P.D. between P and T will be
^Q that on the instrument, whilst the P.Ds. between P and s, and
p and R will be ^ and ^ respectively. Such an arrangement
is called a " Volt Box," or " Ratio Box," and by connecting
p and T to the " potentiometer " the range of the instrument
could be increased tenfold. Volt boxes with ratios i, 2, 5, 10, 20,
50, 100, 200, and 500 are frequently met with, whilst other boxes
are made with ratios I, 3, 10, 30, 100, and 300. A universal shunt,
such as shown in Fig. 165, serves admirably as a volt box for
P.Ds. which will not produce undue heating of the coils.
In another form
of volt box the con-
nections from the
potentiometer re-
main fixed, and
those from the volt-
meter (or P.D.)
altered to suit the
pressure to be
measured. A volt
box with ratios I. 3,
10, 30, 100, and
300, would have
terminals marked 1-5, 4-5, 15, 45, 150, and 45° volts respec-
tively, as indicated diagrammatically in Fig. 251. An outside
view of a Paul ratio box is given in Fig. 252 and its connections
Fig. 252.— Outside View of Ratio Box.
414 PRACTICAL ELECTRICITY
in Fig. 25 2a. By means of this and a potentiometer, any
voltage up to 1,500 can be determined.
Potential differences greater than the E.M.F. of a standard
cell and capable of supplying steady currents can be measured
5OO CsJ 5,OOOte) IO.OOOCD 25,000(s) 5O,000(fl> •
POSITIVE
MAIN
TERMINALS
READ VOLTS TOI5(DIRLCT) I5O(DIRECT) 3OO(RLADING*3 750(*E*,D'NG) I£OOf DIRECT)
450 (o) X— \ 4,500 fo> X~X 5,000 <fl) X">.lS.OOO (a) X^\25,000 (a) S~\
AA/\AM()VV\/W^
+ FOR —
POTENTIOMETER
O COMMON
5O OHMS
' MAIN
Fig. 2520. — Paul's Ratio Box. Diagram of Connections.
without the aid of a storage cell to send the current through the
potentiometer, by using the P.D. to be measured to produce the
potentiometer current, and varying resistance in series with it
until balance is obtained with the standard cell, the slider being
placed at the reading corresponding with its E.M.F. A diagram
of the arrangement is given in Fig. 253. It is necessary, of
course, in this case, to know the value of R in terms of the
resistance of the potentiometer wire.
S.C Q
rl (?)
X
1
1
ED.'to be
measured
R
1 w
Fig. 253.
181. Standard Resistances for Current Measurements. — As
the P.D. between the terminals of a resistance R ohms with a
current / amperes passing through it is IR volts, we can deter-
mine / by measuring V when R is known. V can be readily
measured on a potentiometer, as described above, and the
combination of a known resistance and potentiometer form
a very convenient arrangement for measuring currents with
LOW RESISTANCE STANDARDS
considerable accuracy. The value and form of resistances for
potentiometer measurements of current depend on the magnitude
of the currents to be measured ; usually they are made for a P.D.
drop of o-i to 1-5 volts
when carrying maximum
current, so that the P.D.
can be easily read on a
potentiometer. Resist-
ances intended for large
currents are frequently
arranged to give smaller
P.D. drops than those for
small currents, for other-
wise the power (I2 R) dissipated in the resistance and the resulting
rise of temperature becomes excessive, unless the dimensions are
very great, or artificial cooling is employed. For a given P.D. drop
F,g. 254. — Standard Low Resistance crooi Ohm to
carry 120 Amperes.
Fig. 255. — Standard Low Resistance o'ooi Ohm to carry 1,500 Amperes. (Elliott Bros.)
the power spent is proportional to the current, so a resistance for
1000 amperes has 1000 times as much heat generated in it per
second as one for one ampere. Resistances for carrying large
416
PRACTICAL ELECTRICITY
currents are usually made from sheet metal having a large surface,
so that the heat produced may be readily got rid of by radiation
and convection.
Two forms of standard resistance are shown in Figs. 254 and
255. Both are fitted with " potential terminals " as well as main
terminals, and their resistances are measured between the
potential terminals. The re-
sistances here illustrated are
intended for maximum currents
of 120 amperes and 1,500
amperes respectively, and P.D.
drops of 0*12 and 1-5 volts.
Fig. 256 shows a particular
arrangement of the Ayrton-
Mather universal shunt useful
for potentiometer work. It
consists of six resistances
suitable foi maximum currents
of 2, 4, 10, 20, 40, and 100
amperes respectively, each of
which gives a drop of one volt
with maximum current. The
arrangement is thus equiva-
j lent to six separate resist-
ances, and possesses the
further advantage that no
changes of connections, either
to mains or potentiometer, are
required over the whole range
of current from 0-2 to 100 amperes. With this resistance and an
ordinary potentiometer reading ^ of a volt to i part in 1000,
any current between 0-2 and 100 amperes can be measured to
an accuracy of one-tenth per cent. When the switch is placed as
in diagram, Fig. 2560, the arrangement is suitable for currents up
to 20 amperes ; M and M being the main current terminals,
and p! and p2 the potential terminals.
182. Calibration of Ammeters. — Fig. 257 shows an arrange-
ment for this purpose ; a known resistance, R, whose current-
carrying capacity is suited to the range of the ammeter, being
joined in series with the ammeter A, a separate source of current,
and a variable resistance, Rx. The variable resistance Rx may
conveniently consist of a number of carbon plates in, but insulated
from, a metal frame, Fig. 258, pressed together more or less by a
screw. Movable metal plates, with terminals attached, enable the
Fig. ^.-
AMMETER CALIBRATION
417
number of carbon plates in circuit to be varied, and also permit of
their being arranged in two or more' parallel circuits when very
large currents are required. For small currents, say below two
amperes, the form shown in Fig. 259 is very useful. It is built up
of numerous discs of sail-cloth, c c c c, carbonised at a very high
temperature, whereby the flexibility and elasticity of the cloth
^Potentiomeber
terminals
Fig. as6a, — Diagram of Connections of Universal Shunt for Strong Currents.
are retained. Brass plates plt p3, pt at top and bottom of the
pile and at an intermediate place serve to make contact with the
discs, and more or less pressure is exerted on the discs by the nut
n and wooden washer e. An insulating sleeve is slipped over
the brass rod h to prevent the discs being short circuited.
183. Comparison of Resistances by Potentiometer. — If two
resistances be connected in series and a steady current passed
through them the P.D. drops on them will be proportional to
their resistances. If, therefore, we measure the P.Ds. on a po-
tentiometer we get the ratio of their resistances directly, and if
one be of known value the other is determined. The method,
illustrated diagrammatically in Fig. 260, is particularly useful
for low resistances having potential terminals, for the resistances
of contact between the leads and the main terminals, which may
be quite considerable compared with the whole resistance, are
2 B
4i8
PRACTICAL ELECTRICITY
eliminated in this test. To obtain satisfactory results the two
resistances should be of the same order of magnitude, and when
Fig. 257. — Calibration of Ammeter by Potentiometer.
an ordinary potentiometer, used in the ordinary way, is employed,
it is desirable that the smallest P.D. be not less than about
half a volt.
There is no necessity, however, to have the current through
the potentiometer of strength sufficient to give a drop of o-i
volt per 100 divisions of the wire, for whatever the current
passing, so long as it is steady, the P.Ds. are proportional to the
potentiometer readings which give balance, and in cases where
Fig. 258.— Carbon Plate Rheostat.
P.Ds. of the order J a volt on either of resistances to be com-
pared would produce undue heating, the current through the
potentiometer wire may be reduced by putting resistance in
series with the storage cell supplying the current. This reduction
COMPARISON OF RESISTANCES
419
must not be carried too far, otherwise the galvanometer ma)7
not indicate an appreciable movement of the slider.
When the two resistances to be compared are very unequal, the
Fig. 259. —Carbon Cloth Rheostat.
P.Ds. to be measured on the potentiometer differ widely, so one
reading would be small and difficult to observe accurately. A
way of improving the conditions of the test is to shunt the largest
of the two resistances, by a volt -box, universal shunt, or similar
subdivided high resistance, Fig. 261, and measure a fraction of
the P.D. on the larger resistance, which is of the same order
Fig. a6o. — Comparison of Resistances by Potentiometer.
of magnitude as that on the smaller resistance. To increase
the potentiometer readings one of two things is necessary,
either the current through the two resistances to be compared
420
PRACTICAL ELECTRICITY
must be increased, or, if this be not permissible, the current
through the potentiometer must be diminished by inserting
resistance at RX) Fig. 261. Shunting R will slightly reduce the
K
AWWwJ
o 66 66 ~ 09 oo oo .
Fig. 261. — Comparison of Resistance by Potentiometer.
P.D., but the error will often be negligible. When this is not
the case a proper correction can easily be made.
184. Measurement of Power. — We have already explained
how P.D. and current can be measured by a potentiometer, and
as the power used in a given circuit is the product of the two, it
is evident that the instrument can be employed for power
measurements. A scheme of connections for this purpose is
given in Fig. 262, where T1 and T2 are the terminals of the part
of a circuit in which the power is to be determined. Here R
represents the volt box and R1 the resistance by means of
which the current is measured.
185. Advantages and Disadvantages of Potentiometer Measure*
ments. — Amongst the advantages of the instrument must be
mentioned the universal nature, and the wide range over which
measurements can be made, for with a potentiometer and its
adjuncts, a ratio box and standard resistances, pressures, currents,
resistances and powers can be determined with great facility and
high accuracy. For pressures and currents the range is very large,
suitable ratio boxes and resistances enabling values of either
quantity, from a fraction of a volt (or ampere) to many hundreds
of volts (or amperes) to be measured.
For testing resistances, the elimination of the resistances of
contacts and connections which the method renders possible, is
of great importance, and on this account it is much used for low
POTENTIOMETER MEASUREMENTS 421
resistance measurements. Moderate or large resistances can be
more easily compared by bridge methods.
The disadvantage of the potentiometer, as compared with the
bridge for measuring resistances, lies chiefly in the fact that two
sources of current, both of which must be very steady for appre-
ciable times, are required with the former instrument, whilst a
bridge needs only one battery, the current from which need nol
T,
R
To poCenDiometer
Fig. 262. — Measurement of Power by Potentiometer.
To potentiometer
be exactly constant. Two adjustments and two readings are
also necessary in the former case, and only one in the latter.
Another disadvantage is that the effects of thermo-electric forces
are not so easily eliminated as in bridge measurements.
Example 179. — In comparing an unknown resistance with a
standard of o-i ohm, without using a standard cell, the potentio-
meter readings were 1-265 and 0-832 respectively; find the value
of the resistance tested. Answer.— 0-1521 ohm.
Example 180. — Two unequal resistances are compared and the
standard (o-oi ohm) is shunted by a ratio box of 100 ohms
total, a 10 ohm section of which is used for the potentiometer
measurement. The readings obtained on the unknown and
known are 0-0430 and 0-0695 respectively. What is the value of
the unknown resistance uncorrected for shunting error of the
standard ? Answer. — 619 microhms.
Example 181. — What is the approximate error in the resistance
measured caused by shunting the standard in the last example ?
Answer. — i part in 10,000, i.e. 0-06 microhms approx.
422 PRACTICAL ELECTRICITY
Example 182. — Supposing the shunt of 100 ohms in Example
1 80 to have been placed across the unknown resistance and
that the potentiometer readings on the known and unknown
resistances were 0-0541 and 0-0928 respectively ; calculate the
unconnected value of the latter, and the approximate correction
for shunting.
Answers. — 0-1715 ohm,and 0-0003 approximately.
CHAPTER X
INDUCED CURRENTS
186. Introduction — 187. Direction of Induced Currents due to Magneto-
Electric Induction — 188. Lenz's Law : Fleming's Rule — 189. Rela-
tion between Quantity Induced and the Resistance of the Circuit —
190. Determination of Constant of Ballistic Galvanometer by Earth
Inductor Method — 191. Distribution of Magnetism in a Bar Magnet —
192. Flux Density over Cross-section, and over Surface of Magnet —
193. Mutual Induction — 194. Unit of Mutual Induction : Henry —
195. Self-Induction — 196. Induction Coil — 197. Induction of Currents
in Parallel Wires.
1 86. Introductory. — Of the several means of producing electric
currents, batteries, thermopiles, frictional machines and dynamos,
the last-named is by far the most important, for without this
method of transforming mechanical energy into electrical energy,
electric lighting, electric traction on tramways and railways, the
electric driving of workshops and factories, the electric refining of
copper, the production of aluminium, and other electrochemical
products, would be commercially impossible. The subject of
induced currents on which the action of the conversion depends is,
therefore, of prime importance to electrical engineers. Faraday in
1831 discovered that electric currents could be produced in wires
and coils by the relative motion of magnets and wires. Had the
principle of " conservation of energy " been thoroughly under-
stood when Romagnosi in 1802 and Oerstedt in 1819 observed
the effect of electric currents on a magnetic needle, Faraday's
discovery would probably have been anticipated by many years.
For the deflection of a magnetic needle by a wire conveying a
current, proved that mechanical energy could be produced by the
action of a current on a magnet, and, conversely, the mechanical
energy used in moving a magnet, near wires should, on the prin-
ciple of conservation of energy, produce equivalent effects in the
electric circuit. This we now know to be the case, and the history
of the progress and developments which have led from the mere
shifting of a compass needle when a wire connected with two
plates of metal dipping in a liquid was brought near it, to the
building of 40, ooo -horse -power dynamo machines, capable of
423
424 PRACTICAL ELECTRICITY
lighting a whole city, forms one of the most interesting examples
of the beneficial association of science and engineering.
187. Direction of Induced Currents due to Magneto -Electric
Induction. — The laws of magneto -electric induction may be
investigated, both as regards direction of currents and quantities
of electricity produced, by the apparatus shown in Fig. 263, which
consists of a bar magnet, a coil of wire, some resistances, and a
simple galvanometer. First as regards direction of current, we
find out by means of a small cell the direction in which the needle
of the galvanometer deflects when a current is sent through it in
Fig. 263. — Magneto-Electric Induction Apparatus.
a known direction. To make matters definite we will suppose the
deflection is clockwise when the terminal TJ is positive. On
connecting the coil terminals to those of the galvanometer by
means of long wires,* and bringing the magnet near the coil, the
galvanometer needle is deflected momentarily and returns to
zero if the motion of the magnet ceases. Removing the magnet
produces a swing in the opposite direction. If the magnet
be turned end for end, and the experiment repeated, the effects
are as before, except that the directions of deflection are reversed.
From these tests we learn — ist. Induced currents flow only when
relative motion is taking place ; 2nd. The direction of the current
is reversed when the relative motion is reversed ; and 3rd.
Reversal of the magnet causes the current induced by any given
relative motion to be reversed ; 4th. By observing the polarity
of the magnet, the direction of its motion and of the deflection
of the needle, and the direction of winding of the coil, we find that
bringing the north -seeking pole of the magnet towards a face of
coil, a counter-clockwise current flows in the coil when looking at
the face to which the pole is brought near ; the same is true on
removing a south -seeking pole, whilst if the north pole be removed
or a south pole approaches the coil, the currents induced are in a
clockwise direction in the coil when looking at the face concerned
* To enable the coil to be placed far enough from the galvanometer so
that the magnet, when near the coil, has no direct action on the needle.
LENZ'S LAW 425
Now the lines of force produced by a magnet are regarded as
emanating from the north pole and entering the south pole, so
that when we look at the face of the coil when a north pole is
brought towards this face, we are looking along the lines of force,
i.e. in the direction of the magnetic force. Further, when the
pole of the magnet is near the face of the coil, a number of the
lines of force of the magnet will link through the winding of the coil,
and we may say that as the magnet approaches the coil, the number
of lines linked with the coil increases. The results of the above
observations may therefore be expressed as follows ; When we look
along the lines of force and the number of lines of force linked with
a coil increases, a counterclockwise current is induced in the coil.
This rule can be applied practically to all cases of magneto -electric
induction. When the number of linkages decreases, the current
will, of course, appear clockwise in direction, when we look at the
coil along the lines of force.
188. Lenz's Law ; Fleming's Rule. — As we have already men-
tioned (Section 5) when an electric current passes round a coil
on an iron bar the iron becomes a magnet, and even if the iron
is removed the coil exhibits magnetic properties. A simple test
with a compass needle shows that the north pole of the coil is
that end or face looking at which the current circulates counter-
clockwise, so we see that when a north pole approaches a coil
and induces a current in it, the direction of this current is such
as to produce north polarity at the end of the coil nearest the
north pole of the magnet. The magnet will therefore be repelled.
The experimental facts may be summed up in the statement that
the direction of induced currents produced by relative motion oj
coil and magnet is such as to oppose the motion producing it. This
is one way of stating Lenz's Law of induced currents.
Another rule by which the directions may be remembered was
stated by Prof. J. A. Fleming and is known as Fleming's Right-
hand rule. Here we consider the magnet stationary and the coil
moving, and confine our attention to a small part of the wire
crossing the lines of force. Put the thumb, the index finger, and
middle finger of the right hand mutually perpendicular, and place
the hand so that the index finger points along the lines of force,
the thumb in the direction in which the wire moves, the middle
finger will then point in the direction of the induced E.M.F. in
the portion of wire considered.
If the magnet and coil in Fig. 263 be placed co-axial, and the
magnet be steadily passed through the coil, and away on the other
side, the induced current will be in one direction until the magnet
is halfway through ; it will then reverse, increase, and diminish as
426 PRACTICAL ELECTRICITY
the magnet moves away from the coil. If the movement of the
magnet be rapid the galvanometer needle will give a kick in
one direction and then jerk back, and when the motion is
extremely quick no perceptible movement of the needle will
occur. This proves that the quantity of electricity which
passes round the circuit as the magnet approaches the coil is
equal and opposite to the quantity that passes as the magnet
goes away. This is equally true whether the magnet passes right
through the coil, or is brought up to the coil and then taken
back to its initial position.
189. Relation between Quantity Induced and Resistance of
the Circuit. — A ballistic galvanometer connected with the coil
in Fig. 263 enables the quantities of electricity produced by
definite relative movements of the coil and magnet to be measured.
Suppose we place the magnet inside the coil, so that the middle of
the magnet is at the middle of the coil and the two are coaxial.
Withdrawing the magnet to a distance quickly, so as to approxi-
mate to the condition that the whole quantity passes through
the galvanometer before the needle has moved appreciably from
its zero position, will produce a swing of the needle, from which
the quantity of electricity can be calculated by the formula
c
Q — k sin —(see Section 149),
where k is the " constant " of the instrument.
Instead of withdrawing the magnet from the coil by one move-
ment, we may do it in two or more steps, and observe the swing
produced by each step. When this is done, and the several quan-
tities of electricity added together, it is found that their sum is
equal to that produced by the withdrawal in a single movement,
provided the initial and final positions are the same in the two
cases. Expressed symbolicallv, we have
Q = 0i + <?2 + QB + etc.,
Qi> (?2> (?3» etc., being the quantities produced by the several
steps. This is a most important result, for it proves that the
quantity of electricity induced depends on the initial and final
relative positions of the magnet and coil, and not on the inter-
mediate positions they may have occupied.
The above statement presupposes that the resistance of the
circuit remains constant during the experiments. Changing the
resistance alters the quantity produced by any given movement.
To enable the law of variation to be experimentally determined,
resistances are placed between the mercury cups shown to the
right in Fig. 263, and more or less resistance can be included in
LAW OF INDUCED QUANTITIES 427
the circuit by altering the position of the copper bridge piece b.
To simplify the experiment, each of these resistances is made equal
to the sum of the resistances of galvanometer, coil, and connecting
leads, which, in the apparatus shown, is about 2 ohms. The
experiment is best carried out by placing the magnet centrally
within the coil, and, when the galvanometic needle is quite at rest,
suddenly withdrawing it, and observing the swing. This is done
first with no added resistances in the circuit, and then with i, 2,
3, etc., coils inserted. It is convenient to tabulate the results
observed and calculations made from them as indicated below
Total Resistance
of Circuit.
First Swing
Produced.
Sine of
Half Swing.
Product of
Total Resistance
and Sine
of Half Swing,
2
4
etc.
when it will be found that the numbers in the last column are
practically equal. In this way we can prove that the quantity
of electricity induced in a circuit by a given relative movement oj
magnet and coil, is inversely proportional to the resistance of the
circuit* Putting this statement into symbols we have
R
where N is some constant.
We may now enquire what the constant N represents ? In the
experiment as performed, the initial and final positions of the
magnet relatively to the coil have been maintained constant. Now
in the initial position, there were a certain number of lines of
force of the magnet linked with the coil, and in the final position
another number (generally zero if the magnet has been taken far
enough away) were linked with it, so we see that the change of
linkages has been maintained constant in the several experiments.
Further we know that if the magnet be only partly withdrawn,
the swing, and therefore the quantity, will be less than before,
* The small differences from equality obtained in a carefully made
set of observations arise from the change produced in the " damping "
of the galvanometer swings when the resistance of the circuit is altered,
the damping decreasing as the resistance increases. The differences dis-
appear if we determine the decrements and correct the several quantities
by multiplying by 1 1 4 — 1 (see Section 148).
428 PRACTICAL ELECTRICITY
so we are led to the conclusion that the constant N in the above
equation represents the change of linkages of lines of force with
the coil.* The expression
is similar in form to the usual method of writing Ohm's Law, and
it is easily remembered by regarding it as the Ohm's Law of in-
duced quantities. In fact, it can be deduced directly from Ohm's
Law and the definition of E.M.F. given in Section 550, viz. the
rate of cutting of lines of force.
Suppose at the beginning of a short time, t, the number of lines
linked with the coil to be Nv and at the end of the interval N2.
The change of linkage is N1 — N2, and the average rate of cutting
of lines will be - - - -, which may be written
SN
t '
where m = N - N
2,
where E is the average E.M.F. during the interval t.
Writing Ohm's Law as
'=!•
T 8N
we have / = — ,
§«-!
but It = quantity,
or the quantity of electricity which passes in a given time equals
the change of number of lines which occurs in that time divided
by the resistance of the circuit. As this is true for any short
interval of time, it is true for the whole time, so the whole quantity
equals the whole change of lines, divided by the resistance, i.e.,
* When the coil consists of many turns, as is usually the case, the
number of lines linked with one turn may differ from the number linked
with another, and the whole number of linkages is the sum of numbers
of lines linked with the several turns.
EARTH INDUCTOR METHOD 429
This relation between Q, N, and R is of great importance, and
should be thoroughly understood. If the quantity is to be
expressed in coulombs and R is in ohms, we have (since I
coulomb = -fo C.G.S. unit of quantity, and i ohm = io9 c.G.s.
units of resistance)
Q = -£— 5 coulombs, (no)
N being expressed in C.G.S. lines.
190. Determination of Constant of a Ballistic Galvanometer
by Earth-Inductor Method. — The value of N in the last equation
may be found by observing the swing produced on a ballistic
galvanometer, if we know, or can determine, the constant of the
instrument. This may be done by measuring the periodic time
of the needle, the sensitiveness to steady currents, and the decre-
ment as described in Sections 146-8, or by discharging through it a
known quantity from a condenser of known capacity charged to a
known P.D. But the simplest way of determining the constant
is the Earth Inductor Method, in which a coil of known area and
number of turns is connected by flexible leads with the galvano-
meter and suddenly turned through 180° in the earth's magnetic
field, the strength of which may be measured by methods de-
scribed in Chapter II. (see Section 27).
Values of the horizontal component H of the earth's magnetic
field for several places in England are given in Table III, Section
36. These strengths are rather small, and in this country, where
the " magnetic dip "* approximates to 70°, it is convenient to
make use of the vertical component of the earth's magnetic force.
Calling the vertical component U we have
U = H tan d,
where d is the angle of dip, and when H and d are known,
U can be calculated. Taking H = 0-185 an^ ^ = ^7°'5> the
approximate values for undisturbed areas near London, we get
U = 0-185 tan 67°-5 = 0-447,
approximately.
A coil suitable for use as an earth inductor is shown in Fig.
264. It is wound with 100 turns No. 18 wire of mean area 1000
square centimetres approximately. When held in a horizontal
plane, each turn is linked with 0-447 x IOO° unes °f f°rce due to
the vertical component of the earth's field, so the total linkage
is 447 x 100 or 44,700. When the coil is turned upside down,
* The "magnetic dip" is the inclination to the horizontal at which a
truly balanced, freely-suspended needle sets itself when magnetised
(see Section 15).
430
PRACTICAL ELECTRICITY
there are an equal number of linkages in the opposite direction
as regards the coil, the lines now passing from face B to face A
instead of from face A to face
B. The total change of linkage
is therefore,
zUnA,
where n is the number of turns
in the coil and A the average
area, so in this case
N = 89,400.
If the circuit of the coil be
Fig. 264.-Simple Earth- Inductor. completed through the galvanO-
meter when the movement is made, a quantity of electricity
_. 89,400
Q = -2g — x io~8 coulombs
K
will pass through the circuit, where R is the total resistance.
Since
Q — k sin — ,
where s is the swing produced, we have,
89,400 x io"8
and R and s being known, k is determined.
Should the resistance of the circuit when the earth inductor is
used be made the same as when the observations were made in
measuring N (formula no), there is no need to calculate k, for
in one case
DISTRIBUTION OF MAGNETISM 431
Either of the last two formulae may be written
N1 = A7' sin S-± , (in)
where N' is the change of linkages which would produce a swing
of 180° under the then existing conditions. The constant N' may
be termed the linkage constant of the ballistic galvanometer
and circuit. For a reflecting galvanometer, as in Section 149,
the " linkage constant " may be taken as the change of linkage
which produces a swing of one division.
Fig. 265. — Apparatus for Testing the Distribution of Magnetism in a Bar Magnet
191. Distribution of Magnetism in a Bar Magnet. — The induc-
tion of electric currents due to change of linkage of magnetic
lines may be used to find the distribution of magnetism in a
magnet. A convenient form of apparatus is illustrated in Fig.
265. The cylindrical bar magnet B, 30 centimetres long and
1-67 centimetres diameter, is supported in a vertical position in
a wood block, w. A coil of 100 turns of fine wire on a thin brass
tube which fits closely, but slides freely on the bar, is connected
by long flexible leads to a ballistic galvanometer, G. The brass
sleeve s, on which the coil rests, can be clamped at any point
of the bar, and the bar is graduated in centimetres, so that the
distance of the centre of the coil from the middle of the bar can
be easily read off.
In commencing the experiment we fix s so that the centre of
the coil c is at the middle of the bar when c rests on s. On
suddenly sliding the coil off the bar a swing is produced on G,
from which the quantity of electricity may be determined, the
constant of the galvanometer being conveniently found by the
earth inductor method described above. As the coil fits close
432 PRACTICAL ELECTRICITY
to the magnet and is of small axial length and radial depth,
we may say that the lines of force linked with each turn will be
approximately the same, and if we designate by $0* the number
of lines of force passing through the central section of the bar, the
change of linkage produced by sliding the coil off and away from
the magnet will be 100 $0, and the quantity of electricity which
flows round the circuit, in consequence of this change, is given by
TOO ^
Q = — ^— - x io~8 coulombs,
K
and <1>0 = R sin -° x io8,
100 2
s0 being the swing produced, and R the resistance of the circuit.
By moving the sleeve s one centimetre higher, and repeating
the experiment, *x the flux through the cross-section of the bar
which is one centimetre from the centre can be found. Similarly
the fluxes at distances 2, 3, 4, etc., . . 15 centimetres, from the
centre may be determined and a curve plotted showing the
relation of flux to distance from the centre.
Turning the bar upside down, the distribution in the other half
of the bar can be investigated, and a curve for the complete
magnet obtained, such as is shown in Fig. 266.
Instead of turning the bar the other end up, it is advisable in
testing the second half of the bar to fix it by the upper end and
slide the coil off the bottom, for reversing the bar in the earth's
field affects the magnetism slightly and the curves for the two
halves do not quite join at M, Fig. 266. A better plan still is to
support the magnets in a horizontal direction perpendicular to the
magnetic meridian, so that the earth's field will have little or no
effect on the longitudinal magnetisation of the bar.
192. Flux Density over Cross-sections and over Surfaces of a
Magnet. — On dividing the values of the flux obtained in the experi-
ment just described, by the area of the cross-section in square
centimetres, numbers are obtained to which the name " Flux
density " or " Induction density " are given, this quantity being
usually denoted by B. If we consider the fluxes through two
sections at one centimetre apart, say for definiteness $3 and <fr4,
we see from the curve, Fig. 266, that $3 is greater than <&4, and as
lines of force always form closed curves, a number of lines,
<I>3 — $4, must have emerged from the cylindrical surface of the
bar between the two cross-sections, and if we divide this difference
* The number of lines of force passing through a given area is often
spoken of as the magnetic flux through that area, so in this case $>0 is the
flux through the middle section of the magnet.
FLUX DENSITY IN MAGNET
433
c
0)
"T3
X
CL
^
*^*
!
rf
k-
F
*"*x
^
•1
/•
^
N
^s
|
w
i
<
t\
y
s
7
/
\
/
s,
1
r
/
\
.
/
\
/
1
I
/
\
/
\
/
\
J
!
\
I
14 12 10 8 64 2+0-24 o 8
Distances from centre of magnet
Fig. 266. — Distribution of Magnetism over cross sections of bar.
10 12
Tauu
<^1
•c8 +<&OO
*t-
"S
J_
3
s
1
<0
s
"^
•A
•j_ HOO
~o
X
E
\
g>
s
V
%
V
03 n
^
S °
X
3i
^>s
X3
>*
"s
<O
c
K
"X
<D
V
"^
"" —IOO
X
^
"^
3
.—
^s
<
u_
X
X
-2OO
L
1
V
14 12 IO 8 6 4 2+0-2 4 6 8 IO 12 14
Distances from centre of magnet
Fig. 267, — Distribution of Magnetism over cylindrical surface of bar.
2C
434 PRACTICAL ELECTRICITY
#3 — $4> by the area of the cylindrical surface of the bar between
these cross-sections, viz. TT d, where d is the diameter of the bar in
centimetres, the resulting number will be the average density of
lines emerging from this portion of the magnet's surface. This value
may be plotted on a vertical line at a distance representing 3 J centi-
metres from the centre of the bar, and by obtaining a number of
such points a curve showing the approximate surface distribution
of flux along the length of the bar can be drawn. To avoid great
inaccuracy in the values of surface density, owing to their being
calculated from the difference of two relatively large quantities,
either or both of which may be in error, the difference should
be taken from the smooth curve drawn amongst the points,
shown in Fig. 266, instead of from the numerical values
obtained. When this is done a curve like that shown in Fig.
267 results.
As the cross-section of the bar is the same throughout its length,
the curve in Fig. 266 represents the flux, as well as the flux-density
at different cross-sections of the bar, to a certain scale, and from
it we conclude that the flux, and also the flux density, in a
cylindrical bar magnet, is greatest about the middle, and falls off
rapidly towards the ends, whilst Fig. 267 shows that the surface
density is very small in the centre and increases as the ends are
approached. It is interesting to notice that the greatest density
over the middle cross-section is about 4,600, whilst the greatest
surface density is approximately 200 lines per square centimetre,
in this particular case.
Example 183. — The swing produced on a ballistic galvano-
meter was 15° when the earth inductor, Fig. 264, was quickly
turned over in the earth's field. Find the " constant " of the
galvanometer, having given that the resistance of the galvano-
meter is 0-87 ohm and of the earth inductor and leads, 1-65
ohms. (U — 0-447) Answer. — &= 0-00272.
Example 184. — Withdrawing the magnet from the centre of
the coil c in Fig. 263 causes a swing of 80 degrees on the galvano-
meter when the total resistance of the circuit is 2 ohms. Cal-
culate the flux through the magnet, having given that the coil
has 400 turns and that the galvanometer is constant 0-00283.
Answer. — 910 C.G.S. lines.
Example 185. — What is the induction density at the centre of
the magnet mentioned in Example 184, its dimensions being
15 X 1-6 X 0-25 centimetres.
Answer. — 2,280 c.G.s. lines per square centimetre, approxi-
mately.
MUTUAL INDUCTION
435
Example 186. — Find the " linkage constant " of the galvano-
meter and circuit- in Example 184.
Answer. — 910 X 400 -f- sin 40° = 566,000.
193. Mutual Induction. — Not only can currents be induced in
circuits by bringing a magnet near them, but, as a coil carrying a
current has magnetic properties, relative motion of two circuits,
one of which is conveying an electric current, produces a current
B
T, T
Fig. 268. — Mutual Induction Apparatus.
in the other circuit. This is called mutual induction, and the
two circuits are generally spoken of as the primary and secondary
circuits respectively. Relative motion is, however, not essential
in this case, for stopping the current in the primary is equivalent
to removing it far away from the secondary circuit. A change
of current in the primary circuit thus produces an induced current
in the secondary. This only occurs when the relative position
of the two circuits is such that some of the lines of force produced
by the primary are linked with the secondary circuit.
The laws of mutual induction may conveniently be studied
by the apparatus shown in Fig. 268. Cj and C2 are two coils, the
former of which can be placed inside the latter. The winding on
G! (the primary coil) is connected with a storage battery B,
through a number of resistances, indicated diagrammatically
on the board, by means of which the current through cx may be
varied in known proportions. In the actual apparatus, the
current can have values proportional to i, 2, 4, 6, 8, 10, by
placing the copper bridge piece b in the proper mercury cups.
The direction in which the current flows through Cj may be found
by an examination of the winding, and testing or observing the
polarity of the battery ; and the relation between the direction
of deflection of the galvanometer G, and that of the current in C2,
which is connected by long wires to G, but is entirely insulated
from the primary circuit, can be determined as explained in
Section 187.
436 PRACTICAL ELECTRICITY
If we place cx inside C2 and then complete the primary circuit,
the galvanometer needle will give a swing and then return to
zero. It will remain at zero so long as the primary current
remains constant and the position of the two coils is unaltered.
From the direction of the first swing of the needle it will be
seen that the current induced in the secondary coil is in the oppo-
site direction to that started in the primary. On stopping the
primary current, a swing equal in magnitude to the previous one
but in the opposite direction, will be produced. These tests show,
first, that starting a current in the primary circuit causes a transient
inverse* current in the
secondary ; second, stop-
ping a current in the
primary causes a trans-
ient direct current in the
secondary.
Strengthening the pri-
mary current acts in the
same way as starting a
current, whilst weaken-
ing the primary current is
268*. Fig. *». qualitatively equivalent
to stopping a current.
When the above-mentioned experiment is performed with
currents of several strengths, and the several quantities of
electricity produced determined by a ballistic galvanometer, it
is found that the quantity induced in the secondary circuit is
proportional to the strength of the primary current which is started
or stopped.
The quantity induced depends not only on the strength of the
inducing current but also on the relative position of the two coils.
If the primary coil be placed centrally inside, Fig. 2680, the second-
ary effect is greatest, and stopping the current in this case is
exactly equivalent to withdrawing the primary suddenly to a
distance with the current still flowing. When the coils are placed
co -axial but the primary above the secondary, Fig. 2686, the
effects are much reduced, whilst when the coils are placed
with their axes at right angles and intersecting as shown in Fig.
268c, the effect of starting or stopping the primary current is
nil. These facts are expressed by saying that the mutual in-
duction of the two coils in position 2680 is zero, that in 2686,
small, and in Fig. 2680 the mutual induction is a maximum.
* A current in direction opposite to the primary current is called an
inverse current, and one in the same direction a direct current.
MUTUAL INDUCTION 437
When the two coils are standing side, by side, close together,
Fig. 268^, the mutual induction is small and negative, for
starting a current in Cj causes a direct current in c.2.
Another important fact may be demonstrated by interchanging
the two coils, i.e. using C2 as primary and Cj as secondary. When
Fig. 268c. Fig. 268^.
this is done, and the resistance of the secondary circuit made the
same as before, experiment shows that the induced quantity pro-
duced by a given change of current is exactly the same in the two
cases, whatever the sizes, shapes, or numbers of convolutions in
the two coils. This fact suggested the name mutual induction.
The mutual induction of two coils is much affected by the
presence of iron, and to show this an iron core I, Fig. 268, which
can be fixed inside the coil clf forms part of the apparatus.
When this is inserted the quantities induced by a given change
of current are greatly increased, and to prevent damage to the
galvanometer it is necessary to increase the resistance of the
secondary circuit when the iron core is being used. A core of
the size and proportion shown (six inches long and f inch dia-
meter) when employed in this apparatus, multiplies the effects
about twenty times, and in making tests with it, care must be
taken to avoid direct action between the core, which becomes
an electro -magnet, and the needle of the galvanometer.
194. Unit of Mutual Induction : Henry. — A quantitative mean-
ing is given to the expression mutual induction by defining it as the
linkage of lines of force with one coil due to unit current in the
other coil. It can be measured by observing the quantity induced
in one of the coils by stopping or starting a measured current
(say /' C.G.S. units) in the other. We have then
Linkage N = I'M', (112)
where M' is the mutual induction, or co-efficient of mutual
induction as it is called ; and since
N
Q = — x io~8 coulombs (Section 189),
438 PRACTICAL ELECTRICITY
I'M'
we have Q = — — x icr8 coulombs ;
K
and as /' = - — , when 7 is the current in amperes,
10
we get Af' = 5L, x io9 C.G.S. units, (113)
where Q is in coulombs, R in ohms, and 7 in amperes.
The practical unit of mutual induction is for convenience taken
at io9 C.G.S. units, and is called the " henry," so the above
expression becomes, on writing M x io9 for M',
M = —- henrys,
, . s
k sin —
ry
or M= — — R henrys, (114)
k being the constant of the ballistic galvanometer in coulombs.
195. Self -Induction. — Lines of magnetic force may be linked
with a coil not only by its being placed in a magnetic field, or
near another circuit conveying a current, but also by a current
in the coil itself. In this case the linkage is said to be due to
self-induction, and the co -efficient of self-induction* of a coil
is denned as the number of linkages due to unit current in the
coil. The henry is the unit of self-induction as well as of mutual
induction, and a coil whose inductance is one henry is such that
the linkages of lines of force with the coil when one c.G.s. unit
of current is passing through it, is io9, and the number due to one
ampere, io8.
196. Induction Coil. — The quantity of electricity which passes
round a circuit of fixed resistance due to a given change of linkage
is, as we have already seen, independent of the time in which the
change occurs. The current being a transient one, must increase
and then decrease again, and its average value must be greater
the shorter its duration, so that although the quantity is in-
dependent of the time in which the change of linkage occurs,
this is by no means true of the current. The quicker the change
of linkage, the greater the current, and also the greater the
E.M.F. If, therefore, we can make a given change of linkage
very quickly, a large E.M.F. can be induced in a circuit. Further,
the change of linkage and therefore the E.M.F. can be increased
" The word " inductance " is now commonly used for the expression,
" coefficient of self-induction."
INDUCTION COIL
439
by increasing the number of convolutions of wire through which
the lines of force link, so that to produce a high E.M.F. a large
number of lines of force, linked with a large number of convolu-
tions of wire, combined with rapid change of lines, is required.
A common piece of apparatus in which these principles are
made use of is the " induction coil " of which tens of thousands
T, T2
Fig. 269. — Diagram of Induction Coil.
are in daily use for motor-car ignitions, X-ray work, wireless
telegraphy, etc.
A diagrammatic view of a simple form of induction coil
is given in Fig. 269. It has two circuits, a primary p p con-
sisting of a comparatively few turns of copper wire wound
round a bundle of iron wires 1 1, and a secondary coil, s s, of a very
great number of turns wound outside the primary and entirely
insulated from it. The primary circuit is completed through a
contact breaker, A, which acts like the armature of a trembling
bell, a switch c and battery B. On closing the switch the current
flows round the primary coil and makes the iron core, 1 1, into an
electromagnet and therefore produces a large number of lines of
force linking through both primary and secondary. The core
being an electromagnet, attracts the piece of iron rod or hammer
H supported on the spring s, and breaks the contact between
a piece of platinum on the spring and the platinum-tipped screw p,
440
PRACTICAL ELECTRICITY
thus stopping the current and causing the magnetism in the
core to change very rapidly. This produces a high E.M.F. in
both coils, but especially in the secondary circuit, which has a
large number of turns. Transient E.M.F. 's of tens of thousands
of volts can be produced in this way, between the secondary
terminals, TJ and T2, using a battery whose E.M.F is only a few
T2
Fig. 270.— Marconi zo-inch Induction Coil.
volts, say 2 to 10, and on this account the secondary winding
and secondary terminals must be exceptionally well insulated.
The action of an induction coil is improved by shunting the
break with a condenser K, as shown dotted in Fig. 269, for by this
means the sparking at the platinum contacts is lessened, and the
current stopped more quickly. Induction coils are often provided
with rocking commutators by which the direction of the primary
can be reversed without altering any wires. One is seen at c,
Fig. 270, which represents a form of Marconi coil much used for
Wireless Telegraphy.
The reason for using a bundle of iron wires as the core instead of
a solid iron rod, is to obtain quicker magnetisation and demagneti-
sation of the core. If the core were solid, the change of magnetic
flux which occurs on making or breaking the circuit would induce
electric currents in the material of the core, the direction of which
would, by Lenz's Law, oppose, and therefore delay the change.
These currents would flow in planes at right angles to the axis
INDUCED CURRENTS 441
of the core, and by using a bundle of ,wires the resistance of the
current paths in these directions is enormously increased, and the
eddy currents thereby prevented. Another advantage is also
obtained, viz. that heating of the core due to eddy currents
is practically eliminated, and a considerable waste of energy
avoided.
197. Induction of Currents in Parallel Wires. — If we have two
wires near each other, and a current is started in one of them,
there will be an induced E.M.F. in the adjacent wire, and if the
circuit of this wire is closed, a current will, in general, flow in this
circuit. The induced current is an inverse one on starting
or increasing the primary current, and direct when the current
is stopped or decreased in strength. This phenomenon led to
considerable inconvenience in telegraph and telephone lines
which ran side by side for long distances, until means were
taken to reduce the effect by twisting or crossing the wires at
intervals, so that the mutual induction between the circuits was
positive in some parts and negative in others.
Example 187. — What would be the average E.M.F. generated
in the earth inductor described in Section 190 supposing it to be
turned through 180°, about a horizontal axis, in T\j of a second.
Answer. — The change of linkage in TXQ of a second is 2 UnA (see
Section 190) .'. the average rate of change = 10 x 2 UnA, and
this equals the E.M.F. in C.G.S. units.
.'.Average E.M.F. =10x2x0-447x100x1000 c.G.s. units.
=8-94 X io5 c.G.s. units,
and dividing by io8 to bring it to volts, since I volt is eaual to
io8 C.G.S. units of E.M.F., we have
Average E.M.F. = 8-94 X io~* volts,
or =8-94 millivolts.
Example 188. — Calculate the mutual induction between the
coils of Fig. 268, when q is inside C2, having given that stopping
a current of 2 amps, in Cj caused a swing of 50° of the galvano-
meter, and the swing due to the earth inductor, connected
directly with the galvanometer, is 17°. Resistance of galvano-
meter, coil C2, and earth inductor being 1-91, 1-44, and 1-64 ohms
respectively. Answer. — 0-0012 henry, approximately.
Example 189. — Find the E.M.F. in the secondary circuit
(Example 188), assuming the primary current to fall at constant
rate from 2-5 amperes to zero in a ten-thousandth of a second.
Answer. — 30 volts.
CHAPTER XI
MAGNETISATION OF IRON
198. Lifting Magnets — 199. Relation between Lifting Force and Current-
Turns — 200. Lifting Force and Flux Density — 201. Magnetic Satura-
tion— 202. Magnetic Field produced by a Current in a Straight
Conductor — 203. Magneto-Motive Force — 204. Testing Magnetic
Properties by the Ballistic Method — 205. Permeability — 206. Hys-
teresis of Iron — 207. Remanent Magnetism, Coercive Force — 208.
Loss of Energy due to Hysteresis; Mechanical Analogy — 209. The
Magnetic Circuit ; Reluctance.
198. Lifting Magnets. — One of the properties of the electric
current mentioned in the early part of this volume was that
an iron rod becomes magnetic when a current passes round it
(Fig. 5), and in Fig. 14 a horseshoe electromagnet is shown
supporting a weight. This property is now employed in many
ironworks and shipyards for handling and transporting material.
A properly designed electromagnet hanging from a crane hook
can be used to pick up material such as bars, sheets, plates,
or rails without using slings or ropes, and can be deposited
in any desired position by stopping the current circulating
in the electromagnet. A lifting magnet employed in this way
is shown in Fig. 271, the advantages of which are that loading
and unloading are done simply by starting and stopping
the current, thus effecting considerable saving in time and
labour.
199. Relation between Lifting Force and Current-Turns. —
An apparatus such as shown in Fig. 272 may be conveniently
used for rinding out how the force of attraction between
an electromagnet and its armature depends on the strength
of the current and on the number of turns of wire em-
ployed.
The current may be measured by an ammeter and the force of
detachment by a spring balance. To obtain consistent results
great care must be taken to have the surfaces of contact as
perfectly plane as possible, and that the armature is put on
in exactly the same way each time. In carrying out a series
of tests on the apparatus shown in Fig. 272, the numbers given
442
LIFTING MAGNETS
443
Fig. 271. — Witton Kramer Magnet Lifting Pig Iron,
in the following table were obtained by students of the City
Guilds College. They are plotted in a curve in Fig. 273.
Strength of Current in Amperes. Magnetic Pull in Pounds.
0-3 •• 0-25
0-5 1-5
0-6 3'5
0-8 5'0
i-o 6-9
i-3 975
1-5 12-25
2-0 1575
2'5 20-0
From the shape of the curve we may conclude that when the
current is small, the pull increases more rapidly than the current,
and when the current is large the pull increases more slowly
than the current, for the curve tends to bend over to the right.
444
PRACTICAL ELECTRICITY
The coils on the iron in Fig. 272 are wound in twelve sections,
numbered i to 12, each of 100 turns, connected with the mercury
Fig. 272. — Apparatus for measuring Magnetic Pull,
cups shown at 1 1', 2 2', etc., Fig. 2720, and the coils may be joined
in series or parallel, as indicated in Figs. 2720 and 2726 respect-
ively, or partly in series and partly in parallel as in Fig. 272^,
which shows three coils in series and four in parallel. With
the coils all in series, Fig. 2720, a current passing from T
to x' will pass 1,200 times round the iron core, whereas
T'
*%nrmnn
Fig. 2720.— Mercury Board. All Coils in Series.
when arranged as in Fig. 272^, it will go 100 times round,
and in Fig. 2720, 300 times. Experiment shows that to
produce a certain pull the current required when the coils
LOWER DIVISION
MAGNETIC PULL
445
are all in parallel is twelve times as great as that necessary when
the coils are in series, and we are thus led to the conclusion that
the pull depends on the product of the current and the number
of times it passes round the core. In other words, the pull
depends on current x turns, and, as current is usually measured
T'
* */* * 4 * * t * *
I2<
Fig. 272*.— Mercury Board. All Coils in Parallel.
in amperes, we may say that the pull depends on the ampere-
turns ; the composite word " ampere-turns " meaning amperes
multiplied by turns. From this it will be understood that the
winding of the magnet may be either a large number of turns of
thin wire for carrying a small current, or a small number of
turns of thick wire for carrying a large current, depending on
the source of current available.
Further, from the shape of the curve in Fig. 273, coupled with
the fact that the electric power spent in heating a given winding
of the magnet varies as the square of the current, it is evident
that the pull per watt expended diminishes rapidly when the
pull becomes large. The efficiency of the magnet as a lifting agent ,
efficiency being measured by pull per watt expended, goes down
T?
Fig. 272C. — Mercury Board, showing 3 coils in Series and 4 in Parallel.
after a certain excitation (measured in ampere-turns) is reached,
so we are led to enquire whether the pull can be augmented in any
way other than by increasing ampere-turns. An obvious thing
to try is to increase the thickness of the iron core, and thereby
increase the area of cross -section of the magnetic material ; and
another is to try cores of different lengths. Experiments carried
out on these lines prove that for a given length of core and a
446
PRACTICAL ELECTRICITY
given number of ampere -turns, the pull is proportional to the area
of the core, and that for different lengths of core of given cross-
section, and with a given number of ampere-turns, the pull
diminishes as the length increases. We are thus led to the con-
clusion that the core of a lifting magnet should be short and of
large cross-section.
zo
15
I
§10
E
I 5
A
0-5 I-o 1-5 2-0
Current in amperes
Fig. 273.— Relation between Magnetic Pull and Current.
2-5
200, Lifting Force and Flux Density. — We may now enquire
into the relation that exists between magnetic pull p, and flux
density B. From analogy with electrostatic attraction, page
82, we may surmise that the force will be proportional to the
square of the flux density, and this is found by experiment
to be the case. The apparatus described in the preceding
section (Fig. 272) may be used to prove this statement, for by
winding a few turns of fine wire round the poles of the magnet,
and connecting this coil with a ballistic galvanometer, the change
of flux caused by reversing the current in the main winding can
be found. As the flux is reversed by reversing the current, the
actual flux is half the change produced by reversal. On plotting
the values of half the change of flux measured ballistically,
and the square root of the pull per pole produced by the corre-
sponding current, the points lie approximately on a straight
line passing through the origin, Fig. 274.*
Hence v p : : ft
P :: B*
* In this curve half the change of flux is divided by the area of the iron,
so as to get flux density.
LAW OF MAGNETIC PULL
447
Combining with this the statement made in the last section,
that pull is proportional to the area a of the core, we have
or p = B2 A, x a constant.
If, therefore, we can determine the value of this constant,
we shall have an important formula relating to lifting magnets.
If we measure the area of the cores in the electromagnet in
-3
•6
O 2 4 6 3 IO 12
Flux densiby in bhousa/nds
Fig. 274. — Relation between *J Pull and Flux Density.
Fig. 272, then by aid of the curve in Fig. 274 an approximate
value of the constant can be found. The actual area of each
core is 6.74 square centimetre, and from Fig. 274 we get
0-00037 Bt
Expressed in dynes, we get
, _ 453'6 x 981 y 3-7
~*~"
and as the area A of the two cores is 1-48 square centimetres,
p = 0-041 B2 x 1-48 dynes,
= 0-041 B2 A.
This does not differ greatly from the theoretical value I .
( = 0-0398) given below ; in fact, the agreement is within the
448
PRACTICAL ELECTRICITY
possible error of experiment, which, in pull tests, is fairly
large.
That the pull between two surfaces of area A, over which
the flux density is B, may be expressed by the formula
B*A
p (dynes) = — - ,
can be seen from the following considerations : —
Let L M and L' M', Fig. 275, represent sections of the surface
of poles s and N, between which a magnetic flux of density B
lines per square centimetre exists. Imagine the surfaces separated
by an air gap of in-
finitesimal length,shown
M1 much magnified in the
figure. A unit pole
placed in the middle of
M the gap would experi-
ence a force of B dynes,
for there are B lines
per square centimetre,
Now the unit
S
Fig. 275.
and the strength of the field is therefore B.
pole (unit quantity of magnetism) will be repelled by the pole
N just as much as it is attracted by s, and as the sum of these
forces is B dynes, we may regard it as being attracted by s
r>
with a force equal to — dynes ; and as the lines of force in the
gap will be parallel to each other, this force will be the same
whether the unit pole is at the middle or not. Consider now a small
surface of area a' on the face of N. The number of lines emanat-
ing from this area will be B a', and, as unit pole (unit quantity
of magnetism) emits 4 K lines, the quantity of magnetism on
this area equals
Ba'
and the force exerted on the magnetism
on this area by the pole s will be an attraction of
Ba' B
• . — dynes,
J
i.e., - dynes.
O 7T
This is true of every small area* of the pole N.
* For parts very near the edges this will not be exactly true, but when
the gap is infinitesimal the error introduced by these parts will be
inappreciable.
CALCULATION OF PULL
449
So the force for the .whole area A is1 given by
B2
or p= —^ dynes, (115)
O 7T
= - — — — kilogrammes weight approximately,
= 4-05 B*A io-8 „ „ „ (116)
If the area of the pole be expressed in square inches instead
of square centimetres, B still being expressed in lines per square
12,000
10,000
8,000
4,000
2,OOO
o O-5 i-o 1-5 2-0 z-z
Current in amperes
Fig. 276. — Relation between Flux Density and Current.
centimetre, we have
^(inlbs.) = 577#2.4"io-8.
When B = 10,000 the formula becomes
(117)
so that the magnetic pull per square inch of surface is roughly
half a cwt. when the flux density is 10,000 C.G.S. lines per square
centimetre. As iron cannot readily be magnetised to a higher
flux density than 20,000, we may say that the maximum pull
per square inch is about two cwts.
2 D
450
PRACTICAL ELECTRICITY
201. Magnetic Saturation. — Having now obtained a relation
between pull, flux density and area of polar surface, we can
replot the curve of Fig. 273 in terms of flux density instead of
pull. This has been done in Fig. 276, from which we see that
when the current is large, a given increase in current produces
only a small increase in the flux density. The effect is more
Fig. 377.— Iron Filings diagram, showing lines of force around a straight
wire, carrying a current.
marked in Fig. 284, where the flux density is carried to a higher
value. This phenomenon is described as " magnetic saturation."
for the greater the flux density is, the greater the increase in
exciting current required to produce a given change of flux ;
and the curve between B and / becomes nearly parallel to the
current axis. This fact is of great importance in electrical
engineering, as it seriously limits the flux densities that can
be economically employed in practice. The flux density at
which iron becomes practically saturated differs with different
specimens, but as a rough rule we may say that the values lie
between 10,000 and 21,000 lines per square centimetre, the
lower value being for cast iron, and the higher for wrought iron
or mild steel.
202. Magnetic Field produced by Current in a Straight Con-
ductor.— From the well-known fact that a small magnet tends
to set itself at right angles to a wire conveying a current, and
MAGNETIC FIELD
from considerations of symmetry, we may conclude that the
lines of force produced are concentric circles with the axis of the
wire as centre. This can also be shown experimentally by
mapping out the field either with iron filings, Fig. 277, or by the
Fig. 278.— Lines of Force (Circles) and Equipotential Surfaces (Planes), due to long straight
current of 7-95 i.e., 1^ amperes.
compass needle method. Equipotential surfaces being every-
where perpendicular to the lines of force, they will, in this case,
be planes containing the axis of the wire and at equal angles apart,
Fig. 278. As the work done in conveying unit pole from one
equipotential surface to an adjacent one is of fixed amount,
it follows that the magnetic force varies inversely as the distance
from the axis of the wire, because the distance between adjacent
equipotential surfaces measured along a line of force is pro-
portional to the distances from the axis.
Further, Work = force X distance,
and as the work is constant, the force must vary inversely as
distance.
452
PRACTICAL ELECTRICITY
From the definition of current strength stated in Section 8,
the magnetic force is proportional to the current, so at a point
at distance c centimetres from a straight current, we have
or
Z,
N,
R
where k is a constant.
To find the value of the constant k,
suppose unit pole to be placed at the
point P, Fig. 279, distance c centi-
metres from the long wire zl Z2 carry-
ing a current of / amperes, and
consider the force exerted on the pole
by a very short length M N of the
current, which subtends a very small
angle M p N at P. The current may
be supposed to be resolved into two
components, one along and the other
at right angles to P Q, the point Q
being at the middle of M N. Only
the latter component will exert a
force at P, and this force will be
normal to the plane containing Z1 Z2
and P, and its magnitude is given
by
10 PQ2
Draw a quarter of a circle O Q2 R
Fig. 279.— calculation of Magnetic with centre at P, and a line P R
in Parallel to oz, ; we may write
/ _
/
MiNt
J •
IO
I
PQ
M2N2
=
10
I
I
* If
IO
I
IO i
PO2
-2 M3 N3 ;
PO
I
PO
M2 N2 COS 0, V P Q2 =
where M3 N3 is the projection of M2 N2 on p R, and p o = c.
FIELD OF STRAIGHT CURRENT 453
From this we see that the force exerted on unit pole at P by
a length M N is equal to -- - multiplied by the length M3 N3
derived from M N as shown in Fig. 279. A similar construction
can be used for any part of the wire o zlt and as all the forces
are perpendicular to the plane of the paper, to get the total force
we add them together.
Denoting by Fl the total force due to the part o zl (supposed
to be very long), we have
but the sum of all the parts M3 N3, etc., will equal p R when
o Zj is very long, and therefore will equal p o.
Hence F, = - . p o
10 c'2
IOC2
I
c,
TO C
For the part o Z2 of the wire, the force will be equal to that
produced by o zlf
• F - 7
2~i^'
and the force due to the whole long wire zl Z2 is the sum of Fj
and F2,
2
The constant k is therefore — , and as the magnetic force exerted
on unit pole is taken as the measure of the strength of a magnetic
field (page 36) we learn that the strength of field at a distance c
centimetres from a long straight wire carrying a current of I amperes
is equal to
IOC
If the current be expressed in C.G.S. units, then we have
F=^l. (119)
454 PRACTICAL ELECTRICITY
203. Magneto-Motive Force. — The magnetic force at a distance
27
c from a long straight wire being — , as shown above, the work
io o
done on unit pole in moving once round the wire at a distance
c from it will be
2!
X 2 7T C,
IO C
for the force will be the same at every point of the path and
the length of the path is 2 TT c.
Fig. 280. — Lines of Force around a Straight Wire carrying 12? amperes.
XT ?! 4 7T 7
NOW X 2 7T C = ,
IO C TO
which is independent of c. Hence we see that the work done
on unit pole in travelling once round a long conductor conveying a
current 7 amperes, is the same whatever the path followed, and
is equal to -- . 7 . (120)
MAGNETO-MOTIVE FORGE 455
The pole may be moved along circular paths of radii o PX,
o P2, o P3, Fig. 280, or along the irregular path, pl R s Pt or any
other path whatever ; the work done will be exactly the same
if the initial and final positions of the pole lie in the same
radial plane, and the path passes round the wire once only.
This is a 'most important result. The same conclusion follows
from a consideration of the work done
when a conductor carrying current /' (C.G.S.
units) cuts $ lines of force (see Section 550).
There it was shown that
W = I'* ergs.
Now if unit pole moves once round a very
long wire, or the wire moves once round the
pole, the whole of the lines of force eman-
ating from the pole will cut the wire, and if
the wire be conveying current /' the work SS' tlm^with' Sciiil
done will be /' *. But for unit pole
* = 4 TT (Section 24),
W = 4 TT /', ergs per unit pole,
= 4 TC— , ergs per unit pole.
Now this is true not only of a long straight wire but of any
closed circuit conveying a current /, for if unit pole is moved
through the circuit to its starting-point, all the lines of force
emanating from the pole will have cut the circuit, and the work
done will be I' $ as before, and therefore for unit pole
IO
If the circuit has several convolutions in it, Fig. 281, and the path
of the pole links once with one or more of them (say s), the work
done will be s times as great, so in this case
TT7 4 TT s I .
W = — - , ergs per unit pole, (121)
s / being the total current through the closed curve which forms
the path of the pole.
Now in the electric circuit, Fig. 187, the work done when unit
quantity of electricity passes once round the circuit is called
the electromotive force in that circuit, and by analogy the name
magneto-motive force* is given to the magnitude - - , the
* Magneto-motive force is often written M.M.F. for brevity.
456 PRACTICAL ELECTRICITY
work done on unit quantity of magnetism (unit pole) in passing
once round a path through which a current / links s times.
Magneto-motive force (M.M.F.) is a quantity of fundamental
importance in magnetic work, just like E.M.F. in electrical prob-
lems, and its meaning should be thoroughly mastered by the
student. The name is not a happy one, for M.M.F. is'not really
a force, but a line integral of the force acting on unit pole,
i.e. Z force X distance, or work per unit pole.
Fig. 282. — Iron Ring Wound for Ballistic Tests;
Magneto-motive force is sometimes called Magnetic potential,
and magnetic lines of force (magnetic flux) pass between points
which have a difference of magnetic potential between them,
just as electric currents tend to flow between points whose
electric potentials differ. Difference of magnetic potential there-
fore causes magnetic flux (lines of force), whilst difference of
electrical potential (P.D.) tends to cause a flow of electric current.
Current actually flows when the two points considered are joined
by a conductor, but does not flow if no conducting path exists.
On the other hand, magnetic lines of force always exist where a
difference of magnetic potential exists, for no substance which
acts as a magnetic insulator has yet been discovered.
Some substances, however, such as iron, nickel and cobalt,
allow magnetic flux to pass through them much more readily
than others, and are, in consequence, regarded as good magnetic
conductors. But there is this great difference between electric
conductors and so called magnetic conductors, viz. that heat is
generated, and therefore power wasted, in an electric conductor
whenever electric current passes through it, whereas a constant
magnetic flux may pass through a substance without causing a
TESTING IRON
457
waste of energy. Nevertheless, the similarity between the two
phenomena of electric current and magnetic flux has led to the
conception of the magnetic circuit, the laws of which are closely
related to those of the electric circuit.
204. Testing Magnetic Properties of Iron by the Ballistic
Method. — If we take a uniform ring of iron and wind it uniformly
with a single layer of insulated wire of s, turns, Fig. 282, we have
an arrangement in which the M.M.F. is easily calculated, for
when a current I is passing,
47TSJ/
M.M.F. =
10
and the M.M.F. per unit length of iron
10 / '
where / is the mean length of the lines of force in the iron in
centimetres.
Fig. 283. — Primary and Secondary Circuits on Iron Ring.
Now M.M.F. per unit length is, in such an arrangement as
just described, equal to the magnetic force,* or strength of field
inside the winding, and this is called the " magnetising force "
because it is that which causes magnetic flux in the iron.
It is customary to use the letter H to represent the magnetising
force, so we may write
fi = 4^f (i22)
10 /
and this is often expressed in words as
H = - — times the ampere turns per centimetre length.
If another winding, say of s2 turns, has been put on the ring
below the one above mentioned, we can connect this with a
ballistic galvanometer G, and resistance box R, as in Fig. 283.
In this .figure, to avoid confusion between the two windings, they
* This follows from the fact that M.M.F. = S magnetic force x distance.
PRACTICAL ELECTRICITY
are shown on separate halves of the ring, but it should be under-
stood that the winding of Sj turns, called the primary winding,
covers the whole ring, and it is advisable, although not essential,
that the other winding of s2 turns, called the secondary winding,
should do so too.
In both figures, c represents a "rocking commutator," the
"rocker" of which resembles that shown at B in Fig. 286.
When placed to the right, the current passes in the direction shown
by the arrows, Fig. 282, and when turned over to the left-hand
side the current round the winding of the ring is reversed, whilst
its direction in the ammeter circuit remains unchanged.
When a current flows through the primary P, Fig. 283, mag-
netic flux passes round the iron ring, and links with the secondary
winding s, and if the primary current be reversed, by means of the
commutator c, the flux is reversed, and a quantity of electricity,
TV 2 BAs2
Q = j; — g = -ft — g- coulombs, (see Section 189),
where B is the flux density in the iron, A the area of cross-section
of the iron, and R the total resistance' of the secondary circuit,
will pass round the secondary circuit, if the key K be closed when
the reversal occurs. The multiplier 2 arises from the reversal
of the flux. As s 2 is known and A and R can be measured, the
above equation enables B to be calculated when the constant of
the ballistic galvanometer is known. Assuming for convenience
that a reflecting galvanometer is used, and that its constant is
k', we have (Section 149),
Q = k's',
where s' is the first swing, and therefore
p _ k'Rio* ,
LJ — - - - S .
Now for a given arrangement, the quantity - is a
constant, and may be calculated once for all and called k", the
formula then reducing to
B = k"sr.
A series of observations may be made by using different primary
currents, starting with small values and then increasing to large
ones, the corresponding values of H and B being calculated from
4?r sl
n — — -— 2 ,
10 /
and B = k"s', respectively.
Plotting H and B on squared paper we get a curve such as is
B-H AND B-M CURVES
459
O 2
10 M 20 30
Values of H
Fig, 284. — Magnetisation Curve for soft Iron Ring.
given in Fig. 284, which represents observations made on a
wound iron ring, having the following constants :
A = 1-974 square centimetres,
si = 145,
s2 = 50,
/ = 31-4 centimetres.
I5poo
JO
'in
I
x
10,000
o
5,000
500
1,000
Values of jju
Fig. 285. — Permeability of soft Iron Ring.
2,000
460 PRACTICAL ELECTRICITY
The secondary circuits had a resistance of 3,500 ohms, and for
the ballistic galvanometer k' = 1-571 X 10 ~ 8 coulombs.
From these data we get,
4-2<_145 7 g/
10 x 31-4
B — I — s' = 27-0 s' approximately.
2 X 1-974 X 50
The rapid rise in B for values of H between i and 4 is well
shown in the curve, as well as the bending over for larger values
of H, due to approaching " saturation."
205. Permeability. — The presence of the iron core inside
the winding, Figs. 282 and 283, greatly increases the magnetic
flux, for if there were only air present, the flux density would be
equal to H, the magnetising force. If we draw a straight line
o H through the origin o, Fig. 284, satisfying the equation B = H*
the intercept M' M between o H and o x will represent the flux
density which would exist in the space within the winding if
no iron were present. The effect of the iron for H = OM, is
therefore to increase the flux density from a tenth of MM' to MP.
This increase is usually very large, and the ratio in which the flux
density is increased under any given conditions by the presence of
the iron, is called the " permeability " of the iron under those con-
ditions, and is denoted by the Greek letter /*. Hence we have —
Flux density
Permeability =
magnetising force '
or A* = jj, (123)
We may also write B = jmH.
Calculating the values of fi for several points on the curve in
Fig. 284 we get the results plotted in Fig. 285, which show that
as B increases from zero upwards, jut. first increases and then
decreases, reaching, in the particular specimen here dealt with, a
maximum value of 2,360 when B = 8000.
206. Hysteresis of Iron. — The ballistic method of investigating
the magnetic properties of iron, although one of the most accurate,
is more troublesome, and perhaps less easy for beginners to under-
stand, than the magnetometer method, described below, which
is suitable for specimens in the form of wire. The wire i is bent
into the form of a long hairpin, the legs of which pass through
two long thin coils or solenoids Cj clf C2 C2, Fig. 286, into two soft
* In the actual figure the vertical ordinates of the line o H are made
ten times too large, otherwise o H would not be distinguishable from o x.
HYSTERESIS OF IRON
461
iron spheres Sj S2. A short magnetic needle, with pointer
attached, is suspended at N in the plane of the coils, and equidistant
from the spheres SA and S2, and the apparatus is placed so that the
solenoids lie in the magnetic meridian. When the iron wire is re-
moved the pointer stands at zero on the scale, but when the wire
is replaced and magnetised by passing current through the magnet-
ising coils Cj C2, the needle is deflected. The tangent of the
deflection is an approximate measure of the flux emanating from
the spheres, and, therefore, of the flux through the iron wire. To
Fig. 286.— Simple Apparatus for Testing Hysteresis.
simplify the experiment, a set of resistances with mercury cups
is provided, whereby currents of known values pass through the
magnetising coils, when two storage cells (E.M.F. 4 volts) are
joined to the terminals TJ T2, as described in Section 193.
Placing the copper bridge pieces p successively between the
middle mercury cup o and the holes marked abode, the corre-
sponding currents are 0-2, 0-4, 0-6, 0-8, i-o ampere.
Having first demagnetised the iron, say by heating it to redness
in a flame, the wire is put in position, a small current 0-2 ampere
passed through the coils by inserting the copper bridge piece p in
the holes o and a, and the steady deflection of the magnetometer-
needle observed. The current is next increased to 0-4 ampere by
placing a bridge piece between o and b without removing the
first one, and the corresponding deflection noted. The object
of this procedure is to ensure that the current shall not decrease
between the two readings. Further increments of current can be
obtained in a similar manner, until the maximum value i-o ampere
is reached. Plotting the results we obtain a curve o c, Fig. 287,
resembling the lower part of Fig. 284.
If now we decrease the current from i-o to 0-8 ampere, the
deflection will be found to be larger than that produced by the
462
PRACTICAL ELECTRICITY
same current in the preceding set, and from a series of obser-
vations with gradually decreasing currents from 0-8 to zero, the
curve c D, Fig. 287, is obtained instead of c o.
This shows a very important property of iron, viz. that the
flux produced by decreasing values of current is greater than that
produced by the same current when increasing ; in other words,
the flux produced by a given
magnetising force depends on
whether the force has risen or
fallen to the value in question,
i.e. it depends on the previous
history of the iron. The name
Hysteresis * has been given to this
phenomenon.
After bringing the iron to the
state represented by the point D,
Figs. 287 and 288, the direction
in which current is passed through
5 _ / the magnetising coil may be re-
versed by rocking the commutator
B, f Fig. 286, from left to right in
the cups at c, and inserting a
bridge piece at a, thus causing
a current of- — 0-2 ampere to pass
through G! and C2. This gives a
Currenb in amperes
i-2point between D and E, Fig. 288,
Fig. 287. — Curve showing effect of
Previous Historv.
and by increasing the current step
by step as before, up to its
maximum negative value — i-o,
points on the curve D E F are obtained, so that by changing
the current continuously from -{- i-o ampere to — i-o ampere
we get the curve c D E F. Reducing the current from — i-o
to zero gives points on F G, Fig. 288, and by rocking the
commutator from right to left, and increasing the current step
by step from zero to + i-o ampere, the curve G K c results. We
have now subjected the iron to a complete cycle of magnetising
force by reducing the current from + i-o ampere to zero, from
zero to — i-o ampere, then — i-o ampere to zero, and from zero
to + i-o ampere, its original value, the result of which is the
curve c D E F G K c, which encloses an appreciable area, and is
called the " Hysteresis Loop."
If the cycle of operation be repeated, points lying on the curve
c D E F G K c will be again obtained, provided the maximum
* Meaning lag, f Rocker shown resting on board at B.
HYSTERESIS LOOP
463
value of the current remains unchanged, thus showing that the
loop represents a Definite property of the particular specimen of
iron.
Considering the points p and p' on the curve we notice that for
the same value of the magnetising force, -f- o L, there are two values
of the flux density,
viz. + L P and — L P',
the positive values
occurring after the irtm
has been previously
magnetised more
strongly in a -f direc-
tion, and the negative
value after its being
magnetised in a nega-
tive direction, the
arrowheads on the
curve showing the
direction in which the
cycle of operation has
been carried out. The
points Q and Q' show
that the same flux
density -j- M Q may
result from two widely
different magnetising
forces, viz. + o M and
- o M' respectively.
These facts show that,
unlike an electrical
circuit (in which a
definite electromotive
force always causes a
definite current to
flow), the flux in a
magnetic circuit depends not only on the magneto -motive force
existing at the time, but also on that which existed previously.
On this account the calculations relating to magnetism are some-
what more complicated than analogous electrical problems.
Loops such as that shown in Fig. 288 can only be obtained when
the length of the iron is very great compared with its cross-section,
or when the iron is formed into practically closed rings.
207. Remanent Magnetism : Coercive Force. — In the hysteresis
loop, Fig. 288, the points D and G and E and K are of consider -
-B
Fig. 288. — Hysteresis Loop.
464 PRACTICAL ELECTRICITY
able importance. The length o D represents the flux density in
the iron when the magnetising force has been reduced to zero
from the value existing at c, and the name " residual magnetism''
or " remanent magnetism" is given to this flux density, whilst
that represented by N c is spoken of as " temporary magnetism.'"
Similarly the points F and G show the temporary and residual
magnetism respectively, when magnetised in the negative
direction.
At the point E the flux density is zero, but the magnetic or
magnetising force has a negative value represented by the length
OE, so we see that after the iron has been magnetised in one
direction it is necessary to apply a magnetising force in the other
direction in order to reduce the magnetism to zero. Similarly
after the iron has been magnetised in a negative direction,
represented by the point F, a positive magnetising force OK
is required to remove the negative magnetism previously existing.
The magnitude of this force is a criterion of the magnetic
" hardness " of the iron, and the name " coercive force " is given
to the lengths o E and o K when expressed in terms of H. In
soft iron the coercive force is small, whereas in hard iron, and
more especially in hardened steel, it has large values.
On the other hand, soft iron when forming continuous magnetic
circuits, generally has a larger ratio of remanent to temporary
magnetism than hard iron or steel, but a comparatively small
demagnetising force will reduce the residual magnetism to a very
small value.
For the construction of permanent magnets, a material having
high coercive force and large residual magnetism is desirable, but
of the two properties, the former, high coercive force, is the
more essential, otherwise comparatively weak magnetising
forces, such as the earth's field, will greatly affect the mag-
netism of a magnet of bar or horse-shoe form. Hardened steel
is the material hitherto found most suitable for use as
permanent magnets, and of the many kinds of steel tried,
Tungsten steel has proved most satisfactory.
208. Loss of Energy due to Hysteresis, Mechanical Analogy.—
The fact that when iron is subjected to a cycle of magnetising
force, the magnetism lags behind the force, gives rise to a loss of
energy, for the material behaves as if it were imperfectly elastic.
Before showing how the loss may be calculated, it may be helpful
to describe a mechanical analogue of the phenomenon. Suppose
we take a piece of poor indiarubber, a commodity of very common
occurrence when rubber is dear, and that we stretch it a given
amount / by applying a gradually increasing force and then
LOSS OF ENERGY
465
1200
allow it to contract again. It will not contract to its original
length but will be somewhat longer ; the amount of contraction
is therefore less than /, and as the distance through which
the stretching force acted is greater than that over which
the contractile force operates, the work done in stretching the
rubber is greater
than that done by
the rubber in con-
tracting. Conse-
quently there is
some work or energy
lost in the process.
If we now compress
the rubber with a
force equal to the
stretching force
previously used, and
then gradually re-
duce the compress-
ing force to zero,
the rubber will not t2OQ
quite recover the
length it had before
compression, and
energy will again be
lost. From this we
see that when im-
perfectly elastic
rubber is subjected
to a cycle of opera-
tions, viz. stretch-
ing, release, com-
pression, release,
more work is done
On the rubber than Fig> ^-"Calculation of Energy Loss by Hysteresis.
the rubber gives out again, the difference being the loss already
referred to above. When any imperfectly elastic substance is
subjected to a cycle of mechanical forces, a loss of energy will
occur ; this lost energy generally appears as heat in the sub-
stance.* Similarly when iron or steel is subjected to a cycle of
magnetic forces, a loss of energy is caused by the imperfect
magnetic elasticity of the material, as shown by the hysteresis
* A familiar instance occurs in the tyres of racing automobiles, which
become very hot at high speeds,,
466 PRACTICAL ELECTRICITY
loop, and, as a matter of fact, the area of the hysteresis loop is a
measure of the energy lost during a complete cycle.
To prove this statement let the closed curve c D E F G K c,
Fig. 289, represent the relation between $, the total flux in, and
I Sit the ampere turns acting on a specimen of iron during a com-
plete cycle, and consider what occurs when the iron changes from
the state represented by the point P to that represented by P',
the two points being near together on the curve. The flux
changes from o M to o M', whilst the ampere-turns change from
o L to o L'. There is, therefore, a change of linkage with the
magnetising coil of st . M M' lines, whilst a current, whose mean
value is — , is flowing. Now in Section 550, we have
2Sj
shown that when a conductor carrying a current /', C.G.S. units,
cuts $ lines of force, the work done is given by
W = I' $ ergs,
so in this case the work done when the flux changes from o M
to o M' is given by
W=— S!.MM'
10
-
. M M',
. MM'
IO
O L + O I/
2 x 10
Now - - is the mean length of the figure PMM'P',
and M M' is its height, so x M M' is equal to the area
2
of the figure, therefore
w _ area of figure P M M' P'
10
Hence we see that the work done by the iron* during the change
is proportional to, and may be represented by, the figure
p M M' p'. By employing similar reasoning to other parts of the
* That the work done by the iron and not on the iron may be seen
from the consideration that the iron is being released from a strained
condition, and therefore doing work against the straining force.
It may also be seen from Lenz's Law, Section 188, for the change of
linkage will induce a current in the magnetising circuit in a direction
tending to prevent the change, i.e. it will tend to increase the magnetising
current ; consequently the iron will act as a current generator, and therefore
give out energy.
HYSTERESIS LOSS 467
line c D, it is seen that the work done during the change of state
from c to D is represented by the area of the figure c R D c.
During the change of state represented by the part D E F of
the loop, the flux is changing in the same direction as before, but
the current is in the reverse direction. This means that work
is being done on the iron, and the area of the figure D E F s D
represents the amount. In the stage F G, which is like c D, the
work done by the iron is given by the area F s G F, whilst for the
stage G K c, the area G K c R G represents the work done on the
iron.
Consequently, during the whole cycle we have
Work done by the iron = area CRDC + FSGF,
and work done on the iron = area DEFSD + GKCRG;
and as the latter is greater than the former by the area of the
figure c D E F G K c, the net work done on the iron is represented
by the area of the hysteresis loop.
When the hysteresis curve is plotted in terms of flux and
ampere turns, the area expressed in these units, divided by 10
($ee formula 124) gives the work lost in the whole specimen in
ergs. If, however, B and H be plotted, since
10 I
BHIA
and H =
we have — : — - =
10 4^
IA
i.e. Total loss == area of (B-H) loop x — .
47U
Now IA is the total volume of the magnetic material, so the loss
per unit volume per cycle is given by
area of (B-H) loop , . , . , .
— ergs per cubic centimetre. (125)
209. The Magnetic Circuit : Reluctance. — In Section 203,
and near the end of Section 206, the analogies between electro-
motive force and magnetomotive force, and between an electric
circuit and the path traversed by magnetic flux, have been
mentioned. If we consider the case of a ring uniformly wound
with wire, such as that shown in Fig. 282, and imagine the iron
replaced by air, we have a simple case of a magnetic circuit in
which the path of the flux is in air, the flux density B being equal
to the magnetising force H.
468 PRACTICAL ELECTRICITY
Now, in such a case
TT 47T/S,
H= -^, page 454-
and if A be the area of cross-section of the path inside the
solenoid, the total flux
***BA
= HA, in this case.
10 r
IO /
A
M.M.F.
This expression is similar to Ohm's Law, written in the
ordinary form of
" flux " taking the place of " current," magnetomotive force
that of " electromotive force," and " — " the place of " resistance."
A
In Chapter VI., Section 97, we have shown that electrical
resistance of uniform conductors is proportional to their length
and inversely as their sectional area, and the fact that
I _ Magnetomotive force
A magnetic flux
suggested the name " magnetic resistance " for — (in air). As,
A
however, an electric current flowing through a resistance always
generates heat, whilst magnetic flux can pass continuously
through air without dissipation of energy, the name " magnetic
resistance " has been abandoned and the word reluctance adopted.
We therefore write
Magnetomotive force
Magnetic flux = - 5 - (126)
Reluctance
an expression which is sometimes called " the law of the magnetic
circuit."
THE MAGNETIC CIRCUIT 469
Lines of magnetic force always form closed curves, so the flux
in a magnetic circuit is the same at every cross-section, just as the
current in an electric circuit is the same at every cross-section.
The cross-section of the magnetic path may, however, differ
at different parts, just as the cross-section of an electric circuit
is not necessarily uniform. In the electric circuit the total
resistance is equal to the sum of the resistances of the several
parts, so also in a magnetic circuit the total reluctance is the sum
of the reluctances of the several parts, and to find the reluctance
of a magnetic circuit the cross-section of which is not uniform,
we suppose it divided into parts each of which is of nearly
uniform section, calculate the reluctance of each part, and add
them together.
If we now suppose the iron core to be inside the coil in Fig. 282,
the flux will be increased in the proportion of ^ to i, where //,
is the permeability of the iron under the given conditions, for the
flux density would become B, where B = /* H. The same M.M.F.
would therefore produce n times the previous flux, so the pre-
sence of the iron reduces the reluctance of the circuit. If, there-
fore, the reluctance of an air path of length / and sectional area A
be represented by — , the reluctance of an iron path of the same
A.
dimensions will be — - , where /x is the permeability.
A/*
Further, when the path is made up of several parts of lengths
/! 12 13, etc., and sectional areas A^A2A^, etc., the total reluct-
ance of the whole path will be equal to
*i . ^ _i_ ^ , f
" " ~T~ . I M \ CLLx.
where /ilf jtx2, /w3, etc., are the permeabilities of the several parts.
For air /* = i, so the formula giving the magnetic flux in a
circuit consisting of air and iron can be written
M.M.F.
A A „
si •**• i A^i
where / and A refer to the air parts, and llt A± and ^ to the
iron parts of the circuit.
An exactly similar formula is true for the electric circuit if
the air parts be omitted,* and we write conductivities instead of
* Otherwise no current would flow, as air is a very perfect electric
insulator.
470 PRACTICAL ELECTRICITY
permeabilities of the other parts of the circuit. There is, how-
ever, this difference between the two cases, viz. that the electric
conductivity of conductors does not depend on the current
density, whereas the permeability of magnetic materials is greatly
influenced by flux density, as will be seen from Fig. 285,
Section 205.
Example 190. — Calculate the strength of field at points distant
i, 2, 5, 10, 50, and 100 inches from a long straight conductor
carrying a current of 100 amperes.
Answers. — 7-87, 3-94, 1-57, 0-787, 0-157 and 0-0787.
Example 191. — Find the area of the poles of a lifting magnet
to support one ton, assuming the flux density to be 18,000 lines
per square centimetre.
Answer. — 77-3 sq. cm. or 12 sq. inches.
Example 192. — The iron of the magnet in example 191 has a
mean length of 40 centimetres, and at a flux density of 18,000
the permeability is 100. How many ampere turns will be
required to produce the required flux, neglecting the reluctance
between the surfaces in contact ? Answer. — 5,730.
Example 193. — An iron ring whose mean diameter is 12 cms.
is uniformly wound with 200 turns of wire. Find (a) , the magneto-
motive force when a current of 10 amperes flows through the
winding, (b) , the current required to produce a magnetising force
of 200. c.G.s. units.
Answer. — (a) 2,512.
(b) 30 amperes.
Example 194. — A thin circular ring of iron 2 cms. in diameter,
is threaded over a long straight wire carrying 50 amperes, and
placed concentric with it. Find (i), the magnetomotive force
acting on the ring, (2), the magnetising force, and (3), the flux
density in the ring, assuming the permeability of the iron to be
1,300.
(i) M.M.F. = 62-8
Answers. — (2) H = 10
(3) B = 13,000.
Example 195. — If a narrow air gap 0-2 m/m wide be made
across the ring in example 194, calculate the value of B in the iron,
assuming the permeability of the iron to remain unaltered, and
that the flux density in the air gap is the same as in the iron.
Answer. — The total M.M.I7, is — x 50, i.e. 62-8, and this is
spent partly in the iron and partly in the air gap. The M.M.F.
EXAMPLES
D
in the air gap is B x length of air path, and that in the iron is —
x length of iron path. Taking this length as 2 n we have
HT nr T» • • B X 2 7T
M.M..b. in iron = --
1,300
.«. 62-8 = B X 0-02 + B X — —
1,300
5 = 2,530 approx.*
Example 196. — Find the magnetic reluctance of a square iron
bar 20 centimetres long and of 4 centimetres side, the permeability
being taken as 1000. T
Answer. — — C.G.S. units.
800
Example 197. — Calculate the approximate reluctance of an
air gap 2 m/m long in a magnetic circuit the cross-section of
which is 10 square cms. Answer. — 0-02 C.G.S. units.
Example 198. — Assuming the ring in Example 194 to be a
cylinder i m/m thick and 15 m/m long, find its reluctance.
6-28
Answer. -- =0-0322 (approx.).
o-i x 1-5 x 1300
Example 199. — Find the relation of the reluctance of the air
gap in Example 195 to that of the iron portion of the magnetic
circuit, assuming the lines of force to be circles (i.e. spreading of
lines at the gap is to be neglected) and that the permeability of
the iron is 1880. _/
A in* ,.
Answer. -- - — = "y* = 6 approx.
Example 200. — In a uniformly wound magnetic circuit of
uniform cross-section express the magnetic force H in terms of
" ampere turns per centimetre length, "f
4nsww.— -Here #=—-—, and as
10 /
— = ampere turns per c.m. length,
we have H — 1*256 X " ampere turns per cm. length."
* Observe that the introduction of an air gap of only one-fifth of a
millimetre, in an iron circuit of over 60 millimetres long, reduces the
flux density from 13,000 to about 2,500.
t " One ampere turns per centimetre length," is frequently used as a
unit of magnetising force. " One ampere turn per half-inch length " is
also a convenient approximate unit, when the length of the magnetic
circuit is given in inches.
472 PRACTICAL ELECTRICITY
Example 201. — Find the ratio of the C.G.S. unit of magnetising
force to " one ampere turn per half inch."
Answer. — H = 1*256 x ampere turns per cm. length,
*2.
H = 1-256 x X ampere turns per half inch,
2'54
= x ampere turns per half inch.
I-OII
Hence one C.G.S. unit of magnetising force is about I per cent-
greater than " one ampere turn per half inch."
APPENDIX I
SHORT HISTORY OF THE ABSOLUTE
UNIT OF RESISTANCE,* AND OF THE ELECTRICAL
STANDARDS OF THE BOARD OF TRADE
IN 1821 Sir Humphry Davy published the results of his experi-
ments, proving that metals varied in their power of conducting
electricity, and that this conducting power diminished as the
temperature rose ; but the idea of resistance being a property
of a conductor was due to Ohm, who published the mathematical
proof of his famous law in 1827. The writers, however, who
immediately followed Ohm did not employ a unit of resistance,
but contented themselves with reducing, by calculation, the
resistance of all parts of a heterogeneous circuit to a given length
of some part of that circuit, so that Lenz, for example, in his
paper of 1833, calls the resistance of a conductor its " reduced
length."
The next step, when comparing different circuits, was naturally
to refer these " reduced lengths " to the length of some one
standard wire, although the wire might not form part of any of
the circuits under test, and to consider the resistance of unit
length of this standard wire as the unit resistance : thus, we
find Lenz, in 1838, stating that one foot of No. n copper wire
was his " unit " of resistance — a unit, however, which he appeared
to have selected at random, and without any idea of suggesting
that it should be used by others.
In 1840 Wheatstone constructed the first instrument by which
definite multiples of a resistance unit could at will be added to
or subtracted from, a circuit. And in 1843 he proposed that
the resistance of one foot of copper wire weighing 100 grains,
which was selected with reference to the British standards of
* The earlier part of this History is abstracted from a " Report to the
Royal Society on the New Unit of Electrical Resistance, &c.," by the late
Prof. Fleeming Jenkin, which, together with the Reports from 1862 to
1869 of the British Association Committee on Standards of Electrical
Resistance, and with his Cantor Lectures, were issued by him in 1873 in
the form Of a very useful book.
473
474 PRACTICAL ELECTRICITY
length and weight, should constitute the standard of resist-
ance. Later on other wires were proposed as units of
resistance ; and, to avoid the inconvenience arising from the
multiplicity of standards, Jacobi, in 1848, sent a certain copper
wire to Poggendorff and to others, requesting that copies might
be taken of it. For Jacobi pointed out that the mere definition
of a standard of resistance in terms of the length and weight
of a wire of some material was not sufficiently definite, and that
good copies of a standard, even if that standard had been origin-
ally chosen at random, would be more exact.
Until about 1850, measurements of resistance were confined,
with few exceptions, to the laboratory, but about that time
underground wires, followed shortly afterwards by submarine
cables, began to be employed ; and, since it was impossible to
ascertain the position of a defect in such a telegraph line by
inspection, electrical methods of " localising the position of a
fault " by measuring the resistance of the wire between the
testing station and the faulty spot, had to be developed. As
early as 1847 C. F. Varley is said to have used a rough method
of distance testing, while in 1850 Werner Siemens published two
methods, and in 1852 Charles Bright patented a plan for deter-
mining the position of a fault by the use of resistance coils.
The first effect of this commercial use of resistance was to turn
the " foot " of the laboratory into the " mile " : thus, the unit
of resistance in England became that of a mile No. 16 copper
wire ; in Germany, of a German mile of No. 8 iron wire ; and in
France, of a kilometre of iron wire 4 millimetres in diameter.
Next, Marie Davy and De la Rue pointed out that, as it was
possible by chemical cleaning and subsequent distilling to remove
practically all impurities from mercury, this metal was specially
suitable for selection as a standard substance ; and in 1860
Werner Siemens constructed standards in which his unit was
the resistance of a column of chemically pure mercury i metre
long, i square millimetre in cross-section at a temperature of
o°C.
The definition of the " Siemens unit " of resistance was a very
simple one ; and, since mercury in a nearly pure state is not
very difficult to obtain, it might be thought that the unit pro-
posed by Siemens would have been finally adopted. The sim-
plest way, however, to obtain a column of mercury of uniform
cross-section is to place mercury in a tube of uniform bore, and
the cross-section of the bore of such a tube can be most accurately
determined by finding the weight of mercury that is contained
in a given length, and deducing the volume from a knowledge
ABSOLUTE SYSTEM OF UNITS 475
of the specific gravity of mercury. Although, then, the defini-
tion of Siemens units is apparently based simply on length, cross-
section, and temperature, it really depends on weight, specific
gravity, and temperature.
In the specimens of this unit originally issued there was an
error of 2 per cent., and even in later issues an error of over one-
quarter per cent, was introduced up to 1873, through Werner
Siemens having adopted 13-557 as the specific gravity of mercury
instead of 13-596. The labour, however, bestowed by the late
Werner von Siemens on perfecting electrical measurements merits
special recognition, as it materially helped in introducing strict
accuracy.
All the preceding units of resistance are based on the more or
less arbitrary size and weight of some more or less suitable
material ; but measurements of resistance can be conceived and
carried out entirely without reference to the special qualities of
any particular material. In 1849 Kirchhoff effected a measure-
ment of this kind ; but it is to W. Weber that we owe the
first distinct proposal, made in 1851, of a system of electrical
and magnetic measurement in which an electrical resistance
would be expressed as an absolute velocity, were " magnetic
permeability " a simple numeric.
Previous to this, Gauss, desiring to make precise measurements
of the distribution of terrestrial magnetism, found it necessary
at the outset to decide on a unit of force which, unlike the weight
of a given mass, should not be affected by the position of the place
at which the experiment was made, and on a magnetic pole whose
strength should be independent of any molecular change in steel.
He therefore devised what has since become well known as
Gauss's " absolute unit of force," and the " unit magnetic pole,"
the former being defined as the force which, acting on unit mass,
generates unit acceleration, and the latter as the pole which repels
an exactly similar pole at unit distance with unit force. : -•
Following Gauss's nomenclature, Weber called the two systems
of units to which he was led the " absolute electromagnetic " and
the " absolute electrostatic " systems ; but the name " derived "
would have conveyed the meaning better than " absolute,"
since the essence of Weber's system consisted in the various
electrical and magnetic units being derived from those of length,
mass, and time.
As soon as the proposal of Weber appeared, W. Thomson (now
Lord Kelvin) accepted and extended it by showing that the abso-
lute unit of work formed part of the same system. And ten
years later, at the meeting of th^ British Association in 1861,
476 PRACTICAL ELECTRICITY
W. Thomson proposed that a Committee of that Association
should be formed to determine the best standard of electrical
resistance.
This Committee, which consisted of only six names at the
outset, gradually increased its numbers as it enlarged the scope
of its work. A few of the members of thirty years ago*are still
taking an active part in the labours of the Committee on Elec-
trical Standards of to-day, but the Committee has lost by death
Clerk Maxwell, Cromwell Varley, Fleeming Jenkin, Joule,
Matthiessen, and others whose names are distinguished for the
active part they took in the development of electrical science.
The principle of the method employed by the British Associa-
tion Committee in 1863 for the determination of the unit of resist-
ance was, briefly, as follows : If a coil like that of a tangent
galvanometer — for example, c c (Fig. 62, page 91) — be spun in a
uniform magnetic field round a vertical axis passing through the
centres of the coil and of the needle, an E.M.F. is induced in the
coil, this E.M.F. reaching its maximum when the plane of the
coil is parallel to the lines of force, and becoming zero when
it is perpendicular to the lines of force. If, then, the coil be short-
circuited a current will be induced in it, and, although the E.M.F.
reverses its direction each time the plane of the coil is perpen-
dicular to the lines of force, and although, therefore, as regards
the coil the current flows in opposite directions during the two
halves of its revolution, it flows in the same direction as regards
the needle. Hence, for a uniform speed of rotation of the coil
there will be a constant mean value of the deflecting force exerted
on the suspended needle, and, therefore, if the time taken by
the coil to make one revolution is small compared with the
time of vibration of the needle, the needle will remain steadily
deflected as if it were acted on by a perfectly constant deflecting
force.
Further, since for a given angular velocity of the coil the aver-
age value of the induced current is directly proportional to the
strength of the uniform magnetic field, while the controlling force
exerted on the needle is also directly proportional to the strength
of this magnetic field, it follows that the magnitude of the deflec-
tion is independent of the field. And, as proved in Section 15,
the deflection is also independent of the strength of the needle.
In fact, when the equations connecting the various electric and
magnetic magnitudes are written in their simplest forms, without
the introduction of useless coefficients, it can be shown that, to the
first degree of approximation,
* Written in 1896. -T, M.
BA. UNIT OF RESISTANCE 477
7T2 r n2 co
tan a = — - -- ;
where d is the angular deflection produced by a coil of mean
radius r, wound with n convolutions of wire, and having a resist-
ance R, when spun with a uniform angular velocity <w, in a medium
the magnetic permeability of which is taken as the simple numeric
unity, without dimensions.
The product rco in the last equation equals v, the linear
velocity of a point, c (Fig. 62), at the end of a horizontal diameter ;
hence
**
R=
tan d
or the resistance of the coil equals a number multiplied into the
linear velocity of a point at the end of a horizontal diameter.
If v be measured in centimetres per second, then R will be
expressed in absolute electromagnetic units of resistance* This
unit would, however, be inconveniently small for practical pur-
poses, since, for example, i mile of copper wire, y^n °^ an mc^
in diameter, has a resistance of about fourteen thousand million,
14 x io9, of such units, and a Siemens mercury unit equals about
94 x io7 absolute electromagnetic units. It was, therefore,
decided to call io9 of these new units I B.A. unit ; and, in order
to familiarise people with its use, Sir Charles Bright and Mr.
Latimer Clark proposed the distinctive name of " ohmad," which,
in its abbreviated form of ohm, was finally adopted.
Twenty-six coils having nearly the form shown in Fig. 167,
wound with platinum silver wire, and adjusted so as to have
i B.A. unit of resistance, were distributed gratuitously in 1865
to the directors of public telegraphs in various countries, and
to other important people. Also the Committee announced
that they would furnish similar coils at the price of £2 los.
apiece, and would undertake to " verify at a small charge any
coils made by opticians, as is done for thermometers and barome-
ters at Kew." The expression " opticians " is interesting as
indicating that the electrical instrument maker — of which so
many exist to-day — was unknown in 1865.
A platinum-silver alloy containing 33 per cent, of platinum
and 66 per cent, of silver was used for the wire of these copies
of the standard, in consequence of the results obtained by Mat-
thiessen in his excellent work " On the Variation of the Electrical
* In the main part of this book the name C.G.S. electromagnetic unit
has been used instead ot "absolute" electromagnetic unit, as its meaning
is more definite. — T. M.
478 PRACTICAL ELECTRICITY
Resistance of Alloys due to Change of Temperature. On the
Electrical Permanency of Metals and Alloys," &c., &c., accounts
of which formed part of the Committee's Reports for 1862, 1863,
1864, and 1865.
The Report for 1863 was also remarkable in containing a most
valuable article by Clerk Maxwell and Fleeming Jenkin " On the
Elementary Relations between Electrical Measurements." To
that article the author is indebted for nearly all his early ideas on
the subject of exact electrical measurement, for at the time that
it appeared there existed no one of the hundred text-books of
the present day dealing with the quantitative science of electricity,
as distinct from the qualitative effects obtainable with glass-
legged stools and electrified heads of hair. Indeed, ten years
later Fleeming Jenkin said, in the preface to his book on " Elec-
tricity and Magnetism," published in 1873 : — " In England at
the present time it may almost be said that there are two sciences
of Electricity — one that is taught in ordinary text-books, and
the other a sort of floating science known more or less perfectly
to practical electricians, and expressed in a fragmentary manner
in papers by Faraday, Thomson, Maxwell, Joule, Siemens,
Matthiessen, Clark, Varley, Culley, and others. . . A student
might have mastered De la Rue's large and valuable treatise,
and yet feel as if in an unknown country and listening to an
unknown tongue in the company of practical men. It is also
not a little curious that the science known to the practical men
was, so to speak, far more scientific than the science of the text-
books."
In the 1863 Report of the B.A. Committee the " absolute
electromagnetic unit of current " is defined as the current which,
flowing through unit length placed along the circumference of a
circle of unit radius, exerts a unit of magnetic force at the centre.
Methods are also described for measuring a current in absolute
units by employing Weber's " electro-dynamometer" an instru-
ment in which the torque is measured that is exerted between two
coils conveying the current in question. The weight of water that
is decomposed per second, as well as the weight of zinc that is
deposited per second by this absolute unit of current are stated
when the centimetre, gramme, and second are taken as the funda-
mental units of length, mass, and time. But of so little practical
importance was a unit of current at that time that no multiple
of the absolute unit was chosen for commercial purposes, and,
therefore, no name corresponding with that of the ohm was given
to a unit of current.
Defining, however, the " absolute electromagnetic unit of differ-
ERROR IN B.A. UNIT 479
ence of potentials " as that which sends the absolute electromag-
netic unit of current through the absolute electromagnetic unit
of resistance, then Lord Kelvin had shown in 1851 (see Section 83,
page 205) that the E.M.F. of a Daniell's cell was about io8
absolute electromagnetic units of P.D. in the centimetre, gramme,
second system, a result that was subsequently confirmed by
Bosscha. There was, therefore, a good reason for giving a dis-
tinctive name to io8 absolute electromagnetic units of P.D.
and in 1862, at the suggestion of Sir Charles Bright and Mr.
Latimer Clark, the name volt was adopted for this purpose.
Passing over the Reports for the next three years, we come to
that for 1867, which, in addition to containing an account of
Fleeming Jenkin's first determination of the capacity of a con-
denser, and a most interesting description of various electrometers
constructed by Lord Kelvin, is remarkable in that there is given
in it the results obtained by Joule of a very accurate determina-
tion of the mechanical equivalent of heat carried out electrically.
Joule remarks, in connection with these results : " The equiva-
lents obtained in the two foregoing series of experiments are as
much as one-fiftieth in excess of the equivalent I obtained in
1849 by agitating water."
The significance of the preceding remark was not appreciated
at the time, for it was supposed that the discrepancy arose from
the inherent difficulties met with in such experiments, and it
was not even suspected that the difference between the value of
the mechanical equivalent of heat obtained mechanically by
Joule in 1849 and electrically by his experiments conducted be-
tween 1865 and 1867 really indicated that the B.A. unit of resist-
ance had a value something like 2 per cent, less than the ideal
value it was intended to possess.
The British Association Committee had aimed at choosing the
absolute unit of resistance and the absolute unit of current, so
that when a centimetre, gramme, and second were taken as the
fundamental units of length, mass, and time, the power given to
any circuit stated in ergs per second should be equal to the pro-
duct of the resistance of the circuit in absolute units into the
square of the current in absolute units. Hence, by sending a
current, the absolute value of which Joule himself measured with
great accuracy, through a resistance the value of which was deter-
mined by direct comparison with the B.A. unit, he was able to
calculate the power in ergs per second given to the circuit. If
however, the B.A. unit was, say, 2 per cent, too small, then Joule
would oz^-estimate the total energy given to his calorimeter by
2 per cent., and, consequently, would arrive at a value for the
4So PRACTICAL ELECTRICITY
mechanical equivalent of heat 2 per cent, larger than the true
value.
Thinking, however, that the high value of his result arose from
imperfections in his apparatus, Joule effected a number of im-
provements, and then carried out a fresh series of determinations.
But, in spite of all the precautions which he took, his final value
of the mechanical equivalent of heat obtained electrically was
about 1.4 per cent, higher than the value which he had previously
obtained by stirring water.
No explanation of the discrepancy just referred to was forth-
coming, and the B.A. unit was employed as the practical unit of
resistance in Great Britain during the next ten years ; the
Siemens unit, however, continued to be used as the standard in
Germany and some other countries.
Redeterminations of the ohm were carried out by Lorenz in
1873, by F. Kohlrausch in 1874, by H. F. Weber in 1877, ^Y
Rowland in 1878, and by Rayleigh and Schuster in 1881 ; and
although the methods of experimenting employed in these five
investigations were radically different, the results all agreed in
showing that the resistance of the B.A. standard coil was some-
thing like i per cent, too small. There could be no doubt then
that some mistake must have been made by the British Associa-
tion Committee either in carrying out the experiment described
in page 476, or in reducing the results of the measurements.
In 1881 the International Electric Exhibition was held in
Paris, and the modern industry of electrical engineering may
almost be said to date its existence from that year. A subsidy
of £8,000 was given by the French Government towards the
expenses of this Exhibition, and a further subsidy of £4,000 for
defraying the cost of holding an International Congress of Elec-
tricians. The Congress was divided into three sections, one of
which was entirely occupied with the consideration of the steps
to be taken to secure the general adoption of "an international
system of electrical units."
Everything depended on the selection of the unit of resistance,
and, consequently, the attention of Section I. of the Congress
was mainly devoted to the discussion of this unit. The German
representatives strongly urged that the unit proposed and con-
structed by Siemens possessed the great merits of simplicity of
definition and comparative facility for being reproduced if
destroyed, and, therefore, that the Siemens unit was the one that
ought to secure universal recognition. On the other hand, the
English representatives, while admitting that the difficulties
connected with the absolute determination of the ohm had led
DECISIONS, PARIS CONGRESS, 1881 481
to the introduction of an error of oven per cent, in the concrete
standard issued by the British Association, advocated the import-
ance of the system in which the unit of resistance was based on
the fundamental units of length, mass, and time, and not on the
qualities of some special material like mercury.
Of the other nations represented at the Congress, some sup-
ported the Germans, while others sympathised with the system
that had been developed by the British Association ; and ulti-
mately, after a week's animated debating, the following resolu-
tions were unanimously adopted as the result of a very happy
compromise which was arrived at : —
" i. For electrical measurements, the fundamental units,
the centimetre for length, the gramme for mass, and the second
for time (C.G.S.), are adopted.
" 2. The practical units, the Ohm and the Volt, are to retain
their existing definitions — io9 for the Ohm and io8 for the Volt.
" 3. The unit of resistance (Ohm) is to be represented by a
column of mercury one square millimetre in section at the tem-
perature of zero Centigrade.
"4. An International Commission is to be appointed to
determine, by fresh experiments, the length of a column of mer-
cury one square millimetre in section at a temperature of zero
Centigrade, which for practical purposes is to represent the Ohm.
"5. The current produced by a Volt through an Ohm is to
be called an Ampere.
" 6. The quantity of electricity given by an Ampere in a
second is to be called a Coulomb.
" 7. The capacity defined by the condition that a Coulomb
charges it to a Volt is to be called a Farad."
By the adoption of the preceding seven resolutions the Con-
gress agreed —
(a) To accept the British Association system of absolute units
for international use, but not the concrete standard which the
B.A. Committee had issued as representing io9 C.G.S. absolute
electromagnetic units of resistance.
(b) To meet the wishes of the Germans by employing as the
practical standard the resistance of a column of mercury i square
millimetre in section at o° C. ; but, instead of selecting a purely
arbitrary length like that of i metre, as Siemens had done,
to ascertain by a series of fresh experiments the length of such a
column which had the resistance of io9 c.G.s. units.
(c) To pay a graceful compliment to the French nation, in
whose country the Congress was held, by employing for the
future the names o.( two French experimenters, Ampere and
2 F
482 PRACTICAL ELECTRICITY
Coulomb, for the units of current and quantity, to which no names
had been previously given* with general consent.
The next " International Conference for the Determination
of the Electrical Units " was held in Paris in October, 1882.
Professor Mascart described the various methods that were
known for determining the length of the column of mercury i
square millimetre in cross-section, which had a resistance of io9
c.G.s. units at o° C. ; and the relative values of these methods,
together with the results that had been obtained by their use,
were considered.
Professor von Helmholtz expressed the view that, of all the
investigations on the value of the ohm of which accounts had
been published up to that time, those of Lord Rayleigh appeared
to be the only ones that had been carried out with the necessary
degree of accuracy.
According to the experiments of Lord Rayleigh, I B.A. unit
equalled 0-9867 Xio9 C.G.s. units, and the required length of the
column of mercury, which under the specified conditions had a
resistance of i ohm, was 106-24 centimetres. Other experi-
menters, however, obtained a somewhat shorter length ; and the
Commission of 1882 was ultimately led to the following resolu-
tions : —
FIRST RESOLUTION.
" The Commission considers that the determinations made up
to the present time do not possess the amount of agreement that is
necessary to fix the numerical value of the Ohm in terms of the
mercury column.
" It considers, therefore, that further experiments should be
carried out.
" Without being able to give an authoritative opinion regarding
the different methods that have not yet been put into practice, it is
of opinion that the following are suitable for giving very exact
result : —
" i. Induction of a current in a closed circuit. (Kirchhoff.)
" 2. Induction by the earth. (W. Weber.)
"3. Damping of the motion of swinging magnets. (W.Weber.)
" 4. British Association apparatus.
" 5. Lorenz method.
" In addition, it is desirable that a new determination should be
made of the quantity of heat produced by a current of known strength,
this experiment being for the purpose either of controlling the value
* The names of Weber for the unit of quantity, and Weber per Second,
or Oerstedt, for the unit of current had been used by some writers.
RESOLUTIONS, 1882 CONFERENCE 483
of the ohm or of settling with greatet accuracy the mechanical
equivalent of heat."
SECOND RESOLUTION.
" The Conference expresses the wish that the French Govern-
ment will take the necessary steps to place the same standard, or
several standards, of resistance at the service of experimenters who
are engaged in absolute measurements, in order to render it easy
for comparisons to be made.
" The Commission is of opinion that, as soon as the results
of the different investigations show an agreement which permits of
an approximation to the one-thousandth part being arrived at, it
will be desirable to stop at this approximation, and use it to fix
the practical standard of resistance.
" In conclusion the Commission expressed the wish :
" That the French Government will see fit to transmit to each
of the Governments represented at the Conference its desire that,
in view of the importance and urgency of a practical solution, it
will take the necessary steps to encourage the researches of its people
relating to the determination of the electrical units."
By 1884 a number of new determinations of the value of the
ohm had been carried out, and the results, as far as they were
generally known at the holding of the second session of the Inter-
national Conference at Paris in April, 1884, were as follow : —
I. — MEAN ACTION ON A MAGNETIC NEEDLE OF A CURRENT
INDUCED IN A ROTATING FRAME.
Value of the Ohm Expressed in
Centimetres of Mercury i Square
Date and Observer. Millimetre in Section at o° C.
1865. Committee of the British Association . . . . 104-83
1881. Lord Rayleigh and Schuster .. .. .. ..105-96
1882. Lord Rayleigh . . . . ., . . . . . . 106-27
1882. H.F.Weber 106-13
II. — CALORIMETRIC METHOD.
1866. Joule . . . . . . . . 106-22
III. — STEADY P.D. INDUCED IN A ROTATING Disc BALANCED
AGAINST THE P.D. PRODUCED BY A BATTERY.
1873 Lorenz (preliminary) . . . . . . . . . . 107-10
1883 Lord Rayleigh and Mrs. Sidgwick 106-22
1884. Lorenz . . . . . . . . . . . . . . 106-19
1884. Lenz 106-13
484 PRACTICAL ELECTRICITY
IV. — DISCHARGE INDUCED IN A FRAME WHEN TURNED
THROUGH AN ANGLE OF l8o° IN A MAGNETIC FlELD.
Value of the Ohm Expressed in
Centimetres of Mercury i Square
Date and Observer. Millimetre in Section at o° C.
1874. F. Kohlrausch.. .. .. . ,. .. 105-91
1884. Mascart, De Nerville, and Benoit . . . . . . 106-31
1884. G. Wiedemann . . . . . . . . . . 106-19
V. — DISCHARGE INDUCED IN A COIL BY ALTERING THE
CURRENT IN ANOTHER COIL.
1878. Rowland . . . . . . . . . . . . 105-76
1882. Glazebrook . . . . . . . . . . . . 106-28
1884. Mascart, De Nerville and Benoit 106-31
1884. H. F. Weber 105-37
1884. Roiti 105-90
VI. — DAMPING OF THE VIBRATION OF A MAGNET,
1882. Dorn 105-46
1884. Wild 105-68
1884. H. F. Weber 105-26
Mean of the whole i°5-97
In consequence of the difference in the methods adopted in
carrying out the investigations as well as in the skill possessed
by the various experimenters, it would have been right, scientifi-
cally, to give different weights to the results before taking the
mean. This, however, was regarded as too delicate a matter to
undertake ; and although it seemed pretty certain that the true
number exceeded 106-2 centimetres, it was thought better to
accept 106, the nearest whole number to the arithmetic mean, in
order to avoid all question of national jealousy. The resolutions
of the 1884 Conference were, therefore, as follows : —
" I. The legal ohm is the resistance of a column of mercury
of a square -millimetre cross-section and 106 centimetres in length,
at a temperature of melting ice.
" 2. The Conference expresses the wish that the French Govern-
ment will transmit this resolution to the various States, and recom-
mend the international adoption of it.
"3. The Conference recommends the construction of primary
standards in mercury in accordance with the resolution previously
adopted, and the concurrent employment of sets of secondary resist-
ances in solid alloys which shall be frequently compared amongst
one another, and with the primary standard.
PARIS CONGRESS OHM 485
" 4. The ampere is the current the absolute value of which is
io-1 in electromagnetic units.
"5. The volt is the electromotive force which maintains a current
of one ampere in a conductor the resistance of which is one legal
ohm."
Many people considered at the time that the Conference had
come to a decision too hurriedly, and that, had the matter been
postponed for a year or two, the value of the ohm in terms of the
column of mercury might have been stated with greater accuracy.
Experience, however, showed that unless the decision had been
postponed for several years, nothing would have been gained
from the delay ; for the results obtained up to the end of the
next year, 1885, were as follow : —
Method No. II., 1885. Fletcher 105-95
„ No. III., 1884. Rowland, Kimball, and
Duncan
„ No. III., 1885. Lorenz
„ No. V., 1884. Rowland and Kimball
„ No. V., 1885. Himstedt
Mean . . 106-09
And the mean of these five new determinations does not differ
much from the mean of the results published before the holding
of the 1884 Conference.
The next step was the resolution arrived at by the British
Association Committee on Electrical Standards at their meeting
in Birmingham, in 1886, to recommend to the English Govern-
ment—
" (i) To adopt for a term of ten years the Legal Ohm of the
Paris Congress as a legalised standard sufficiently near to the
absolute Ohm for commercial purposes.
" (2) That at the end of the ten years' period the Legal Ohm
should be defined to a closer approximation to the absolute
Ohm.
" (3) That the resolutions of the Paris Congress with respect
to the Ampere, the Volt, the Coulomb, and the Farad be adopted.
" (4) That the Resistance Standards belonging to the Com-
mittee of the British Association on Electrical Standards now
deposited at the Cavendish Laboratory at Cambridge be accepted
as the English Legal Standards comformable to the adopted
definition of the Paris Congress."
486 PRACTICAL ELECTRICITY
The English Government, however, decided that it was prema-
ture to take any action in the matter, and, therefore, although
many resistance boxes graduated in " legal ohms " were
constructed in England during the next few years,* neither this
unit of resistance nor any other of the electrical units just
referred to had any legal value in Great Britain.
Although the 1884 Conference thought it politic to take the
arithmetic mean of all the results given in the table on pages 483-
85 in order to arrive at the value of the legal ohm, there was no
question that certain of the numbers were much less trustworthy
than others. For example, an examination of the calculations
that had been made by the British Association Committee in
1865 brought to light certain errors that had not previously been
detected.
Further, in 1885 Prof. Mascart pointed out that the method of
determining the value of -the ohm by measuring the damping of
the vibration of a magnet, when swinging in a closed coil, was
likely to give erroneous results from the alteration of the perman-
ent magnet by the currents induced in the coil ; and he proved
that this error so introduced would tend to make the length of the
column of mercury corresponding with the ohm appear to be too
small.
Later on, when discussing the results obtained by Ro'iti and
Himstedt from their measurements of the effect of a series of
currents induced in a secondary coil, on starting and stopping a
current in a primary coil, Prof. Mascart drew attention to the
fact that, since it was necessary to disconnect the galvanometer
from the secondary coil at every make, or else at every break, of
the current in the primary coil, there was considerable risk of part
of the induced current being lost. And he pointed out that such
a diminution in the mean value of the induced secondary current
would make the required length of the mercury column appear to
be too low.
It is further to be noticed that Lord Rayleigh's 1882 result
* In accordance with the resolution passed at the meeting of the Electri-
cal Standards Committee of the British Association in 1884, the " legal
ohm " coils constructed in England were intended to represent the resist-
ance of a column of mercury 106 centimetres in length. But, as a matter
of fact, they were made equal to 1-0112 B.A. unit, for in 1884 it was
believed that the specific resistance of mercury at o° C. was 0-9540 x io~ 4
B.A. unit ; and, therefore, that 1-0112 B.A. unit was equal to the resistance
at o° C. of 106 centimetres of mercury i square millimetre in cross-section.
Subsequent measurements, however, showed that the specific resistance of
mercury was more nearly 0-9535 x io~4 B.A. unit ; hence a " legal ohm"
constructed in England really represented the resistance of 106-05 centi-
metres of mercury, and was, therefore, about 5 parts in 10,000 too large.
DISCUSSION OF RESULTS 487
may be regarded as superseding that found by Schuster and
himself in 1881, that the value obtained by Lorenz in 1873 was
professedly but a provisional one, and that the value arrived at
by H. F. Weber, using method No. V., was manifestly too
small.
We shall, therefore, obtain a more accurate mean if we neglect
the first and second results of method No. I., the first result of
method No. III., the fourth result of method No. V., all the results
obtained with method No. VI., and the last result in the table
on page 485. When this is done, and when Rowland's 1878
value of 10576 centimetres is replaced by 106-16, which was
afterwards found to represent more accurately the result of his
test, we obtain as the mean of all the remaining values 106-17
centimetres.
In 1890 the account of a very accurate " Determination of the
Specific Resistance of Mercury in Absolute Measure," by Lorenz's
method, was communicated by J. Viriamu Jones to the Royal
Society, from which it followed that 106-307 centimetres was the
required length of the mercury column which represented the
ohm.* This number, in consequence of the great care that had
been taken by Prof. Jones in arriving at it, may with safety
be used to discriminate between the various lengths previously
published, and it is seen that it is closely in accord with the result
106-29 obtained by Rowland, Kimball, and Duncan in 1884 by
the use of this same method No. III., as well as with the value
106-31, which represents the result obtained in each of three
separate investigations also carried out in 1884 — viz. by Mas-
cart, De Nerville, and Benoit, using method No. IV., by the same
experimenters using method No. V., and by Rowland and Kim-
ball, also using method No. V. It also differs but little from
Lord Rayleigh's result, 106-27, obtained by using method No. L,
or from that deduced by Glazebrook, 106-28, from the employ-
ment of method No. V.
There is, then, a very strong reason for believing that the
length of 106-3 centimetres is correct to the first four figures.
• The bobbin of the coil used in this investigation was made of brass,
and yielded a little when it was being turned. This caused it to acquire
a slightly elliptical shape with a difference in the lengths of the axes of about
i part in 1,300. The value 106-307 given above for the ohm was decided
on the assumption that the coil was truly circular, but, in a communication
made to the Physical Society in May, 1896, Professor Jones has proved
that the correction for the ellipticity is about 7 parts in 100,000. Hence,
this determination of the specific resistance of mercury leads to the result
that the length of the mercury column i square millimetre in cross-section,
which has a resistance of i ohm at o° C., is 106-300 centimetres (see page 497).
488 PRACTICAL ELECTRICITY
In December, 1890, the Board of Trade appointed the repre-
sentatives of the Board of Trade, General Post Office, Royal
Society, British Association, and Institution of Electrical Engin-
eers, whose names are given below,* "to be a Committee
to consider and report whether any, and if so, what, action
should be taken by the Board of Trade under Section 6 of the
Weights and Measures Act, 1889, with a view of causing new
denominations of Standards for the measurement of electricity
for use for trade to be made and duly verified."
The first report of this Committee was issued in July, 1891.
It contained sixteen resolutions, of which the following were the
most important : —
" i. That it is desirable that new denominations of standards
for the measurement of electricity should be made and approved
by Her Majesty in Council as Board of Trade standards.
" 3. That the standard of electrical resistance should be
denominated the ohm, and should have the value of 1,000,000,000
in terms of the centimetre and second.
" 4. That the resistance offered to an unvarying electric
current by a column of mercury of a constant cross-sectional
area of one square millimetre, and of a length of 106-3 centi-
metres, at the temperature of melting ice, may be adopted as
one ohm.
" 5. That the value of the standard of resistance constructed
by a committee of the British Association for the Advancement
of Science in the years 1863 and 1864, and known as the British
Association unit, may be taken as -9866 of the ohm.
" 6. That a material standard, constructed in solid metal,
and verified by comparison with the British Association unit,
should be adopted as the standard ohm.
" 9. That the standard of electrical current should be denom-
inated the ampere, and should have the value one-tenth (o-i) in
terms of the centimetre, gramme, and second.
" 10. That an unvarying current which, when passed through
a solution of nitrate of silver in water, in accordance with the
Specification attached to this Report, deposits silver at the rate
of 0-001118 of a gramme per second, may be taken as a current
of one ampere.
" 12. That instruments constructed on the principle of the
* This Committee consisted of Sir Courtenay Boyle, Mr. Hopwood and
Major Cardew representing the Board of Trade ; Mr. Preece and the late
Mr. Graves representing the Postal Telegraph Department ; Lord Kelvin
and Lord Rayleigh the Royal Society; Prof. Carey Foster and Mr.
Glazebrook the British Association ; and Dr. J. Hopkinson and the
Author the Institution of Electrical Engineers.
BOARD OF TRADE COMMITTEE 489
balance, in which, by the proper disposition of the conductors,
forces of attraction and repulsion are produced, which depend
upon the current passing, and are balanced by known weights,
should be adopted as the Board of Trade standards for the
measurement of current, whether unvarying or alternating.
" 13. That the standard of electrical pressure should be
denominated the volt, being the pressure which, if steadily applied
to a conductor whose resistance is one ohm, will produce a current
of one ampere.
" 14. That the electrical pressure at a temperature of 62° F.
between the poles, or electrodes, of the voltaic cell known as
Clark's cell, may be taken as not differing from a pressure of
1-433 volts, by more than an amount which will be determined
by a sub-committee appointed to investigate the question, who
will prepare a specification for the construction and use of the
cell.
" 16. That instruments constructed on the principle of Sir
W. Thomson's Quadrant Electrometer used idiostatically, and,
for high-pressure, instruments on the principle of the balance
electrostatic forces being balanced against a known weight,
should be adopted as Board of Trade standards for the measure-
ment of pressure, whether unvarying or alternating."
Next followed the Specification (see pages 494 and 495) which
was referred to in Resolution 10, and a Draft Order in Council
proposed by the Committee for Her Majesty's signature.
In August, 1892, on the occasion of the meeting of the British
Association at Edinburgh, there was a conference of its Committee
on Electrical Standards with Professor von Helmholtz, the director
of the Imperial Physico-Technical Institute of Berlin, Dr.
Guillaume, of the Bureau International des Poids et Mesures of
France, and Professor Carhart, of the University of Michigan,
U.S.A.
Professor von Helmholtz pointed out that in order to measure
the bore of a narrow glass tube we must fill it with mercury and
weigh it (see page 474), and therefore that it would be better to
specify the weight than the cross-section of the column of mercury
106-3 centimetres in length that at o° C. represented the ohm.
He stated that from experiments carried out in his laboratories
this weight was found to be 14-452 grammes, which, therefore,
he had already recommended the German Government to adopt.
He also mentioned that in the recommendations to his Govern-
ment he had taken 15° C. as the standard temperature for the
490 PRACTICAL ELECTRICITY
specification of the E.M.F. of the Clark's cell, and that at 15° C.
the value was 1.434 volts.
The British Association Committee accordingly adopted resolu-
tions in conformity with the preceding, and transmitted these
resolutions to the Board of Trade. In consequence of this, the
Committee of the Board of Trade, after further deliberation, issued
a supplementary report, in November, 1892, in which their former
Resolution 4 was replaced by —
" 4. That the resistance offered to an unvarying electric
current by a column of mercury, at the temperature of melting
ice, 14-4521 grammes in mass of a constant cross-sectional area,
and of a length of 106-3 centimetres, may be adopted as one
ohm,"
and their former Resolution 14 by —
"14. That the electrical pressure at a temperature of 15°
Centigrade between the poles, or electrodes, of the voltaic cell,
known as Clark's cell, prepared in accordance with the Specifica-
tion attached to this report, may be taken as not differing from
a pressure of 1-434 volt by more than one part in one thousand."
Then followed the Specification referred to in Resolution 14,
which will be found in full on pages 495-7.
This substitution of the Centigrade for the Fahrenheit scale of
temperature was only made after some discussion ; for this
supplementary report was the first document issued by the
Board of Trade in which the Centigrade scale was officially
recognised in Great Britain.
By 1892, then, both the English and the German Governments
were advised to adopt the resistance, at o° C., of a column of mer-
cury 106-3 .centimetres long, of uniform cross-section, and weighing
14-4521 grammes, as the value of the ohm ; whereas the French
Government, some nine years before, had legalised as the ohm the
resistance, at o° C., of a column of mercury only 106 centimetres
in length. Hence, before any material unit of resistance could
receive international support, it was necessary to summon another
international congress. The United States Government was,
therefore, advised to utilise the occasion of the holding of the
World's Fair at Chicago in 1893 by inviting the other Govern-
ments to co-operate with it in sending representatives to constitute
a " Chamber of. Delegates " for selecting the units of electrical
measure. Five delegates were nominated by America, and the
Governments of Great Britain, Germany, and France were each
asked to nominate an equal number, while three, two, and in some
instances one, were allotted to other countries.
CHICAGO CONGRESS, 1893 491
Ten countries, as enumerated in Sect. 52, were actually repre-
sented in the Chamber, and, after many sittings, it was agreed to
adopt certain units, to each of which the name international
was to be affixed. The definitions of the international ampere
and international volt were the same as those recommended by the
Committee of the Board of Trade in the previous year, and the
definition of the international ohm only differed from that of the
Board of Trade ohm in that, while the latter had been defined
as having " the value 1,000,000,000 in terms of the centimetre
and second," coupled with the statement " that the resistance
offered. ... by a column of mercury. . . . may be adopted as
one ohm," the international ohm was denned as " based upon the
ohm equal to io9 units of resistance of the c.G.s. system of electro-
magnetic units, and is represented by the resistance offered
.... by a column of mercury," &c.
Hence, the resistance of the specified column of mercury, which
is the secondary definition of the ohm in the Board of Trade sys-
tem, was taken as the primary definition in the international
system. As, however, the specification of the mercury column
was exactly the same in the two cases, and as the resistance of
this column is believed to represent the ideal ohm to a high degree
of accuracy, no practical difference in the value of the ohm was
introduced by this variation in the form of the definition.
From the preceding pages it will be seen that, while the Paris
Congress and the British Association Committee on Electrical
Standards had defined and named the units of quantity and
capacity as well as those of resistance, current, and E.M.F., the
Board of Trade had confined its attention to the three latter.
The Chamber of Delegates, on the other hand, embodied in their
definitions not only the five units just referred to, but also the
unit of work — the joule (see Sect. 117) — the unit of power — the
watt (seeSect. 120) — and a new name, the " henry," page 437, for
the unit of self-induction (see Section 195), this name being se-
lected partly because some of the earliest work on self-induction
had been carried out by Prof. Henry, of America, and partly out
of compliment to the nation at whose invitation the Chamber
had been summoned.
The definition adopted for this sixth unit was : — " As the unit
of induction, the henry, which is the induction in a circuit when
the electromotive force induced in this circuit is one international
volt, while the inducing current varies at the rate of one ampere
per second."
The recommendations made by the Chamber of Delegates and
adopted by the International Congress in August, 1893, were
492 PRACTICAL ELECTRICITY
passed by both Houses of Congress in July, 1894, in the form of
an " Act to define and establish the Units of Electrical Measure,"
which, after receiving the signature of the President of the United
States, became law in that country.
In August, 1894, the Committee of the Board of Trade sub-
mitted their final report to the President of the Board, and stated
that since the International Congress held in Chicago had adopted,
almost without change, the definitions proposed by the committee
in 1892, " they saw no reason for further delay in the legalization
of standards." The committee appended to this report a revised
Draft Order in Council, which they had prepared, and Mr.
Glazebrook's Notes to the Specification of the Clark cell.
An Order in Council in the suggested form was made by Her
Majesty on the 23rd August, 1894, and so became law. The
following is the text : —
AT THE COURT AT OSBORNE HOUSE,
ISLE OF WIGHT,
The 23rd day of August, 1894.
PRESENT.
THE QUEEN'S MOST EXCELLENT MAJESTY
IN COUNCIL.
WHEREAS by " The Weights and Measures Act, 1889," it
is among other things enacted that the Board of Trade
shall from time to time cause such new denominations of
standards for the measurement of electricity as appear to them
to be required for use in trade to be made and duly verified.
And whereas it has been made to appear to the Board of
Trade that new denominations of standards are required for use
in trade based upon the following units of electrical measurement
viz. : —
1. The Ohm, which has the value io9 in terms of the centimetre
and the second of time and is represented by the resistance
offered to an unvarying electric current by a column of mercury
at the temperature of melting ice 14-4521 grammes in mass of a
constant cross-sectional area and of a length of 106-3 centimetres.
2. The Ampere, which has the value ^ in terms of the centi-
metre, the gramme and the second of time and which is represented
by the unvarying electric current which when passed through a
ORDER IN COUNCIL, 1894 493
solution of nitrate of silver in water, in accordance with the specifi
cation appended hereto and marked A, deposits silver at the rate
of 0-001118 of a gramme per second.
The Volt which has the value io8 in terms of the centimetre,
the gramme and the second of time, being the electrical pressure
that if steadily applied to a conductor whose resistance is one
ohm will produce a current of one ampere, and which is represented
by -6974 (fifl) of the electrical pressure at a temperature of
15° C. between the poles of the voltaic cell known as Clark's cell
set up in accordance with the specification appended hereto and
marked B.
And whereas they have caused the said new denominations
of standards to be made and duly verified.
NOW, THEREFORE, Her Majesty, by virtue of the power
vested in Her by the said Act, by and with the advice of Her
Privy Council, is pleased to approve the several denominations
of Standards set forth in the schedule hereto as new denominations
of Standards for electrical measurement.
C. L. PEEL.
SCHEDULE.
I.— STANDARD OF ELECTRICAL RESISTANCE.
A standard of electrical resistance denominated one Ohm being
the resistance between the copper terminals of the instrument
marked " Board of Trade Ohm Standard Verified 1894 " to the
passage of an unvarying electrical current when the coil of insu-
lated wire forming part of the aforesaid instrument and connected
to the aforesaid terminals is in all parts at a temperature oi
15-4° C.
II.— STANDARD OF ELECTRICAL CURRENT.
A standard of electrical current denominated one ampere being
the current which is passing in and through the coils of wire
forming part of the instrument marked " Board of Trade Ampere
Standard Verified 1894 " when on reversing the current in the
fixed coils the change in the forces acting upon the suspended
coil in its sighted position is exactly balanced by the force exerted
by gravity in Westminster upon the iridioplatinum weight
marked A and forming part of the said instrument.
III.— STANDARD OF ELECTRICAL PRESSURE.
A standard of electrical pressure denominated one Volt being
one hundredth part of the pressure which when applied between
494 PRACTICAL ELECTRICITY
the terminals forming part of the instrument marked " Board of
Trade Volt Standard Verified 1894," causes that rotation of the
suspended portion of the instrument which is exactly measured
by the coincidence of the sighting wire with the image of the
fiducial mark A before and after application of the pressure and
with that of the fiducial mark B during the application of the
pressure, these images being produced by the suspended mirror
and observed by means of the eyepiece.
In the use of above standards the limits of accuracy attainable
are as follows : —
For the Ohm, within one-hundredth part of one per cent.
For the Ampere, within one-tenth part of one per cent.
For the Volt, within one-tenth part of one per cent.
The coils and instruments referred to in this schedule are
deposited at the Board of Trade Standardising Laboratory, 8,
Richmond Terrace, Whitehall, London.
SPECIFICATIONS referred to in the foregoing Order in Council.
SPECIFICATION A.
In the following specification the term silver voltameter means the
arrangement of apparatus by means of which an electric current is passed
through a solution of nitrate of silver in water. The silver voltameter
measures the total electrical quantity which has passed during the time of
the experiment, and, by noting this time, the time average of the current,
or if the current has been kept constant, the current itself, can be deduced.
In employing the silver voltameter to measure currents of about i ampere
the following arrangements should be adopted. The kathode on which
the silver is to be deposited should take the form of a platinum bowl not
less than 10 centimetres in diameter, and from 4 to 5 centimetres in depth.
The anode should be a plate of pure silver some 30 square centimetres
in area and 2 or 3 millimetres in thickness.
This is supported horizontally in the liquid near the top of the solution
by a platinum wire passed through holes in the plate at opposite corners.
To prevent the disintegrated silver which is formed on the anode from
falling on to the kathode, the anode should be wrapped round with pure
filter paper, secured at the back with sealing-wax.*
The liquid should consist of a neutral solution of pure silver nitrate,
containing about 15 parts by weight of the nitrate to 85 parts of water.
The resistance of the voltameter changes somewhat as the current passes.
To prevent these changes having too great an effect on the current, some
resistance besides that of the voltameter should be inserted in the circuit.
The total metallic resistance of the circuit should not be less than 10 ohms.
Method o'f making a Measurement^
The platinum bowl is washed with nitric acid and distilled water, dried
by heat, and then left to cool in a desiccator. When thoroughly dry it is
weighed carefully.
* See page 511.
f This procedure properly carried out gives very satisfactory results for
ordinary purposes. For work of the highest precision, however, the
Appendix to the Report of the International Conference on Electrical
Units and Standards, 1908, should be consulted.— T. M.
ORDER IN COUNCIL, 1894 495
It is nearly filled with the solution, and connected to the rest of the
circuit by being placed on a clean copper support to which a binding screw
is attached. This copper support must be insulated.
The anode is then immersed in the solution so as to be well covered by it
and supported in that position ; the connections to the rest of the circuit
are made.
Contact is made at the key, noting the time of contact. The current
is allowed to pass for not less than half an hour, and the time at which con-
tact is broken is observed. Care must be taken that the clock used is
keeping correct time during this interval.
The solution is now removed from the bowl, and the deposit is washed
with distilled water and left to soak for at least six hours. It is then rinsed
successively with distilled water and absolute alcohol and dried in a hot-air
bath at a temperature of about i6o°C. After cooling in a desiccator it is
weighed again. The gain in weight gives the silver deposited.
To find the current in amperes, this weight, expressed in grammes, must
be divided by the number of seconds during which the current has been
passed, and by o-ooiiiS.
The result will be the time-average of the current, if during the interval
the current has varied.
In determining by this method the constant of an instrument the current
should be kept as nearly constant as possible, and the readings of the
instrument observed at frequent intervals of time. These observations
give a curve from which the reading corresponding to the mean current
(time-average of the current) can be found. The current, as calculated by
the voltameter, corresponds to this reading.
SPECIFICATION B«
ON THE PREPARATION OF THE CLARK CELL.
Definition of the Cell.
The cell consists of zinc, or an amalgam of zinc with mercury, and of
mercury in a neutral saturated solution of zinc sulphate and mercurous
sulphate in water, prepared with mercurous sulphate in excess.
Preparation of the Materials,
1. The Mercury. — To secure purity it should be first treated with acid
in the usual manner, and subsequently distilled in vacuo.
2. The Zinc. — Take a portion of a rod of pure redistilled zinc, solder to
one end a piece of copper wire, clean the whole with glass-paper or a steel
burnisher, carefully removing any loose pieces of the zinc. Just before
making up the cell dip the zinc into dilute sulphuric acid, wash with dis-
tilled water, and dry with a clean cloth or filter paper.
3. The Mercurous Sulphate. — Take mercurous sulphate, purchased as
pure, mix with it a small quantity of pure mercury, and wash the whole
thoroughly with cold distilled water by agitation in a bottle ; drain off the
water, and repeat the process at least twice. After the last washing, drain
off as much of the water as possible.
4. The Zinc Sulphate Solution. — Prepare a neutral saturated solution of
pure (" pure re-crystallised ") zinc sulphate by mixing in a flask distilled
water with nearly twice its weight of crystals of pure zinc sulphate, and
adding zinc oxide in the proportion of about 2 per cent, by weight of the
zinc sulphate crystals to neutralise any free acid. The crystals should be
dissolved with the aid of gentle heat, but the temperature to which the
solution is raised should not exceed 30° C. Mercurous sulphate treated as
described in 3 should be added in the proportion of about 12 per cent. b\
496 PRACTICAL ELECTRICITY
weight of the zinc sulphate crystals to neutralise any free zinc oxide re-
maining, and the solution filtered, while still warm, into a stock bottle.
Crystals should form as it cools.
5. The Mercurous Sulphate and Zinc Sulphate Paste. — Mix the washed
mercurous sulphate with the zinc sulphate solution, adding sufficient crys-
tals of zinc sulphate from the stock bottle to ensure saturation, and a small
quantity of pure mercury. Shake these up well together to form a paste
of the consistence of cream. Heat the paste, but not above a temperature
of 30° C. Keep the paste for an hour at this temperature, agitating it
from time to time, then allow it to cool ; continue to shake it occasionally
while it is cooling. Crystals of zinc sulphate should then be distinctly
visible, and should be distributed throughout the mass ; if this is not the
case add more crystals from the stock bottle, and repeat the whole process.
This method ensures the formation of a saturated solution of zinc and
mercurous sulphates in water.
To set up the Cell.
The cell may conveniently be set up in a small test-tube of about 2
centimetres diameter, and 4 or 5 centimetres deep. Place the mercury
in the bottom of this tube, filling it to a depth of, say, .5 centimetre. Cut
a cork about .5 centimetre thick to fit the tube ; at one side of the cork
bore a hole through which the zinc rod can pass tightly ; at the other side
bore another hole for the glass tube which covers the platinum wire ; at
the edge of the cork cut a nick through which the air can pass when the
cork is pushed into the tube. Wash the cork thoroughly with warm water,
and leave it to soak in water for some hours before use. Pass the zinc
rod about i centimetre through the cork.
Contact is made with the mercury by means of a platinum wire about
No. 22 gauge. This is protected from contact with the other materials of
the cell by being sealed into a glass tube. The ends of the wire project
from the ends of the tube ; one end forms the terminal, the other end and
a portion of the glass tube dip into the mercury.
Clean the glass tube and platinum wire carefully, then heat the exposed
end of the platinum red hot, and insert it in the mercury in the test-tube,
taking care that the whole of the exposed platinum is covered.
Shake up the paste and introduce it without contact with the upper part
of the walls of the test-tube, filling the tube above the mercury to a depth
of rather more than i centimetre.
Then insert the cork and zinc rod,^passing the glass tube through the
hole prepared for it. Push the cork gently down until its lower surface is
nearly in contact with the liquid. The air will thus be nearly all expelled,
and the cell should be left in this condition for at least twenty-four hours
before sealing, which should be done as follows : —
Melt some marine glue until it is fluid enough to pour by its own weight,
and pour it into the test-tube above the cork, using sufficient to cover
completely the zinc and soldering. The glass tube containing the platinum
wire should project some way above the top of the marine glue.
The cell may be sealed in a more permanent manner by coating the
marine glue, when it is set, with a solution of sodium silicate, and leaving
it to harden.
The cell thus set up may be mounted in any desirable manner. It is
convenient to arrange the mounting so that the cell may be immersed in
a water bath up to the level of, say, the upper surface of the cork. Its
temperature can then be determined more accurately than is possible when
the cell is in air.
In using the cell sudden variations of temperature should as far as possible
be avoided.
The form of the vessel containing the cell may be varied. In the H
form, the zinc is replaced by an amalgam of 10 parts by weight of zinc to
AMERICAN SPECIFICATION, 1895 497
90 of mercury. The other materials should be prepared as already described.
Contact is made with the amalgam in one leg of the cell, and with the mer-
cury in the other, by means of platinum wires sealed through the glass.
The drafting of the American " Specification of the Practical
Application of the Definitions of the Ampere and Volt " was
deputed by Congress to the National Academy of Sciences, and
in February, 1895, the President of the Academy submitted to
the Home Secretary of the United States the report drawn up by
the committee appointed by the Academy.
The specification so prepared for the use of the silver volta-
meter was nearly identical with that recommended by the Com-
mittee of the Board of Trade (see pages 494 and 495), but that
ior the Clark's cell dealt exclusively with the H form. Otherwise,
the American and English specifications were generally in accord.
In the note on page 487 it was explained that, after correcting
for the ellipticity of the coil used by Professor Viriamu Jones in
his determination of the specific resistance of mercury in absolute
measure, it followed that the length of the mercury column one
square millimetre in cross-section, which had a resistance of i ohm
at o° C., was 106-300 centimetres. On using, however, Lorenz's
apparatus to test the resistance of a copy of the Board of Trade
standard ohm, and assuming that this really represents the resist-
ance at o° C. of a column of mercury i square millimetre in cross-
section and 106-3 centimetres in length, Professor Jones finds
that the true ohm must have a value equivalent to that of 106-326
centimetres of mercury, or 106-319 after allowing for the ellip-
ticity of the coil.
We are, therefore, not yet sure of the fifth figure in the preceding
number, but it is to be expected that this will shortly be obtained
with accuracy by the employment of the very carefully-made
Lorenz's apparatus that has been constructed for Professor Cal-
lendar of the McGill University, Montreal, and which is now —
September, 1896 — being tested at the City and Guilds En-
gineering College, by Professor Viriamu Jones and the author.
ADDENDUM, 1910. Revised 1921*
THE Lorenz method of determining resistance absolutely is
acknowledged to be the one capable of the highest accuracy, and
as electromagnetic E.M.F. and mutual induction have now been
defined (see Sections 550 and 194), the method can be explained.
2G
498 PRACTICAL ELECTRICITY
In this method a metal disc is rotated about an axis coinci-
dent with the axis of a circular coil of wire through which a
current passes, and the speed is varied until the E.M.F. produced
in the disc by its rotation balances the P.D. between the
terminals of a resistance carrying the same current as the coil.
When a current /' c.G.s. units flows through the coil, a certain
number of lines of force (I'M) will pass through the disc, where
M is the coefficient of mutual induction between the coil and a
circle coincident with the periphery of the disc.*
During one revolution each radius of the disc cuts all the lines
of force, viz. I'M, and at n revolution per second, the rate of
cutting, and therefore the E.M.F. produced between the centre
and the circumference of the disc is given by
E = n I'M.
The P.D. between the terminals of a resistance R C.G.S. units
when a current I' passes through it is
V = I'R
and when n is varied until the E.M.F. equals the P.D.f we then
have
I'R = nl'M,
or R = nM,
a very simple relation whereby the resistance of a conductor can
be determined in terms of length and time, without reference to
any other quantity.
The result of the tests referred to in the last paragraph of the
late Professor Ayrton's " Short History, etc." page 497, was
communicated to the British Association at the Toronto meeting
in 1897, viz. :
i Board of Trade Ohm = 1-00026 true ohms.
Assuming that the Board of Trade Ohm was at that time equal
to the resistance of a column of mercury 106-3 centimetres long
and i square millimetre in cross-section, the above figure gives
106-273 centimetres of mercury as equivalent to one true ohm,
a value in close agreement with that of Lord Rayleigh in 1882,
and of Dr. Glazebrook in the same year (see pages 483-84) .
A new determination made in 1912 by Mr. Albert Campbell, of
the National Physical Laboratory, using alternating currents,
gave the length of the column of mercury 106-27 cms. The re-
sult of later and more elaborate experiments carried out by
* M can be calculated from the dimensions of the coil and disc, and
expressed as a length of so many centimetres.
f This can be observed by arranging the two voltages in opposition in
a circuit containing a sensitive galvanometer.
ST. LOUIS AND BERLIN CONFERENCES 499
Mr. F. E. Smith, F.R.S., of the National Physical Laboratory,
was 106-245 +_ 0-004 cms.* In the latter research a modified
form of Lorenz's Apparatus, designed at the Laboratory, was
employed by Mr. Smith, and the precision attained was much
in advance of previous determinations. Funds for the con-
struction of the apparatus were generously provided by the
Drapers' Company in memory of the late Professor Viriamu
Jones, and by the late Sir A. Noble.
The latest determination yet published (May, 1921) is one
by Messrs. E. Gruneisen and E. Giebe,t who compared a capacity
(calculated) with an inductance (also calculated), by Maxwell's
method, the result being 106-246 centimetres of mercury, a
result in close agreement with Mr. Smith's 1914 value. J
Subsequent to the legalisation of the Electrical Units in Eng-
land, in 1894, differences amounting to several parts in 10,000
between the units as realised in different countries were found
to exist. These differences led to inconvenience in commercial
transactions. With a view to removing the inequalities the
subject was fully considered at the International Electrical
Congress held in St. Louis, U.S.A., in 1904, and the Chamber of
Delegates reported, amongst other things, that " It appears
from papers laid before the International Congress and from the
discussions, that there are considerable discrepancies between the
laws relating to the electric units or their interpretations in the
various countries represented, which, in the opinion of the Cham-
ber, require consideration with a view to securing practical
uniformity," and advised the appointment of an International
Commission to deal with the subject.
An informal conference of the Commission was held in Char-
lottenberg, in June, 1905, at which it was decided that " the
International Ohm defined by the resistance of a column of mer-
cury, and the International Ampere, denned by the deposition of
silver, are to be taken as the fundamental electrical units."
" The International Volt is that electromotive force which pro-
duces an electric current of one International Ampere in a con-
ductor whose resistance is one International Ohm."
" The West on Cadmium Cell shall be adopted as the Standard
Cell."
* Phil. Trans., Vol. 214, A., 1914, p. 106.
t Ann. der Physik, 1920, No. 18.
I The actual agreement is not so close as the apparent agreement
because the International ohm as realised in Germany is slightly greater than
that produced at the National Physical Laboratory.
5oo PRACTICAL ELECTRICITY
At that time the information regarding the electrochemical
equivalent of silver was considered insufficient to enable any
alteration in the formerly accepted value of the ampere to he
proposed, so this question, as well as the E.M.F. of the stan-
dard cell, was left for consideration by an Official Conference
to be held in London in October, 1906. This meeting was post-
poned, first to 1907 and then to 1908, to permit of the necessary
experiments being carried out.
In 1908 the International Conference on Electrical Units and
Standards was held in the Rooms of the Royal Society, London,
from October 12 to October 23, on the conclusion of which the
following report was adopted : —
INTERNATIONAL CONFERENCE ON
ELECTRICAL UNITS AND STANDARDS, 1908.
REPORT.
The Conference on Electrical Units and Standards for which
invitations were issued by the British Government, was opened
by the President of the Board of Trade, the Right Hon.
Winston S. Churchill, M.P., on Monday, I2th October, 1908, at
Burlington House, London, S.W.
Delegates were present from twenty-one countries, and also
from the following British Dependencies, namely, Australia,
Canada, India, and the Crown Colonies.
It was decided by the Conference that a vote each should be
allowed to Australia, Canada and India, but a vote was not
claimed or allowed for the Crown Colonies.
The total number of delegates to the Conference was forty-six,
and their names are set out in Schedule A to this Report.*
The officers of the Conference were : —
President :
The RIGHT HON. LORD RAYLEIGH, O.M., President of the Royal Society.
Vice-Presidents :
PROFESSOR S. A. ARRHENIUS. M. LIPPMANN.
DR. M. EGOROFF. DR. S. W. STRATTON.
DR. VIKTOR EDLER VON LANG. DR. E. WARBURG.
Secretaries :
MR. M. J. COLLINS. MR. C. W. S. CRAWLEY.
MR. W. DUDDELL, F.R.S. MR. F. E. SMITH.
The Conference elected a Technical Committee to draft specifica-
tions and to consider any matter which might be referred to the
Committee, and to report to the Conference.
* Schedule A is not reprinted here.
LONDON CONFERENCE, 1908 501
The Conference and its Technical Committee each held five
sittings.
As a result of its deliberation the Conference adopted the resolu-
tions and specifications attached to this report and set out in
Schedule B, and requested the Delegates to lay them before their
respective Governments with a view to obtaining uniformity in
the legislation with regard to Electrical Units and Standards.
The Conference recommend the use of the Weston Normal
Cell as a convenient means of measuring both electromotive
force and current when set up under the conditions specified in
Schedule C.
In cases in which it is not desired to set up the Standards pro-
vided in the resolutions Schedule B, the Conference recommends
the following as working methods for the realisation of the
International Ohm, the Ampere and the Volt.
1. For the International Ohm.
The use of copies, constructed of suitable material and of
suitable form and verified from time to time, of the
International Ohm, its multiples and submultiples.
2. For the International Ampere.
(a) The measurement of current by the aid of a current
balance standardized by comparison with a silver
voltameter ;
or (b) The use of a Weston Normal Cell whose electromotive
force has been determined in terms of the International
Ohm and International Ampere, and of a resistance of
known value in International Ohms.
3. For the International Volt.
(a) A comparison with the difference of electrical potential
between the ends of a coil of resistance of known value
in International Ohms, when carrying a current of
known value in International Amperes ;
or (b) The use of a Weston Normal Cell whose electromotive
force has been determined in terms of the International
Ohm and the International Ampere.
The duties of specifying more particularly the conditions under
which these methods are to be applied has been assigned to the
Permanent Commission, and pending its appointment, to the
Scientific Committee to be nominated by the President (see
Schedule D), who will issue a series of Notes as Appendix to this
Report.
The Conference has considered the methods that should be re-
502 PRACTICAL ELECTRICITY
commended to the Governments for securing uniform administra-
tion in relation to Electrical Units and Standards, and expresses
the opinion that the best method of securing uniformity for the
future would be by the establishment of an International Elec-
trical Laboratory with the duties of keeping and maintaining
International Electrical Standards. This Laboratory to be
equipped entirely independently of any National Laboratory.
The Conference further recommends that action be taken in
accordance with the scheme set out in Schedule D.
Signed at London on 2ist October, 1908.
By the Delegates of the Countries above written.
SCHEDULE B.
RESOLUTIONS.
I. The Conference agrees that as heretofore the magnitudes
of the fundamental electric units shall be determined on the
electro-magnetic system of measurement with reference to the
centimetre as the unit of length, the gramme as the unit of mass
and the second as the unit of time.
These fundamental units are (i) the Ohm, the unit of electric
resistance which has the value of 1,000,000,000 in terms of the
centimetre and second ; (2) the Ampere, the unit of electric
current which has the value of one tenth (o-i) in terms of the centi-
metre, gramme, and second ; (3) the Volt, the unit of electro-
motive force which has the value 100,000,000 in terms of the
centimetre, the gramme, and the second ; (4) the Watt, the unit
of Power which has the value 10,000,000 in terms of the centi-
metre, the gramme, and the second.
II. As a system of units representing the above and sufficiently
near to them to be adopted for the purpose of electrical measure-
ments and as a basis for legislation, the Conference recommends
the adoption of the International Ohm, the International Am-
pere, and the International Volt defined according to the following
definitions.
III. The Ohm is the first Primary Unit.
IV. The International Ohm is defined as the resistance of a
specified column of mercury.
V. The International Ohm is the resistance offered to an un-
varying electric current by a column of mercury at the temper-
ature of melting ice, 14-4521 grammes in mass, of a constant cross
sectional area and of a length of 106*300 centimetres.
To determine the resistance of a column of mercury in terms
RESOLUTIONS OF 1908 CONFERENCE 503
of the International Ohm, the procedure lo be followed shall be
that set out in Specification I. attached to these Resolutions.
VI. The Ampere is the second Primary Unit.
VII. The International Ampere is the unvarying electric
current which, when passed through a solution of nitrate of silver
in water, in accordance with the Specification II. attached to these
Resolutions, deposits silver at the rate of 0-00111800 of a gramme
per second.
VIII. The International Volt is the electrical pressure which,
when steadily applied to a conductor whose resistance is one
International Ohm, will produce a current of one International
Ampere.
IX. The International Watt is the energy expended per second
by an unvarying electric current of one International Ampere
under an electric pressure of one International Volt.
SPECIFICATION I.
SPECIFICATION RELATING TO MERCURY STANDARDS OF RESISTANCE.
The glass tubes used for mercury standards of resistance must be made
of a glass such that the dimensions may remain as constant as possible.
The tubes must be well annealed and straight. The bore must be as nearly
as possible uniform and circular, and the area of cross-section of the bore
must be approximately one square millimetre. The mercury must have a
resistance of approximately one ohm.
Each of the tubes must be accurately calibrated. The correction to be
applied to allow for the area of the cross-section of the bore not being
exactly the same at all parts of the tube must not exceed 5 parts in 10,000.
The mercury filling the tube must be considered as bounded by plane
surfaces placed in contact with the ends of the tube.
The length of the axis of the tube, the mass of mercury the tube contains,
and the electrical resistance of the mercury are to be determined at a
temperature as near to o° C. as possible. The measurements are to be
corrected to o° C.
For the purpose of the electrical measurements, end vessels carrying
connections for the current and potential terminals are to be fitted on to
the tube. These end vessels are to be spherical in shape (of a diameter
of approximately four centimetres) and should have cylindrical pieces
attached to make connections with the tubes. The outside edge of each
end of the tube is to be coincident with the inner surface of the correspond-
ing spherical end vessel. The leads which make contact with the mercury
are to be of thin platinum wire fused into glass. The point of entry of the
current lead and the end of the tube are to be at opposite ends of a diameter
of the bulb ; the potential lead is to be midway between these two points.
All the leads must be so thin that no error in the resistance is introduced
through conduction of heat to the mercury. The filling of the tube with
mercury for the purpose of the resistance measurements must be carried
out under the same conditions as the filling for the determination of the
mass.
The resistance which has to be added to the resistance of the tube to
allow for the effect of the end vessels is to be calculated by the formula
0-80 /i i \ ,
A = — •? I — ^ I ohm
1063 Ir\r1 T r,7
504 PRACTICAL ELECTRICITY
where r , and r , are the radii in millimetres of the end sections oi the bore
of the tube.
The mean of the calculated resistances of at least five tubes shall be taken
to determine the value of the unit of resistance.
For the purpose of the comparison of resistances with a mercury tube
the measurements shall be made with at least three separate fillings of the
tube.
SPECIFICATION II.
SPECIFICATION RELATING TO THE DEPOSITION OF SILVER.
The electrolyte shall consist of a solution of from 15 to 20 parts by weight
of silver nitrate in 100 parts of distilled water. The solution must only be
used once, and only for so long that not more than 30 per cent, of the silver
m the solution is deposited.
The anode shall be of silver, and the kathode of platinum. The current
density at the anode shall not exceed 1/5 ampere per square centimetre
and at the kathode 1/50 ampere per square centimetre.
Not less than 100 cubic centimetres of electrolyte shall be used in a volta-
meter.
Care must be taken that no particles which may become mechanically
detached from the anode shall reach the kathode.
Before weighing any traces of solution adhering to the kathode must be
removed, and the kathode dried.
SCHEDULE C.
WESTON NORMAL CELL.
The Weston Normal Cell may be conveniently employed as a
standard of electric pressure for the measurement both of E.M.F.
and of current, and when set up in accordance with the following
Specification, may be taken, provisionally, as having, at a
temperature of 20° C., an E.M.F. of 1-0184 volts.*
SPECIFICATION RELATING TO THE WESTON NORMAL CELL.
The Weston Normal Cell is a voltaic cell which has a saturated aqueous
solution of cadmium sulphate (CdSO 4 8/3 H 2O) as its electrolyte.
The electrolyte must be neutral to Congo Red.
The positive electrode of the cell is mercury.
The negative electrode of the cell is cadmium amalgam consisting of 12-5
parts by weight of cadmium in 100 parts of amalgam.
The depolariser, which is placed in contact with the positive electrode,
is a paste made by mixing mercurous sulphate with powdered crystals of
cadmium sulphate and a saturated aqueous solution of cadmium sulphate.
The different methods of preparing the mercurous sulphate paste are
described in the notes. f One of the methods there specified must be carried
out.
For setting up the cell, the H form is the most suitable. The leads
passing through the glass to the electrodes must be of platinum wire,
which must not be allowed to come into contact with the electrolyte. The
amalgam is placed in one limb, the mercury in the other.
The depolariser is placed above the mercury and a layer of cadmium
sulphate crystals is introduced into each limb. The entire cell is filled with
a saturated solution of cadmium sulphate and then hermetically sealed.
* This has now been changed to 1-0183 volts, at 20° C.
t See pages 506-7, for method and procedure adopted at the National
Physical Laboratory.
RECOMMENDATIONS OF CONFERENCE 505
The following formula is recommended for the E.M.F. of the cell in
terms of the temperature between the limits o° C. & 40° C.
Et =E 2 0 — 0-0000406 (t — 20°)— 0-00000095 (t — 20°) * + 0-00000001 (t— 20)»
SCHEDULE D.
1. The Conference recommends that the various Governments
interested establish a permanent International Commission for
Electrical Standards.
2. Pending the appointment of the Permanent International
Commission the Conference recommends that the President,
Lord Rayleigh, nominate for appointment by the Conference a
scientific Committee of fifteen to advise as to the organisation of
the Permanent Commission,* to formulate a plan for and to direct
such work as may be necessary in connection with the mainten-
ance of standards, fixing of values, -f inter-comparison of Stand-
ards and to complete the work of the Conference. J Vacancies on
the Committee to be filled by co-optation.
3. That Laboratories equipped with facilities for precise electri-
cal measurements and investigations should be asked to co-oper-
ate with this Committee and to carry out, if possible, such work
as it may desire.
4. The Committee should take the proper steps forthwith for
establishing the Permanent Commission, and are empowered
to arrange for the meeting of the next Conference on Electrical
Units and Standards, and the time and place of such meeting
should this action appear to them to be desirable.
5. The Committee or the Permanent International Commission
shall consider the question of enlarging the functions of the Inter-
national Commission on Weights and Measures, with a view to
determining if it is possible or desirable to combine future Con-
ferences on Electrical Units and Standards with the Inter-
national Commission on Weights and Measures, in place of holding
in the future Conferences on Electrical Units and Standards.
At the same time it is the opinion of the Conference that the
* In accordance with the above, Lord Rayleigh has nominated the
following Committee, which has been approved by the Conference, viz. : —
Dr. Osuke Asano, M. R. Benoit, Dr. M. N. Egoroff, Prof. Eric Gerard,
Dr. R. T. Glazebrook, Dr. H. Haga, D. L. Kusminsky, Prof. Lindeck,
Prof. G. Lippmann, Prof. A. R6iti, Dr. E. B. Rosa, Dr. S. W. Stratton,
Mr. A. P. Trotter, Prof. E. Warburg, Prof. Fr. Weber.
f This will include the reconsideration from time to time of the E.M.F.
of the Weston Normal Cell.
J With this object the Committee are authorised to issue as an Appendix
to the Report of the Conference Notes detailing the methods which have
been adopted in the Standardising Laboratories of the various countries
to realise the International Ohm and the International Ampere, and to set
up the Weston Normal Cell.
5o6 PRACTICAL ELECTRICITY
Permanent Commission should be retained as a distinct body,
which should meet at different places in succession.
NOTES RELATING TO THE WESTON NORMAL CELL.
PROCEDURE AT THE NATIONAL PHYSICAL LABORATORY.
(i) Preparation of the Materials.
(a) Mercury. — Commercially pure mercury is squeezed through wash-
leather and passed, in the finely divided condition in which it emerges,
through dilute nitric acid (one-part of acid to six parts of water) and mer-
curous nitrate solution, and afterwards through distilled water. The
mercury is then distilled twice in vacuo.
(b) Cadmium amalgam. — Two methods have been used, (i) A current is
passed from a thick rod of commercially pure cadmium to distilled mercury,
the electrolyte being cadmium sulphate solution rendered slightly acid by
the addition of a small quantity of sulphuric acid. The cathode is weighed
before and after electrolysis, and the percentage of cadmium calculated
from the two weights. More than the requisite quantity of cadmium is
deposited, and the percentage reduced to 12^ by the addition of mercury.
The anode is conveniently contained in a filter-paper cup. The amalgam
with dilute sulphuric acid flooding its surface is melted over a water bath
and stirred to ensure homogeneity. It is then ready for use. (2) Com-
mercially pure cadmium is added to mercury in the proportion of one to
seven by weight and the mixture fused. It is freed from oxide by flooding
its surface with dilute sulphuric acid.
(c) Cadmium sulphate. — Commercially pure cadmium sulphate is
dissolved in water, and re-crystallised by evaporation at a temperature
not exceeding 50° C. The re-crystallised salt is washed with successive
small quantities of distilled water, and part of it is dissolved to form a
saturated solution. If the solution is not neutral to congo red, the pro-
cedure is repeated until it is so.
(d) Mercurous sulphate. — Fifteen cubic centimetres of pure strong nitric
acid are added to 100 grams of mercury, and the mixture placed on one
side until the chemical action is practically over. The mercurous nitrate
thus formed, together with the excess of mercury, is transferred to a beaker
containing about 200 cc. of dilute nitric acid (i part of acid to 40 parts of
water). A clear solution results. About one litre of dilute sulphuric
acid (one part of acid to three of water) is prepared, and while the mixture
is hot the acid mercurous nitrate solution is added to it as a very fine
stream from the narrow orifice of a pipette, the mixture being violently
agitated during the mixing. Mercurous sulphate is precipitated. The
hot clear liquid is decanted, and the precipitate washed twice by decanta-
tion with dilute sulphuric acid (one part of acid to six parts of water).
The precipitate is then filtered, washed three times with dilute sulphuric
acid (one to six of water), and afterwards six or seven times, with neutral
saturated cadmium sulphate solution to remove the acid. When these
operations are complete the mercurous sulphate is flooded with saturated
cadmium sulphate solution and left for one hour ; the solution is then
tested with congo red paper. In general no acid is detected, and the mer-
curous sulphate is ready for use.
The following electrolytic method is also sometimes employed, the pre-
paration being conducted in a darkened room. Mercury forms the anode,
and platinum foil the cathode, the electrolyte being dilute sulphuric acid
(one of acid to five of water, by volume). The mercury is placed in the
bottom of a large flat-based beaker, and about 20 times its volume of the
dilute acid is added. Contact with the mercury is made by a platinum
wire passing through a glass tube, and the cathode is suspended in the upper
portion of the liquid, A current density of 0*0 1 ampere per square centi-
BOARD OF TRADE STANDARDS 507
metre of anode surface is generally used, and the electrolyte is continually
stirred during electrolysis. The mercurous sulphate so prepared is filtered,
and the greater part of the mercury is removed ; it is then washed with
dilute sulphuric acid and saturated cadmium sulphate solution in a manner
already described for the previous preparation.
(e) The Paste. — The mercurous sulphate is mixed with about one-fourth
its volume of powdered cadmium sulphate and about one-tenth its volume
of mercury. To the mixture sufficient saturated cadmium sulphate solu-
tion is added, so that when well mixed the whole forms a thin paste.
(2) Setting up the Cell.
Fig. 132, page 206, shows the form of cell used. It is of the H form, and
the lower end of each limb is slightly constricted. The platinum wires
inside the vessel are amalgamated by passing an electric current from a
platinum anode through an acid solution of mercurous nitrate to each of
the wires in turn as cathode. The vessel is washed out twice with dilute
nitric acid and several times with distilled water ; it is dried in an oven.
The amalgam is fused and its surface flooded with dilute sulphuric acid ;
sufficient of it to cover the amalgamated platinum wire completely is then
introduced by means of a pipette into one of the limbs of the H vessel.
To free from acid the amalgam is remelted and washed with distilled water.
Into the other limb of the vessel sufficient mercury is introduced to cover
the amalgamated platinum wire completely. Then the paste, finely pow-
dered crystals of cadmium sulphate, and saturated cadmium sulphate solu-
tion are added in the order named. The upper surfaces of the layers of
cadmium sulphate crystals are on a level with the constrictions in the tubes.
The cell is left in a warm room for about three weeks and is then hermeti
cally sealed. It is portable and may be sent through the post.
From Schedule B of the above report it will be observed that no
essential changes were made in the definitions of the International
units, but that two zeros were added to the numbers 106-3 and
o-ooiii8 in the definitions of the ohm and ampere respectively.
These " distinctions without a difference " were made to give
greater precision to the definitions.
The International Watt is also defined in Schedule B (No. IX.)
in terms of the ampere and volt.
Experiments carried out by Mr. F. E. Smith, of the National
Physical Laboratory, and the writer on the Board of Trade
Ampere Standard early in 1908 showed that the Board of Trade
Ampere will deposit 1-1179 milligrammes of silver per second.
It is therefore equal to the International Ampere within I part
in 10,000, or — of I per cent. This result after 14 years' use
is highly satisfactory, both as regards permanency and accuracy
of original calibration.
Tests on the Board of Trade Ohm, however, have disclosed
differences exceeding i part in 10,000 from the International Ohm
which it was intended to represent within this limit of accuracy.
The resistance of the coil was found to be too low at the stated
temperature 15-4° C., but at 16-4 its value is very nearly equal to
5o8 PRACTICAL ELECTRICITY
one international ohm. To bring the British standard of resistance
into substantial agreement with the International one, the
Order in Council dated 23rd August, 1894, has been revoked
and an amended Order in Council (see below) issued dated the
xoth day of January, 1910.
AT THE COURT AT BUCKINGHAM PALACE,
The loth day of January, 1910.
PRESENT.
THE KING'S MOST EXCELLENT MAJESTY IN COUNCIL.
WHEREAS by " The Weights and Measures Act, 1889," it
is, among other things, enacted that the Board of Trade
shall from time to time cause such new denominations
of standards for the measurement of electricity as appear to
them to be required for use in trade to be made and duly verified.
And whereas by Order in Council dated the 23rd day of August,
1894, Her late Majesty Queen Victoria, by virtue of the power
vested in Her by the said Act, by and with the advice of Her
Privy Council, was pleased to approve the several denominations
of standards set forth in the Schedule thereto as new denomina-
tions of standards for electrical measurement.
And whereas in the said Schedule the limits of accuracy attain-
able in the use of the said denominations of standards are stated
as follows : —
For the Ohm within one hundredth part of one per cent.
For the Ampere within one tenth part of one per cent.
For the Volt within one tenth part of one per cent.
And whereas, at an International Conference on Electrical
Units and Standards held in London in the month of October,
1908, the International Electrical Units corresponding with the
said denominations of standards were defined as follows : —
The International Ohm is the resistance offered to an unvarying
electric current by a column of mercury at the tempera-
ture of melting ice 14-4521 grammes in mass of a constant
cross sectional area and of a length of 106*300 centimetres.
The International Ampere is the unvarying electric current
which when passed through a solution of nitrate of silver
in water deposits silver at the rate of 0-00111800 of a
gramme per second.
ORDER IN COUNCIL, 1910 509
The International Volt is the electrical pressure which when
steadily applied to a conductor whose resistance is one
International Ohm will produce a current of one Inter-
national Ampere.
And whereas it has been made to appear to the Board of Trade
to be desirable that the denominations of standards for the
measurement of electricity should agree in value with the said
International Electrical Units within the said limits of accuracy
attainable.
And whereas the denominations of standards made and duly
verified in 1894 and set forth in the Schedule to the said Order
in Council have been again verified.
And whereas the Board of Trade are advised that the said
denominations of standards agree in value with the said Inter-
national electrical units within the said limits of accuracy attain-
able, except that in the case of the Ohm the temperature should
be 16-4 C. in place of 15-4 C. as specified in the Schedule to the
said Order in Council.
And whereas it has been made to appear to the Board of Trade
that the said denominations of standards should be amended so
that the aforesaid exception may be remedied.
NOW, THEREFORE, His Majesty, by virtue of the power
vested in Him by the said Act, by and with the advice of His Privy
Council, is pleased to revoke the said Order in Council dated the
23rd day of August, 1894, and is further pleased to approve the
several denominations of standards set out in the Schedule
hereto as denominations of standards for the measurement of
electricity.
ALMERIC FITZROY.
SCHEDULE ABOVE REFERRED TO.
I. Standard of Electrical Resistance.
A standard of electrical resistance denominated one Ohm agreeing in
value within the limits of accuracy aforesaid with that of the International
Ohm and being the resistance between the copper terminals of the instru-
ment marked " Board of Trade Ohm Standard Verified, 1894 and 1909,"
to the passage of an unvarying electrical current when the coil of insulated
wire forming part of the aforesaid instrument and connected to the afore-
said terminals is in all parts at a temperature of 16*4 C.
II. Standard of Electrical Current.
A standard of electrical current denominated one Ampere agreeing in
value within the limits of accuracy aforesaid with that of the International
Ampere and being the current which is passing in and through the coils of
wire forming part of the instrument marked " Board of Trade Ampere
Standard Verified, 1894 and 1909," when on reversing the current in the
fixed coils the change fn the forces acting upon the suspended coil in to
510 PRACTICAL ELECTRICITY
sighted position is exactly balanced by the force exerted by Gravity i&
Westminster upon the iridioplatinum weight marked A and forming part
of the said instrument.
III. Standard of Electrical Pressure.
A standard of electrical pressure denominated one Volt agreeing in value
within the limits of accuracy aforesaid with that of the International Volt
and being one hundredth part of the pressure which when applied
between the terminals forming part of the instrument marked " Board
of Trade Volt Standard Verified, 1894 and 1909," causes that rotation of
the suspended portion of the instrument which is exactly measured by the
coincidence of the sighting wire with the image of the fiducial mark A
before and after application of the pressure and with that of the fiducial
mark B during the application of the pressure these images being
produced by the suspended mirror and observed by means of the eyepiece.
In the use of the above standards the limits of accuracy attainable are
as follows : —
For the Ohm, within one hundredth part of one per cent.
For the Ampere, within one tenth part of one per cent.
For the Volt, within one tenth part of one per cent.
The coils and instruments referred to in this Schedule are deposited at
the Board of Trade Standardizing Laboratory, 8, Richmond Terrace,
Whitehall, London."
THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER AND THE
E.M.F. OF STANDARD CELLS.
Relating to the second independent electrical unit, the ampere,
the most reliable determinations of the electrochemical equivalent
of silver are given to five significant figures in the following
table—
Electro-chemical Equivalent of Silver to five significant figures
in milligrammes per coulomb.
1884 . . . . Mascart . . . . . . . . 1-1156*
1884 . . . . Kohlrausch . . . . . . 1-1183
1884 . . . . Rayleigh and Sidgwick . . 1-1179
1890 . . . . Pellat and Potier . . . . 1-1192*
1899 . . . . Kahle 1-1183
1903 . . . . Pellat and Leduc . . . . 1-1195*
1904 . . . . Van Dijk and Kunst . . . . 1-1182
1906 ,, „. Guthe .. .. .. .. 1-1182
1907 . . . . Smith and Mather . . . . 1-1183
1908 . . . . Janet, Laporte, and de la Gorce 1-1182
1909 . . . . Laporte . . . . . . 1-1183
from which it will be seen that, with the exception of the three
marked with asterisks, the results are in good agreement. Com-
paring these with 1-11800, the value taken as defining the Inter-
national Ampere, it will be noticed that the true ampere exceeds
REGENT DETERMINATIONS 5"
the International Ampere by about i part in 5,000, a difference
much greater than the possible experimental error in the
determinations.
In the experiments by Mr. Smith and the writer the ampere was
determined by weighing the attraction between coils of wire
conveying the current to be measured, the " current weigher "
or " Ampere Balance," as it is sometimes called, having been
designed at the Central Technical College in 1898 by the late
Professor J. Viriamu Jones and the authors. It has proved
to be an instrument of remarkable precision, most of the
observations made with it agreeing within I part in 100,000.
The actual value obtained for the electrochemical equivalent of
silver, the mean of a large number of consistent experiments
by Mr. F. E. Smith and the writer, was 1-11827 milligrammes
per coulomb. A more recent research by Prof. J. Laporte, of the
Laboratoire Centrale d'Electricite, gave 1-11829, so the two results
are in very close agreement.
By means of the same current weigher Mr. Smith and the
authors determined in 1905-6-7 the E.M.F. of the Weston
Normal Cell with very great accuracy, the result, expressed in
terms of the ampere, as given by the current weigher, and the
International Ohm. as realized at the National Physical Labora-
tory, being 1-01830 at 17° C.
The Normal Clark Cell expressed in terms of the same unit
has an E.M.F. of I>432 g at I5<> C.
In accordance with the recommendations made in Schedule D
of the Report of the 1908 Conference, § 3, page 505, further
investigations were made in several National Laboratories,
notably at the Bureau of Standards, Washington, the Laboratoire
Central d'Electricite, Paris, the Physikalisch-Technischen, Reichs-
anstalt, Charlottenburg, and the National Physical Laboratory,
Teddington, on Resistance Standards, Silver Voltameter and
Weston Cadmium Cell. For the better co-ordination of the re-
searches it was agreed that the chief experimenters of the several
laboratories should work together and carry out joint and separ-
ate experiments on the cadmium cell and silver voltameter.
Further work on the silver voltameter had been shown to be
necessary on account of differences in values for the E.M.F. of
cadmium cells in International volts as determined in the several
laboratories.
The persons entrusted with the work were as follows : —
512 PRACTICAL ELECTRICITY
Dr. E. B. Rosa and Dr. F. A. Wolff, of the Bureau of Standards,
Washington ; Prof. F. Laporte, of the Laboratoire Central
d'Electricite, Paris ; Dr. W. Jaeger, of the Physikalisch-Tech-
nischen Reichsanstalt, Charlottenburg ; Mr. F. E. Smith, of the
National Physical Laboratory, Teddington. About seven weeks
(April 4th to May 25th, 1910) was devoted to the work, which
resulted in the definite recommendation (i) that 1-0183 be taken
as the E.M.F. of the Weston cell in International volts at
20° C, and (2) that further experiments on the mercury
ohm, silver voltameter and standard cells were necessary
before completing or changing the specifications of the London
Conference (1908).
In the Washington determinations (1910) the mean of the
mercury ohms as realised at the Reichsanstalt and N. P. L.
respectively was taken as the International ohm. These differ
by about i part in 100,000, the former being the greater.
The work on the silver voltameter indicated that the use of
filter paper caused a slight increase in the deposit.
Combining the latest absolute determination of the ohm at
the National Physical Laboratory (page 499) with the Wash-
ington results we get the following relation between the In-
ternational units and absolute units.
i International ohm = i-ooo52 true ohm (approx.)
i „ ampere = 0-99988 „ ampere „
i „ volt = i-ooo40 „ volt „
i „ watt = i-ooo28 „ watt „
A determination of the E.M.F. of the Weston cell of great pre-
cision was published in 1914 (Phil Trans., Vol. 214) by Mr. A.
Norman Shaw, who used a Maxwell- Weber bipolar electro-
dynamometer to measure the current, and the international
ohm as resistance standard. The result obtained was
1-01831 semi-absolute volts at 20° C.,
the semi-absolute volt being taken as the P.D. between the
'terminals of an international ohm (Britain, America and
Germany) when it is traversed by a current of one absolute
ampere.
REGENT DETERMINATIONS 5i3
NOTE re Silver Voltameter (New Form).
In the latest and most satisfactory form of voltameter yet devised the
anode, instead of being wrapped in filter paper as described on page 494, rests
in a shallow glass cup, having a ground edge supported clear of the platinum
bowl. A glass cylinder is ground to fit over the edge of the cup, thus forming
a chamber for the anode. Before making an experiment the anode and
cathode chambers are filled with pure electrolyte to the same level, and the
glass cylinder raised until its lower edge is just below the surface of the liquid.
After the deposit has been made the glass cylinder is lowered so as to close
the anode chamber before removing the anode system from the platinum
bowl. The anode is coated with electro-deposited silver before use in a
determination.
PRACTICAL ELECTRICITY
APPENDIX II
COMPARISON OF C.G.S. AND BRITISH SYSTEMS OF UNITS.
FUNDAMENTAL UNITS.
Unit.
C.G.S. System.
British System
Relations between the Units.
Length
Mass
I centimetre
i gramme
i foot
i pound*
{i cm. =0-03281 ft. = 0-3937 inch,
i ft. =30-48 cms., i inch = 2-54
cms.
/I gramme = 0-0022046 Ib.
\i Ib. =453-6 grammes.
Time
i second
i second
* British engineers commonly take g pounds as the unit of mass (where
g = 32-19 approximately), so that the weight of a pound may be used as
the unit of force.
DERIVED UNITS (MECHANICAL).
C.G.S. System.
British System.
Ratios of Units.
C.G.S. Unit.
British Unit.
British Unit.
C.G.S. Unit.
Velocity
i cm. per second . .
i foot per second
0-03281
30-48
Acceleration
i „ „ „ per sec.
i „ „ „ per sec.
0-03281
30-48
i' i gramme moving at
i pound moving at a^
Momentum
I a velocity of i cm.
velocity of i foot per I
0-00007233
13,826
I per second.
second. j
(i dyne, gfcT of the
11 poundal (53^3 the )
weight of a gramme
weight of a pound at j
0-00007233
13,826
Force
4 at London, approx.
London (approx.)
i pound, the weight of }
( Ditto
a pound at London >
0-000002247
445,000
(approx.)
Moment of
1*
force about
axis or of a
i dyne centimetre
(i poundal foot
i pound footf
0-000002373
0-00000007372
421,400
13,560,000
couple
.
Work or
energy
j- i cm. dyne or i erg 3
f i foot poundal. .
( i foot pound . .
0-000002373
0-00000007372
421,400
13,560,000
Powee
1i erg per second
i watt, or 10 7 ergs
per second
J i foot poundal per sec.
\ i foot pound per sec.
1 1 foot poundal per sec.
i foot pound per sec. . .
•1 i horse-power, or \
33,000 foot pounds per >
\ minute, 1
0-000002373
0-00000007372
0-000002373
0-00000007372
0-001340
421,400
13,560,000
421,400
13,560,000
746
t The " pound-inch " or " inch-pound " is frequently employed as the unit of moment of z
force about an axis, or moment of a couple.
$ Another unit of work frequently used by engineers is the kilogramme-metre or metre-
kilogramme, whose relations with the foot-pound are as follows : —
i metre-kilogramme = 7*233 foot-pounds,
i foot-pound = 0*13826 metre-kilogrammes,
also i horse-power hour = 1,980,000 foot pounds.
,, ,. ,, = 273,700 metre-kilogrammes,
and i kilowatt hour = 2,654,000 foot pounds.
,, „ = 366,900 metre-kilogrammes.
RELATIONS OF UNITS
APPENDIX III
RELATIONS BETWEEN THE PRACTICAL, C.G.S. ELECTROMAGNETIC, AND
C.G.S. ELECTROSTATIC UNITS.
Unit of
Practical Unit.
C G.S. Electro-
magnetic Unit.
C G.S. Electrostatic Unit.
Current
i ampere
10 amperes
I , Y
^ x 10 9 amPere laPProx-J
Resistance
i ohm
— j ohm
9 x 10 n ohms „
P.D., or )
E.M.F. }
Quantity
i volt
i coulomb
—a volt
10 8
10 coulombs
3°° volts „
3xio» ^ouluuib „
Energy
i joule
i
— -. joule
I07 J
7o^ j°ule »
Power
i watt
-L watt
I07
— 7 watt „
Capacity
i farad
10 • farads
i f A^
9X 10 u
Inductance
i henry
i
9Xion henry „
APPENDIX IV
SPECIFIC GRAVITIES, SPECIFIC RESISTANCES, AND SPECIFIC CONDUCTIVITIES
OF MIXTURES OF PURE SULPHURIC ACID AND DISTILLED WATER.
Percentage
H,S04
by weight.
Percentage
H.SO,
by volume.
Specific gravity
at i8°C.
Specific resistance
per centimetre
cube (ohms).
Specific conductiv-
ity per centimetre
cube at 18° C.
5
2-7
•03
4-8!
0-208
10
5-7
•07
2-56
0-391
15
8-7
•II
•84
0-543
20
I2-O
•M
•53
0-653
25
I5'3
•18
•40
0-717
30
18-9
•22
•35
0-739
35
22-6
•26
•39
0-721
40
26-6
•31
•47
0-680
50
35-2
•40
•85
0-540
60
44'9
•50
2-68
o-373
70
55'9
I'6l
4-63
0-216
80
68-3
i-73
9-00
O-III
90
83
1-82
9-26
0-108
IOO
IOO
1*4
52-6
0-019
516 PRACTICAL ELECTRICITY
APPENDIX V
SHOWING THE DIMENSIONS OF WIRES ACCORDING TO THE
APPROXIMATE RELATIONS BETWEEN LENGTHS,
WIRE AT A
i
q
CO
DIAMETER.
AREA.
LENGTH AND RESISTANCE.
0.4.WNN | S.W.G. NO. |
Mils.*
Milli-
metres.
loooths ol
a sq. inch.f
Square
millimetres.
Feet
per ohm.
Metres
per ohm.
Ohms per
1000 feet.
Ohms per
kilometre.
i
2
3
4
5
300
276
252
232
212
7-62
7-01
6-40
5-89
5-38
70-6
59'8
50-0
4?-3
35-3
45-6
38-6
32-2
27'3
22-8
8870
7500
6250
5300
4420
2700
2290
1900
1610
1340
0-113
0-133
0-160
0-189
0-226
0-370
0-437
0-526
0-621
0-746
6
I
9
10
192
176
160
144
128
4-88
4-47
4-06
3-66
3-25
29-0
24-3
2O-I
16-3
12-9
18-7
15-7
13-0
10-5
8-30
3640
3050
2520
2050
1620
IIOO
930
770
621
49i
0-275
0-327
0-397
0-487
0-617
0-909
1-08
1-30
1-61
2-04
6
7
8
9
IO
ii
12
13
14
15
116
104
Io
72
2-95
2-64
2'34
2-03
1-83
10-6
8-5
6-65
5-03
4-06
6-82
5-48
4-29
3'24
2-63
1320
1060
832
603
5io
405
325
254
192
156
0-758
0-943
1-20
1-66
1-96
2-47
3-08
3'94
5-21
6-45
ii
12
13
14
15
16
17
18
19
20
%
48
8
1-63
1-42
1-22
roi6
0-914
3-21
2-46
1-81
1-26
1-02
2-08
X'59
1-17
0-8 1 1
0-657
404
309
226
158
128
123
94-0
69-1
48-0
38-9
2-47
3-24
4-42
6-32
7-8i
8-13
10-6
14-5
20-8
25-7
16
11
19
20
21
22
23
24
25
ii
24
22
20
0-813
0-711
0-610
o-559
0-508
0-804
O-6l5
0-452
0-380
0-314
0-519
0-397
0-292
0-245
0-203
IOI
76-2
56-8
47-6
39*4
30-7
23-6
17-3
I4'5
I2-O
9-90
I3'i
17-6
21-0
25-4
32-6
42-4
57-8
69-0
83-3
21
22
23
24
25
26
27
28
29
30
18
l6-4
14-8
13-6
12-4
0-457
0-417
0-376
0-345
0-315
0-254
O-2II
0-I72
0-145
0-120
0-164
0-136
0-111
0-0937
0-0779
3i'9
26-5
21-6
18-2
I5'2
9'73
7-89
6-59
5'53
4-61
31-3
37-7
46-3
54'9
65-8
103
127
152
181
217
26
27
28
29
30
31
32
33
34
35
n-6
10-8
10-0
r;
0-295
0-274
0-254
0-234
0-213
0-106
0-092
0-0785
0-0665
0-0554
0-0682
0-0591
0-0507
0-0429
0-0358
I3'2
n-7
9-85
8-32
6-95
4-04
3'50
3-00
2-54
2-II
75-8
85-5
JOI
I2O
144
248
286
333
394
479
31
32
33
34
35
36
I?
39
40
36
H
39
4°
7-6
6-8
6-0
n
0-193
0-173
0-152
0-132
O-I22
0-0452
0-0363
0-0282
O-02I2
O'OlSl
0-0293
0-0234
0-0182
0-0137
0-0117
5'7o
4'55
3'55
2-66
2-26
i'73
1-48
1-08
0-8 1
0-69
175
220
282
376
442
578
676
926
1230
1450
41
4*
43
44
45
~4*~
S
49
50
4'4
g
3
O-II2
O-IO2
0-0914
0-0813
O-O7II
0-0152
0-0126
0-OI02
0-0080
0-O06I5
0-00982
0-00811
0-00656
0-00519
0-00397
1-90
1-58
1-28
i-oi
0-762
0-58
0-48
0-389
0-307
0-236
526
633
78i
990
1310
1720
2080
2570
3260
4240
4i
42
43
44
45
2'4
2-0
1-6
fa
I'D
0-0610
0-0508
0-0406
0-0305
0-0254
O-OO452
O-003I4
O-O02OI
O-OOII3
0-00078
0-00292
0-00203
0-00129
0-00073
0-00051
0-568
0*394
0-252
0-142
• 0-098
0-173
O-I2O
o«o77
0-0432
0-0300
1760
2640
3970
7040
IOIOO
578o
8330
13000
23100
33300
46
47
48
49
50
* A mil is a thousandth of an inch (o-ooi inch), and eauals ^ of a millimetre (approx.).
f This column also shows the carrying capacity of the wires on the basi? of 1000 amperes pec
square- inch.
COPPER WIRE TABLE 517
APPENDIX V (continued}.
BRITISH STANDARD WIRE GAUGE (S.W.G.) AS WELL AS THE
RESISTANCES, AND WEIGHTS OF PURE COPPER
TEMPERATURE OF 15° C.
0
£
0
£
C/j
I
2
3
4
5
RESISTANCE AND WEIGHT.
WEIGHT AND LENGTH.
1'
o"
£
c/i
Ohms
per pound.}
Pounds
per ohm.§
Pounds per
1000 feet.
Grammes
per metre.
Feet
per pound.
Metres
per gramme.
0-000417
0-000581
0-000833
0-00116
0-00167
2400
1720
1200
860
600
272
230
192
163
136
407
3*4
286
242
202
3-64
4-35
5-21
6-14
7-35
0-00246
0-00291
0-00350
0-00413
0-00495
i
2
3
4
5
6
7
8
9
10
0-00247
0-00351
0-00515
0-00781
0-0127
405
285
194
128
79
112
93-7
77-4
62-8
49'6
1 66
140
116
93-5
73-9
8-93
10-7
12-9
15-9
20-2
0-00602
0-00714
0-00862
0-0107
0-0135
6
I
9
10
ii
12
13
14
15
0-0185
0-0286
0-0472
0-0820
0-125
54
35
21'2
12-2
8-0
40-8
32-6
25-6
19-4
15-7
60-7
48-5
38-2
28-8
23'3
24-5
30-7
39-1
5i-5
63-7
0-0165
0-0206
0-0262
0-0347
0-0429
ii
12
13
14
15
16
17
18
19
20
0-202
0-344
0-633
1-31
2-O
4-96
2-91
1-58
0-765
0-50
12-4
9-5
7-o
4-85
3-92
18-5
14-1
10-4
7-20
5-85
80-6
105
143
206
255
0-0541
0*0709
0-0962
0-139
0-171
16
17
18
19
20
21
22
23
24
25
3*23
5-49
10-2
14*3
2I'I
0-310
0-182
0-098
0-070
0-0475
3-10
a-37
1-74
1-46
I-2I
4-62
3-54
2-60
2-18
1-81
323
422
III
826
0-217
0-283
0-385
0-459
0-553
21
22
23
24
25
26
27
28
29
30
32-3
46-5
70-4
99-0
144
0*0310
0-0215
0-0142
o-oioi
0-00696
0-98
0-8I5
0-662
0-560
0-466
1-46
1»2I
0-988
0-835
0-693
IO2O
I23O
1510
1790
2I5O
0-685
0-826
I-OI
1-20
i-44
26
27
28
29
30
31
32
33
34
35
I84
244
336
474
680
0-0054
0-0041
0-00298
0-0021 1
0-OOI47
0-406
0-353
0-303
O-256
O-2I4
O-607
0-525
0-451
0-382
0-3I7
2460
2830
3300
3910
4670
1-65
1-91
2-22
2-62
3-i6
31
32
33
34
35
36
%
39
40
IOIO
1560
2600
4610
6410
0-00099
O-OOO64
0-000385
O-OOO2I7
O-OOOI56
0-175
0-I40
O-IO9
' 0-0820
0-0698
O-26O
0-208
O-l62
0'122
0-104
5710
7140
9170
I220O
14300
3-85
4-8x
6-17
8-2O
9-62
36
37
38
39
40
4i
42
43
44
45
8930
13100
20000
32400
54900
O-OOOII2
O-OO00765
O-OOOO5OO
O-O0003O9
O'OOOOl82
0-0585
0-0485
0-O392
0'03IO
0-0237
0-087
0-072
0-0585
0-0462
0-0354
I7IOO
2060O
2250O
32300
4220O
1 1-5
13-9
17-1
21-7
28-3
41
42
43
44
45
46
47
48
49
50
IOIOOO
210000
518000
I62OOOO
340OOOO
O'O000099
0-00000476
O-OOOOOI93
0-00000o6l8
0'O00000294
0-0174
O-OI2I
0-00775
0-00436
0-O0303
0-0260
O'OlSo
0-0116
0-0065
0-0045
57500
82600
129000
229000
330000
38-5
55-6
86-2
154
222
46
»
49
50
To get "ohms per kilogramme" (approx.) double the numbers in this column and add 10%.
To get " kilogrammes per ohm " (approx.) halve the numbers in this column and deduct 10%.
NOTES re APPENDIX VI
These tables are based on data relating to covered copper wires
contained in list of London Electric Wire Co. & Smith's, Ltd.
Each wire in a winding is assumed to occupy a square whose
side is equal to the diameter of the covered wire, as indicated
in the figure
, and that no bedding occurs.
For the resistance columns a temperature of 15° C. is taken,
one foot of I mil copper wire having at this temperature a resist-
ance of 10-15 ohms. As the thicknesses of coverings are liable,
to appreciable variation the calculated values are only given
to three significant figures.
APPENDIX VI (a)
ORDINARY COTTON COVERED (SINGLE).
Approximate
Approximate
Approximate Resistance i
Mils
Wires per
Number of Wires
Ohms
Dia. in
„*
Dia. +
S.W.G.
Mils.
OI
Covg
Covg.
Lineal
inch.
Lineal
cm.
Per sq.
inch.
Per sq.
cm.
Per cu. inch
Per cu. cm.
10
128
8
136
7'35
2-89
54-i
8-38
0-00279
0-000170
IO
II
116
8
I24
8-06
3-17
65-0
IO-I
0-00408
0-000249
II
12
104
8
112
8-93
3-52
79-7
12-4
0-00623
0-000380
12
13
92
8
IOO
10-0
3'94
IOO
15-5
O-OIOO
0-000610
13
M
80
8
88
11-4
4'49
129
2O'O
0-0170
0-00104
14
*5
72
8
80
12-5
4'92
156
24-2
0-0255
0-00156
15
16
64
7
7i
14-1
5'55
198
30-7
0-0409
0-00250
16
*7
56
7
63
15-9
6-26
252
39-1
0-0679
0-00415
17
18
48
6
54
18-5
7-28
343
53'2
O-I26
0-00768
18
19
40
6
46
21-7
8-54
472
73-2
0-250
0-0152
19
20
36
6
42
23-8
9'37
567
87-9
0-370
O-O225
20
21
32
6
38
26-3
10-4
692
107
0-572
0-0349
21
22
28
6
34
29-4
n-6
865
!34
0-933
0-0569
22
23
24
6
30
33'3
13-1
IIIO
172
1-63
0-0996
23
24
22
6
28
35'7
14-1
1280
198
2-23
0-136
24
25
20
6
26
38-5
15-2
1480
229
3'I3
0-191
25
26
18
6
24
41-7
16-4
1740
269
4-53
0-277
26
,27
16-4
6
22-4
44-6
17-6
1990
309
6-26
0-382
27
28
I4-8
6
20-8
48-1
18-9
2310
358
8-92
0-545
28
29
I3-6
6
19-6
51-0
2O-I
2600
4°3
11-9
0-727
29
30
12-4
6
18-4
54'4
21-4
2950
458
16-2
0-991
30
31
n-6
6
17-6
56-8
22-4
3230
500
20-3
1-24
31
32
10-8
5
15-8
63-3
24-9
4010
621
29-0
1-77
32
33
10-0
5
15-0
66-7
26-3
4440
689
37-6
2-29
33
34
9-2
5
14-2
70-4
27.7
4960
769
49-5
3-02
34
35
8-4
4
12-4
80-6
31-7
6500
1010
78-0
4-76
35
36
7-6
4
n-6
86-2
33'9
7430
1150
109
6-64
36
37
6-8
4
10-8
92-6
36-5
8570
1330
57
9-57
37
38
6-0
4
10-0
IOO
39'4
10000
1550
235
4'3
38
39
5'2
4
9-2
109
42-9
11800
1830
370
2-6
39
40
4-8
4
8-8
114
44'9
12900
2OOO
474
8-9
40
WINDINGS TABLE
APPENDIX VI (b)
ORDINARY COTTON COVERED (DOUBLE).
Approximate
Approximate
Approximate Resistance in
Dia. in
Mils
_*
Dia. J-
Wires per
Number of Wires
Ohms
S.W.G.
Mils.
OI
Covg.
Covg.
Lineal
inch
Lineal
cm.
Per sq.
inch.
Per sq.
cm.
Per cu. inch.
Per cu. cm.
10
128
M
I42
7-04
2'77
49-6
7-69
0-00256
0-000156
IO
ii
116
14
I30
7-69
3-03
59-2
9-l8
0-00372
0-000227
II
12
104
M
118
8-48
3-34
71-8
II-I
0-00561
0-000343
12
13
92
M
1 06
9-43
3-7I
89-0
13-8
0-00889
0-000543
13
14
80
T4
94
10-6
4-17
H3
17-6
0-0149
0-000912
14
15
72
J4
86
n-6
4'57
135
2I'O
0-O22O
0-00134
15
16
64
12
76
13-2
5-20
173
26-8
0-0357
0-00218
16
I7
56
12
68
14-7
5'79
216
33-5
0-0583
0-00356
17
18
48
10
58
17-2
6-77
297
46-1
O-IO9
0-00673
18
19
40
IO
50
20-0
7-88
400
62-0
O-2II
O-OI29
19
20
36
IO
46
21-7
8-55
472
73'2
0-308
0-0188
20
21
32
IO
42
23-8
9'37
567
87-9
0-468
0-0286
21
22
28
10
38
26-3
10-4
692
107
0-747
0-0455
22
23
24
10
34
29-4
n-6
865
134
1-27
0-0775
23
24
22
10
32
3i-3
12-3
977
151
1-71
0-104
24
25
20
IO
30
33-3
13-1
IIIO
172
2-35
0-143
25
26
18
IO
28
35'7
14-1
1280
198
3*33
0-203
26
27
16-4
IO
26-4
37'9
14-9
1440
222
4-5i
0-275
27
28
I4-8
10
24-8
40-2
15-8
1630
252
6-28
0-383
28
29
13-6
10
23-6
42-4
16-7
1800
.278
8-21
0-501
29
30
12-4
10
22-4
44'7
17-6
1990
309
II-O
0-669
30
31
n-6
IO
21-6
46-3
18-2
2140
332
13-5
0-822
31
32
10-8
9
19-8
50-5
19-9
2550
395
18-5
I-I3
32
33
10-0
9
19-0
52-6
20-7
2770
429
23-4
i'43
33
34
9-2
9
18-2
54'9
21-6
3020
468
30-2
1-84
34
35
8'4
8
16-4
61-0
24-0
3720
576
44-6
2-72
35
36
7-6
8
15-6
64-1
25-2
4110
637
60-2
3-67
36
37
6-8
8
14-8
67-6
26-6
4560
708
83-5
5-10
37
38
6-0
8
14-0
71-4
28-1
5100
791
1 20
7-32
38
39
5-2
8
13-2
75-8
29-9
574°
890
1 80
n-o
39
4°
4-8
8
12-8
78-1
30-8
6100
946
224
137
40
520
PRACTICAL ELECTRICITY
APPENDIX VI (c)
SPECIALLY FINE COTTON COVERED (SINGLE),
Dia. in
Mils
of
Dia. +
Approximate
Wires per
Approximate
Number of Wires
Approximate Resistance in
Ohms
Mils.
Covg.
Covg.
Lineal
inch.
Lineal
cm.
Per sq.
inch.
Per sq.
cm.
Per cu. inch.
Per cu. cm.
10
128
7
135
7-41
2-92
54'9
8-5I
0-00283
0-000173
10
II
116
7
123
8-13
3*20
66-1
10-2
0-00415
0-000253
ii
12
104
7
III
9-01
3'55
81-2
12-6
0-00635
0-000387
12
13
92
7
99
IO-I
3-98
102
I5'8
O-OI02
O-OOO622
13
J4
80
7
B?
"•5
4'53
132
20-5
0-0175
0-00106
14
15
72
7
79
12-7
5-00
160
24-8
O-O26I
0-00159
15
16
64
6
70
14-3
5-63
204
3f6
0-O42I
0-00257
16
J7
5<5
6
62
16-1
6-34
260
4°'3
O-O7O2
0-00428
17
18
48
5
53
18-9
7'44
356
55-2
O'I3I
0-00797
18
19
40
5
45
22-2
8-74
494
76-6
O-26I
0-0159
19
20
36
4
40
25-0
9-85
625
96-9
0-408
0-0249
20
21
32
4
36
27-8
io-9
772
I2O
0-637
0-0389
21
22
28
4
32
31-3
12-3
977
152
1-05
0.0643
22
23
24
4
28
35'7
14-1
1280
198
I-87
0-114
23
24
22
4
26
38-5
15-2
1480
229
2-59
0*158
24
25
20
4
24
41-7
16-4
1740
269
3-67
0-224
25
26
18
4
22
45-5
17-9
2070
320
5'39
0-329
26
27
16-4
4
20-4
49-0
19-3
2400
373
7-56
0-461
27
28
14-8
4
18-8
53'2
21-0
2830
439
10-9
0-667
28
29
13-6
4
17-6
56-8
22'4
3230
500
14-8
0-901
29
30
12-4
4
16-4
61-0
24-0
3720
576
20-5
1-25
30
31
n-6
4
15-6
64-1
25-2
4110
637
25-8
1-58
31
32
10-8
4
14-8
67-6
26-6
4560
708
33-i
2-O2
32
33
IO-O
4
14-0
71-4
28-1
5100
791
43'3
2-64
33
34
9-2
4
13-2
75-8
29-9
574°
890
57'3
3-50
34
35
8-4
3
ix-4
87-7
34'5
7700
1190
92-2
5-63
35
36
7-6
3
10-6
94'4
37-2
8900
1380
130
7-96
36
37
6-8
3
9-8
102
40-2
10400
1610
190
n-6
37
38
6-0
3
9-0
III
43'7
12400
1910
290
17-7
38
39
5'2
3
8-2
122
48-1
14900
2300
465
28-4
39
40
4-8
3
7-8
128
50-4
16400
2550
604
36-8
40
WINDINGS TABLE
521
APPENDIX VI (d)
SPECIALLY FINE COTTON COVERED (DOUBLE).
S.W.G.
Dia. in
Mils.
Mils
of
Covg.
Dia. +
Covg.
Approximate
Wires per
Approximate
Number of Wires
Approximate Resistance in
Ohms
s.w.c.
Lineal
inch.
Lineal
cm.
Per
sq.inch.
Per
sq. cm.
Per
cu. inch.
Per
cu. cm.
IO
II
12
13
H
15
128
116
104
92
80
72
10
10
10
IO
IO
IO
138
126
114
102
90
82
7-25
7'94
8-78
9-80
n-i
I2'2
2-85
3'13
3-46
3-86
4'37
4-80
52'5
63-0
77-0
98-0
124
149
8-14
9-76
II-9
14-9
I9-I
23-I
0-00271
0-00396
0-0060I
0-00960
0-0163
0-0242
0-000165
0-000241
0-000367
0-000586
0-000995
0-00148
10
II
12
13
M
15
16
17
18
19
20
64
56
48
4°
36
9
9
8
8
6
73
65
56
48
42
13-7
15-4
I7-9
20-8
23-8
5-39
6-06
7'°5
8-19
9'37
188
237
319
434
567
29-1
367
49'4
67-3
87-9
0-0387
0-0638
O-II7
0-230
0-370
0-00237
0-00389
0-00714
0-0140
O-0226
16
11
19
20
21
22
23
24
25
32
28
24
22
20
6
6
6
6
6
38
34
30
28
26
26-3
29-4
33'3
35'7
38-5
10-4
n-6
13-1
14-1
15-2
692
865
IIIO
1280
1480
107
*34
172
198
229
0-572
0-933
1-63
2-23
3-13
0-0349
0-0569
0-0996
0-136
0-I9I
21
22
23
24
25
26
27
28
29
30
18
16-4
I4-8
13-6
12-4
6
6
6
6
6
24
22-4
20-8
19-6
18-4
41-7
44'7
48-1
51-0
54'4
16-4
17-6
18-9
2O-I
21-4
1740
1990
2310
2600
2950
269
300
358
4°3
458
4'53
6-27
8-92
11-9
16-3
0-277
0-382
0-545
0-727
0-991
26
27
28
29
30
31
32
33
34
35
n-6
10-8
10-0
9-2
8-4
6
6
6
6
5
17-6
16-8
16-0
15-2
I3H
56-8
59-6
62-5
65-8
74-6
22'4
23-5
24-6
25-9
29H
3230
354°
3910
4330
557°
500
549
605
671
863
20-3
25-7
33'i
43'3
66-8
1-24
i-57
2-02
2-64
4-08
31
32
33
34
35
36
37
38
39
40
7-6
6-8
6-0
5'2
4-8
5
5
5
5
5
12-6
n-8
II-O
IO-2
9-8
19A
04-0
90-9
98-0
IO2
31-3
33-4
35-8
38-6
40-2
6300
7180
8260
9610
10400
976
IIIO
1280
1490
1610
92-2
131
194
301
382
5'63
8-02
11-9
18-4
23-3
36
37
38
39
40
PRACTICAL ELECTRICITY
APPENDIX VI (e)
SILK COVERED (SINGLE).
S.W.G.
Dia. in
Mils
of
Dia. +
Approximate
Wires per
Approximate
Number of Wires
Approximate Resistance in
Ohms
Mils.
Covg.
Covg.
Lineal
Lineal
Per sq.
Per
Per
Per
S.W.G.
inch.
cm.
inch.
sq. cm.
cu. inch.
cu. cm
10
128
3
131
7-64
3-01
58-3
9-03
0-00301
0-000183
IO
II
116
3
119
8-4I
3-31
70-6
II-O
0-00444
0-000271
II
12
104
3
107
9-35
3-68
873
13-5
0-00683
0-000417
12
13
92
3
95
10-5
4-14
III
I7-2
o-oin
0-000675
13
14
80
3
83
I2-I
4-76
145
22-5
0-0191
O-OOII7
15
72
3
75
13-3
5-24
I78
27-6
0-0290
0-00177
15
16
64
3
67
I4-9
5-87
223
34'5
0-0460
O-OO28I
16
17
56
3
59
I6'9
6-66
287
44-5
0-0774
0-00473
17
18
48
2
50
2O-O
7-88
400
62-0
0-147
0-00896
18
19
40
2
42
23-8
9-37
567
87-9
0-300
0-0183
19
20
36
2
38
26-3
10-4
692
107
0-452
0-0275
20
21
32
2
34
29-4
n-6
865
134
0-715
0-0436
21
22
28
2
30
33-3
13-1
IIIO
172
I-2O
0-0731
22
23
24
2
26
38-5
15-2
1480
229
2-17
0-133
23
24
22
75
2375
42-1
16-6
1770
275
3-10
0-189
24
25
20
•75
21-75
46-0
18-1
2110
328
4'47
0-273
25
26
18
•75
19-75
50-6
19-9
2560
397
6-69
0-408
26
27
16-4
•75
18-15
55-1
21-7
3040
9-55
0-583
27
28
14-8
'75
16-55
60-4
23-8
3650
566
14-1
0-860
28
29
13-6
•75
15-35
65-2
25-7
4240
658
19-4
1-18
29
30
12-4
'5
13-9
72-0
28-4
5180
802
28-5
1-74
30
31
n-6
•5
13-1
76-3
30-1
5830
903
36-6
2-24
31
32
10-8
•5
12-3
81-3
32-0
66lO
1020
47'9
2-93
32
33
IO'O
•5
87-0
34-3
7560
1170
64-0
3-90
33
34
9-2
•5
10-7
93'4
36-8
8730
1350
87-3
5'33
34
35
8-4
•5
9-9
101
39-8
10200
1580
122
7'45
35
36
7-6
•5
9-1
no
43-3
I2IOO
1870
I77
10-8
36
37
6-8
•5
8-3
121
47-6
14500
2250
266
16-2
37
38
6-0
•5
7'5
133
52-4
17800
2750
417
25-5
38
39
5'2
'5
6-7
149
58-7
223OO
3450
697
42-5
39
40
4-8
•5
6-3
159
62-6
25200
3910
925
56-5
40
41
4'4
•5
5'9
169
66-6
28700
4450
I25O
76-6
41
42
4'°
•25
5-25
191
75'2
36300
5620
I92O
117
42
43
3-6
•25
4-85
206
81-1
42500
6590
2780
169
43
44
3-2
•25
4'45
225
88-6
50500
7830
4170
255
44
3-o
•25
4'25
235
92-6
554°°
8580
5200
45
2-8
•25
4-05
247
97-3
6IIOO
9470
6590
402
45
46
2'4
•25
3-65
274
108
75100
11600
IIOOO
673
46
47
2-O
•25
3-25
308
121
94700
14700
2OOOO
I22O
47
48
1-6
•25
2-85
35i
138
I230OO
19100
40700
2480
48
1*4
•25
2-65
377
148
I42OOO
22IOO
61400
3750
49
1-2
1-25
2-45
408
161
167000
25800
97800
597°
49
50
VO
1-25
2-25
445
175
198000
30600
167000
10200
50
WINDINGS TABLE
523
APPENDIX VI (/)
SILK COVERED (DOUBLE).
Approximate
Approximate Number
Approximate Resistance in
Dia. in
Mils
_.r
Dia. +
Wires per
of Wires
Ohms
S.W.G.
Mils.
OI
Covg.
Covg.
Lineal
inch.
Lineal
cm.
Per sq.inch.
Per sq. cm.
Per cu. inch.
Per cu. cm.
S.W.G.
10
128
4'5
I32'5
7'55
2-97
57'°
8-83
0-00294
0-000179
IO
n
116
4'5
120-5
8-30
3-27
69-0
10-7
0-00433
0-000264
II
12
104
4'5
108-5
9-22
3-63
85-0
13-2
0-00664
0-000405
12
13
92
4'5
96-5
10-4
4-10
107
16-7
0-0107
0-000655
13
M
80
4'5
84-5
n-8
4'65
I40
21-7
0-0185
0-00113
M
15
72
4'5
76-5
13-1
5'i6
171
26-4
0-0278
0-00170
15
16
64
4'5
68-5
14-6
5'75
213
33-o
0-0440
0-00270
16
17
56
4'5
60-5
16-5
6-50
274
42-4
0-0737
0-00449
17
18
48
3'5
51-5
I9'4
7-64
377
58-4
0-139
0-00845
18
19
40
3'5
43'5
23-0
9-06
528
81-9
0-279
0-0170
19
20
36
3'5
39'5
25-3
9-96
641
99'3
0-418
0-0255
20
21
32
3'5
35'5
28-2
n-i
793
123
0-655
0-0400
21
22
28
3'5
3i-5
3i-8
12-5
IOIO
156
1-09
0-0663
22
23
24
3'5
27-5
36-4
M-3
1320
205
1-94
0-119
23
24
22
3
25
40-0
15-8
1600
248
2-80
0-171
24
25
2O
3
23
43-5
17-1
1890
293
4-00
0-244
25
26
18
3
21
47-6
18-7
2270
352
5-92
0-361
26
27
16-4
3
19-4
5i-6
20-3
2660
412
8-36
0-510
27
28
14-8
3
I7-8
56-2
22-1
3160
489
12-2
°*744
28
29
13-6
3
16-6
60-2
23-7
3630
562
16-6
I-OI
29
30
12-4
2'5
14-9
67-1
26-4
4500
698
24-8
«'5i
30
31
n-6
2'5
14-1
70-9
27-9
5030
780
31-6
1-93
31
32
10-8
2'5
I3-3
75'2
29-6
5650
876
41-0
2-50
32
33
10-0
2'5
12-5
80-0
31-5
6400
992
54'i
3-31
33
34
9-2
2'5
11-7
85-5
33-7
73io
1130
73-0
4-46
34
35
8'4
2-5
10-9
91-8
36-2
8420
1310
101
6-16
35
36
7-6
2-25
9-85
102
40-2
10300
1600
151
9-22
36
37
6-8
2-25
9-05
110
43'3
12200
1890
223
13-6
37
38
6-0
2-25
8-25
121
47'7
14700
2280
345
2I-I
38
39
5'2
2-25
7'45
T34
52-8
I8OOO
2790
564
34'4
39
40
4-8
2-25
7-05
142
55-9
20IOO
3120
739
45-1
4°
41
4*4
2-25
6-65
150
59'i
22600
35io
988
60- 1
41
42
4-0
2
6-0
167
65-8
27800
43io
1470
89-6
42
43
3-6
2
5-6
179
7°-5
31900
4940
2080
127
43
44
3-2
2
5'2
192
75-6
37000
5730
3060
187
44
3-0
2
5'0
200
78-8
4OOOO
6200
3760
230
45
2-8
2
4-8
208
81-9
43400
6730
4700
287
45
46
2'4
2
4'4
227
89-4
51600
8000
7580
463
46
47
2-0
2
4-0
250
98-4
62500
9690
13200
807
47
48
1-6
2
3-6
278
109
77200
I2OOO
25500
1560
48
I '4
2
3'4
294
116
86500
13400
37300
2280
49
1-2
2
3'2
313
123
97700
15200
574°o
35oo
49
50
1-0
2
3-0
333
131
IIIOOO
17200
94000
574°
5°
l
1
524
PRACTICAL ELECTRICITY
APPENDIX VI fe)
ENAMEL INSULATED.
S.W.G.
Dia. in
Mils.
Mils
of
Covg.
Dia. +
Covg.
Approximate
Wires per
Approximate
Number of Wires
Approximate Resistance in
Ohms
S.W.G.
Lineal
inch.
Lineal
cm.
Per
sq. inch.
Per
sq. cm.
Per
cu. inch.
Per
cu. cm.
16
17
18
19
20
64
56
48
40
36
2'5
2'5
2-5
2-25
2-25
66-5
58-5
50-5
42-25
38-25
I5-0
I7-I
19-8
23-7
26-1
5-91
6-73
7-80
9-33
10-3
226
292
392
560
685
35-i
45-3
60-8
87-0
106
0-0467
0-0788
0-144
0-296
0-447
0-00285
0-00481
0-00878
0-oi8l
0-0272
16
17
18
19
20
21
22
23
24
25
32
28
24
22
20
2-0
2-0
i'75
i'75
i'75
34'°
30-0
25-75
23-75
21-75
29-4
33-3
38-8
42-1
46-0
n-6
13-1
I5-3
16-6
18-1
865
IIIO
1510
1770
2110
134
172
234
275
328
0-715
I-I2
2-21
3-10
4'47
0-0436
0-0731
0-135
0-189
0-273
21
22
23
24
25
26
27
28
29
30
18
16-4
14-8
13-6
12-4
i'75
i'5
i*5
i'5
1-25
19-75
17-9
16-3
I5-I
i3<65
50-6
55-9
61-4
66-2
73-3
19-9
22-0
24-2
26-1
28-9
2560
3120
3760
4380
537°
397
484
583
680
832
6-69
9-8l
H-S
20-0
29-5
0-408
0-599
0-887
1-22
1-80
26
27
28
29
30
31
32
33
34
35
n-6
10-8
10-0
9-2
8'4
1-25
1-25
1-25
i
i
12-85
12-05
11-25
IO-2
9'4
77-8
83-0
88-9
98-0
1 06
30-6
32-7
35-o
38-6
41-7
6060
6890
7900
9610
11300
94°
1070
1230
1490
1750
38-1
5O-O
66-8
96-0
136
2-32
3-05
4-08
5-86
8-28
31
32
33
34
35
36
37
38
39
40
7-6
6-8
6-0
5'2
4-8
i
i
i
o-75
o-75
8-6
7-8
7-0
5'95
5'55
116
128
*43
168
1 80
45-7
50-4
56-3
66-2
70-9
13500
16400
20400
28300
32500
2IOO
2550
3160
4380
5030
198
300
479
883
1190
I2-I
18-3
29-3
53-9
72-7
36
37
38
39
40
41
42
43
44
45
4'4
4-0
3-6
3-2
2-8
o-75
o-75
o-75
o'75
o-75
5-i5
4'75
4'35
3'95
3'55
194
211
230
253
282
76-4
83-1
90-6
99-6
in
37700
44300
52800
64100
79400
5840
6870
8190
993°
12300
1650
2340
3450
5290
8560
100
143
210
323
522
41
42
43
44
45
WINDINGS TABLE
525
APPENDIX VI (h)
ENAMEL-INSULATED AND COTTON COVERED (SINGLE).
S.W.G.
Dia. in
Mils.
Mils
of
Covg.
Dia. +
Covg.
Approximate
Wires per
Approximate
Number of Wires
Approximate Resistance
in Ohms
S.W.G.
Lineal
inch.
Lineal
cm.
Per
sq. inch.
Per
sq. cm
Per
cu. inch.
Per
cu. cm
16
J7
18
19
20
64
56
48
40
36
9
9
8
8
8
73
65
56
48
44
137
I5H
17-9
20-8
22-7
5'4°
6-06
7'°5
8-19
8-94
1 88
237
319
434
519
29-1
36-7
49'4
67-3
80-4
0-0387
0-0638
0-II7
0-230
0-339
0-00236
0-00389
0-00714
0-0140
O-O2O6
16
17
18
19
20
21
22
23
24
25
32
28
24
22
20
8
8
8
8
8
40
36
32
30
28
25-0
27-8
3I3
33'3
35'7
9-85
10-9
12-3
13-1
14-1
625
772
977
IIIO
1280
96-9
I2O
152
I72
198
0-516
0-832
i-43
1-94
2-70
0-0314
0-0508
0-0875
0-118
0-165
21
22
23
24
25
26
27
28
29
30
18
l6'4
I4-8
13-6
12-4
8
7
7
7
7
26
23-4
21-8
20-6
19-4
38-5
42-7
45-9
48-6
51-6
15-2
16-8
18-1
19-1
20-3
1480
1830
2110
2360
2660
229
283
326
365
4I2
3-86
5'75
8-12
10-8
14-6
0-236
o-35i
0-496
0-658
0-892
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
n-6
10-8
10-0
9-2
8-4
6
6
6
6
5
17-6
16-8
16-0
15-2
13-4
56-8
59-5
62-5
65-8
74-6
22-4
23-4
24-6
25-9
29-4
3230
354°
3910
4330
557°
500
549
606
671
863
20-3
25-7
33-1
43'3
66-8
1-24
i'57
2-O2
2-64
4-08
31
32
33
34
35
7-6
6-8
6-0
5'2
4-8
5
5
5
5
5
12-6
n-8
II-O
IO-2
9'8
79'4
84-8
91-0
98-0
102
31-3
33'4
35-8
38-6
40-2
6300
7180
8260
9600
10400
976
IIIO
1280
1 1490
1 1610
92-2
131
194
301
382
5-63
8-02
n-9
18-4
233
36
37
38
39
40
526
PRACTICAL ELECTRICITY
APPENDIX VI (t>
ENAMEL INSULATED AND COTTON COVERED (DOUBLE).
Mils
Approximate
Wires per
Approximate
Number of Wires
Approximate
Resistance in Ohms
Dia. in
r>f
Dia. +
S. W.G.
Mils.
OI
Covg.
Covg.
Lineal
inch.
Lineal
cm.
Per
sq. inch.
Per sq. cm.
Per
cu. inch.
Per
cu. cm.
S.W.G.
16
64
T4
78
12-8
5-04
!64
25H
0-0339
O-OO2O7
16
17
56
*4
7°
H'3
5-63
204
3I-6
0-0550
0-00336
17
18
48
12
60
16-7
6-58
278
43-o
O-IO2
0-00623
18
19
40
12
52
19-2
7-56
370
57'4
0-196
O-OII9
19
20
36
12
48
20-8
8-19
434
67-3
0-284
0-0173
20
21
32
12
44
22-7
8-94
519
80-4
0-427
0-026I
21
22
28
12
40
25-O
9-85
625
96-9
0-675
0-0412
22
23
24
12
36
27-8
10-9
772
I2O
I-I3
0-0695
23
24
22
12
34
29-4
n-6
865
134
i'5i
0-0920
24
25
2O
12
32
31-3
12-3
977
152
2-06
0-126
25
26
18
12
30
33'3
13-1
IIIO
172
2-91
0-178
26
27
16-4
II
27-4
36-5
14-4
1330
207
4-19
0-256
27
28
I4-8
II
25-8
38-8
I5-3
1500
232
5-83
0-356
28
29
I3-6
II
24-6
40-7
16-0
1650
257
7-56
0-461
29
30
12-4
II
23H
427
16-8
1830
283
IO-O
0-610
3°
31
n-6
IO
21-6
463
18-2
2140
333
13-4
0-818
31
32
10-8
IO
20-8
48-I
18-9
2310
358
16-4
I-OO
S2
33
IO'O
IO
2O-O
50-0
19-7
2500
388
21-2
1-29
33
34
9-2
IO
19-2
52-1
20-5
2710
421
27-2
1-66
34
35
8-4
9
17-4
57'5
22-6
3300
5"
39-6
2-42
35
36
7-6
9
16-6
6O-2
23^
3630
562
53-o
3-24
36
37
6-8
9
15-8
63-3
24-9
4010
620
73'3
4'47
37
38
6-0
9
15-0
66.7
26-3
4440
689
104
6-35
38
39
5'2
9
14-2
70-4
27-7
4960
768
156
9-48
39
40
4-8
9
13-8
72-5
28-6
5250
812
193
11-8
4°
TABLE OF SYMBOLS
527
APPENDIX VII.
TABLE OF SYMBOLS.
ADOPTED BY THE INTERNATIONAL ELECTROTECHNICAL COMMISSION, 1913.
Name of Quantity Symbol
1. Length / L \ For
2. Mass . . . . . . m M Y Dimensional
3. Time . . . . . . t TJ Equations.
4. Angles . j$ . - . . a, (3, 7
5. Acceleration of gravity . . g
6. Work < . . . . A or W
7. Energy . . . . . W or U
8. Power . . . , .- P
9. Efficiency . . ... rj
10. Number of turns in unit time . n
11. Temperature Centigrade . t or 0
12. Temperature absolute . ; . T or @
13. Period . . . . T
14. Angular frequency, 2:r/T . w
15. Frequency . . . . /
1 6. Phase displacement . . • 9
17. Electromotive force . . . E The alternative
18. Current / sym^olH isf recofm-
D menAed for the
19. Resistance . . . * . R cage in which the
20. Resistivity . . • • * • P principal symbol
21. Conductance . . , . G is not suitable.
22. Quantity of electricity . Q
23. Flux-density, electrostatic « . D
24. Capacity . . . . C
25. Dielectric constant . • . . e
26. Self inductance . . . . L or 5?
27. Mutual inductance . . . M ,, *d*
28. Reactance . . . . X ,, *£
29. Impedance . . . . . Z „
30. Reluctance . . . . S „ 8%
31. Magnetic flux . • . $ ,, *fl
32. Flux-density, magnetic . . B ,, J$
p//?
33. Magnetic field . . . » • H „' &&
34. Intensity of magnetisation . •/*»»*'
35. Permeability . . ^
36. Susceptibility . . . . K|
37. Difference of potential . . V
4"-
INDEX
Absolute system of units (see C.G.S.
system)
— , unit of resistance, short
history of, 473
Accumulator (see Storage cells)
Acid voltameter, Ayrton's form of,
31
, McMichael's form of, 33
Action of zinc, local or prejudicial,
i?5
Air condenser, standard, 391, 392
— , dielectric strength of, 373
— , specific inductive capacity
of.- 37i
Alcohol, specific inductive capacity
of, 371
Alloys, resistance of, 231, 234
Aluminium, annealed, resistance of,
233- 235
— , relative conductivities of, 244
Amalgamating zinc, method of, 176
Ambers, resistivity of, 374
, specific inductive capacity of,
371
American specification defining
electrical standards, 497
Ammeter, Ayrton and Mather's
moving coil, 115
— , Ayrton and Perry's perman-
ent magnet, no
— , calibration of, by potentio-
meter, 418
— , damping device in Nalder, 120
, Evershed and Vignole's new,
122
— , Evershed gravity control, 121
— , hot-wire, 124
— , hot-wire, Hartmann and
Braun, 123, 124
— , moving coil, 112
-, Nalder Bros, and Thompson's
moving coil, 114
— , Nalder gravity control, 119
— , needle, pointer and staff of
Ayrton and Perry's permanent
magnet, no
Ammeter, shunted voltmeter used
as, 164
— , Weston's moving coil, work-
ing parts of, 113
Ammeters and current voltmeters,
resistances of, 158
— , calibration of, 416
— , definition of, 108
, permanent magnet, 109
, soft iron, 118
— , spring and gravity control,
118
— , used as voltmeters, 158
— , voltmeters used as, 163
Ampere, definition of, 18
Ampere-hour, definition of, 24
- meter, Ferranti, 305, 306
— meters, 302
Amperes, value in, of deflection of
tangent galvanometer, 96
Ampere-turns, definition of, 121
Analogies, mechanical, of con-
densers, 363
Analogue, hydraulic, of condensers,
364
Angle of dip, definition of, 37
Anode, definition of, 20
Antimony, 233, 235
Apparatus for measuring heat
equivalent of electric energy,
280
for testing strength of mag-
netic field along axis and in
plane of circular coil, 74
Armature, rotating, of Elihu Thom-
son energy meter, 298
Aron energy meter, 293
— energy meter, connections of,
294
- supply meter, differential
gearing of the Aron, 292
Astatic needle, 352
Atomic weights, international, 22
(footnote)
Ayrton and Mather's moving coil
ammeter, 115
— and Mather's non-inductive
resistance coil, 148
2 I
529
530
INDEX
Ayrton and Mather's reflecting elec-
trostatic voltmeter, 157
— and Mather's reflecting gal-
vanometer, 353
- and Mather's shunt for strong
currents, 263
and Mather's shunt, principle
of, 260
— and Mather's universal shunt,
260, 416, 417
— and Mather's universal shunt
box, plan of, 261
— arid Mather's zero electro-
meter or zero electrostatic volt-
meter, 135-137
- and Perry's gold-leaf electro-
scope, 167
- and Perry's original gaining
clock joulemeter, 291
Ayrton's form of sulphuric acid
voltameter, 31
B
B.A. unit, error in, 479
- unit of resistance, 477
Back, E.M.F., 315, 316 -
Balance for finding strength of mag-
nets, 63
- for finding strength of pole,
62
— , Hibbert's magnetic, 61
Balata, specific inductive capacity
of, 371
Ballistic galvanometer, 349
galvanometer, constant of,
359, 427
galvanometer, correction for
damping, 356
— galvanometer, determination
of constant of, by earth induc-
tor method, 429
— galvanometer, measurement
of quantity by, 353
— galvanometer, needle and
coils of, 349
— galvanometer, reading tele-
scope and scale for, 352
— galvanometer, reflecting, 350
— galvanometer, reflecting, lamp
stand and scale for, 351
— galvanometer, simple, 349
— tests, iron ring wound for,
456
Bar magnet, lines of force of a, 57
magnets, lines of force with
two, 69
- pattern of bridge, 271
Bast-ian energy meter, 302 w
Batteries, cells and, 3
Battery, finding resistance of, by
condenser method, 398
Battery, Muirhead's telegraph, 182
— resistance of, key and con-
denser for testing, 398
Berlin conference on electrical
standards, 499
Bismuth, 233, 235, 244, 406
Bleeck-Love cell, 333
Blue Bell cell, 198
Board of Trade committee on
electrical standards, composition
of, 488
— of Trade electrical standards,
short history of, 473
— of Trade (1894) form of
Clark's cell, 202
— of Trade unit of energy,
294
Box, shunt, method of constructing,
253
Boxes, ratio, 413
— , resistance, 145
— , volt, 413
Branch circuits in parallel, 245
Brass, relative conductivities of,
244
Bridge, bar pattern of, 271
Bridge, British Association, 222
— galvanometer, meaning of
deflection on, 227
— key, 225, 226
— , metre, 222
— , metre, circular, 223
— , metre, diagram of, 222
— , portable, with switch con-
tacts, 272
— , three-wire, 224
— , use of shunt with, 227
— , Wheatstone's bar pattern of,
270
— , Wheatstone's diagram of, 219
— , Wheatstone's dial pattern of,
270, 271
— , Wheatstone's method of con-
structing, 221
— , Wheatstone's, ordinary forms
of, 265
— , Wheatstone's portable, dia-
gram of connections of, 270
, Wheatstone's portable forms
of, 269
— , Wheatstone's portable, with
battery and galvanometer com-
bined, 269
Wheatstone's Post Office,
267, 268
— , Wheatstone's, principle of,
218
— , Wheatstone's top of a com-
mercial, 266
Wheatstone's, use of, 221
Bridges, coil, 267
INDEX
British Association bridge, 222
— Association's recommenda-
tions on electrical standards, 485
Bronze, silicum, resistance of, 233
Bunsen's cell, 183, 184
C
Cadmium cell, Weston's, 206
, relative conductivities of, 244
Calculation of capacity of conden-
sers, 368
- of E.M.F. of cell from energy
liberated by chemical action, 206
— of magnetic field strength, 452
— of magnetic pull, 448
Calibrating deflectional voltmeter,
1 60
- potentiometer wire, 404
potentiometer wire, knife
edges for, 404
- voltmeter by using ammeter
and one known resistance, 161
- voltmeter by using several
known resistances with known
current passing through them,
162
- wire by differential galvano-
meter, 405
Calibration, absolute, of tangent
galvanometer, 42
— curve, method of plotting, 45
— curve of galvanometer, 45
- curve of tangent galvano-
meter, 84
, definition of, 34
— of ammeters, 416
- of ammeter by potentiometer,
418
— of detector by comparison
with tangent galvanometer, 44
- of galvanometer, absolute, 39
of galvanometer by direct
comparison with tangent gal-
vanometer, 43
of galvanometer, relative^ 39
- of voltmeter by potentio-
meter, 411
potential divider for volt-
meter, 412
Callaud cell, 178
Calorimeter for measuring coils of
wire, 237
Calorimeter used in measuring heat
equivalent of electric energy,
279
Canada balsam, resistivity of, 374
Capacity, absolute measurement of,
387
• , calculation of, 368
, measurement of specific in-
ductive, 388
Capacity of conductors, 362
— of several condensers, com-
bined, 382
— , specific inductive, of di-
electrics, 370
— , unit of, and farad, relation
between, 367
— , units of, 365
— , variation of, of condenser, 364
Carbon cloth rheostat, 419
— dioxide, specific inductive
capacity of, 371
- plate rheostat, 418
— , resistance of, 233
— , specific resistance and tem-
perature variation of, 241
" Carsak " cell, 192
Castor oil, specific inductive capa-
city of, 371
Cathode, definition of, 19
Cell and circuit, simple, 2
- arranged for experiments on
polarisation, 171
- arranged for proving inde-
pendence of E.M.F. , 180
— , Bleeck-Love, 333
— , Blue Bell, 198
— , Board of Trade (1894) form
of Clark's, 202
, Bunsen's, 183, 184
, calculation of E.M.F. of,
from energy liberated by chemi-
cal action, 206
, Callaud, 178
— , " Carsak," 192
— , Clark's, 200
— , Columbia, 198
— , Dania dry, 196
— , Daniell's porous pot, 174
— , Daniell's two-fluid, 173
— , dry, 193
— , Edison-Lalande, 199, 200"
— , Exchange Telegraph Com-
pany's form, 178
— , Extra-Sec, 198
— , Fery's modified Leclanche,
193
1 86
Fuller's mercury bichromate,
— , G.E.C., 197
— , " Gassner's " dry, 195
— , Grove's, 183, 184
— , Hellesen dry, 196
— , inert, 199
— , Kahle's modification of Ray-
leigh's H-form of Clark's, 203
— , L,acombe central zinc, 192
— , L,eclanche, 189
— , Leclanche agglomerate, 190
— , I/ord Rayleigh's H-form of
Clark's, 201
532
INDEX
Cell, Meidinger, 177
• , Minotto's, 179, 1 80
• , Obach dry, 197
, portable Clark's (Muirhead's
form), 204
• , potassium bichromate, 185
• , " Six Block Agglomerate,"
191
• , Tinsley, 208
, Weston's, 200
- — — , Weston's cadmium, 204
• , Weston's cadmium (F. E.
Smith's form), 205, 206
Cells and batteries, 3
, arrangement of, to give maxi-
mum power to external circuit of
fixed resistance, 325
— , charge and discharge curves
of storage, 188-
— , discharge curves for Icelandic,
194
— , B.M.P. of Daniell's, 181
— , galvanic, 170
— , gravity, Daniell's, 178
- joined in parallel, 326
— joined in series, 325
— joined, partly in parallel and
partly in series, 326
minimum number of, required
to give fixed amount of power to
given external circuit, 329
— , mixed grouping of, 327
— , modifications introduced in,
333
— , polarisation in simple, 170
— , resistance of, 180, 182, 183,
196, 313
— , standard, 200
storage or secondary, 187
Cellulose, dry, resistivity of, 374
Central station, load diagram of, 284
C.G.S. and British systems of units,
comparison of, 512
Chamberlain and Hookham quan-
tity meter, 303, 304
Charge and discharge curves of
storage cells, 189
Charged condensers stores for
energy, 384
Chatterton's compound, specific
inductive capacity of, 371
Chemical action in simple voltaic
element, 1 70
- property of a current, 7
Chicago conference on electrical
standards, 490
Circuit, cell and, simple, 2
, diagram of, for testing Ohm's
law, 142
, differential galvanometer,
diagram of, 216
Circuit, distribution of power in elec-
tric, 318
— , electric, linked with paths of
pole, 455
— , external, receiving maximum
power from current generator, 319
— •, magnetic, 467 et seq.
of condenser in which charge
and discharge are measured, 379
— of condenser in which charge
only is measured, 379
Circuits, complete, application of
Ohm's law to, 149
— in parallel, 257
— in parallel, branch, 245
— , primary and secondary, on
iron ring, 457
Circular plate condenser, 389
Clark's cell, 200
— cell, Board of Trade (1894)
form, 202
— cell, I^ord Rayleigh's H-form,
201
- cell, portable
form), 204
(Muirhead's
Coercive force, definition of, 464
Coil, Ayrton and Mather's non-
inductive resistance, 148
- bridges, 267
— , induction, 438
— , lines of force due to circular,
carrying current, 59
— , lines of force due to current in
circular, 58
— of a tangent galvanometer,
adjusting, 84
— , standard ohm (Reichsanstalt
form), 265
Coils of wire used in apparatus for
measuring variation of resistance
with temperature, 236
— , resistance, 145
— , standard resistance, 264
- used for testing the resistance
of conductors in parallel, 246
Columbia cell, 198
Combined resistance, 245
Comparing E.M.F.'s. by condenser,
397
K.M.Fs., Poggendorff's method
of, 400
Comparison of condensers, 379
— of quantities, 361
of resistances by potentio-
meter, 417
— of resistances by substitution
method., 214
Compass needle, mapping out lines
of force with, 71
— needle, weighted, for measur-
ing strength of magnetic field, 73
INDEX
533
Condenser, charge and discharge
key, 379
— circuit in which both charge
and discharge are measured, 379
— circuit in which charge only is
measured, 379
circular plate, 389
— , comparing E.M.Fs. by, 397
— , cylindrical standard air, 395
— , diagram of connections for
testing guard ring, 394
— , diagrammatic representation
of a, 365
— , energy wasted in charging,
from a source of constant P.D.,
386
— , hydraulic analogue of, 364
improved form of plate air,
392
— , rectangular plate, 388
— , simple diagram of, 365
— , standard air, 393, 394
— , standard spherical, 396
— , variation of capacity of, 366
Condensers, calculation of capacity
of, 368
— , capacity of spherical and
plate air, 368
— , charged, stores of energy,
384
• , combined capacity of several,
382
— , comparison of, 379
— , construction of, 374
, cylindrical, capacity of, 369
— , for large potential differences,
376
, in parallel, 383
— , in series, three, 383
— , mechanical analogies, 363
Conditions for maximum power, 320
Conductance and conductivity, 242
- of conductors in series and in
parallel, 244
Conductivities, approximate rela-
tive table of, 244
, comparison of electric and
heat, 243
— , specific, of mixtures of sul-
phuric acid and water, 515
Conductivity and conductance, 242
Conductor, conditions affecting
resistance of, 228
, definition of, 3
Conductors and insulators, 3
, capacity of, 362
— , currents in parallel, 248
- in parallel, resistance of, coils
used for testing, 246
in series and in parallel, con-
ductance of, 244
Conductors in series, 243
- in series and in parallel, resist-
ance of, 244
— of large specific resistance and
small temperature coefficients,
239
Connection between E.M.F. and
P.D. of battery, 313
Connections of ratio box, 413
Constant cell, 4
— cell, definition of, 3
- of ballistic galvanometer, 359,
429
Constantan, resistance of, 233, 235
, specific resistance and tem-
perature variation of, 241
Construction of condensers, 374
— of Wheatstone's bridge, 221
— of shunt box, 253
Controlling force, definition of, 36
Copper, annealed, resistance of, 233,
235
— , hard drawn, resistance of , 233,
235
— , relative conductivities of, 244
- wire tables, 518 et seq.
Coulomb, definition of, 24
Coulomb meter (see Ampere-hour
meters)
Crompton potentiometer, 408, 409
Current and flux density, relation
between, 449
, apparatus for showing pro-
perties of, 9
— , chemical property of, 7
— , defining strength of, 12
— , direction of, 26
— , electric, definition of, i
, electric, direction of flow of,
i, 26
, electric measurement of, i
, electric method of production
of, 2
— , electric properties of, 4
— , field of straight, 453
— , generator, E.M.F. of, 314
, generator, external circuit
receiving maximum power from,
319
, generator, power developed
by, 312
— , heat produced by, 277
— , heating property of, 7
, increase of, produced by
applying shunt, 255
— , magnetic property of, 7
— , measurement of, by galvano-
meter, 36
- measurements, reason for
using low resistance galvano-
meters for, 154
534
INDEX
Current measurements, resistances
for, 416
, measuring effects of, 8
• , measuring, with copper volta-
meters, 14
— measuring, with electro-mag-
net, 15
— measuring, with galvano-
scopes, 14
measuring, with sulphuric acid
voltameters, 13, 17
— — measuring, with thermometer,
16
- method of comparing poten-
tial differences and resistances,
153
strength, 7
- turns and lifting force, rela-
tion between, 442
unit, definition of, 18
voltmeters and ammeters,
resistances of, 158
— , work done by, 273
Currents, electro-dynamometer for
measuring, 104
, induced, direction of, 424
, induced, introductory remarks
on, 423
, induction of, in parallel wires,
441
in parallel conductors, 248
, shunt for strong, Ayrton and
Mather's, 263
Curve, calibration of a tangent
galvanometer, 84
, calibration, of galvanometer,
45
connecting current and num-
ber of cells in series when cells
and external resistance are fixed,
327
connecting power received
by an external circuit and its
resistance, 322
connecting rate of flow of
water with loss of head, 139
, discharge, for Icelandic cells,
194
, magnetisation, for soft iron
ring, 459
showing value of current
giving maximum power to
external circuit, 320
Curves, charge and discharge of
storage cells, 188
, hysteresis, 462, 463
- of flux density in magnet, 433
— , practical value of drawing, to
record results of experiments, 147
Cylindrical condensers, capacity of, I
369
Damped vibration, diagrams of, 358
Damping, correction of ballistic
galvanometer for, 356
— devices in ammeters, 120
Dania dry cell, 196
Daniell's cells, K.M.F. of, 181
— cells, gravity, 178
— cells, resistance of, 180
- porous pot cell, 174
— two-fluid cell, 173
- use of depolariser, 173
Decrement, determination of, 357
— , logarithmic, 357
Definition of ampere, 18
— of ampere-hour, 24
— of Board of Trade Unit, 294
— of coulomb, 24
— of direction of current, 26
— of electric current, i
— of electromagnetic unit quan-
tity of electricity, 24
— of electrostatic unit of quan-
tity of electricity, 81
of farad, 266
of henry, 438
of joule, 277
of ohm, 143
of volt, 148-153
of watt, 284
Deflection of magnet by conductor
carrying current, 4
— of tangent galvanometer,
value in amperes of, 96
- on bridge galvanometer,
meaning of, 227
Deflectional voltmeter, calibrating,
1 60
- wattmeter, Elliott's, 289
Desiccator used with silver volta-
meter, 20
Detector, calibration of, by direct
comparison with a tangent gal-
vanometer, 44
Determination of decrement, 357
Dial pattern of bridge, 270
— potentiometer, 409
- potentiometer, N.C.S., 410
Dielectric constant, 389
- strength of insulators, 372
Difference of potentials, 126
Differential galvanometer, 216
— galvanometer-circuit, diagram
of, 216
— gearing of the Aron supply
meter, 292
Dip, angle of, definition of, 37
Direct reading scale, 51
Direction of current, de^m'tion of,
26
INDEX
535
Discharge curves for Leclanche
cells, 194
Distribution of gas pressure,
apparatus for testing, 130, 131
— of magnetism, 431
— of water pressure, apparatus
for testing, 128, 130
Divider, potential, 380
Dry cell, Dania, 196
- cell, " Gassner's," 195
- cell, Hellesen, 196
- cell, Obach, 197
- cells, 193 et seq.
Earth inductor, simple, 430
— , potential of, arbitrarily called
nought, 132
Ebonite, dielectric strength of, 373
— , resistivity of, 374
— , specific inductive capacity
of, 37i
Economy in transmission of energy,
346
Edison-I,alande cell, 199, 200
Efficiency, 335 et seq,
- of electric transmission of
energy, 339
Electric and heat conductivities,
comparison of, 243
- circuit linked with paths of
poles, 455
current and its measurement, i
- current, definition of, i
current, direction of flow of, i
- current, methods of produc-
tion of, 2
- current, properties of, 4
- energy (see Energy, electric)
— energy and power, 273
- flow, water analogy of, 129
- lines of force (see lyines of
force, electric)
- power (see Power, electric)
- pressure (see Potential differ-
ence)
- quantity, measurement of,
348
— quantity, units of, 394
- transmission of energy, 308
- unit of energy, the joule, 277
Electrical efficiency of transmission
and ratio of power received to
power receivable, 343
— measurements, decisions of
(1882) conference on, 482
— measurements, decisions of
Paris Congress on, 481
Electrically lighted house, part of
plan of, 257
Electricity, definition of unit quan-
tity of, 24
Electro-chemical equivalents, 21
Electrodynamometer, 102
for measuring very small
currents, 104, 142
- , Siemens, 102
, simple, 1 02
— , zero for small currents, 142
Electrolysis, 21
Electromagnetic and electrostatic
units, relations between, 515
- definition of E.M.F., 151
Electromagnets, measuring current
with, 15
Electrometer, 134
, Ayrton and Mather's zero,
135, 137 .
Electromotive force, 150-153
of current generator, 314, 315
— of standard cells, 204, 205
Electroscope, Ayrton and Perry's
gold-leaf, 167
, gold-leaf, 1 66, 362
, gold-leaf, a deflectional
gravity voltmeter, 168
— , gold-leaf, sensibility of, 168
Electrostatics, electric lines of force
and, 8 1
Elliott's deflectional wattmeter, 289
E.M.F. and P.D. of battery, con-
nection between, 313
- back, 315, 316
— , cell arranged for proving in-
dependence of size, 1 80
— , electromagnetic definition of,
— of any current generator, 314
— of cell from energy liberated
by chemical action, calculation
of, 206
- of Daniell's cells, 181
E.M.Fs., comparing by condenser,
397
, Poggendorff's method of
comparing, 400
Energy, Board of Trade, unit of,
294
charged condensers, stores for,
384
— , electric, and power, 273
— -, electric, measuring heat
equivalent of, 278
— , electric, table of heat equiva-
lent of, 281
— , electric transmission of, 308
-, electric transmission of, effi-
ciency of, 339
— loss, calculation of, by hyster-
esis, 465
— , loss of. due to hysteresis, 464
536
INDEX
Energy meter, Aron, 293
— meter, Aron, connections of,
294
— meter, clock form, 290
meter, law of, 301
meter, motor form, 296
- meter, Thomson, 298, 299,
300
— stored in condensers, 384
, transmission of, economy in,
346
transmission of, mechanical
of
analogies illustrating, 311
, transmission of, table
results achieved, 342
wasted in charging condenser
from a source of constant P.D.,
386
Equipotential surface, 79
- surfaces and lines of force due
to long straight current (dia-
gram), 451
— surfaces due to circular cur-
rent, 79
Equivalents, electro-chemical, 21
Eureka, resistance of, 233, 235
, specific resistance and tem-
perature variation of, 241
Evershed and Vignole's new am-
meter, 122
gravity control ammeter, 121
- Megger, 214
ohmmeter, diagram of con-
nections of, 213
Exchange Telegraph Co.'s gravity
Daniell's cell, 178
Experiment, graphically recording
results of, 44
"Extra-Sec" cell, 198
Farad, 365
— and unit of capacity, relation
between, 367
, definition of, 364
Farads, capacity of spherical and
plate air condensers in, 368
Ferranti ampere-hour meter, 305,
306
Fery's modified Leclanche cell, 193
Field, magnetic (see Magnetic field)
— of straight current, 453
Fields, magnetic (see Magnetic
fields)
Five-wire potentiometer, 403
Fleming's rule, 425
Flow, electric, water analogy of, 129
— of electric current, direction
of, i
Flux density and current, relation
between, 449
Flux density and lifting force,
relation between, 446
— density in magnet, 432
— density in magnet, curves of,
433
Force, coercive, 464
— -, controlling, definition of, 36
— , laws of magnetic, 61
— , lines of (see Lines of force)
— , magnetic lines of, 56
— , magnetomotive, 454
Fuller's mercury bichromate cell,
1 86
Galvanic cells, 170
Galvanometer, absolute calibra
tion of, 39
— , Ayrton and Mather's reflect
ing, 353
— , ballistic, 349
— , ballistic, constant of, 359
— , ballistic, reflecting, 350
— , calibrating, by direct com-
parison with tangent galvano-
meter, 43
— , calibrating wire by differ en
tial, 405
— , calibration curve of, 45
— — circuit, diagram of differen-
tial, 216
— , differential, 216
for measuring potential dif-
ferences, 153
— , high resistance, with highly
insulated coils, 390
— , meaning of deflection on
bridge, 227
-, measurement of current by,
36
- needles, pivot and fibre sus-
pensions of, 99
— , Paul's single-pivot, 117
— , reflecting, 35
— , relative calibration of, 39
- scale, constructing a, 50
- scale, protractor used in
subdividing a, 50
— , section of, with pivot and
fibre suspension, 100
— , sine, 100
— , single-pivot, core, coil and
pole pieces of, 1 1 8
— , single-pivot moving coil, 117
— , single-pivot section of, 118
— , tangent, 36, 86
— , tangent, absolute calibration
of, 42
— , tangent, adjusting coil of, 84
— , tangent, calibration curve of,
84
INDEX
537
Galvanometer, tangent, comparison
with a voltameter, 40
— , tangent, constructing scale
for, 87
— , tangent, showing modes of
supporting fibre, 37
, tangent, testing laws of varia-
tion of sensibility, apparatus for,
9i
, tangent, value in amperes of
deflection of, 96
, tangent, variation of sensi-
bility of, 90
, torsion, n
— , Walmsley and Mather's pro-
portional, 1 06
Galvanometers, high resistance,
reason for using for potential
difference measurements, 154
, low resistance, reason for
using for current measurements,
J54
of invariable sensibility, 107
, proportional, construction of,
105
, voltameters and, relative
advantages of, 33
Galvanoscope, definition of, 34
(footnote)
, measuring current with, 14
Gas pressure apparatus for testing
distribution of, 130, 131
" Gassner's " dry cell, 195
G.E.C. cell, 197
Generator, E.M.F. of current, 314
— , power absorbed in circuit
exterior to, 315
, power developed by current,
312
Geometrical construction for find-
ing strength of field at point on
axis of circular coil, 77
German silver, relative conductivi-
ties of, 244
, resistance of, 233, 235
specific resistance and tem-
perature variation of, 241
Glass, dielectric strength of, 373
— , resistivity of, 374
, specific inductive capacity of,
371
Glow lamp, 8, 10, 87
Gold, annealed, resistance of, 233,
235
, hard drawn resistance of,
233. 235
, relative conductivities of,
244
Gold-leaf electroscope, 166, 362
— electroscope, Ayrton and
Perry's, 167
Gold-leaf electroscopes, sensibility
of, 168
Gold-silver alloy, resistance of,
233. 235
alloy, specific resistance and
temperature variation of, 241
Gravitational potential gradient, 80
Gravities, specific, of mixtures of
sulphuric acid and water, 515
Gravity control ammeter, 118
— control ammeter, Evershed,
121
— control ammeter, Nalder, 119
— , Daniell's cell, 178
Grove's cell, 183, 184
Guard ring condenser, diagram of
connections for testing, 394
Gutta-Percha, resistivity of, 374
, specific inductive capacity
of, 37i
H
Hartmann and Braun hot-wire
ammeter, 123
Heat and electric conductivities,
comparison of, 243
— equivalent of electric energy,
measuring, 278
— equivalents of energy, table
of, 281
- produced by current, 277
Heating property of a current, 7
Hellesen dry cell, 196
Henry, definition, 437
Hibbert's magnetic balance, 61
High E.M.F. for large powers,
importance of low resistance and,
— insulation shunt box, 254
— resistance galvanometer, rea-
son for using for potential
difference measurements, 154
resistance galvanometer with
highly insulated coils, 390
Hoffman's sulphuric acid volta-
meter, 27
Horse-shoe magnet with curved
iron pole pieces, 70
Hot-wire ammeter, 124
- ammeter, Hartmann and
Braun, 123, 124
House, electrically lighted, part
of plan of, 257
— service energy meter, Thom-
son, 300
Hydraulic analogue of condenser,
364
Hydrogen, specific inductive -capa-
city of, 371
Hysteresis, apparatus for testing,
461
538
INDEX
Hysteresis, calculation of energy
loss by, 465
- curves, 462, 463
loop, 463
, loss of energy due to, 464
of iron, 460
I
India rubber, resistivity of, 374
, specific inductive capacity of,
Induced currents, direction of, 424
, Fleming's rule on, 425
currents, introductory re-
marks on, 423
- currents, Lenz's law of, 425
Induction apparatus, magneto-
electric, 424
coil, 438
- coil, diagram of, 439
- coil, Marconi lo-inch, 440
, mutual, 435
, mutual, unit of, 437
- of currents in parallel wires,
441
Inductivity, 389
Inductor, simple earth, 430
Industrial forms of potentiometer,
406
"Inert " cell, 199
Insulator, definition of, 4
Insulators, conductors and, 3
, dielectric strength of, 372
, resistivity of, 373, 388
International atomic weights, 22
(footnote)
conference (1908) on units
and standards, 500
- ohm, 144
Ion, definition of, 26
, electronegative, definition of,
27
27'
-, electropositive, definition of,
Iron, annealed, resistance of, 233, 235
filings, diagram showing lines
of force round straight wire
carrying current, 450
, hysteresis, of, 460
— , magnetisation of, 442 et seq.
, relative conductivity of, 244
— ring, magnetisation curve for
soft, 459
- ring, permeability of soft, 459
- ring, primary and secondary
circuits on, 457
ring wound for ballistic tests,
456
rod picking up nails when
current flows through wire coiled
round it, 5
Iron, specific resistance and tem-
perature variation of, 241
Ja Ja, resistance of, 233, 235
, specific resistance and tem-
perature variation of, 241
Jars, Leyden, 376, 377
Joule, definition of, 277
Joule's law, 285
Joulemeter, or energy meter, clock
form, 290
, Ayrton and Perry's original
gaining clock, 291
Jute, resistivity of, 374
, specific inductive capacity
of, 37i
K
Kahle's modification of Rayleigh's
H-form of Clark's cell, 201
Kelvin's law, 346
Key, bridge, 225, 226
Kilowatt hour, 295
Kirchhoff's rules, 248
Kruppin, resistance of, 233, 235
Lacombe central zinc cell, 192
Lamp, glow, 7
stand and scale for ballistic
galvanometer, 351
Law, Joule's, 285
— , Kelvin's, 346
— , Lenz's, 425
— of energy, meter, 301
of magnetic pull, 446
— , Ohm's, 138
— , Ohm's apparatus for testing,
140
, Ohm's, verification of, 141
— , sine, apparatus for testing,
101
, tangent, 89
— , tangent, improved apparatus
for testing, 90
— , tangent, simple apparatus for
testing, 89
Laws of magnetic force, 61
— of resistance, 210
— of variation of sensibility of
tangent galvanometer, 91
Lead, pressed, resistance of, 233,
235
, relative conductivities of,
244
Leclanche agglomerate cell, 190
— cell, 189
— cell with porous pot, 190
— cells, discharge curve for, 194
Lenz's Law, 425
INDEX
539
Leyden jars, 376, 377, 378
Lifting force and current turns,
relation between, 442
- force and flux density, rela-
tion between, 446
magnet, Witton Kramer, 439
magnets, 442
Lines of force and equipotential
surfaces due to long straight
current (diagram), 451
— of force of bar magnet, 57
— of force due to circular coil
carrying current, 59
of force due to circular cur-
rent, 79
of force due to current in
circular coil, 58
— of force, electric, and elec-
trostatics, 8 1
of force, mapping out with
compass needle, 71
of force round straight wire,
454
— of force with two bar magnets,
69
- of magnetic force, 56
Linkage Constant, 431
— lines of force, 425, 429
Linseed oil, specific inductive
capacity of, 371
Load diagram of central station,
284
Logarithmic decrement, 357
London conference on electrical
units and standards, 500
Loop, hysteresis, 463
Lord Rayleigh's H-form of Clark's
cell, 201
Loss of energy due to hysteresis, 464
Low resistance and high B.M.F. for
large powers, importance of, 333
- resistance galvanometers,
reason for using, for current
measurements, 154
- resistance, standard, 415
M
McMi chad's form of acid volta-
meter, 33
Magnet ammeters, permanent, 109
- apparatus for testing distri-
bution of magnetism in a bar, 431
— , curves of flux density in, 433
— deflected by conductor carry-
ing current, 4
— , distribution of magnetism in
bar, 431
— , flux density in, 430
— , horse-shoe, with curved iron
pole pieces, 70
lines of force of bar, 57
Magnet, tinsel coiling itself round,
when current flows through the
tinsel, 5
Magnetic balance, Hibbert's, 61
— circuit, 467 et seq.
- field, 51, 55, 66, 68-78, 451
- field, absolute measurement
of, 66
— field, apparatus for testing
strength of, along axis and in
plane of circular coil, 74
field, arrangement for neutra-
lising uniform, 72
field, comparing by magneto-
meter method, relative strengths
of different parts of, 74
— field, comparing by vibration
method, relative strengths of
different parts of, 72
field curve of variation of
strength along axis of coil, 75
- field, earth's, 37, 38, 97
— field, geometrical construc-
tion for finding strength of,
at point on axis of circular coil,
field strength, calculation of,
452
— field, weighted compass needle
for measuring, 73
fields, 53
- fields, magnetometer method
of measuring, 55
— fields, mapping, 68
— fields, measuring, 5
- flux, 432
— flux density, 432
— force, laws of, 61
- induction, 424
- lines of force, 56
— linkage, 425
— moment, 63
- moment, absolute measure-
ment of, 66
— moment, measuring, 65
- moment, torsion apparatus
for measuring, 64
— needle, 10, 39, 73
— needle, astatic, 352
needles, time of vibration of,
67
poles, strength of, 59
potential difference, definition
of, 79
potential gradient, 80
- properties, testing, by ballistic
method, 457
- property of a current, 7
- pull, 442
— pull, apparatus, for testing,
' 444
540
INDEX
Magnetic pull, law of, 446
- pull, mercury board for test-
ing apparatus, 444, 445
— reluctance, 468
— saturation, 450
Magnetisation, curve for soft iron
ring, 459
— of iron, 442
Magnetism, distribution of, 431
— , remanent, 463
Magneto-electric induction appara-
tus, 424
Magnetometer, measuring magnetic
moment by, 65
— method, comparing relative
strength of different parts of
magnetic field by, 74
- method of measuring mag-
netic fields, 55
, principle of, 56
, reflecting, 57
- with pointer, 56
Magnetomotive force, 454
Magnets, balance for finding
strength of, 63
— , lifting, 442
, lines of force with two bar,
69
Manganese peroxide, 189-191
Manganin, relative conductivity
of, 244
, resistance of, 233, 235
— , specific resistance and tem-
perature variation of, 241
Mansb ridge condensers, 376
Mapping magnetic fields, 68
— out lines of force with compass
needle, 71
Marble resistivity of, 374
— , specific inductive capacity
of, 371
Marconi lo-in. induction coil, 440
Mather, Ayrton and (see Ayrton
and Mather)
Mather's form of sulphuric acid
voltameter, 32
— ratio-switch, 270, 272 (dia-
gram)
- simple apparatus for testing
iron, 461
Maximum power, conditions for, 320
Measurement by potentiometer,
advantages and disadvantages
of, 420
— of capacity, absolute, 387
— of electric current, i
- of E.M.P., 151
— of potential difference, 133
of power, 283, 286 et seq.,
420
of resistance, 210
Measurement of specific inductive
capacity, 388
- of specific resistance, 231,
373
Measurements, decisions of 1882
conference on electrical, 482
— , electrical, decisions of Paris
congress on, 481
Measuring effects of a current, 8
— electric current, 13-17
Mechanical analogies illustrating
transmission of energy, 311
- analogies of condensers, 363
- analogy illustrating hysteresis,
464
— equivalent of heat, 144, 278
Megger, 214
Meidinger cell, 177
Mercury bichromate cell, Fuller's,
1 66
— ohm, 143
- — , resistance of, 233, 235
— , specific resistance and tem-
perature variation of, 241
Metals, resistance of, 231, 234
— , tables of resistance of, 233,
235
Meter, Aron energy, 293, 294
— , Aron energy, connections of,
294
— , Ayrton and Perry's, 290
— , Bastian, 302
— , Chamberlain and Hookham,
303, 304
— , Ferranti, 305, 306
, Thomson, 297 et seq.
Meters, energy, motor form, 296
— , quantity, or ampere-hour, 302
Metre bridge, 222
- bridge, circular, 223
- bridge, diagram of, 222
Mica, dielectric strength of, 373
— , resistivity of, 374
— , specific inductive capacity
of, 371
Micanite, dielectric strength of, 373
— , resistivity of, 374
Microfarad, definition of, 366
Miiiotto's cell, 179, 1 80
Model of electric circuit composed
of current generator and external
resistance, 312
Moment, magnetic (see Magnetic
moment)
Motor form of energy meters, 296
Moving coil ammeter, Ayrton and
Mather's, 115
- coil ammeter, Nalder Bros,
and Thompson's, 114
coil ammeter, • West on 's,
working parts of, 113
INDEX
Moving coil ammeters, 112
— coil, galvanometer, single-
pivot, 117
— coil voltmeter, 160
Muirhead's telegraph battery, 182
- telegraph battery, composite
copper and zinc plates for, 183
Mutual induction, 435
- induction apparatus, 435
— — induction, unit of, 437
N
Nalder ammeter, damping device
in, 120
- Bros, and Thompson's moving
coil ammeter, 114
— gravity control ammeter, 119
Naut, definition of, 384
N.C.S. dial potentiometer, 410
Needle, astatic, 352
- time of vibration of, 67
Needles, galvanometer, pivot, and
fibre suspensions of, 99
Negative and positive potentials,
132
Neutralising uniform magnetic field,
arrangement for, 72
Nichrome, resistance of, 233, 235
Nickel, annealed, resistance of, 233,
235
• , relative conductivities of, 244
, specific resistance and tem-
perature variation of, 241
Nickelin, resistance of, 233, 235
, specific resistance and tem-
perature variation of, 241
Non-conductor, or insulator, defini-
tion of, 4
Obach dry cell, 197
Ohm coil, standard (Reichsanstalt
form), 265
, definition of, 143
— , international, 144
— , Paris (1884) Congress, 484
, the unit of resistance, 143
Ohmmeter, diagram of, 212
— diagram of connections of
Evershed, 213
Ohmmeters, 211
Ohm's law, 138
- law, apparatus for testing,
140
- law, application of to com-
plete circuits, 149
— law, diagram of circuit for
testing, 142
— law, verification of, 141
Olive oil, specific inductive capa-
city of, 371
Order in Council defining electrical
standards, 492
Paper, resistivity of, 374
Paper, specific inductive capacity
of, 37i
— , squared, use of, 44
Paraffin oil, specific inductive capa-
city of, 371
Paraffined paper, dielectric strength
of, 373
Parallel conductors, currents in, 248
— resistance, 246
— , three condensers in, 383
- wires, induction of currents
in, 441
Paris (1884) Congress ohm, 484
Paul's ratio box, diagram of con-
nections, 414
— single-pivot galvanometer,
117
P.D. and E.M.F. of battery, con-
nection between, 313
Pendulum, law of, 67
Periodic time of vibration, 64-67,
355
Permanent magnet ammeter, 109
- magnet ammeter, Ayrton and
Perry's, no
Permeability, 460
- of soft iron ring, 459
Phosphor bronze, resistance, 233,
235
bronze, specific resistance and
temperature variation of, 235
Pitch, specific inductive capacity of,
371
Pivot and fibre suspension of
galvanometer needle, 99
- and fibre suspension, section
of galvanometer with, 100
Plate-air condenser, of improved
form, 392
Plates,, composite copper and zinc
for Muirhead's telegraph battery,
183
Platinoid, relative conductivity of,
244
, resistance of, 233, 235
, specific resistance and tem-
perature variation of, 241
Platinum, annealed, resistance, 233,
— , relative conductivities of, 244
— , specific resistance and tem-
perature variation of, 241
Platinum-iridium, alloy, resistance
of, 233, 235
— , alloy, specific resistance and
temperature variation of, 241
542
INDEX
Platinum -silver alloy, resistance of,
233, 235
alloy, specific resistance and
temperature variation of, 241
Platinum thermometer, 239
Plotting calibration curve, method
of, 45
Poggendorff 's method of comparing
E.M.F.s, 400
- method, using only one gal-
vanometer, 402
Polarisation, cell arranged for
experiments in, 171
— in simple cell, 170
Pole, balance for finding strength
of, 62
- pieces, horse-shoe magnet
with curved iron, 70
Poles, magnetic strength of, 59
Porcelain, dielectric strength of,
373
, resistivity of, 374
, specific inductive capacity of,
3?i
Portable bridge with switch con-
tacts, 272
Positive and negative potentials,
132
Post Office Wheatstone's bridge, 267
Potassium bichromate cells, 185
Potential difference, effects of, 133
— difference, gravitational, 80
difference, magnetic, defini-
measurement of,
tion of, 79
— difference,
133
- difference, measurements,
reason for using high resistance
galvanometers for, 154
— difference, ratios of practical
units of, 148
- differences and resistances
current method of comparing,
153
differences, condensers for
large, 375
- differences, diagram of ar-
rangement for obtaining two, of
known ratio, 381
differences, galvanometer for
measuring, 153
divider, 380
— divider for voltmeter calibra-
tion, 412
divider, simple dial, 381
- divider, two-dial, 382
gradient, magnetic, 80
magnetic, difference of : equi-
potential surface, 79
of Earth arbitrarily called
nought, 132
Potentials, positive and negative,
132
Potentiometer, calibration of am-
meter, by, 418
— , calibration of voltmeter by,
411
, comparison of resistances by,
417 et seq.
— , Crompton, 408, 409
— , dial, 409, 410
— , five-wire, 403
-, industrial form of, described,
406
measurement, advantages
and disadvantages of, 420
— , N. C. S., 409, 410
— , principle of, 403
— , simple form of, 407
— , wire, calibrating, 404
-, wire, materials for, 406
Power absorbed in circuit exterior
to generator, 315
— „ conditions for maximum,
320
— , definition of, 282
— developed by current genera-
tor, 312
— , distribution of, in electric
circuit, 318
— , electric, 282
— , electric energy and, 273
— , instruments for measuring,
286-289
— , measurement of, by potentio-
meter, 420
, transmission of, with an end-
less belt, 310
unit of electric, 283
Practical units of potential differ-
ence, ratios of, 148
Presspahn, dielectric strength of,
' 373
- — , resistivity of, 374
Pressure, electric (see Potential
difference)
Primary and secondary circuits 011
iron ring, 457
cells (see Cells)
Production of an electric current,
methods of, 2
Properties of a current, apparatus
for showing, 9
— magnetic, 457-467
Proportional galvanometers, con-
struction of, 105
- galvanometer, Walmsley and
Mather's, 106
Protractor used in subdividing a
galvanometer scale, 50
Pull, calculation of magnetic, 448
, magnetic, 442
INDEX
543
Quantities, comparison of, 361
Quantity, electric, measurement of,
348
— , induced, and resistance of
circuit, relation between, 426
— , measurement of, by ballistic
galvanometer, 353
— meters, 302
— , ratio of units of, 394
Quartz, resistivity of, 374
— , specific inductive capacity of,
Ratio boxes, 413, 414
— of units of quantity, 394
Ratios of practical units of resist-
ance, 145
Rayleigh's, Lord, form of Clark's
cell, 201
Reading scale, direct, 51
- telescope and scale for re-
flecting galvanometer, 35, 350,
352, 353
Recording results of an experiment
graphically, 44
Rectangular plate condenser, 388
Reflecting ballistic galvanometer,
350
— electrostatic voltmeter, Ayr-
ton and Mather's, 157
-- - galvanometer, Ayrton and
Mather's, 353
- galvanometer, making, 35
-- , high resistance, 390
— magnetometer, 57
Reichsanstalt form of standard ohm
coil, 265
Relative advantages of voltmeters
and galvanometers, 33
Reluctance, 467
Remanent magnetism, 461
Residual magnetism, 464
Resin, resistivity of, 374
- oil, resistivity of, 374
- oil, specific inductive capacity
of, 37i
- , specific inductive capacity of,
37.1
Resistance, 142
- - , absolute unit of, short history
of, 473
- - , arrangement of cells to give
maximum power to external
circuit of fixed, 325
- , B.A. unit of, 477 «>
- boxes, 145, 146
- coil, Ayrton and Mather's
non-inductive, 148
Resistance coils, 145
— coils, standard, 264
— combined, 245
, conductors of large specific,
and small temperature coeffi-
cient, 239
, its laws and measurement,
210
- of battery, finding, by con-
denser method, 398
— of cell condenser method of
measuring, 398
of cells, 1 80
of circuit, relation between
quantity induced and, 426
- of conductors, conditions
affecting, 228
— of conductors in series and in
parallel, 244, 246
- of insulator, 373, 393
- of metals and alloys, 231, 234
- of metals, tables of, 233, 235
— ohm, the unit of, 143
- parallel, 246
- ratios of practical units of, 145
- specific, 233
- standard low, 415
— , variation of, with cross-
section, 230
— , variation of, with length,
229, 230
., variation of, with material,
231
— , variation of, with tempera
ture, 236, 241
Resistances, comparing, by substi-
tution method, 214
— , comparing, voltmeter and
ammeter methods, 210
— , comparison of, by potentio-
meter, 417 et seq.
— , potential differences and, cur-
rent method of comparing, 153
, specific, of mixtures of sul-
phuric acid and water, 515
, standard, for current measure-
ments, 416
Resistivity of conductor, 233
- of insulators, 373, 388
Rheostat, carbon cloth, 419
— , carbon plate, 418
Rolled condenser, 376
Rubber covered cable, dielectric
strength of, 373 •
— , dielectric strength of, 373
Rule, Fleming's, 425
Rules, Kirchhoff's, 248
Sagging wire magnifying system ot
hot-wire ammeter, 124
544
INDEX
St. Ivouis conference on electrical
standards, 495
Saturation, magnetic, 450
Scale, constructing galvanometer, 50
- - , direct reading, 5 1
- - for reflecting galvanometer,
351, 352
- - for tangent galvanometer,
constructing, 87
Secondary cells, 187
Self-induction, 438
Sensibilities, comparing two volt-
meters of very different, 160
Sensibility, invariable, galvano-
meters of, 107
-- of tangent galvanometer, vari-
ation of, 90
Series, cells in, 3, 325
-- , condensers in, 383
— , resistances in, 244
Shellac, resistivity of, 374
- , specific inductive capacity of,
371
Shunt, Ayrton and Mather's, 416
— for strong currents, Ayrton
and Mather's, 263
, increase of current produced
by applying, 255
, principle of
Ayrton and
Mather's, 260
- with bridge, use of, 227
- box, high insulation, 254
- box, method of constructing,
253
box, plan of Ayrton and
Mather's universal, 261
box, top of, showing parallel
arrangement of shunts, 254
- box, top of, showing series
arrangement of shunts, 255
• box, universal, advantages of,
260
- box, universal, construction
of, 260
- box, universal, recent form
of, 262
Shunted voltmeter used as amme-
ter, 164
Shunts, 251
— , multiplying power of, 252
, universal, principle of, 259
Siemens dynamometer, 103
Silicum bronze, resistance of, 233
Silver, annealed, resistance of, 233,
235
, hard drawn, resistance of,
233, 235
Sine law, apparatus for testing, 101
- galvanometer, 100
Single pendulum, time of vibration
of, 67
Single-pivot galvanometer, 117, in
" vSix Block Agglomerate " cell, 198
Slate, resistivity of, 374
Soft iron ammeters, 118
Specific gravities, resistances and
conductivities of mixtures of
sulphuric acid and water, 515
- inductive capacity, measure-
ment of, 388
— resistance of insulation, 374
— resistance of metals, 233, 235,
241
Sperm oil, specific inductive capa-
city of, 371
Spherical condenser, standard, 396
Spring control ammeters, 118
Squared paper, use of, 44
Standard air condenser, 391, 392,
395
cells, 200
— low resistance, 415
— ohm coil (Reichsanstalt) form,
265
resistance coils, 264, 265
- resistances for current meas-
urements, 416
spherical condenser, 396
Standards, Board of Trade elec-
trical, short history of, 473
et seq.
— , Board of Trade committee
on electrical, composition of, 488
— , British Association's recom-
mendations on electrical, 485
— , Chicago conference on elec-
trical, 490
— , electrical, American specifi-
cation defining, 497
— , electrical, Berlin conference
on, 499
, electrical, Orders in Council
defining, 492
, electrical, St. Louis confer-
ence on, 499
, international conference
(1908) on units and, 500
Steel, relative conductivities of, 244
Storage cell, Edison nickel-iron, 188
Storage cells, 187
- cells, charge and discharge
curves of, 189
Straight current, field of, 453
Strength, calculation of magnetic
field, 452
— •-, current, 7
— of current, defining, 12
— of magnetic poles, 59
- of magnetic pull, 51, 66
Substitution method, comparing
resistances by, 214
Sulphur, resistivity of, 374
INDEX
545
Sulphur dioxide, specific inductive
capacity of, 371
, specific inductive capacity of,
37i
Sulphuric acid voltameter, Ayrton's
form of, 31
— acid voltameter, description
of practical forms of, 31
acid voltameter, Mather's
form of, 32
acid voltameter, McMichael's
form of, 33
Supply meters, 292
Surface, equipotential, 79
Swinburne wattmeter, 288
Symbols, Table oi, 527
Tables, calibration, 43
, chemical decomposition, 41
, dielectric strengths, 373
, dimensions of wires, etc., 517
, fundamental units, 514
, intensity of earth's field, 97
— , materials for resistances, 241
, ratio of practical units of
resistance, 145
— , ratio of practical units of
P.D., 148
, relations between electro-
magnetic . and electrostatic unit,
515
— — , relative conductivities of
metals, 244
: resistance, weight, length, 235
, resistivity of insulators, 314
• , specific gravities, etc., of
mixtures of pure sulphuric acid
and distilled water, 515
• , specific inductive capacities,
371
, specific resistances of metals,
233
. , symbols, 527
. , temperature co-efficients of
copper, platinum, mercury, 238
. , values of ohm, 483-485
, windings, 518, 526
Tangent galvanometer, 36, 83, 84
galvanometer, absolute cali-
bration of, 42
galvanometer, calibrating any
galvanometer by direct com-
parison with, 43
galvanometer, comparison
with a voltameter, 40
galvanometer, constructing
scale for, 87
• galvanometer, scale for, 86
galvanometer sensibility,
examples, 93 et seq.
2J
Tangent galvanometer, showing
modgs of supporting fibre, 37
- galvanometer, value in am-
peres of deflection of, 96
— galvanometer, variation of
sensibility of, 90, 91
law, 89
law, improved apparatus for
testing, 90
— law, simple apparatus for
testing, 89
Telegraph battery, Muirhead's, 182
Telescope, reading and scale for
reflecting galvanometer, 352
Temperature, variation of resist-
ance with, 236, 241
Temporary magnetism, 464
Testing hysteresis, 457, 461 .
- magnetic properties by bal-
listic method, 457
- of copper, 238
- Ohm's law, apparatus for,
140
, polarisation, 171
resistance of cells, 181, 313
resistivity, 388
— sine law, apparatus for, 101
specific inductive capacity,
388
- tangent law, apparatus for,
89, 90
temperature coefficient of
wires, 241
variation laws of sensibility
of tangent galvanometer, appara-
tus for, 91
Thermo-electric currents, 406 (foot-
note)
Thermometer, platinum, 239
Thermometers, measuring current
with, 1 6
Thomson energy meter, 299
Three-wire bridge, 224
Tin, pressed, resistance of, 233, 235
, relative conductivities of, 244
Tinsley cell, 208
Torque, definition of, 35 (footnote)
Torsion apparatus for measuring
magnetic moment, 64
electrostatic voltmeter, 135,
137
galvanometer, n
Transmission, electrical efficiency
of, and ratio of power received
to power receivable, 343
of energy, economy in, 346
of energy, efficiency of elec-
tric, 339
of energy, electric, 308
of energy, mechanical ana-
logies illustrating, 311
546
INDEX
Transmission of energy, table of
results achieved, 342
- of power with an endless belt,
310
Tungsten, resistance of, 233, 235
Turpentine, oil of, specific inductive
capacity of, 371
U
Unit current, definition of, 18
of capacity and farad, rela-
tion between, 367
• of energy, Board of Trade,
294
of energy, electric, the joule,
7
of mutual induction, 437
— of potential difference, 148
— of power, electric, 283
— of resistance, 143
— of resistance, absolute, short
history of, 473
quantities of electricity, rela-
277
tion between, 24
— quantity, 24
Units and standards, International
Conference (1908) on, 500
, C.G.S. and British systems of
comparison of, 514
, electromagnetic and electro-
static relations, between, 515
of capacity, 365
- of quantity, ratio of, 394
of resistance, ratios of prac-
tical, 145
Universal shunt, Ayrton and
Mather's, 259, 260 et seq., 416,
417
shunt box, advantages of, 260
— shunt box, construction of,
260
shunt box, plan of Ayrton
and Mather's, 261, 262
• shunts, principle of, 259
Variation of resistance with cross-
section, 230
of resistance with length, 229
of resistance with material,
231
of resistance with tempera-
ture, 236
of sensibility of tangent gal-
vanometer, 90
of strength of magnetic field
along axis of coil, curve of, 75
Vibration, diagram of moderately
damped, 358
, diagram of undamped, 358
, diagram of well-damped, 358
Vibration method, comparing rela-
tive strength of different parts of
magnetic field by, 72
Voltaic element, chemical action in
simple, 170
Voltameter, Ayrton's form of sul-
phuric acid, 31
, comparison of, with a tangent
galvanometer, 40
— , Hoffman's sulphuric acid,
27
, McMichael's form of acid, 33
— , Mather's form of sulphuric
acid, 32
, silver, desiccator used with,
— , silver, for measuring cur-
rents, 19 (footnote)
Volt-boxes, 411
Voltmeter and ammeter method of
comparing resistances, 210
, Ayrton and Mather's reflect-
ing electrostatic, 157
, Ayrton and Mather's zero
electrostatic, 135-137
, calibrating, 160, 161, 162
, calibration of, by potentio-
meter, 411
— , calibration, potential divider
for, 412
, explanation of, 155
, moving coil, 160
— , shunted, used as ammeter,
164
Voltmeters, ammeters used as, 158
— , comparing two, of very dif-
ferent sensibilities, 160
» current, and ammeters, re-
sistances of, 158
used as ammeters, 163
W
Walmsley and Mather's pro-
portional galvanometer, 106
Water analogy of electric flow, 129
, curve connecting rate of flow
of, with loss of head, 139
pressure, apparatus for testing
distribution of, 128, 130
, specific inductive capacity of,
371
Watt, definition of, 284
Wattmeter, commercial forms of,
288
, diagram of, 287
, Elliott's deflectional, 289
, Swinburne, with cover re-
moved, 288
Wattmeters, 286
Wax, paraffin, resistivity of, 374
INDEX
547
Wax, paraffin, specific inductive
capacity of, 371
, sealing resistivity of, 374
, sealing specific inductive
capacity of, 371
Weights, international atomic, 22
(footnote)
Weston's cadmium cell, 205
cadmium cell (F. E.
Smith's form), 206, 207
cell, 200
moving coil ammeter, work-
ing parts of, 113
Wheatstone's bridge, bar pattern
of, 270
bridge, construction of, 211
bridge, diagram of, 219
bridge dial, pattern of, 270,
271
bridge, ordinary forms of, 265,
266
bridge, portable, diagram of
connections of, 270
bridge, portable forms of, 269
bridge, portable, with battery
and galvanometer combined, 269
bridge, Post Office, 267, 268
bridge, principle of, 218
— bridge, use of, 221
Windings table for cotton- covered
(double) wire, 519
table for cotton - covered
(single) wire, 518
table for enamel insulated
wire, 524
table for enamel insulated and
cotton-covered (double) wire, 526
table for enamel insulated and
cotton-covered (single) wire, 525
table for silk covered (double)
wire, 523
Windings table for silk covered
(single) wire, 522
table for specially fine cotton
(double) wire, 521
table for specially fine cotton
(single) wire, 520
tables, 518 et seq.
Wire, calibrating, by differential
galvanometer, 405
, lines of force round straight,
454
, relation between lengths,
resistances and weights of pure
copper, 516
Wires, dimensions of, according to
British Standard wire gauge
(S.W.G.), 516
, windings tables for, 518 et seq.
Witton-Kramer magnet, 443
Work done by current, 273
, electric unit of, 277
Zero electrodynamometer for small
currents, 142
electrometer, Ayrton and
Mather's, 135-137
Zinc, 233, 235, 244
amalgam, 201
, amalgamating, 176
— , consumption of, in batteries,
185, 186, 200
, local or prejudicial action of,
175
, pressed, resistance of, 233,
235
, relative conductivities of, 244
, resistance of, 233, 235
, sulphate of, heat of forma-
tion of, 207
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CASSELL & COMPANY, LIMITED, LA BELLE SAUVAGB,
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