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ADVANCED LABORATORY PRACTICE

ELECTRICITY AND MAGNETISM

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Vect.
T.

ADVANCED
LABORATORY PRACTICE

ELECTRICITY

AND

MAGNETISM

EARLE MELVIN TERRY, Ph.D.

ASSOCIATE PROFESSOR OF PHYSICS, UNIVBRSITY OP WISCONSIN

First Edition

I So^hl

It « a ^

McGRAW-HILL BOOK COMPANY, Inc.
NEW YORK: 370 SEVENTH AVENUE

LONDON: 6 & 8 BOUVERIE ST., E. C. 4
1922

THE MAPLE PRESS - YORK PA

PREFACE

In preparing this book, the author has had in mind particularly
the needs of those students who have at their disposal only one
year to devote to the study of electricity and magnetism in addi-
tion to the work covered in an elementary course in general
physics. It has been his aim to include, in addition to the usual
work in electrical measurements, a sufficient study of the dis-
charge of electricity through gases, radio activity, and thermionics
to enable those who cannot pursue special courses to gain an
idea of the fundamentals of these newer branches.

The subject matter covers the work given to third year students
in electrical engineering at the University of Wisconsin. Follow-
ing the elementary work of the first nine chapters, a number of
the complex bridge methods for precise measurements of induc-
tance and capacitance are discussed, together with descriptions of
the various sources of alternating currents which have been
developed in recent years for energizing bridge circuits. A dis-
cussion of the more modern instruments for detecting the balance
condition of bridges, together with their individual merits, has
been included. This is preceded by an elementary study of
"transients," in which the fundamental phenomena of reactance,
necessary for an understanding of bridge methods, are set forth.

The electron tube, Taecause of the multiplicity of its uses, finds
many applications, not only in the art of radio communication,
but also in engineering practice and in the general research
laboratory. Considerable space has been devoted to this device,
as well as to the fundamentals of the electron theory and the
passage of electricity through gases.

The author is a firm believer in the laboratory method of
instruction, and each exercise is preceded by a discussion of the
theories involved sufficient to enable the student to understand
clearly the relation of each experiment to the general field in
which it lies. It is believed that with the material given in the
text and the references to standard works, which have been
included, the student can pursue the subject without the aid of

y| PHKFACE

funtml IccturoH, although at the present time the writer is devot-
ing one h<»ur pi'r week to a lecture-conference. Experience
aiiowii that the average student performs fourteen of these
exerriue* per semester and the topics herewith presented accord-
ingly |M»rnjit of soMje little choice.

In !«t*Iecting material, advantage has been taken not only of the
original sources, but also of the standard texts in the special
fioldn reprcsfntecl. Being a collection of laboratory exercises,
thi« l»ook makes no claim to originality of the subject matter
included, and the author hereby acknowledges his indebtedness
to the many IxKiks and special articles referred to in the foot-
not«4 throughout the text. He is indebted, also, to the Leeds-
Northrup Company, J. G. Biddle Company, Queen & Company,
General Radio Company, Tinsley & Company, and other manu-
facturers of electrical apparatus for supplying the cuts which have
l»en used. In particular, he wishes to express his gratitude to
Dr. H. B. Wahlin and L. L. Nettleton, instructors in physics at
the I'niversity of Wisconsin, who have read the entire manu-
script and made many vauable suggestions during its
preparation.

E. M. Terry.
UNiA'ERaiTJ- or Wisconsin,
Madison, Wis.
June, 1922.

CONTENTS

Page

Preface v

CHAPTER I. — General Directions — Electrical Units 1

Preparation — Connections — Keys and Switches — Rheostats —
Switch Board — Care of Apparatus — Notebooks — Electrostatic
System of Units — Electromagnetic System of Units — Practical
System of Units — Ratios of the Electrical Units — Rationalized
Practical System of Units.

CHAPTER II.— Galvanometers 17

Thomson Galvanometer — D'Arsonval Galvanometer — Galvano-
meter Sensitivity — Figure of Merit — Ballistic Galvanometer —
Constant of Ballistic Galvanometer — Flux Meter — Checking
Devices.

CHAPTER III. — Measurement op Resistance 35

Ohm's Law — Specific Resistance — Temperature Coefficient of
Resistance — Wheatstone Bridge — Measurement of Low Resistance
— Measurement of High Resistance — Internal Resistance of Cells —
Battery Test.

CHAPTER IV. — Measurement of Potential Difference 55

Description of Potentiometer — Direct Reading Potentiometer —
Leeds and Northrup Potentiometer — Wolff Potentiometer —
Tinsley Potentiometer — Weston Standard Cell — Volt Box.

CHAPTER V. — Measurement of Current 70

Kelvin's Balance — Siemen's Electrodynamometer — Ammeters and
Voltmeters — Adjustment of Ammeters and Voltmeters — Measure-
ment of Current by the Potentiometer.

CHAPTER VI. — Measurement of Power 82

Wattmeters, Types of — Compensation of Wattmeters — Calibra-
tion of an Indicating Wattmeter.

CHAPTER VII. — Measurement of Cai'Acitance 86

Condensers — Grouping of Condensers — Standard Condensers —
Comparison of Condensers — Flemming and Clinton Commutator.

CHAPTER VIII.— Magnetism 95

Strength of Pole — Strength of Field — Magnetic Moment —
Magnetic Induction — Permeability and Susceptibility —
Demagnetization due to Ends of a Bar Magnet — Magnetic
Circuit — Magnetic Units — Magnetization Curves — Hysteresis.

vii

viii CONTENTS

Page

CHAITKR IX. — Self AND Mutual Inductance 117

(leneral Principles — Definition of Units of Inductance — Standards
of Inductance — Measurement of Self-inductance — Measurement
of Mutal Inductance.

CHAPTER X. — Elementary Transient Phenomena 126

Time Constant — Circuit Containing Resistance and Inductance —
Circuit Containing Resistance and Capacitance — Circuit contain-
ing R&sistance, Inductance and Capacitance — Non-oscillatory
Discharge of a Condenser — Aperiodic Discharge of a Condenser —
Oscillatory Discharge of a Condenser — Logarithmic Decrement —
Harmonic E. M. F. acting on a Circuit Containing Resistance,
Inductance, and Capacitance — Vector Diagrams — Series
Resonance — Parallel Resonance — Measurement of Inductance
and Capacitance by Resonance — Effective Value of an Alternating
Current — Power Consumed by a Circuit Traversed by an
Alternating Current.

CHAPTER XI. — Sources of Electromotive Force and Detecting

DEVICE.S FOR Bridge Methods 151

Sechometer — Wire Interrupter — Motor Generator — Microphone
Hummer — Audio Oscillator — Vreeland Osallator — Electron Tube
Oscillator — Telephone Receiver — Thermo Galvanometer — Vibra-
tion Galvanometer — Alternating Current Galvanometer.

CHAPTER XII.— Alternating Current Bridges 168

General Considerations — Maxwell's Bridge — Anderson's Modifica-
tion of Maxwell's Bridge — Stroude and Oates' Bridge — Trow-
bridge's Method — Heydweiller's Network — Heavieside's Bridge —
Maxwell's Bridge for Mutual Inductance — Mutual Inductance
Bridge — Frequency Bridge — Circuits of Variable Impedance —
Motional Impedance of a Telephone Receiver — Power Factor
and Capacitance of Condensers — Resistance of Electrolytes.

CHAPTER XIII. — Conduction op Electricity through Gases . . 198
Electrons — Conductivity of Gases — Structure of the Atom — Ioni-
zation Current — Resistance of a Discharge Tube — Phenomena of
the Discharge Tube — Theory of the Discharge — Field Strength at
Various Points in the Discharge — Cathode Rays — Velocity and
Ratio of the Charge to the Mass of an Electron — Radioactive
Substances — .\lpha Rays — Beta Rays — Gamma Rays — Radio
active Transformations — Ionization by Radio Active Substances.

CHAPTER XIV.— Electron Tubes 218

Free Electrons — Electron Emission — Two Element Electron Tube
— Voltage Saturation — Space Charge — Characteristics of the
Two Element Electron Tube — Three Element Electron Tube —
Static Characteristics — Amplification Factor — Internal Resistance
of Three Element Electron Tube — Tungar Rectifier.

CONTENTS ix

Page

CHAPTER XV. — Photometer and Optical Pyrometer 239

Intensity of Radiation — Photometers — Lummer-Brodhun Photo-
meter— Stutly of Incandescent Lamps — General Principles of
Radiation — Black Body Furnace — Distribution of Energy in the
Spectrum — Applications to Pyrometry — Optical Pyrometer.

Appendix 252

Calculation of Self Inductance — Calculation of Capacitance.

Index 259

LIST OF EXPERIMENTS

Page

1. Specific Resistance of Materials 42

2. Temperature Coefficient of Resistance 45

3. Insulation Resistance by Leakage 48

4. Internal Resistance of Cells by Condenser Method 52

5. Battery Test 53

6. Comparison of E.M.F. of Cells by the Potentiometer 66

7. Calibration of a Voltmeter by the Potentiometer and Volt Box 68

8. Calibration of an Electrodynamometer 73

9. Electrical Adjustment of an Ammeter and a Voltmeter 77

10. Calibration of an Ammeter by the Potentiometer and Standard

Resistance 80

11. Calibration of a Wattmeter 84

12. Comparison of Capacitances by the Bridge Method 92

13. Capacitance by the Fleming and Clinton Method 93

14. Magnetization Curves by Hopkinson's Bar and Yoke 108

15. Magnetization Curves by the Rowland Ring Method 113

16. Hysteresis Curves 114

17. Comparison of Self Inductances by the Bridge Method 122

18. Mutual Inductance by Carey-Foster's Method 124

19. Measurement of Inductance and Capacitance by Resonance 148

20. Maxwell's Bridge for Self Inductance 171

21. Stroude and Gate's Bridge for Self Inductance 174

22. Trowbridge's Method for Self Inductance 177

23. Heydweiller's Method for Mutual Inductance 179

24. Heaviside's Bridge for Self Inductance 182

25. Maxwell's Bridge for Mutual Inductance 184

26. Comparison of Mutual Inductances 185

27. Bridge Method for Measuring Frequency 187

28. Motional Impedance of a Telephone Receiver 191

29. Phase Difference and Capacitance of a Condenser 193

30. Resistance of Electrolytes 197

31. Resistance of a Discharge Tube 202

32. Variation of Field Strength along the Discharge 207

33. Measurement of — and Velocity of an Electron 211

»34. Ionization by Radio Active Substances 216

35. Characteristics of the Two Element Electron Tube 224

36. Static Characteristics of a Three Element Electron Tube 226

37. Amplification Factor of a Three Element Electron Tube 231

38. Plate-Filament Resistance of a Three Element Electron Tube 235

39. Study of the Tungar Rectifier 237

40. Study of Incandescent Lamps 243

41. The Optical Pyrometer 250

ADVANCED LABORATORY PRACTICE

ELECTRICITY AND MAGNETISM

CHAPTER I

GENERAL DIRECTIONS— ELECTRICAL UNITS

1. Preparation. — If one is to make the best use of his time in
the laboratory, he must understand thoroughly what is to be
done and then proceed in a systematic manner to do it. This
can be accomplished only when preparation for the task has
been made before taking up the experimental work. Assign-
ments accordingly will be made one week in advance, and the
student is expected to enter the laboratory with the following
preparation :

1. An understanding of the theory of the experiment.

2. A knowledge of the working principles of the instruments
to be used.

3. A schedule according to which the data are to be taken.
In order to facilitate the work of the first few periods, the follow-
ing general directions should be carefully read :

2. Connections. — A large portion of the trouble in performing
electrical measurements arises from imperfect connections. All
instruments, to which wires are to be attached, are provided with
binding posts. To secure good contact, remove the insulation
about an inch from the end of the wire, scrape it clean, wrap it
two-thirds around the binding post, and then screw down the
nut. If the wire is too short to reach between the points desiied,
join two or more wires with connectors, having first scraped the
ends clean. Never join wires by twisting their ends together,
as connections of this sort, unless soldered, are entirely unreliable.
Do not coil wires about a rod or a pencil, since then they cannot
be used again. Cut wires to the proper length, thus avoiding a
complicated tangle difficult to trace, which, through leaks,

2 ELECTRICITY AND MAGNETISM

furnishes a source of constant trouble. Never allow one wire to
rest upon another, even though both are covered with insulation.

Before attempting a set-up, make a rough sketch of connec-
tions, arranging the apparatus in a compact and orderly manner.
This will be of great service later in checking connections and
locating faults. In many cases, especially in complicated net
works, a little forethought in the arrangement will save much
time and inconvenience in the performance of the test. Always
make the connection with the source of current supply last,
having first assured yourself as to the correctness of the connec-
tions by comparison with the sketch, or by consultation with your
instructor. As a further precaution, close the main switch at
first only an instant, opening it at once to see if there are any
indications of a short circuit. This is especially important where
the source is a dynamo or a storage battery.

3. Keys and Switches. — Always open and close a switch
quickly, to avoid burning it at the point of contact. If the

Fig. 1. — Reversing switch.

circuit includes mercury cups and connecting links, it should be
broken by means of a knife switch, not by removing the link, as
mercury is especially likely to arc. Ordinary contact keys should
be used only where a small current is to be carried, and where
variations in the resistance of the circuit introduce no serious
error; as, for example, the galvanometer circuit of a Wheatstone
bridge.

The device generally employed for reversing the current
through any portion of a circuit is the Pohl's commutator, which
consists of a double pole double throw switch with two cross
wires, as shown in Fig. 1. It will be seen that when the switch is
closed, as shown by the heavy lines, the current through R flows
upwards, but is reversed when the switch is thrown towards the
right. Since the cross wires in such a commutator are frequently

ELECTRICAL UNITS 3

placed under the block, a double pole double throw switch should
be examined carefully before it is connected in circuit, as the
cross wires of the commutator may produce short circuits,
resulting in serious injury to the apparatus.

4. Rheostats. — A rheostat is a variable resistance capable of
carrying considerable current. It is used primarily as a control-
ling device and its value, in general, need not be accurately
known. When connected in series with a source of electrical

Fig. 2. — Rheostat with fixed steps.

power, the current supplied to any circuit may be varied and
brought to any desired value, within certain definite limits, by
changing the resistance -of the rheostat. Since the energy con-
sumed by a rheostat always appears in the form of heat, the
current carrying capacity for a given resistance depends upon
the provision made for dissipating heat either by conduction,
convection, or radiation.

Many different forms of rheostats are in use and only a few of
the more common types will be mentioned here. Figure 2
illustrates one that is frequently used for controlling relatively
large currents. It consists of a number of copper lugs between
which are connected units of high resistance metal in the form
of a thin ribbon to give as much heat iradating surface as possible.
They are bent back and forth in a zig-zag shape and embedded
in sand. With this arrangement the resistance varies by steps.
By connecting two in series, one having large and the other small
steps, a fairly smooth variation in current may be obtained.

4 ELECTRICITY AND MAGNETISM

Another common form of rheostat is shown in Fig. 3. A hi
resistance wire is wound on an insulating tube. Binding poj
are connected to each end of the wire, and as the sliding contact
moved along, the resistance between it and one end of the wi
changes from zero to a maximum. Wires of various sizes a
frequently wound on the same tube thus giving two or mc

Fig. 3. — Tube rheostat.

ranges for one instrument. For carrying large currents, the en
of the tube are closed and coohng water is passed throu^
Such rheostats are generally wound with a ribbon to improve t
thermal contact between the wire and tube. The figure sho
a high resistance instrument, which may also be used as a pote
tial divider. If the E.M.F. to be divided is connected across t
end binding posts, any desired fraction of this voltage may

Fig. 4. — Carbon compression rheostat.

obtained by "picking off" between one end and the slider, a
moving the latter back and forth.

Another device for controlling current makes use of the h
that the resistance between two carbon surfaces varies with t
pressure. An instrument of this sort is shown in Fig. 4.
consists of series of rectangular carbon plates placed in a trou
and arranged in such a way that they can be subjected to varial
pressures by means of an adjusting screw. Rheostats of t]

ELECTRICAL UNITS 5

type are useful where low voltage currents are to be controlled.
They have the disadvantage of requiring frequent readjustment
since the tension changes with variations of the temperature of
both the carbon plates and metal parts.

6. Switch Board. — A switch board is a necessary adjunct to
any electrical laboratory and is used to distribute electrical
power of different types and voltage to the various working cir-
cuits of the laboratory, and to connect the different circuits with
one another. It consists of a panel of insulating material, usu-
ally marble, on which is mounted a series of pairs of sockets.
The various laboratory and power circuits are joined to these
sockets on the back of the board and connections between them
are made at the front by means of flexible connectors, often called
"jumpers," to the ends of which are attached plugs which fit
snugly into the sockets. Power circuits are distinguished by the
word "Volts." With the exception of the A.C. circuits, each
terminal is labeled plus or minus. If, for example, 10 volts
are desired on circuit 91, connect the positive of a 10 volt set
with the positive of 91, and similarly for the negative, when the
polarity at the laboratory end will be found as indicated. If
some voltage is desired, e.g., 16 volts, for which there is no separate
set, connect two or more sets in series, joining plus to minus as
though connecting cells on a table; then, considering the group
as a single set, connect to the laboratory terminals as above. If
a current larger than the normal rated capacity of the storage
battery is desired, use the dynamo or connect in parallel two or
more sets of equal voltage. To do the latter, join all the positive
terminals, similarly the negatives, and then connect to the labo-
ratory terminals as above. Before making switch board connec-
tions, be sure that the circuit switch in the laboratory is open.
Connect the "dead" ends first, and, before pushing in the final
plug closing the circuit, tap it cautiously against the socket,
quickly withdrawing it. If a spark is seen, some error in connec-
tion has been made which must be located before the circuit is
closed. Never connect in parallel on a battery with some one
else without first obtaining his permission.

6. Care of Apparatus. — Electrical apparatus is delicate and
expensive, and it is necessary to proceed with the utmost caution.
If an instrument is provided with a shunt, use the smallest
resistance first; or, if protected by a series resistance, use the
largest value first decreasing it until the desired value has been

6 ELECTRICITY AND MAGNETISM

reached. If an instrument fails to work, do not replace it in
the case and get another, but report at once to the instructor.
Resistance lx)xes are most frequently injured by carrying too
large currents. Before closing the main switch, look over the
connections and make a rough calculation of the current that will
flow in each box. In no case should the power consumed by a
single coil, given by PR, be more than four watts. Plugs should
be seated by a gentle pressure, accompanied by a twisting motion,
heavy pressure being unnecessary.

Never move galvanometers from one place to another without
first making sure that the weight of the moving system has been
removed from the suspension by means of the arrestment which
is always provided. Standard cells should never be tipped up for
purposes of inspection or otherwise, and should not be used as a
source of current, but merely for balancing potentials; and even
here, a large series resistance should at first be included and cut
out as a balance is approached. Ammeters are instruments for
measuring the total current flowing, and should be connected in
series with the circuit, analogous to a water meter. They are
most frequently injured by the passage of too large currents.
If the arrangement of the apparatus is not such that the current
can approximately be calculated before the circuit is closed, a
sufficiently large rheostat should be included and cautiously cut
out, the instrument being watched in the meantime. Voltmeters
are electrical pressure gauges, indicating the difference of poten-
tial between the points to which their terminals are attached, and
are accordingly connected in parallel with the circuit. Most
voltmeters are provided with two scales; and in such cases, one
should use the larger first, transferring to the smaller one if the
voltage is found to be less than the lower full scale reading.
Before leaving the laboratory, return all apparatus to its proper
place in the cases. Wires less than a foot long should be thrown
in the waste box, and the others returned to their hooks in the
wire cabinet. Switch board connectors should be pulled and
returned to the proper hooks. Leave the laboratory as tidy as
you found it.

7. Notebooks. — All data, as they are taken during an experi-
ment, should be recorded in tabular form in a rough note-
book with bound leaves. Ascertain from the instructor specific
directions regarding the form in which the final report is to be
made, and in its preparation observe the following outline:

ELECTRICAL UNITS 7

1. Give name of experiment and references.

2. Enumerate apparatus used, giving number of each piece.

3. Make a sketch (not a picture) including all instruments,
resistance boxes, switches, etc., which will show the actual path
of the current. (Use a ruler and dividers.)

4. Give the theory of the experiment as fully as possible,
deriving all formulae used.

5. Outline briefly the method of procedure, mentioning special
precautions to be taken and difficulties to be overcome.

6. Tabulate your data, arranging it in compact form. State
the units in which your results are expressed.

7. Plot curves showing your results graphically, using as
ordinates the dependent variable. Choose scales such that the
curves will cover as nearly as possible the entire sheet, labeling
axes and putting the scale along each. Draw in a smooth curve,
striking an average between outstanding points.

8. Give a brief discussion of results, including estimates of
accuracy and sources of error.

9. Answer all questions asked under special directions at the
end of each experiment.

ELECTRICAL AND MAGNETIC UNITS

8. Systems of Units. > — There are two distinct systems of units
used in the measurement of electrical quantities; the electro-
static and the electromagnetic. In the former, the fundamental
unit is determined by means of the repulsion between two similar
charges of electricity, while in the latter, it is based upon the
repulsion of two similar magnetic poles. Both of these systems
may properly be termed "absolute" since all the quantities
involved are directly expressible in terms of the fundamental
units of length, mass, and time. The ratio between correspond-
ing units of these two systems is some power of the velocity of
light. In actual practice, however, neither of these systems is
used, since, in general, the quantities therein defined are not of
such magnitudes as to be convenient working units. A third
system, known as the "practical system," has accordingly been
devised, in which all the units are decimal multiples of the cor-
responding electromagnetic units. The units of this system are

1 Everett, The C. G. S. System of Units, chap. X, XI.
Electrical Meterman's Handbook, chap. II.

8 ELECTRICITY AND MAGNETISM

the only ones to which names have been given, and it has been
the custom of the international conferences by which they have
been defined, to honor scientists, famous in the fields in which the
unit* lie, by giving to them their names. Electrical quantities,
expressed in the electrostatic and electromagnetic systems, are
designated by the letters E.S.U. and E.M.U., respectively.

FUNDAMENTAL ELECTRICAL UNITS

9. Magnetic Units. Magnetic Pole Strength. — The unit mag-
netic pole is a pole of such strength that it repels a like pole at a
distance of one centimeter, in air, with a force of one dyne.

Magnetic Field Strength. — A magnetic field of unit intensity is
a field that acts upon a unit magnetic pole placed in it, with a
force of one dyne.

10. Electrostatic Units. Quantity. — The electrostatic unit of
quantity is of such a magnitude that it repels a like quantity at a
distance of one centimeter, in air, with a force of one dyne.

Current. — The electrostatic unit of current exists when an
electrostatic unit of quantity flows past any plane in a conductor
per second.

Potential Difference. — Unit electrostatic difference of potential
exists between two points when the amount of work required to
carry an electrostatic unit of quantity from one to the other is one
erg.

Resistance. — A conductor possesses the electrostatic unit of
resistance if, when carrying the electrostatic unit of current, the
difference of potential across its terminals is one electrostatic unit.

Capacitance. — A condenser possesses an electrostatic unit of
capacitance if the electrostatic unit of potential difference
across its terminals gives to it the electrostatic unit of charge.

Indv,ctance. — A coil possesses an electrostatic unit of induc-
tance if, when the inducing current is changing at the rate of one
electrostatic unit per second, the induced electromotive force is
one electrostatic unit. This applies both to self and mutual
induction.

11. Electromagnetic Units. Current. — The electromagnetic
unit of current is a current such that, when flowing through an
arc of one centimeter length of a circle of one centimeter radius,
it produces, at the center, a unit magnetic field.

Quantity. — The electromagnetic unit of quantity is that quan-

ELECTRICAL UNITS 9

tity which passes, per second, any plane of a conductor in which
the electromagnetic unit of current is flowing.

Potential Difference. — The electromagnetic unit of potential
difference exists between two points when the amount of work
required to carry the electromagnetic unit of quantity from one
to the other is one erg.

Resistance. — A conductor possesses the electromagnetic unit of
resistance, if, when carrying the electromagnetic unit of current,
the difference of potential across its terminals is one electro-
magnetic unit.

Capacitance. — A condenser possesses the electromagnetic unit
of capacitance if the electromagnetic unit of potential difference
across its terminals gives to it one electromagnetic unit of charge.

Inductance. — A coil possesses an electromagnetic unit of
inductance if, when the inducing current varies at the rate of one
electromagnetic unit per second, the induced electromotive
force is one electromagnetic unit.

12. Practical Units. Current. — An ampere is one-tenth of an
electromagnetic unit of current.

Quantity. — The coulomb is the quantity of electricity which
passes per second any plane of a conductor in which the current
is one ampere.

Potential Difference. — The difference of potential between two
points is one volt when the amount of work required to carry one
coulomb from one to the other is one joule.

Resistance. — A conductor possesses a resistance of one ohm if,
when carrying a current of one ampere, the difference of poten-
tial across its terminals is one volt.

Capacitance. — A condenser possesses a capacitance of one
farad if a difference of potential of one volt across its terminals
gives it a charge of one coulomb.

Inductance. — Two coils possess a mutual inductance of one
henry if, when the primary current is changing at the rate of one
ampere per second, the electromotive force induced in the second-
ary is one volt.

A coil possesses one henry of self-inductance if, when the
current through it is varying at the rate of one ampere per second,
the induced counter electromotive force is one volt. One milli-
henry equals 0.001 henry.

Magnetic Flux. — The total flux in a magnetic circuit is one
maxwell when it possesses one magnetic Une of induction.

10 ELECTRICITY AND MAGNETISM

Magnetic Induction. — The induction in a magnetic circuit is
one gauss when the flux density is one maxwell per square
centimeter.

Magnetomotive Force. — The magnetomotive force of a magnetic
circuit is one gilbert if the work required to carry a unit magnetic
pole once around the circuit is one erg.

Field Strength. — A magnetic field possesses unit strength if the
magnetomotive force is one gilbert per centimeter. (This
definition is identical with that previously given for field
strength.)

Reluctance. — A magnetic circuit possesses a reluctance of one
oersted if a magnetomotive force of one gilbert produces a flux of
one maxwell.

13. Legal Defmitions of the Practical Units. — At the last
International Conference on Electrical Units and Standards,
which met in London, in 1908, the following resolutions were
adopted, which have served as the basis for legislation in the
different countries of the world for fixing the legal definitions
of the fundamental electrical units now in force. The full
report of this Conference, in which 21 different nations were
represented, may be found in The Electrical Review, vol. 63,
(1908), page 738.

RESOLUTIONS

I. The Conference agrees that as heretofore the magnitude of
the fundamental electric units shall be determined on the elec-
tromagnetic system of measurements with reference to the centi-
meter as the unit of length, the gram as the unit of mass, and the
second as the unit of time.

These fundamental units are (1) the ohm, the unit of electric
resistance which has the value of 1,000,000,000 in terms of the
centimeter and second ; (2) the ampere, the unit of electric current
which has the value of one-tenth (0.1) in terms of the centimeter,
gram, and second; (3) the volt, the unit of electromotive force
which has the value of 100,000,000 in terms of the centimeter,
the gram, and the second; (4) the watt, the unit of power, which
has the value of 10,000,000 in terms of the centimeter, the gram,
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

ELECTRICAL UNITS 11

the adoption of the International ohm, the International ampere,
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
unvarying electric current by a column of mercury at the tem-
perature of melting ice, 14.4521 grams in mass, of a constant cross-
sectional area and of a length of 106.300 cm.

To determine the resistance of a column of mercury in terms
of the International ohm, the procedure to 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 cur-
rent 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 gram 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 pre,ssure of one International volt.

The Conference recommends the use of the Weston Normal
Cell as a convenint method of measuring both electromotive
force and current, and when set up under the conditions specified
in schedule C, may be taken, provisionally, as having, at a tem-
perature of 20° C, an E.M.F. of 1.0184 volts.

14. The New Value of the Weston Standard Cell.— Since the
meeting of the London Conference, a large amount of research
has been carried on at the Bureau of Standards at Washington
on the Weston Cell and the electrochemical equivalent of sil-
ver; and it has been found that the electromotive force of this
cell, in terms of the International ohm and International ampere,
is, within one part in 10,000,

E = 1.0183 International volts at 20° C,

and this value was adopted by the Bureau of Standards Jan. 1,

ELECTRICITY AND MAGNETISM

1911. The formula for the temperature coefficient of the Weston
Cell adopted by the London Conference, based on the investiga-
tions of the Bureau of Standards, is as follows :
Et = E20 - .0000406 (t - 20°) - .00000095 (t - 20) 2

+ .00000001 (t - 20)3 (1)

15. Ratios of the Electrical Units. — For convenience of com-
parison the dimensions of the electrostatic and electromagnetic
units are given below. The dimensions of the dielectric constant
and the permeability are unknown and are inserted in the formula
as K and /i, respectively. All that is known concerning the

nature of these quantities is that

■\/kJi

equals v, equals 3 X

10'" cm. per second, the velocity of light in free space. The
last column gives the ratio of the corresponding units in the two
systems, in terms of v.

Unit

Electro-
magnetic

Electro-
static

Electro-
magnetic

E.M.U.

Electro-
static

E.S.U.

Quantity

Current

Pot. diff

Resistance

Capacity

Inductance

[My'L^^n-y2]

[M^'^L'^^T-^K^^]
[M^^IJ^T-^k'^^]
[M^^L^^T-^K-^^]

[Lk]

[L-ir^K-i]

[L-^Tk-'^^^-^-]
[L^T-^K^]

V
V

The following table gives the practical units in terms of the corre-
sponding units of both the electromagnetic and the electrostatic systems:

1 Ampere
1 Coulomb

1 Volt
1 Ohm

= 10-1 E.M.U.'s = 3 X 109 E.S.U.'s
= 10-1 E.M.U.'s = 3 X 10" E.S.U.'s
1

= 108 E.M.U.'s =
= 10" E.M.U.'s =

3 X 102
1

E.S.U.'s

9 X 1011 E.S.U.'s
1 Farad = lO"" E.M.U.'s = 9 X 10" E.S.U.'s

1 Microfarad = 10"" E.M.U.'s = 9 X lO^ E.S.U.'s

1 Henry

= 10* E.M.U.'s =

9X10

n E.S.U.'s

ELECTRICAL UNITS 13

THE RATIONALIZED PRACTICAL SYSTEM OF UNITS

16. Advantages of the Rationalized System. — In the discussion
of the practical system it was pointed out that our present work-
ing units are decimal multiples of the corresponding units of the
electromagnetic system. In fixing these ratios the international
conferences have selected values in such a way that the electrical
quantities commonly measured are expressed by numbers of
ordinary magnitude. This, in reality, constitutes a mixed system
of units, and, as a result, many of the formula? used in every day
calculations contain factors such as 10~\ 10*, 10', etc. Again, a
system based upon the unit magnetic pole and the unit electric
charge as given in paragraphs 9 and 10, respectively, inevitably
leads to many formulae in which the factor 4t appears.

It has been pointed out by Perry^ and by Fessenden^ that by
properly choosing new units for magnetomotive force and field
strength , and by submerging the factor 4x in the arbitrary constants
defining the dielectric and magnetic properties of materials, that
these objectionable factors may be eliminated, and all that
Heaviside sought to accomplish by his "Rationalized System
of Units," realized. In an admirable paper entitled "A Digest of
the Relations between the Electrical Units and the Laws under-
lying the Units," Bennett^ has carried out the suggestions of
Perry and Fessenden and has developed a consistent series of
defining equations and working formulae in which the objection-
able factors are suppressed, and has clearly set forth the relations
between the units of the different systems. In this treatment, a
new unit of force, the "Dyne-seven" (equal to 10^ dynes) has
been introduced. The advantage of this unit is obvious, since,
when acting through one centimeter, it performs one joule of
work.

17. Definitions. 1. Unit Quantity of Electricity. — The method
followed here is similar to that of the electrostatic system in
that the unit of charge is taken as the fundamental unit , and its
magnitude is arrived at by an application of Coulomb's law,
namely,

Q1Q2

F =

kd^

» Perry, Electrician, vol. 27, 1891, p. 355.

» Fessenden, Electrical World, vol. 34, 1899, p. 901.

' Bull. 880, Univ. of Wisconsin.

14 ELECTRICITY AND MAGNETISM

Where Qi and Q2 are two charges of electricity placed d cm. apart.
The quantity A; is a constant depending upon the medium in
which the charges are placed, and for free space is arbitrarily

put equal to n y inu' If Qi = Q2 = 1 and d = 1, then F =

9 X 10*' dyne sevens. Accordingly the coulomb is that quan-
tity of electricity which repels a similar quantity at a distance of
one centimeter in a vacuum with a force of 9 X 10" dyne sevens.
The coulomb, as thus defined, is identical with that defined in the
practical system of Art. 12.

2. Permittivity of a Medium. — The factor k in the expression
for Coulomb's law is called the dielectric constant of the medium.

k
Since in many formulae the factor j- appears, it is expedient to

replace A; by a new medium constant defined by the relation

and Coulomb's law is then written

Q1Q2

F =

4Trp d^

The quantity p is called the permittivity of the medium, and for
free space has the numerical value

^ = S = i^lon = 8-84 X 10-

The relative permittivity of a substance is the ratio of the per-
mittivity of the substance to the permittivity of free space, and is
thus numerically equal to the dielectric constant or specific
inductance capacity as ordinarily defined.

3. Unit of Current; the Ampere. — A current of one ampere is
flowing in a circuit if the quantity passing any plane in the circuit
per second is one coulomb.

4. Unit of Potential Difference; the Volt. — A difference of poten-
tial of one volt exists between two points if the work required to
carry one coulomb from one point to the other is one joule.

5. Unit of Resistance; the Ohm. — A conductor has a resistance
of one ohm if a difference of potential between its terminals of
one volt maintains a current of one ampere.

6. Unit of Capacitance; the Farad. — A condenser has a capaci-
tance of one farad if a charge of one coulomb produces differences
of potential between its plates of one volt.

ELECTRICAL UNITS 15

7. Unit of Inductance; the Henry. — A coil has an inductance of
one henry if a current through it, changing at the rate of one
ampere per second, induces within it an E.M.F. of one volt.

8. Line of Magnetic Intensity. — By a line of magnetic intensity
or a Une of force in a magnetic field is meant any line which is
traced out by the center point of a small plane direction -finding
coil,^ as the coil is moved in the direction pointed out by its
normal axis. Such lines are always found to be closed loops,
which either link with electric currents or pass through
magnets.

9. Magnetic Flux Density. — The magnetic flux density, B, at
a point in a magnetic field is defined as a vector quantity whose
direction is the positive direction along the line of magnetic
intensity passing through the point, and whose magnitude is equal
to the force upon a straight wire one centimeter in length carrying
a current of one ampere, the direction of the wire making a right
angle with a Une of magnetic intensity through the point.

Unit of Flux Density. — The Weber per square centimeter. — If a
wire one centimeter in length carrying a current of one ampere,
in a direction at right angles to the lines of magnetic intensity is
acted upon by a force of one dyne seven, the flux density is one
weber per square centimeter. One weber per square centimeter
equals 10^ gauss.

10. Relation Between the Magnetic Flux Density and the Current
Causing the Field. — Experimental measurements show that at
any point in a field, free from iron, the value of the magnetic flux
density, B, is directly proportional to the value of the current
producing the field. For the special case of an annular ring
uniformly wound with a coil of N turns, carrying a current /,
experimental measurements show that the lines of magnetic inten-
sity are circles lying within the ring as illustrated in Fig. 57 and
that the value of the flux density, B, is uniform along each circle
and has the value

' A direction-finding coil is a small plane circular coil carrying a continuous
current. The coil is so mounted on gimbals that its normal axis is free to
take any direction. The normal axis is a line perpendicular to the plane of
the coil at its center. The positive direction along the normal axis is defined
to bear the same relation to the direction of the current around the coil that
the direction of advance of a right-hand screw bears to its direction of rota-
tion. This is called the right-hand screw convention.

16 ELECTRICITY AND MAGNETISM

in which L is the length of the circle, /u is a constant having the
value 1.257 X 10~* for all except ferromagnetic materials.

11. Permeability. — The constant n, which appears in the equa-
tion expressing the relation between the flux density and the
current, is called the permeability of the medium in which the
magnetic field is set up. It is a constant analagous to conduc-
tivity in the conducting field and to permittivity in the electric
field. This unit is called the weber per ampere turn per centi-
meter and is equal to 47r X 10~^ units of permeability as defined
in the unrationalized practical system.

12. Magnetic Intensity. — The defining equation of (10) may
be written in the form

The expression appears in so many calculations dealing with

magnetic fields that, for the sake of convenience, the name "mag-
netic intensity" or "strength of field" is given to it. It is seen
to be equal to the number of ampere turns per centimeter. This
unit of field strength is called the ampere turn per centimeter and

is equal to j_ gilberts per centimeter.

13. Magneto-motive Force. — The line integral SHdl for any

closed magnetic circuit is called the magneto-motive force for

that circuit. For the simple circuit of Fig. 57 we have J'Hdl =

HL = NI. The unit of magneto-motive force is the "Ampere

47r
Turn" and is equal to ^^ gilberts.

14. Reluctance, Ampere Turn per Weber. — A magnetic circuit
possesses a reluctance of one ampere turn per weber if a magneto-
motive force of one ampere turn produces a flux of one weber.

10'
One ampere turn per weber equals . oersteds.

CHAPTER II
GALVANOMETERS^

18. Description of a Galvanometer. — A galvanometer is an
instrument for the detection and measurement of very small
electric currents. Strictly speaking, when used merely for the
detection of an electric current, as, for example, in determining
the balance condition for a Wheatstone bridge or a ^
potentiometer, it should be called a galvanoscope,
and the term galvanometer restricted to the case in
which it is standardized and used for the accurate
measurement of currents. The fundamental principle
upon which all galvanometers operate is the reaction
between a current and a magnetic field, one of which b
is fixed and the other movable. There are two types ^-E
of instruments, named after their originators, and
known respectively as the Thomson and the
D'Arsonval types. ^Cj

19. Thomson Galvanometer. — The Thomson gal-
vanometer was invented by William Thomson (Lord
Kelvin) and was first used as a detecting instrument - IT w
in connection with the trans-Atlantic cable. It uses
fixed coils and moving magnets, the axes of which fig. 5.
are placed at right angles to the fields produced by Astatic
the current in the coils. A high sensitivity requires, system,
among other things, that the restoring torque on the
moving system should be as small as possible. This is accom-
plished by use of the so-called astatic system which is illustrated
in Fig. 5. A rigid rod, BC, usually a slender glass tube, is
suspended by a very fine quartz fibre AB. This rod carries
two systems of magnets NS placed with their planes accurately
parallel, but with polarities reversed. If the magnetic mom*^nts
of the two groups of magnets are equal, then when the system

* Laws, Electrical Measurements, chap. I.
Brooks and Poyser, Magnetism and Electricity, chap. XIX.
Hadley, Magnetism and Electricity, chap. XVI.
2 17

18 ELECTRICITY AND MAGNETISM

is placed in a uniform magnetic field, it will remain in any
position in which it is placed, since the torque on one group of
magnets is balanced by that on the other.

The fixed coils which carry the current to be measured are
wound in opposite directions so that the reactions of their fields
upon the magnets of the moving system give torques in the same
direction. By making the system very light, e.g., a few milli-
grams, and by using a very fine quartz fibre for suspension, it is
possible, with this type of instrument, to measure currents of the
order of 10~^^ amperes. Since the fields due to currents of such
magnitudes are very weak, slight gradients in the external field
produce relatively large differences in the torques upon the
upper and lower magnet systems, and unsteadiness of the zero
position results. Galvanometers of this type must, therefore, be
carefully shielded magnetically.

Magnetic shields ^ may be either spherical or cylindrical in
shape, but since no openings may be permitted without serious
reduction in effectiveness, the latter form is usually employed.
It has been found that if the iron is all concentrated in a single
cylindrical shell having an outside diameter five times that of the
inner, the effectiveness is 98 per cent of that of a shield having
an infinite thickness. Furthermore, for a given amount of iron,
the effectiveness is greatly increased by using several concentric
cylinders. For extreme sensitivity, Thomson galvanometers are
made very small, and the coils are often mounted in a solid iron
container made by splitting a soft iron rod longitudinally and
drilling small holes in each half to receive the coils.

20. D'Arsonval Galvanometer. — The D'Arsonval galvano-
meter consists of a fixed, permanent horse-shoe magnet and a
light coil suspended between the pole pieces by a fine phosphor-
bronze ribbon, the plane of the coil being parallel to the direction
of the field. The current is led to the coil by the supporting
ribbon and away by a helix of the same material attached at the
bottom. While this type of instrument cannot be made as
sensitive as the Thomson, it has the following special advantages :
(a) The deflections are but little affected by variations in the
external magnetic field ; (6) the instrument may face in any direc-
tion; (c) the moving system may be made aperiodic, thus avoiding
loss of time in waiting for it to come to rest. For these reasons,
except where extreme sensitivity is required, the D'Arsonval

1 Wills, Physical Review, vol. 24, 1907, p. 243.

GALVANOMETERS

galvanometer has practically replaced the Thomson for general
laboratory work.

There are two distinct purposes for which galvanometers are
used: (a) The measurement of small currents, and (6) the
measurement of small quantities of electricity, such as are
obtained by the discharge of condensers. When designed for the
first purpose, they are called "current galvanometers" and for
the second, "ballistic galvanometers."

til

Fig.

High sensitivity galvanometer with cover removed.

21. The Current Galvanometer. — Figure 6 shows a high
sensitivity current galvanometer manufactured by the Leeds and
Northrup Company. The permanent magnet is mounted in a
vertical position and is provided with pole tips shaped so as to
give a nearly cylindrical gap between them. Coaxial with this
gap is placed a cylinder of soft iron and the coil rotates in the
annular space thus formed. The suspension is carried on a rod
supported by a bracket from the magnet. A set screw permits
a vertical adjustment of the coil and the knurled head, which
projects through the top of the case, gives a rough adjustment for
zero position on the scale. A slow motion screw at the base of
the instrument gives the final zero setting The axis of the coil

20 ELECTRICITY AND MAGNETISM

is made to coincide with that of the gap by means of three level-
ing screws which support the instrument. These screws are
turned by heavy vulcanite nuts which give, at the same time,
good insulation from ground. The right hand screw at the top
operates an arresting device by means of which the weight of the
coil may be taken off the suspension when the instrument is
being moved. A cylindrical case, provided with a window to
pass light to and from the mirror, protects the system against air
currents.

22. Galvanometer Sensitivity. — If several galvanometers,
selected at random, are connected in series and a definite current
is sent through them, it will be found that there are marked
differences in the response made by the individual instruments.
Those showing greater responses are said to have higher sensi-
tivities. The indication of a galvanometer is usually read by
means of a beam of Ught reflected from a mirror, attached to the
moving system, on a fixed scale. Obviously, for a given motion
of the system, the indication will be proportional to the distance
from mirror to scale, and so it is customary, when comparing
galvanometers, to place the scale at a distance of one meter,
and to read the deflection in millimeters. The sensitivity of
galvanometers is defined in a number of ways among which the
following are the most common :

(a) Microampere Sensitivity. — This is defined as the deflection
in millimeters of a spot of light on a scale one meter from the
mirror when the deflecting current is one microampere.

(6) Microvolt Sensitivity. — By this is meant the deflection in
millimeters of a spot of light on a scale one meter from the mirror
when an E.M.F. of one microvolt is impressed across the
terminals of the galvanometer.

(c) Megohm Sensitivity. — By this is understood the number of
megohms which must be placed in the galvanometer circuit in
order that with an impressed E.M.F. of one volt there results a
deflection of one millimeter on the scale whose distance is one
meter.

The dependence of the sensitivities, as just defined, upon the
constants of the instrument and the relations between them may
be understood from the following considerations. It will be
assumed that the coil is rectangular in shape and that it is so
supported as to be capable of rotation about a vertical axis of
symmetry. It will also be assumed that the field is radial,

GALVANOMETERS 21

uniform and horizontal, as shown in Fig. 7. A field of this sort is
obtained by means of a cylindrical core between properly shaped
pole pieces, and has the advantage that, for a constant current
through the coil, the torque is independent of its angular position.

Let I be the length of the coil; 6 its width; n the number of
turns; and H the strength of the field in which it is placed.
Calling T the torque on the coil when the current flowing through
it is i, we have, if c,g,s units are used,

T = Hinlh (1)

The quantity nlh, that is, the product of the number of turns
and the area of the coil, is frequently called the "equivalent
winding surface." Designating this by E we have

T = HiE = a (2)

where C, equal to HE, is the torque for unit current and is called
the ''Dynamic Constant" for the instrument.

Fig. 7. — Diagram of moving coil galvanometer.

As the coil rotates, it twists the supporting metallic ribbon
which exerts an elastic counter torque proportional to the angle
of twist, and the coil takes an equilibrium position such that the
two torques balance each other. Designating by r the constant
of the suspension, that is, the restoring torque when it is twisted
through an angle of one radian, the angular deflection 6 for a
given current i satisfies the relation

Td = a (3)

Letting A equal the angular displacement resulting from unit
current we have

^ = ^ = ^ (4)

T T

A is the angular displacement in radians resulting from one
C.G.S. unit of current and is, accordingly, the current sensitivity
in C.G.S. units. The microampere sensitivity, as defined above,
may be obtained from eq. (4) in the following manner: For

22 ELECTRICITY AND MAGNETISM

small deflections, the angular displacement of the system is
proportional to the linear displacement of the spot of light along
the scale. Moreover, the angular displacement of the reflected
beam is twice that of the reflecting mirror. Accordingly, a
deflection A in radians is equivalent to a deflection 2,000il when
expressed as the deflection in millimeters of a spot of light along
a scale at a distance of one meter from the mirror.

Again, if the current is measured in microamperes instead of
C.G.S. units, it follows, since 1 microampere is 10~^ C.G.S. units,
that the right-hand member of eq. (4) must be divided by 10^ for
this case. Therefore, replacing A by its value S divided by 2,000,
where S is the deflection in millimeters due to one microampere,
we have

S = ~ X2X 10-'' (5)

This is the microampere sensitivity and is seen to be .0002 times
the ratio of the dynamic constant to the suspension constant.
It is easily seen that the megohm sensitivity defined above is
numerically equal to the microampere sensitivity just discussed.
For, if S is the deflection in millimeters due to one microampere,
then the current in amperes required for a deflection of milli-
meter is o^TTe- Let M be the megohm sensitivity; that is, the

number of megohms placed in series such that the deflection is 1
millimeter when the E.M.F. is 1 volt. By Ohms law,

* " SlO^ ^ MAO^ whence M = S (6)

To obtain the relation between the microampere and the micro-
volt sensitivity, let it be supposed that a difference of potential
of 1 microvolt is impressed across the galvanometer. The cur-
rent i in microamperes is given by

'• = s <^)

where R is the resistance of the galvanometer. The resulting
deflection V, or the microvolt sensitivity is, accordingly,

V = Si = ^ (10)

Thus the microvolt sensitivity is obtained by dividing the micro-
ampere sensitivity by the resistance of the coil.

23. Figure of Merit. — While the definitions of galvanometer
sensitivity given above are convenient for distinguishing the

GALVANOMETERS

properties of one galvanometer from another, they are not well
suited to the practical case in which the instrument is to be stand-
ardized and used for the measurement of currents. Here it is
simpler to use the relation

(11)

= ^™

WWVVNA-

. If d is the deflection in millimeters at a meter distance and i
is in amperes, F is called the "figure of merit" or simply the
"constant" of the galvanometer and is defined as the current in
amperes required to produce
a deflection of 1 millimeter
at a distance of 1 meter.
The smaller F, the greater is
the sensitivity of the instru-
ment.

To determine the figure of
merit of a galvanometer it is
merely necessary to pass
known currents through the
instrument and measure the
deflections they produce.
These currents may be sup-
pHed through a standardized
variable resistance by a cell
of known E.M.F., and computed by Ohm's law. Since, for
most galvanometers, the required current is very small, the
arrangement shown in Fig. 8 is generally employed. By making
P small, usually 10 or 100 ohms, and Q large, 1,000 or 10,000
ohms, only a small fraction of the E.M.F. of the cell is effective
in sending a current to the galvanometer G, and this current
may be still further reduced by making R large. If i2 + (? is
large in comparison to P, the fall of potential across P is

K-

Fig. 8. — Connection for figure of merit.

where E = E.M.F. of cell read by the voltmeter VM.
current i through the galvanometer is

• - e _ P ^

* R-\-G P + QR i-G

and the constant F is given by

PE 1

F =

iP-]-Q)iR-^G)d

(12)
The

(13)
(14)

24 ELECTRICITY AND MAGNETISM

Since, in no galvanometer, is the deflection strictly proportional
to current, it is necessary, in making a standardization, to use
currents giving deflections over the entire range for which the
instrument is to be used, determining from each a value of F
which, when plotted as ordinates against d, as abscissas, gives a
working curve showing F as a function of the deflections.

24. The Ballistic Galvanometer. — The ballistic galvanometer,
which may be of either the moving coil or the moving magnet
type, differs from the current galvanometer in that its moving
system has a large moment of inertia, giving it a long period of
vibration. If, while the system is at rest, a small quantity of
electricity, such as a condenser charge, is suddenly passed through
it, during the small interval of time that this electricity is flowing,
there will be a torque acting on the system. This torque must
be of very short duration as compared with the time required
for the complete swing of the instrument, and is called an impul-
sive torque. The system is thus given an angular velocity, and
an application of the laws of mechanics shows that the amplitude
of the first ballistic throw is a measure of the impulsive torque
applied, and hence of the quantity of electricity that has passed.
The ballistic galvanometer is, then, an integrating rather than an
indicating instrument. The rotational energy of the moving
system is consumed in two ways: (a) The air surrounding the
system is set in motion; (6) the relative motion of the coil and
magnet induces a current in the coil, if the circuit is closed. Since
the system is thus losing energy, each succeeding swing is less
than the preceding one, the instrument comes gradually to rest,
and the motion is said to be "damped." If the resistance across
the galvanometer terminals is very large, the system will make
several swings before coming to rest. If the resistance is small,
the system will not vibrate at all, but will come to rest slowly.
If, however, it is of the proper value, the motion may be just
aperiodic; that is, it will not swing past zero, but will return to
zero in the shortest time. The instrument is then said to be
"critically" damped, and the resistance required is called the
"critical resistance." In many instruments, the moving coil is
wound on a closed copper form in which currents are induced as
it swings, thus making it nearly aperiodic on open circuit.

25. Constant of a Ballistic Galvanometer. — A study of the
equation of motion of the ballistic galvanometer shows that, no
matter what the damping may be, whether zero or so great that

GALVANOMETERS 25

the motion is aperiodic, the first throw is proportional to the
quantity of electricity discharged through it, the only limitation
being that this discharge must take place before the system moves
appreciably from its zero position. If the throw is small, so
that the tangent is proportional to the angle, this fact may be
expressed thus

Q = Kd (15)

where Q is the quantity of electricity, d the deflection as read by
a mirror and scale, and K a constant depending upon the sensi-
tiveness of the instrument, numerically equal to the quantity
necessary to give unit deflection. The smaller K, the greater is
the sensitiveness of the instrument. If, then, K is known, we
have a means of measuring small quantities of electricity

26. Theory of the Undamped Ballistic Galvanometer. — It
will be assumed that the galvanometer is of the D' Arson val type
and that the field in which the coil moves is radial and uniform.
It will also be assumed that the duration of the discharge is short
compared to the time required for the first ballistic throw to
take place. The conditions under consideration, then, are these:
A small quantity of electricity, such as the charge of a condenser,
is passed through the coil. While the current is flowing, the
reaction between the current and the field produces a torque on
the coil which starts it rotating. Although the duration of this
torque is very short, the coil has, nevertheless, acquired a certain
kinetic energy, and its motion is opposed only by the counter
torque of the suspension, since we are neglecting damping. It
will continue to rotate until its energy has been transferred to
the suspension where it is stored as potential energy of elastic
deformation. The coil then starts swinging in the reverse direc-
tion and when it passes through its zero position, it again pos-
sesses the same kinetic energy that it had originally, and will
continue to oscillate indefinitely.

Let / be the moment of inertia of the coil, co its angular velo-
city, a its angular acceleration, and 6 its angular deflection at any
instant. As in the discussion of the current galvanometer, let C
be the coil constant, that is, the torque produced by unit current,
and let t be the suspension constant, that is, the counter torque
for a twist of one radian. At any instant during the discharge,
the equation of motion for the system is

a -Td = la (16)

26 ELECTRICITY AND MAGNETISM

Since we are assuming that the discharge takes place before the
coil swings appreciably from its zero position, the second term on
the left hand side may be neglected; and, writing for a its value,

-n^> we have

a = / If (17)

Let t' be the time required for the discharge to take place. Then

Carrying out this integration and letting w' be the angular velo-
city at the time t', we have

CQ = /«' (19)

Where Q is the quantity of electricity which passed through the
coil. The kinetic energy thus acquired by the coil is

Energy = y2l<.'' = y2^ (20)

If the coil swings through an angle di, the potential energy of

elastic deformation is

roi

w = T \ dde = yirdi^ (21)

Since this is equal to the initial kinetic energy of rotation,
there results

whence

Q = ^e^ (22)

Inasmuch as the quantities in the coefficient of 6i are not
readily determined, it is simpler to express this quantity in terms
of the figure of merit of the galvanometer and its period of oscilla-
tion T. Since the coil executes an angular harmonic motion, its
period is given by

r = 2.^i

(23)

Substituting from (23) and (11) in (22) there results
The constant K of eq. (15) is thus seen, for the undamped

Q = ^Fd (24)

GALVANOMETERS

ballistic galvanometer, to be ^ times its figure of merit when

used as a current measuring instrument.

The ideal condition, i.e., zero damping, cannot be realized in
practice. Moreover, it would be exceedingly cumbersome to
use, because of the difficulty in bringing the coil to zero and main-
taining it in this position while adjusting other parts of the appa-
ratus in preparation for an observation. Since a certain amount
of damping must necessarily be present, it is usually most con-
venient to increase the damping until the motion is just aperiodic.
In this case, the galvanometer deflects to a certain point and then
returns to zero in the quickest time; and, barring external dis-
turbances, remains in this position indefinitely.

The theory of the damped ballistic^ galvanometer is somewhat
involved and is beyond the scope of this book. It may be shown,
however, that when damping exists, the quantity of electricity
passed through is given by

e = r,^0+l)'^

(25)

where X is called the "logarithmic decrement" and is defined as
the Naperian logarithm of the
ratio of any deflection to the
next one succeeding it in the
same direction. It is thus
seen that damping reduces the
ballistic sensitivity of a gal-
vanometer. Further, if the
galvanometer is standardized
under conditions such that
the damping is different from
what it is in use, the decre-
ment must be determined in
both cases and the difference
allowed for by eq. (25).

27. Determination of the
Constant of a Ballistic Gal-
vanometer.— The standardi-

K "

Fig. 9. — Condenser and standard cell
method for obtaining constant of ballistic
galvanometer.

zation of a ballistic galvanometer consists in passing known
quantities of electricity through it and measuring the deflec-
1 O. M. Stewart, Phys. Rev., vol. XVI, 1903, p. 158.
Laws, Electrical Measurements, chap. II.

28 ELECTRICITY AND MAGNETISM

tions they produce. Two methods are in common use, known
respectively as the "condenser and standard cell method" and
the mutual inductance or "standard solenoid method."

1. The Condenser and Standard Cell Method. — This method
consists in charging a condenser of known capacity by means of
a standard cell, and then discharging this quantity through the
galvanometer. The apparatus is arranged as shown in Fig. 9,
where G is the galvanometer to be standardized, C a standard
condenser, K a charge and discharge key, and S a standard cell.
If V is the E. M. F. of the cell, the quantity stored in the con-
denser when the key is pressed down is

Q = CV (26)

and since

Q = Kd (27)

we have

K = ^ (28)

If C is a subdivided condenser, several different values should be
used, a curve plotted using Q as abscissas and d as ordinates, and
the constant computed from the slope of the straight line. If C
is expressed in farads, V in volts, and d in centimeters, K will be
given in coulombs per centimeter; but if C is in micro-farads, K
will be given in micro-coulombs per centimeter.

2. The Standard Solenoid Method. — This method is especially
applicable to cases in which the galvanometer is used on low
resistance circuits where the damping is large. The known
quantity of electricity discharged through the galvanometer is
obtained from the secondary of a standard mutual inductance
when a measured change in the primary current is produced.
The connections are shown in Fig. 10 where AD is the primary
of the mutual inductance, SS' the secondary coil, and G the
galvanometer to be calibrated.

Let Q = quantity of electricity discharged through the galvano-
meter.

i = instantaneous current in galvanometer.
e = instantaneous E.IVI.F. in secondary coil.
/ = value of primary current.
R = total resistance of secondary circuit.
M = mutual inductance between AD and SS.
T = time required for discharge to take place.

GALVANOMETERS
Then, from the above,

idt

But

Hence,

e dl

i = n a-nd e = ilf -r- from definition
K dt

-m^'-.i>-

MI
R

iC =

Mil

R d

(29)

(30)

(31)
(32)

nMTiJMmim

A.M.

Ill

-vwwww

Fig. 10. — Standard solenoid method for ballistic galvanometer constant.

If one of the coils is uniformly wound and has a length great in
comparison to its diameter, as the primary AD of Fig. 10, it is
called a standard solenoid. The mutual inductance may then
be calculated from the dimensions of the solenoid, and the
number of turns on the coils, as follows :

Let A = area of standard solenoid
L = length of standard solenoid
i\r = number of turns on standard solenoid
H = field strength in standard solenoid
<t> = total fluj^ in standard solenoid
n = turns on secondary of standard solenoid.

30 ELECTRICITY AND MAGNETISM

The coefficient of mutual inductance may be defined, in electro-
magnetic units, as the number of magnetic linkages through the
secondary when unit current is flowing in the primary, where, by
linkages is understood the product of the number of turns and the
total flux. As the secondary coil surrounds the standard solenoid ,
we have

M = n(t> = nHA (33)

= — J — electromagnetic units (34)

Since, however, we wish M expressed in henries, we must divide
by 10^, the number of E.M.U's. required for one henry. Accord-
ingly, our equation becomes

_ ^irNnA I

~ RLIO' d ^'^^^

It is customary to reverse the current through the primary of the
standard solenoid instead of merely "making" it as implied in the
above derivation. The limits of integration in equation (31)
should then be — / and +/ instead of 0 and I, in which case
our formula becomes

_ SttNuA I

In the above derivation, we have assumed that the field
strength at the center of the standard solenoid is given by the
formula

47riV7
H = -f^ (37)

which is true only for an infinitely long solenoid. If the length
of the standard solenoid is fifty times the diameter, the error,
which is due to the demagnetizing effects of the ends, is less than
one-haK of one per cent. We have further assumed that there is
no magnetic leakage between primary and secondary coils, a
condition which is never realized. Our value for M, computed
above, is, therefore, too large; and for very accurate work, a
correction should be made. If we call/ the demagnetization and
leakage factor, our corrected formula for K becomes

In practice, it is customary to obtain a series of deflections
using different values of I, then plot I as abscissas and d as

ordinates, and obtain the ratio -j- from the slope of the line. If

GALVANOMETERS 31

practical units of electrial quantities are used throughout, K will
be expressed in coulombs per centimeter.

28. The Fluxmeter. — It was pointed out above, as a necessary
condition that the ballistic galvanometer should give indications
proportional to the quantity of electricity passed through it, that
this passage must be completed before the moving system swings

Grassot Flux Meter.

appreciably from its zero position. In certain instances, as,
for example, the testing of iron possessing magnetic viscosity,
the induced current which is passed through the galvanometer
persists too long, and hence the ordinary instrument cannot be
used. The Grassot fluxmeter is a modified ballistic galvano-
meter of the moving coil type, in which this difficulty is over-
come. The coil is suspended by a fine silk fibre and is practically
free from restoring forces, the current being led in and out by

32 ELECTRICITY AND MAGNETISM

means of fine helical springs. It is rectangular in shape and is
placed in a field as nearly radial as possible, with respect to its
axis of rotation, the parts involved being similar to those of the
Weston ammeter. The torque, for a given current, is practi-
cally independent of the position of the coil. When connected
to a resistance equal to or less than its critical resistance, the
coil is stationary in any position. When a given quantity of
electricity is discharged through it, it moves to a new position and
the change in position is proportional to the quantity that
passed, no matter how long a time was required. It is standard-
ized and used as an ordinary ballistic galvanometer, except that
some means must be provided for bringing it back to its zero
position. Figure 11 shows the construction of an instrument of
this type.

29. Theory of the Fluxmeter.^ — As originally designed, the
fiuxmeter was intended as an instrument for the direct measure-
ment of magnetic flux density. For this purpose, coils are con-
structed which consist of a definite number of turns wound on a
plate of nonmagnetic material, the area of which must be care-
fully measured. These coils are made very thin so that they may
be inserted in a narrow air gap such as exists between the arma-
ture and pole pieces of a dynamo. The measurement of an un-
known flux density consists then in connecting the test coil by flex-
ible leads directly to the fluxmeter and placing it at right angles to
the flux to be measured. The instrument is brought to zero by
some suitable device. The test coil is then withdrawn from the
flux and the accompanying deflection of the instrument, multiplied
by its constant, measures directly the change in flux through the
test coil.

The direct proportionality between change of flux through the
coil and deflection of the instrument may be shown as follows:

Let <f) = flux through the exploring coil
N = number of turns in exploring coil
L = inductance of exploring and galvanometer coils
R = resistance of exploring and galvanometer coils
C = constant of galvanometer coil = Hnlb
I = moment of inertia of galvanometer coil

1 Laws, Electrical Measurements, p. 124.
M. E. Gbassot, Fluxmfetre, Journal de Physique, 4th series, vol. 3, 1904,
p. 696.

GALVANOMETERS 33

« = angular velocity of galvanometer coil
i = instantaneous current in galvanometer coil
6 = angular deflection due to change of flux

As the test coil is withdrawn from the flux, there is induced in

it an E.M.F. given by jr- This is opposed by the counter

di
E.M.F., L -r.' due to the inductance of the galvanometer coil, and

also by an E.M.F. Cw due to the motion of this coil through the
field of the instrument. Accordingly, the current at any instant
is

■.y d(l> J- di ^

The motion of the coil is given by

_ dw _ CN_ d^ _ CL di _C^ ,

^'' ~ ^ dt ~ R dt R dt R ^^"^

Integrating between the limits o and t, where t is the duration of
the change of flux and consequent motion of the galvanometer
coil, we have

^ mdt = 7 f^ ., + ^ {'^^dt +^- r.dt (41)

R Jodt Jodt R J dt Rjo

Remembering that at both limits the current and angular velocity
are each zero and that y w dt = 6, we have

' <^2 - <Ai = ^ ^ (42)

The change of flux through the test coil is thus seen to be
directly proportional to the angle 6 through which the coil rotates.
This deflection may be read either by a pointer or a mirror and
scale. The fluxmeter may be used for almost any purpose for
which the balUstic galvanometer is suited, but has, in general, a
somewhat lower sensitivity.

30. Checking Devices. — If a ballistic galvanometer is not
critically damped, it is convenient to have some device to check
its motion and to set it accurately at its zero position. If the
instrument is of the D'Arsonval type, this may usually be accom-
plished simply by a short circuiting key. However, since most
keys possess slight thermal E.M.F.'s, the zero with the key
closed will usually be different from the normal zero with the key
open. When the galvanometer is used on a closed circuit, the

ELECTRICITY AND MAGNETISM

device shown in Fig. 12 is much more satisfactory. It consists of
a coil of wire through which a bar magnet may be moved. The
coil is connected in series with the galvanometer and the motion of
the magnet induces in it a small E.M.F,, positive or negative,

Ballistic
Galvanometer

To Apparatus

Fig. 12. — Checking device for ballistic galvanometer.

depending upon the direction of motion. The key must remain
closed except when it is necessary to "get a new hold" on the
galvanometer. With a little experience the instrument may,
with this device, be set on zero very quickly and accurately.

CHAPTER III
MEASUREMENT OF RESISTANCE

31. Ohm's Law. — When a current of electricity is flowing from
one point to another along a conductor, a difference of potential
is found to exist between these points. The magnitude of the
difference of potential depends upon the current and upon a
property of the material in virtue of which it offers opposition to
the passage of current. The relation between potential difference
and current was first given by Ohm, and is known as Ohm's law.
It states that, as long as the physical condition of a conductor
remains unchanged, there is a constant ratio between the current
and potential difference; or, in symbols,

where the proportionality factor R is called the resistance of the
conductor. This law is a result of experiment and has been found
to be true within the limits of the most refined measurements,

32. Specific Resistance. — For a uniform conductor, other con-
ditions remaining the same, the resistance is proportional to the
length and inversely proportional to the area of cross section.
Hence, if I represents the length and a the cross section, we have

R = p[ (2)

where p is a constant depending upon the material of the conduc-
tor. Considering this as a defining equation for p, we see that,
when I and a are unity, p equals R. The constant p is thus the
resistance of a unit cube of the material, and is known as the
Specific Resistance, In tabulating the resistivities of substances,
the specific resistance is a convenient quantity to use, since
knowing it, one can readily compute the resistance of a conductor
of any length and cross section by means of eq. (2). The
value of p depends upon the units employed for the measurement
of length and resistance. Since the resistance of a unit cube of
any metal is a very small quantity, it is customary to express the
specific resistance in microhms per centimeter cube where a

36 ELECTRICITY AND MAGNETISM

microhm is one millionth of an ohm. Alloys, in general, have a
much higher specific resistance than pure metals, and the pres-
ence of even a trace of another metal which, of itself, may be a
good conductor, has a considerable effect upon the resistance; and
hence, copper, for electrical purposes, should be pure.

33. Temperature Coefficient of Resistance. — The resistance of
all conductors is found to change with the temperature. In the
case of the pure metals, the resistance increases with increasing
temperature, while for carbon and electrolytes, the opposite is
true. The former are said to have a positive, and the latter a
negative, temperature coefficient. Experiment shows that, over
relatively large intervals of temperature, the resistance of a given
conductor, at any temperature t, may be expressed by the
equation

Rt = Ro{l +at + ^t^+ ) (3)

where R^ is the resistance at zero degrees and a and (3 are constants
depending upon the material and the temperature interval con-
cerned. Over small ranges of temperature, the change in resis-
tance is nearly proportional to the change in temperature, and
may be represented by the linear relation

Rt = Ro{l+ at). (4)

The coefficient a is called the "Temperature Coefficient," and is
the change in resistance per ohm per degree change in tempera-
ture. Some alloys, such as german silver and manganin, have a
very small temperature coefficient, that of the latter being zero at
some temperatures. Manganin is well suited, for this reason,
for the construction of standard resistances.

34. Measurement of Resistance. — The independent or "abso-
lute" determination of resistance, that is, measurement in terms of
the fundamental units of length, mass, and time, is a matter of
considerable difficulty; and so the establishment of primary
standards is, at the present time, left almost entirely to govern-
ment Bureaus of Standards, which are especially equipped for work
of this character. On the other hand, the comparison of resis-
tances, even to a high degree of accuracy, is relatively simple, and
it is with work of this character only that we are concerned here.

35. The Wheatstone Bridge. — This is the usual method
employed for comparing resistances of ordinary magnitudes,
and its principle may be readily understood from Fig. 13. Four
resistances are connected in the form of a diamond, with current

MEASUREMENT OF RESISTANCE

from the battery entering at A, where it divides in two parts
which unite again at B. The galvanometer G is connected across
the other corners of the diamond.

Since the points P and Q possess potentials intermediate between
those of A and B, it must be possible to make Q have the same
potential as P by suitably choosing R3 and Rt. When this
condition has been established, no current flows through the
galvanometer, as indicated by zero deflection, and the bridge is

Fig. 13. — Wheatstone Bridge.

said to be balanced. Calling the current through ^1 and R^,
Ci, and that through 'R3 and R4, C2, we have the

P.D. between A and P = P.D. between A and Q and
P.D. between P and B = P.D. between Q and B.
By Ohm's law

RiCi = R,C2 (5)

and

RiCr = RiC2 (6)

Whence

Ri _ Rs
Ri R4

This is the law of the Wheatstone bridge, and it is clear that, if

three of these resistances are known the fourth may be computed.

36. The Slide Wire Bridge. — If, in the above equation, Ri

is an unknown and R2 a standard resistance, the former may be

(7)

ELECTRICITY AND MAGNETISM

expressed in terms of the latter by means of the ratio of Rz to Ri,
It is obvious then that the actual values of R3 and R^k need not be
known, their ratio being sufficient. Advantage is taken of this
fact in the construction of the slide wire bridge, which is shown
diagrammatically in Fig. 14 where the corresponding points of
Fig. 13 are indicated by the same letters. Rz and R/^ are replaced

-—t2 CO

Sid

I BIZ

Fig. 14. — Slide wire bridge.

by portions of the slide wire SW and their magnitudes varied by
moving the slider Q. Calling p the resistance of 1 cm. of the wire,
we have

R
whence

p[i

(8)

X = \'R

(9)

In order to increase the accuracy of setting, and to reduce the
relative errors in measuring h and h, especially where X and R
have quite different values, resistances are introduced in place
of the links M and N, which may be measured in terms of p and
expressed, therefore, as a certain number of slide wire units to be
added to h and U.

37. The Post-office Box. — A more compact form of Wheat-
stone bridge is shown in Fig. 15, which is known as the post-office
box, from the fact that it was adopted at an early date by the
British Post and Telegraph Office. The slide wire is replaced by
two series of ratio coils, AQ and BQ, having resistances of 10,
100, 1,000 and 10,000 ohms each, while the third arm is a series

MEASUREMENT OF RESISTANCE

of coils, arranged as in the ordinary resistance box, frequently-
having a total of 100,000 ohms. The unknown X is connected
between B and P. Since the ratio of X to 72 is thus a decimal
number, no calculation is required. With the ratio coils set at

, Q

K 0

0 9

C) C)

(!!) ol

^^>:

OOOO

1000

100

10 1

lOO lOOO

/ffOOO

tMK

Soc

20C

zoo

too

() Ol

o—

/ooo

gooo

2«oe

Sooo

/O0OO p

Fig. 15. — Post-Office box diagram.

1,000:1, resistances up to 100 megohms may be measured; while
with the ratio reversed, resistances of the order of .001 may be
detected. The range is thus great and its advantages are obvious.
In using the box bridge, one should first use a 1:1 ratio, setting
the coils at 100 ohms each, and obtain a rough balance, thus finding

Fig. 16. — Post-office box.

the order of magnitude of the unknown. He should then choose
such a ratio as will cause the balance setting of R to be as large
as possible. For example, suppose C is found to be of the order of
45 ohms. By using a ratio of 1 : 1,000 a balance may be obtained

ELECTRICITY AND MAGNETISM

at 45,638, let us say, giving, as the value of the unknown, 45,638;
while if a ratio of 1 :100 had been used, the result would have been
45 .64 . The higher ratio thus increases the accuracy. Box bridges
of the better class are provided with plugs for interchanging the
ratio arms, by means of which inequalities in the internal connec-
tions of the bridge may be eliminated, and a check obtained upon
the accuracy of the ratio coils. For accurate work, one should
reverse the battery terminals in each case and re-balance, thus
eliminating errors due to thermal and contact differences of
potential. A convenient form of post-office box is shown in
Fig. 16.

38. Measurement of Low Resistance.^ Kelvin's Double
Bridge. — For the measurement of extremely low resistances
such as that of a few feet of trolley wire, cable, bus-bars, etc.,
the Wheatstone bridge is unsuited for two reasons: First, when
the resistances to be compared are very low, the bridge becomes
insensitive; and second, some sort of connectors must be used for

joining the unknown to the
bridge, and these may have a
resistance comparable to that
to be measured. The Kelvin
double bridge avoids both of
these difficulties. The general
scheme of this circuit is shown
in Fig. 17, where X and S are
the unknown and standard re-
sistances, respectively, through
which a large current flows
which need not necessarily be
constant. There are four ratio
coils, a, h, c, and d, arranged in pairs, while the galvanometer is
connected at the points C and D, between each pair. By
properly adjusting the ratio coils, C and D may be brought to
the same potential, when no current flows through the galvano-
meter and the currents in X and S are equal. When the balance
has thus been obtained, let us call / the current through X and S,
7i that through a and c, and h that through b and d. Then, by
Ohm's law,

cli = XI -\- dh and ali = SI -h bh (10)

^ NoRTHRUP, Measurement of Resistance, chap. VI.
Laws, Electrical Measurements, chap. IV.

Fiu. 17. — Diiigriim for Kelvin's
double bridge.

MEASUREMENT OF RESISTANCE

Whence

XI = c7i - dh and SI = ah - bh (11)

XI = c(^h- ^U ) SI = a(/i - ^72) (12)

By the construction of the instrument,

d ^h
c a
which gives, on dividing equations (12),

f = a (1^)

which is the working formula for the instrument. One form of
this bridge devised by Leeds and Northrup, is shown in Fig. 18
where the points corresponding to those of the schematic diagram

Fig. 18. — Laboratory form of Kelvin's double bridge

of Fig. 17, are lettered similarly. The unknown is represented
as a heavy rod with potential taps at A and B, while the standard
consists of the bar MN and the coils with posts numbered 0-9.
Each of the coils, as well as the standard bar, has a resistance of
.01 ohms, and the resistance being used as the standard S, is
that between the slider F and the movable plug E. The standard
thus has a range of 0 to . 1 ohms by infinitesimal steps. The ratio
coils are situated to the right of the standard coils and are
connected to the galvanometer in different ways by means of the
plugs C and D. A little study of the connections will show that
three different ratios are possible; namely, 1:10, 1:1, and 10:1.
The plugs C and D must be placed opposite one another, since a
double ratio must be maintained as indicated by the equations;
that is,

- = - = - (14)

Sab ^^^

42 ELECTRICITY AND MAGNETISM

The resistance X is that portion of the rod between the points A
and B only. When the resistance to be determined is of some
other form than a rod, it must be provided with two sets of leads;
a heavy pair for the current, which are joined to the bridge at S
and T, and a light pair for the potential drop across it, joined at
/ and m. The bridge thus measures the resistance of the con-
ductor between the points to which the potential leads are
attached.

39. Experiment 1. Specific Resistance of Materials. — In
this experiment, the specific resistance of three metals, copper,
brass, and iron, is to be found. The metals are provided in the
form of rods, which are to be clamped in the bridge at S and T.
Make sure that good contact is obtained at A and B by polishing
the bars at those points with emery cloth. Use, as a current
supply, a ten-volt storage battery and include an ammeter and
a reversing switch in this circuit, and a press key in the galvano-
meter circuit. Operate the bridge on 3 amperes. Measure the
resistance of 50 cm. and 100 cm. lengths of each bar, reversing
the current at each setting to eliminate errors due to thermal and
contact potential differences within the instrument. Make at
least four balances for each length approaching the balance
point from both sides. Determine the diameter of the rod by
means of a micrometer gauge, taking the average of ten measure-
ments uniformly distributed over its length.

Report. — 1. Compute the specific resistance of each material
in michroms per centimeter cube.

2. Compare your results with the data given in one or two of
the standard tables of physical constants to be found on the
reference shelves. How do you account for the discrepancies?

40. Carey-Foster Method for Comparing Two Nearly Equal
Resistances. — A very accurate method for comparing two
resistances which are nearly equal to one another has been
devised by Carey Foster.^ It possesses the advantage that
errors arising from the resistance of leads within the bridge, as
well as those due to thermal and contact electromotive forces,
providing they remain constant, are automatically eliminated.
The wiring diagram is shown in Fig. 19. It consists of a slide
wire bridge in which Ri and R2 are ratio coils and A and B are
the resistances to be compared. Let Vi and r^ be the resistances
of the internal bridge leads between the battery and slide wire

1 Phil. Mag., May, 1884.

MEASUREMENT OF RESISTANCE

connections on each side. Then if h and mi are lengths of the
bridge wire at balance, we have

^1 ^ A + ri + ph
R2

Now

(15)

B + r2 + pwi

where p is the resistance of the slide wire per unit length

let A and B exchange places, and let h and ma be the correspond

ing lengths for a new balance. Then

Ri ^ A±^i +A
R2 A -\- r2 -\- prrii

(16)

I m

Fig. 19. — Wiring diagram for Carey-Foster bridge.

Equating the right hand members of equations (15) and (16) and
taking the resulting equation by addition, we have

A + ri + pZi + B + r2 + pmi

B + r2-\- prrii

B -\- ri -\- pU -\- A -\- r^ -\- pnii

(17)

A -{- r2 -{■ prrii

Since li -\- mx = U -\- m^, the numerators of these fractions are
equal; the denominators are therefore also equal, whence

B -\- r2 -\- prrii = A + r2 + pm2
A - B = p(mi - ma) = p(h - h) (18)

The difference in the resistance of the two coils, A and B, is thus
seen to be equal to the resistance of the slide wire between the
two points of balance, before and after the interchange of the
coils. It is to be noted that this result is independent of
the values of Ri and R2

ELECTRICITY AND MAGNETISM

The Carey-Foster bridge is usually employed for the com-
parison of coils whose temperature must be maintained constant
and they are usually immersed in some sort of oil bath for this

Fig. 20. — Coil interchanger for Carey-Foster bridge.

purpose. A convenient device therefore must be provided for
interchanging them without removing them from their baths or
producing any changes in contact resistances by handling them.

Fio. 21. — Complete Carey-Foster bridge.

Figure 20 shows an arrangement for this purpose. The coils are
supported at the ends of heavy copper bars which swing so as to
receive units of different sizes. Contact between resistance

MEASUREMENT OF RESISTANCE 45

terminals and bars as well as between bars and links of the com-
mutator are made by boring cups in the bars and partially filling
them with mercury. The interchange of the coils is effected by
a half turn of the commutator at the center. Adjustable legs
enable the coils to be lowered in the baths to the proper depth.

To adapt the bridge to the comparison of coils of high as well
as low resistances and to secure at the same time a satisfactory
sensitivity, it is important to have several shde wires of different
resistances per unit length. Figure 21 shows a complete bridge
in which any one of three slide wires may be used at will. To
obtain the effect of a very low resistance slide wire, one of ordi-
nary magnitude may be shunted. A link, seen at the front of the
switch board is provided for this purpose.

41. Determination of p. — The Carey-Foster method requires
that the slide wire be of uniform resistance, and that its resistance
per unit length be accurately known. To measure p, the process
of measurement above described may be inverted, using for A
and B two equal coils of known resistance, one of which is shunted
by a known variable resistance. By choosing appropriate
values for the shunt, any desired difference between A and B may
be obtained, and by changing Ri and R2 the balance points may
be shifted to different positions along the slide wire. The con-
stant p is obtained by substituting in eq. (18).

42. Experiment 2. Measurement of Temperature Coefficient. —
The Carey-Foster method is particularly well adapted to the
measurement of the variation of a resistance with temperature.
The process consists in determining the difference between the
resistances of two coils one of which is constant while the other is
changed by holding it at different temperatures. The metal,
whose coefficient is to be measured, is in the form of a wire
wound on a frame which may be placed in an oil bath to secure a
uniform temperature throughout. Place the container in an ice
pack and measure the resistance at a temperature as near as
possible to 0°C. Next place the container in a water bath
heated by an electric heater. Obtain the resistance at 10°
intervals up to 80° C. While each measurement is in progress,
remove the heater and place the bath upon a wooden
stand. Stir the oil continuously and read the thermometer fre-
quently. Settings should be made as rapidly as possible to avoid
temperature changes. After the highest temperature has been
reached, allow the bath to cool and check the readings at three

40 ELECTRICITY AND MAGNETISM

points on the way down. The standard resistance should also
be placed in an oil bath and its temperature maintained constant.

Report. — 1. Plot a resistance temperature curve using resis-
tance as ordinates and temperature as abscissas. Draw a
straight line through these points to strike an average, and from
it determine the values for Rq and R%i,. Compute the tempera-
ture coefficient a from the equation

Rt = 72o(l + oLt)

2. Consult a table of physical constants and see if you can
identify the wire tested from the value of the temperature coeffi-
cient obtained.

43. The Measurement of High Resistance. ^ — In previous
sections, methods for measuring resistances of ordinary magni-
tude and for very small resistances have been considered. The
measurement of very high resistances, such as the insulation
between the bus-bars of a switch board and the ground, the
armature bars and core of an electrical machine, insulation of
cables, etc., requires special consideration. A ready method
commonly employed by engineers, which gives reliable results
for resistances up to several tenths of a megohm, and even
higher, is that in which a voltmeter of known resistance is em-
ployed, the unknown high resistance taking the place of the multi-
plier in the ordinary use of the instrument. Suppose a
voltmeter, of resistance r, is connected across a source of E.M.F.,
and the voltage, which we will call V, is measured. Then let
an unknown resistance R be connected in series with the instru-
ment across the same source. Since the voltmeter measures
the fall of potential across its own internal resistance, which we
will call Vr, while the total voltage across R and r is that origin-
ally measured, i.e., V, we may write, by Ohm's law,

V r

i = rU ^''^

- ^ (20)

Or,
Whence

« = rOL^^. (21)

^ NoRTHRUP, Measurement of Resistance, chap. VIII.
Carhart and Patterson, Electrical Measurements, p. 92.
Gray's, Absolute Measurements in Electricity and Magnetism, p. 253.

MEASUREMENT OF RESISTANCE

If the resistance R is too large, Vr will be insignificant com-
pared with V, and the method evidently will not yield satisfactory
results. For resistances of such magnitude, e.g., several meg-
ohms, recourse is generally taken to some leakage method in
which the high resistance is used as an insulator, and its magni-
tude estimated from the rate at which a known charge leaks
through it. As an example, consider the case in which it is

■AAAA/WW-^

I I I I I 1

Fig. 22. — Insulation resistance by leakage.

desired to measure the resistance of the insulation of a given
length of cable. The cable should be coiled up and placed in a
tank of water, both ends being left outside. This arrangement
may be considered a condenser, one plate of which is the water,
the other the wire, while the insulation is the dielectric. Its
electrical equivalent is shown in Fig. 22. If the wire and water
are charged to a given potential difference and the insulation
were perfect, the charge would remain constant; but, if the insu-
lation possesses a slight conductivity, the charge will gradually leak
through, reducing the potential difference of the condenser.
The rate of leak may be estimated by measuring the residual
charge after leakage has been going on for a definite time and
comparing it with the original charge. The resistance is then
calculated as follows :

48 ELECTRICITY AND MAGNETISM

Let C =

= capacity of the coil

Fo =

= applied voltage

R =

= insulation resistance in

ohms

V =

= instantaneous difference of potential

I --

= instantaneous leakage ^

current

Q-

= instantaneous charge

The charge Q

is given by

Q = CV

and

J dQ dV
^ dt dt

<M = o

Separating the variables

dV dt ^
V ^ CR

Integrating

log.F^J^ = X

(22)
(23)

(24)

(25)

(26)

where X is a constant of integration to be determined from the
initial conditions. For this purpose, reckoning time from the
instant when the leakage begins, the condition to be satisfied
by the equation is, when t = Q, V = Vo. Substituting these
values in eq. (26) , we have log Vo = K. Replacing K by this
value, we have

log. V + -^= log. F„ (27)

Thus,
Solving,

log. F. - loge F = ^ (28)

log.^ = ^ (29)

R = ^-

Clog.

Vo (30)

44. Experiment 3. Insulation Resistance hy Leakage. — Con-
nect the apparatus as shown in Fig. 22 where G is a ballistic
galvanometer, C the coil under test, B a storage battery, and K
a well insulated charge and discharge key. B should have such

MEASUREMENT OF RESISTANCE 49

a voltage that the first throw of the galvanometer is about 15
cms. Since this is a leakage experiment, its success depends upon
having all parts well insulated ; the tank should be placed upon a
glass plate or an insulated stand, and care be taken that no wires
touch each other, the table, or other apparatus. Since the
capacity of the coil does not change with time, the deflections
of the galvanometer are proportional to the voltage across its
terminals. Charge and discharge immediately thus obtaining a
deflection proportional to Vo- Repeat this several times and take
the average. Then charge and place the key on the point marked
"Insulate" and, after allowing 15 seconds for leakage, again
discharge and obtain a deflection proportional to V. Repeat
the operation for the following times of leak: 0.5, 1, 2, 5, 7, 10, 20,
30 minutes. If, in computing the resistance, common logarithms
are used, the modulus for changing to natural logarithms must be
introduced. If t is in seconds, and C in farads, R will be expressed
in ohms; but if C is in microfarads R will be in megohms. The
formula becomes

R = ' J (31)

2.303 C logio^

The capacitance C of the cable may be obtained from the relation

Qo = CFo = M^ (32)

where k, the constant of the galvanometer, is to be obtained by
charging a standard condenser with a standard cell and dis-
charging through the galvanometer, as explained in Art. 27.
Measure Vo by an ordinary voltmeter. Measure the resistance
of the wire of the cable by the voltmeter-ammeter method, using
for this purpose about 20 amperes. Determine also the diameter
of the wire by means of a micrometer gauge.

Report. — 1. Compute the resistance of the cable for each
time of leak from eq. 31, and plot the insulation resistance in
megohms as ordinates, and time of leak as abscissas. It will
be found that the resistance is not constant but increases with
the time during which it was subjected to a voltage, approaching
assymptotically to a limiting value. This is characteristic of
all insulators of this class, and, in stating their resistances,
the time for which it was determined must always be specified.

2. From the data on the resistance and diameter of the wire,
find, by means of a wire table, the length of the cable, and com-

50 ELECTRICITY AND MAGNETISM

pute the insulation resistance per mile for some selected time of
leak. In making this calculation, remember that the insulation
resistance is measured in the direction in which the leakage
current flows, namely, radially from the wire to the outside, and
that the longer the cable the less will be the total insulation
resistance,

46. The Internal Resistance of Cells. ^ — It is a well-known
experimental fact that when a cell is delivering current, the
E.M.F. across its terminals is not the same as on open circuit but
changes with the current, being less the larger the current. This
is true not only for cells, but for all electrical generators contain-
ing internal resistance. Let a cell, having an E.M.F. of E volts
and an internal resistance of r ohms, be connected to an external
resistance of R ohms, and let 7 be the amperes flowing; then the
rate at which energy is delivered by the cell is EI watts. Since
the current must flow, not only through the external resistance,
but also through the internal resistance of the cell, this energy
will be consumed by both of these resistances; I'^R watts in the
former, and /V watts in the latter. Accordingly we have

EI = {R-\- r)P (33)

E = RI -\-rI (34)

This is an equation of E.M.F. 's which states that the total
E.M.F. of the cell is equal to the external plus the internal
potential drops. Putting the terminal P.D. equal to E', we have

E - E' =rl (35)

from which it is seen that the internal resistance may be com-
puted if E, E', and I are known. In fact, this is the method
generally employed for cells which are able to furnish a consider-
able current without polarization; for example, storage cells.
Suppose such a cell, whose internal resistance is to be measured is
connected as shown in Fig. 23, where AM, R, and K are an
ammeter, rheostat, and key, respectively. Let the voltmeter
(V.M.) be an instrument taking no current, e.g., an electrostatic
voltmeter, having an infinite resistance. When K is open, the
voltmeter registers the total E.M.F. of the cell because there
is no fall of potential across r, as no current is flowing. When K
is closed, however, the voltmeter registers, not the total E.M.F.

^ NoRTHRUP, Measurement of Resistance, chap. XI.

Carhart and Patterson, Electrical Measurements, pp. 96-105.

MEASUREMENT OF RESISTANCE

as before, but the terminal F.D. E' = E — rl, the portion rl
being consumed in sending the current through r. Reading now
the current, the internal resistance may be computed from Eq.
35. In practice, an ordinary Weston voltmeter may be used
without appreciable error since the resistance of the voltmeter
is very large compared with that of the cell, and the ir drop
within the cell, which is the quantity by which the indications
of the instrument differ from the total E.M.F., is so small that it
may be neglected, i being the current taken by the voltmeter.

-A/WWWWW\

Fig. 23. — Voltmeter-ammeter method for internal resistance of cells.

If, however, the cell is one that polarizes rapidly, this method
cannot be used, since E and E' will depend upon how long the
current has been flowing. This difficulty may be overcome by
using a known resistance R and taking the voltages so quickly
that Httle or no polarization sets in. The current / is given by

= ^ = K

R+r r' (36)

Substituting either of these values for /, preferably the latter, in

eq. (35), we have

E - E' = E'

and solving for r, we have

= R

E - E'
E'

(37)

(38)

Since the right-hand member of this equation contains a ratio of
voltages, it is not necessary to know actual values, relative
values being sufficient; hence, any device giving indications

ELECTRICITY AND MAGNETISM

proportional to the voltage may be used in place of the volt-
meter; for example, a condenser and ballistic galvanometer.
46. Condenser and Ballistic Galvanometer Method. — The
basis for this method is that the first throw of the galvanometer
is proportional to the quantity of electricity discharged through
it, and that the charge of a condenser is proportional to the
potential difference across its terminals; that is

Q = CE

where C is the capacity of the condenser. Accordingly, if the
voltages E and E' are used to charge the condenser, and these
charges are then passed through the ballistic galvanometer, the
deflections are proportional to the voltages; that is

Q ^ Kid = CE
or

E = kid (39)

Substituting in (38), we have

_ R(E - E') _ {d - d')
^ ~ E' ~ ^ d'

(40)

47. Experiment 4. Internal Resistance of a Cell by Condenser
Method. — Connect the apparatus as shown in Fig. 24, where

B is the cell to be tested, C a con-
denser, G a ballistic galvanometer,
K2 a charge and discharge key, and
R a known variable resistance. First,
with Ki open, press down K2 thus
charging the condenser to the total
E.M.F. of the cell, and discharge by
allowing K2 to rise, obtaining a deflec-
tion proportional to E. Take several
readings in this manner and average.
Then, having set R at a suitable
value, close Ki, charge and discharge
as above, opening Ki as quickly as
possible to avoid polarizing the cell.
The average of several readings taken in this manner measures
E', whence r may be computed. It is well first to practice
operations upon a cell other than the one to be tested, in order
to become expert in manipulating the keys quickly and in
their proper order. In carrying out this experiment, use the

A/wwwyv\

Fig. 24. — Condenser method for
internal resistance of a cell.

MEASUREMENT OF RESISTANCE 53

following values for R: 10, 7, 5, 4, 3, 2, 1, .5 and .2 ohms. The
current from the cell is given by —

where E is the total E.M.F. To obtain E, it is necessary to
determine the voltage constant of the condenser and galvano-
meter system, which is, in reality, the constant of eq. 39. In
other words, we must measure the voltage required to give unit
deflection. For this purpose replace B by a standard cell and,
with Ki open, charge and discharge several times. Substitute in
eq. (39) and solve for kz.

Report. — 1. Compute the internal resistance for each differ-
ent current drawn from the cell and plot the former as ordinates
against the latter as abscissas.

2. How do you account for the fact that the internal resistance
is not constant?

48. Battery Test. — When a primary battery is furnishing
current, it polarizes; that is, hydrogen, which is one of the products
of the reactions going on within, collects on the positive plate.
This, together with other causes, diminishes the activity of the
cell. Indeed, the polarization may become so great as to cause
the E.M.F. to fall to zero. A chemical, called the depolarizer,
is introduced to remove the hydrogen, or to prevent its being
formed. Cells intended for open circuit work contain a depolariz-
ing agent that acts very slowly; thus they polarize rapidly if left
on closed circuit, but recover if left for a time on open circuit.
Cells intended for closed circuit work should polarize very
little, thus the depolarizing agent should act quickly. The
deterioration of a cell, when left on open circuit, due to local action
within the cell, is important, but can best be found by actual use,
since it takes too long to test this in the laboratory. We might also
run an efficiency test by working the cell to exhaustion ; but this,
too, is better found by actual use. What we are interested in,
however, is the behavior of the cell when run on a closed circuit
for a given time as the value of a cell is determined by the rate
of its polarization and recovery as well as by its E.M.F. and
internal resistance.

49. Experiment 5. Battery Test.^ — Study in this experiment
the time variation of: (a) Total E.M.F. on open circuit, (6) the
terminal potential difference on closed circuit, (c) the internal

1 Carhart, Primary Batteries.

54 ELECTRJCITY AND MAGNETISM

resistance, (d) the current, and (e) the rate of recovery from
polarization. The set-up is the same as in Exp. 4 and all
quantities are to be measured by the methods there outUned.
The difference here is that the key Ki is left closed all the time
except for an instant when it is opened to charge the condenser
for measuring the total E.M.F. As above, obtain the readings
for the E.M.F. and terminal potential difference for the initial
condition of the cell. Now close Ki, and at the end of a minuto,
charge the condenser and discharge it through the galvanometer,
thus obtaining the value of the terminal potential difference on
closed circuit after the cell has been delivering current for one
minute. As soon as this is done, open Ki for an instant and
charge the condenser; then set K2 on "insulate" and again close
Ki. This key, Ki, must be opened and closed quickly, also
these two readings, i.e., for total E.M.F. and terminal P.D.,
taken as nearly simultaneously as possible. As soon as the
galvanometer comes to rest, discharge the condenser through it,
obtaining a measure of the E.M.F. on open circuit after the cell
had been furnishing current for one minute. This will give a
measure of the polarization. Repeat these readings every minute
for five minutes and then every five minutes for twenty-
five minutes more. At the end of this time, open Ki and
measure the E.M.F. of the cell as it recovers for another thirty
minutes. As above, take readings at first every minute and then
at intervals of five minutes. Find out from the instructor what
resistance to use for R. Read Exp. 4 before attempting this one.
Practice with another cell as there suggested.

Report. — 1. Compute the total E.M.F. and terminal potential
difference in volts, and internal resistance in ohms.

2. Plot on one sheet, with time as abscissas, the total E.M.F.,
the terminal potential difference, the internal resistance, and the
current as ordinates.

3. Plot also on the same sheet the recovery curve, starting
at the other end of the time axis, running the curve backwards.

CHAPTER IV

MEASUREMENT OF POTENTIAL DIFFERENCE

60. Description of a Potentiometer.' — There is, perhaps, no
single electrical instrument which has so wide a field of usefulness
and which gives, at the same time, such trustworthy results as
the potentiometer. While comparing potentials primarily, it
may, with proper accessories, be adapted to compare currents and
resistances as well, and is so easy to manipulate as to be an
effective instrument even in the hands of a novice. The funda-

+ I-

FiG. 25. — Simple Potentiometer circuit.

mental principle of the potentiometer may be illustrated by Fig.
25, where MN is a wire of uniform resistance, stretched along a
scale with equal divisions and supplied with current from a
battery B, whose E.M.F. must be larger than those to be com-
pared. If the polarity of B is as shown, M will be at a higher
potential than N, and the fall of potential per unit length will be
the same all along the wire. If the difference of potential between
M and N is known, the wire may be regarded as a potential measur-

>Law8, Electrical Measurements, p. 271.
Electrical Meterman's Handbook, p. 208.
Kakapetoff, Experimental Engineering, p. 74.

66 ELECTRICITY AND MAGNETISM

ing rod. To measure an unknown E.M.F., such as the battery X
of the figure, an auxihary circuit MXL is provided, containing a
galvanometer and key. If the battery X were temporarily re-
moved and a short circuiting wire substituded in its place, a por-
tion of the current from B would flow in the shunt circuit from M
to L, causing a deflection of the galvanometer in a particular
direction. If, instead, the battery B were removed, X would cause
a current to flow in the direction XMLG, giving a reverse deflect-
ion. If, however, both batteries are included, and the slider L is
adjusted until the / R drop in the wire due to the current from B is
exactly equal to the E.M.F. of X, no current will flow in the shunt,
indicated by zero deflection of the galvanometer. The current in
the circuit BNM is then just the same as though the shunt were
disconnected. If the potential drop per unit length of the slide
wire is known, X may be directly determined, for we have

X = plU, (1)

where p is the resistance per unit length, and ?i the length required
for balance, pi may be determined by substituting for X a cell
S of known E.M.F. and balancing as before. Let Z2 be the length
required for this balance. Then

S = plh (2)

Whence

pJ = f (3)

and

X = ^^^ (4)

The unknown E.M.F. is thus obtained in terms of *S by a direct
comparison of the lengths li and h. If the fall of potential per
unit length of wire were some decimal fraction of a volt, the
unknown X could be read from the slide wire directly, thus avoid-
ing the calculation indicated. The method of accomplishing
this may be illustrated by the following example: Suppose the
slide wire MN contains 200 divisions, and the fall of potential
between M an iV is 2 volts. The fall of potential per division
is then .01 volt. Let the standard cell have an E.M.F. of
1.0185 volts. Set the slider at 101.85 divisions, include /S in the
shunt circuit, and obtain a balance, not by moving the slider,
but by varying the control resistance C, thereby changing the

current 7. When a balance has been secured, pi = '^-. ^-

MEASUREMENT OF POTENTIAL

.01 volt per division. The potentiometer is now standardized.
Substituted forS and balance by moving L, leaving C unchanged.
If this reading should be 145.63 divisions, the E.M.F. of the
unknown cell would be 1.4563 volts. When used in this manner,
the instrument is said to be a "Direct Reading Potentiometer."
To carry out comparisons with great accuracy, a very long wire,
having a high degree of uniformity, is obviously necessary.
Since such a wire is difficult to obtain, and inconvenient to use, it
is customary to substitute for it two resistances, as shown in
Fig. 26. If the sum of Ri and R2 is kept constant, they together

AAAAAAA

AMAAAAA

+. -

i—i n s

6 6
O O

MNW\f\M-

l^.'

H.lt.

Fig. 26. — Potentiometer constructed from resistance boxes.

are equivalent to a wire of fixed length, and an increase in Ry
accompanied by an equal decrease in R2 is equivalent to moving
the slider of Fig. 25 to the right, while an increase in Rz and a
decrease in Ri moves it to the left. Balances may easily be ob-
tained, the conditions for which are the same as outlined above.
For example, when a balance has been obtained with X in circuit,
we have

X = R,i. (5)

where Ri is the resistance required for balance and i, the current
flowing through the potentiometer, i.e., through CR1R2. This
current, which may be obtained by balancing against the stand-
ard cell S, is given by

S = Ri'i or i = J- (6)

a 1

58 ELECTRICITY AND MAGNETISM

where R'l is the vahic required for balance in this case. Whence,

X = 1^ ^ (7)

It must constantly be borne in mind that the above relations
require that ? should remain constant during the entire process,
which will be true only when the sum of the resistances, i2i+
^2 = C is unchanged, and the E.M.F. of B is constant. This
arrangement may be made "Direct Reading" if i is a known dec-
imal fraction of an ampere. This may be accomplished by giving
to R'l a value having the same significant figures as the E.M.F.
of the standard cell, and balancing by varying C. For example,
suppose, as above, S = 1.0185 volts and the boxes used have
resistances in the neighborhood of 20,000 ohms. If R'l = 10,185
ohms, when a balance has been reached

. _ 1.0185 _ 1

' ~ 10,185 ~ 10,000 ^"^P^^^'

and the fall of potential across each ohm is ^^ ^^^ of a volt.

Replacing now *S by X and balancing again, leaving C undisturbed
and keeping Ri -\- R2 constant,

^ ^ io;ra ^^^

51. Standard Potentiometers. — In order to avoid the necessity
of providing two exactly similar boxes, making the various con-
nections as explained above, and transferring plugs from one to
the other, it is convenient to have a single instrument, including
all resistances, switches, keys, etc., provided with binding posts,
to which the various E.M.F.'s may be connected. A number of
such potentiometers are on the market, three of which will be
described.

1. The Leeds and Northrup Potentiometer. — The arrangement
of this circuit, which is the simplest of those to be studied, is
shown diagrammatically in Fig. 27, in which the letters corre-
spond, as far as possible, to those used in Fig. 26. The potentio-
meter circuit proper, BNMC, consists of 16 coils of 5 ohms each,
and a long slide wire NO, also of 5 ohms. This circuit, in normal
operation, carries one fiftieth of an ampere, giving across each
coil as well as the slide wire, one-tenth of a volt fall of potential.
The circuit, containing the unknown potential is included between
the movable contacts, T and L. The box Ri of the previous

MEASUREMENT OF POTENTIAL

diagram is that part of the circuit lying between T and L,
while Ri consists of two parts, namely, the right-hand portion of
the slide wire, and the resistance to the left of T. The slide
wire, shown in the figure by a single turn, in reality consists of
ten turns wound on a marble cylinder and is about 17 feet in
length. The fall of potential across each turn is thus .01 volt,
and, as the dial circle is divided into 200 parts, the instrument

StdCIii

Fig. 27. — Diagram of Leeds and Northrup potentiometer.

reads to directly .00005 volt. By moving the slider L and the con-
tact T, the difference of potential may be varied by infinitesimal
changes from 0 to 1.6 volts.

The rangG of the instrument, for small electromotive forces,
such as those furnished by thermo-couples, is increased tenfold
by means of a shunt. As seen from the diagram, when the plug P
is inserted in the hole marked .1, the shunt S, which contains a
resistance of one-ninth that of the potentiometer proper, is con-
nected across the entire circuit, so that only one-tenth of the
normal current flows through the potentiometer proper. In
order that the current from the battery B may remain unchanged a
resistance K is automatically included, thus keeping the resis-
tance of the entire circuit the same. A ready means of standard-
izing the potentiometer current is furnished by the extra dial
DE, containing 19 coils of such a resistance, that, with the

ELECTRICITY AND MAGNETISM

normal current flowing, the fall of potential across each is .0001
volt. From the .6 post of the tenth volt dial, a permanent lead
is brought out, and connected, when the selecting switches are
thrown toward the left, through the galvanometer and standard
cell to the contact F. The fall of potential from the .6 plug to
M is 1 volt; from M through the resistance to £' is ,018 volt, and
from E to F &s many ten-thousandths of a volt additional as may
be required to equal the E.M.F. of any normal Weston cell
within the ordinary range of temperature (1.0180-1.0204 volts).

Fig. 28. — Leeds and Northrup potentiometer.

It is thus possible to check the potentiometer current without
re-setting the instrument. The operation then is as follows:
Set the standard cell dial to correspond to the E.M.F. of the cell,
corrected for temperature. Move the selecting switch to the
left, set P in the hole marked 1, and vary the control resistance
C (usually mounted at the right-hand end of the instrument)
until a balance is obtained. One-fiftieth of an ampere, the nor-
mal current is now flowing. Move the selecting switch to the
right, thus including the unknown E.M.F., and vary T and L
until the balance is once more obtained, when the unknown may
be read directly. If it is less than .15 volt, sot P in the .1 hole,
and balance again, when the reading of the instrument must be
divided by 10. The complete instrument is shown in Fig. 28.

2. The Wolff Potentiometer. — The fundamental principle of
this instrument is shown by the simplified connections of Fig.
29. The result which must be secured by any arrangement is

MEASUREMENT OF POTENTIAL

that the resistance of the potentiometer circuit proper, namely,
MN, must be kept constant, while the resistance across which
the auxiliary circuit FL is connected, must be continuously
variable. By moving 7^ and L, changes of 1 ,000 and 100 ohms,
respectively, are obtained, without changing the total resistance
as is at once obvious. The resistance coils connected by the
double sliders are sets with units of 10, 1, and .1 ohms respec-
tively. These double sliders are mounted so as to move together,
but are insulated from each other and connected in circuit as

N' (AVN^vww^ av^wvvwk jvwww^

/\r wvJvwwv' vwwvwsr vvvwwwv-

o o

l-J

©•

■wwwwvw— •

Fig. 29. — Principle of Wolff potentiometer.

shown in the diagram. If the pair at the left is moved one divi-
sion to the right, it-is evident that the resistance between N and
L is increased by 10 ohms, while that between F and L is decreased
by the same amount, thus leaving the resistance between M
and N unchanged. In the same way, the middle pair of sliders
produces changes of 1 ohm each between F and L leaving MN
unchanged, while the right-hand pair produces changes of .1
ohm each. Shifting any one of these sliders, therefore, is equi-
valent to moving the slider L of Fig. 25 by definite amounts.

The actual wiring of the instrument, mounting of the sliding
contacts, connections to accessories, switches, etc., are shown in
Fig. 30. The control resistance K is not included in the instru-
ment. Any ordinary resistance box capable of small variations
will serve for this purpose. The total resistance of the instru-
ment, as sketched, is 19,000 ohms. When carrying the normal
current of one ten-thousandth of an ampere, the difference of

ELECTRICITY AND MAGNETISM

potential across consecutive posts of the first dial is one-tenth
of a volt; of the second dial, one-hundredth of a volt; and of the
last dial, one hundred-thousandth of a volt, while the maximum
voltage directly measurable is one and nine-tenths volts.

In using this instrument, first obtain the E.M.F. of the stand-
ard cell, corrected for temperature, and set the small middle dial
of the upper row at this value. Set the switch in the upper left-
hand corner at NN , and the one in the right-hand corner, which
includes a high resistance in the galvanometer circuit, at its
largest value. Obtain a balance by varying the control resis-

FiG. 30. — Wiring diagram for Wolff i>otentiometer.

tance, cutting out the galvanometer resistance as a balance is
approached. This operation standardizes the current at one
ten-thousandth of an ampere. Now switch to XX and balance
by setting the large dials, when the unknown may be read off
directly. In checking the potentiometer current, which must
frequently be done, it is not necessary to change the dials
from their positions when balanced on the unknown E.M.F.

3. The Tinsley Potentiometer. — The working diagram for this
instrument, which is unique in that it employs an electrical
vernier, is shown in Fig. 31. Seventeen coils, with a resistance
of 5 ohms each, connected in series with a short slide wire of .5
ohm, form the potentiometer circuit proper MN, while the auxil-
iary circuit is FGL. Attached to a movable arm are two sliding
contacts, so spaced that they always rest upon two alternate
posts, leaving one post between them as indicated. This pair of
contacts is connected to a second series of 10 coils of 1 ohm each.

MEASUREMENT OF POTENTIAL

When the normal current of one-fiftieth of an ampere is flowing
through MN, the fall of potential between adjacent posts is .1
volt. However, the fall of potential between the posts connected
by the pair of contacts to the second series of coils is also .1 volt,
since the two coils of the main circuit are now shunted by a resis-
tance equal to their own, giving a resultant resistance between the
contacts equal to that of a single main circuit coil. Between
adjacent posts of the second series there is accordingly .01 volt
fall of potential, and across the slide wire there is also .01 volt
potential difference. This instrument, like the Leeds and North-
rup, is provided with separate connections for the standard cell,

i\r-<I>

Rheostat [

» leisj VoiTs ■

U^.t|U |.y| >4|.i I ■« I '. 1 1*! V I » |V| «s I Vh h 1 1 1 V I « I 5k!£iJ!

I l^l^lalTlalshhUlil.l "

p J .01 W»tT

'—-'—•I ©-J

Pot* ^-^

Fig. 31. — Principle of Tinsley potentiometer.

so that it is not necessary to re-set all of the sliders when checking
the current through the potentiometer circuit proper. A stand-
ard cell lead is permanently attached to post number 7. Across
the ten coils between it and post 17 there is, accordingly, 1 volt
potential difference,' and in series with the main circuit is another
coil shown at the left of 17, of such a value that, with the normal
current flowing, the fall of potential across it is .0183 volts, and to
the other side of this, the second standard cell terminal is attached.
Unhke the Leeds and Northrup instrument, this coil cannot be
varied to compensate for variations in the E.M.F. of the standard
cell due to temperature changes, but the value 1.0183 volts is
sufficiently accurate for ordinary purposes.

The wiring connections, switches, etc., are shown in Fig. 32.
The control rheostat is included in the instrument, and consists
of the dial in the right-hand corner and the slide wire immediately
above it. By moving the plug in the upper left-hand corner to
the hole marked X by .1, the instrument is shunted by a resis-
tance of such a value that all readings should be divided by ten, a

ELECTRICITY AND MAGNETISM

feature of great importance in thermo-couple work. In using
the instrument, set the shunt plug in the hole marked X by 1,
and the two-point switch below the middle dial on SC. The
first dial must be set so as to shunt none of the coils above the
seventh, otherwise, the resistance over which the standard cell is
to be balanced will be reduced effectively by one coil. A good
rule is to set this dial always at zero when balancing on the stand-
ard cell. Obtain a balance by changing first the rheostat and
then the slider above it, which is provided with a slow motion
screw for the final sotting. Use in this connection the key mark

Pot. Kev -S.C. Key

FiQ. 32. — Wiring diagram of Tinsley potentiometer.

SC. The current is now exactly one-fiftieth of an ampere and the
instrument is standardized. To measure the unknown, simply
move the two-point switch to Potl., and balance by setting the
two dials and the lower slide wire. If the unknown is less than .2
volt, use the shunt as explained above, dividing the reading by 10.
In carrying out any measurement, the current through the
instrument should be checked frequently.

52. The Weston Standard Cell.— While the legal definition
of electromotive force is given in terms of the standard current
and resistance by means of Ohm's law, nevertheless, in actual
practice, the volt is specified in terms of the standard cell. After
many years of investigation, the Weston standard cell has been
perfected to such an extent that persons in different parts of the

MEASUREMENT OF POTENTIAL

world, may, by following definite specifications, construct cells
of this type and be sure of securing E.M.F.'s which agree within
less than 1 part in 10,000. This cell is usually set up in an air-tight
H -shaped vessel, as shown in Fig. 33, with platinum wires sealed
through the bottoms for connection with the electrodes. The
positive electrode consists of pure mercury while the negative is
an amalgam of cadmium and mercury. These are placed in the
bottoms of the tubes, and a solution
of CdS04 with a few extra crystals to
insure saturation, forms the electro-
lyte between them. To protect the
mercury against contamination by
the CdS04 and at the same time
prevent polarization, a thick paste,
consisting mainly of mercurous sul-
phate , is placed over the mercury. As
the cell operates, the cadmium ions
from the CdS04 solution displace
some of the ions from the mercurous
sulphate paste and mercury is de-
posited upon the mercury electrode.

One of the advantages of this cell over former types is that its
electromotive force changes but very little with the temperature.
The electromotive force of a cell which has been set up with care is
given, with accuracy sufficient for most purposes, by the equation

Fig. 33. — Weston standard cell.

Ec = E20 - 0.0000406 (t - 20° C).

(9)

That is, the E.M.F. decreases 0.0000406 volt for each degree the
temperature is raised above 20° C, and increases by the
same amount for each degree below 20° C. This quantity is
called the temperature coefficient. Since standard cells are never
used as a source of current, but merely for balancing potentials or
charging condensers, they are made of small size. Those fur-
nished in the laboratory are mounted in a brass tube, with a hard
rubber top provided with binding posts and a hole through which
to insert a thermometer. The E.M.F. of the individual cells
is usually given at 20° C, from which the E.M.F. at the temper-
ature at which they are used may be computed by means of the
formula given above. When used in a potentiometer circuit,
a high resistance should be included and gradually cut out as a
balance is approached-

66 ELECTRICITY AND MAGNETISM

53. Experiment 6. Comparison of Cells by the Potentio-
meter. A. Simple Potentiometer. — Connect the apparatus, as
shown in Fig. 26, Art. 50, omitting the control resistance C,
and using for Ri and R2 two exactly similar boxes. B should be
a cell of constant E.M.F., preferably a portable storage battery.
Obtain from the instructor a standard and several unknown cells
whose E.M.F.'s are to be determined. The high resistance
marked H.R. need not be known accurately, since its purpose is
merely to protect the galvanometer and standard cell from
excessive currents when the potentiometer is far from balance.
It is well to start this at about 10,000 ohms, gradually reducing
it as a balance is approached. Be sure that the double pole
double throw switch for connecting S and X in circuit is not
provided with cross wires, as they would short circuit the cells.
To keep Ri + R2 constant, as required in the theory, start with
all the plugs out of Ri and all in R2, and obtain a balance by
transferring them from their places in one box to the correspond-
ing holes in the other. Ri + 7^2 will then always remain equal to
the total resistance of one box. To test whether the polarity of
the cells is properly arranged in the two circuits, first rock the
double pole double throw switch on X, break the potentiometer
circuit at B, tap the key K lightly, and note the direction of
swing of the galvanometer. Now close again the circuit at B,
remove the wires from the middle posts of the double pole double
throw switch, and join them. The galvanometer should swing
in the opposite direction on tapping the key. First, secure a
balance on X; then rock the switch over and balance on S,
afterwards checking your balance on X, to make sure that the
potentiometer current has not changed during the process.
Reverse the connections at B, also on the auxiliary circuit, and
proceed as before, taking the average of the two results thus
obtained. This is necessary to eliminate errors due to spurious
contact and thermal E.M.F.'s within the potentiometer.

B. Direct Reading Potentiometer. — Include in the potentio-
meter circuit the control resistance C, as shown in Fig. 26.
Determine the temperature of the standard cell and its E.M.F.
corrected for this temperature . Set R 1 to have the same significant
figures as this E.M.F. , using the largest multiple possible, and
put R2 equal to the difference between the total capacity of
one box and Ri. Switch S into the shunt circuit and balance by
varying C. Then rock over on to X and, leaving C fixed, balance

MEASUREMENT OF POTENTIAL

by plugging back and forth between Ri and R2, keeping their
sum constant. The reading of Ri, when properly pointed off,
gives X directly. After each balance on X, the setting on the
standard cell should be checked and C changed, if the current
has not remained constant, which, of course, necessitates a new
balance on X. Now reverse terminals as in Part A, and repeat,
taking the average of the two results.

Report. — 1. Give values of E.M.F. for all cells compared,
and where temperature corrections are known, reduce to 20° C.

2. Suppose a balance has been obtained without H.R. in
circuit. Now include H.R. How will the balance point be
affected? Why?

3. What is the maximum E.M.F. that may be measured
by the direct reading potentiometer, as you have used it in this
experiment?

64. The Volt Box. — In standard poterrtiometers, operated on
normal current, the maximum difference of potential which may

Fig. 34.— Volt box.

be measured directly never exceeds two volts and is usually even
less. When it is desired to measure voltages in excess of this
value, some means must be provided for accurately dividing the
unknown voltage into definite fractions of the total, small enough
to be measured by the potentiometer available. This may be

ELECTRICITY AND MAGNETISM

accomplished by means of the "volt box." This consists of an
accurately adjusted resistance box, of large range, in which the
blocks to which the coils are attached are provided with sockets
for receiving traveling plugs. The voltage to be divided is
impressed across the terminals and the fraction to be measured
is obtained across the traveling plugs, which may be set at any
points desired. By Ohm's law, the voltage across the traveling
plugs is such a fraction of the total voltage as the resistance
between the traveling plugs is of the total resistance. If the
resistance of the volt box is 10,000 ohms, the drop across 1,000
ohms is one-tenth of the total; that across 100 ohms, one-hun-
dredth of the total, and so on. It is simpler to use decimal ratios
wherever practicable. Special boxes are made in which these
ratios are obtained by setting a dial switch or a single plug as
shown in Fig. 34.

V^A\/V^A^^^AA^\^A/WVVVVV^r

<5)

^iiiiiiiiiih

Fig. 35. — Connections for standardizing a volt meter.

55. Experiment 7. Calibration of a Voltmeter by Potentio-
meter and Volt Box.^ — The method, in brief, consists in impress-
ing across the terminals of the voltmeter various voltages and
measuring these voltages by means of a potentiometer provided
with a volt box. The connections for this purpose are shown in
Fig. 35. VM is the voltmeter to be calibrated; LO the volt
box; RS a high resistance rheostat with a sliding contact for

1 Jansky, Electrical Meters, chap. V.
Kabapetopf, Experimental Engineering, vol. I, pp. 51-55.

MEASUREMENT OF POTENTIAL 09

voltage regulation, and B a .storage battery. P and T are the
terminals from the traveling plugs of the volt box which are to be
attached to the potentiometer. The voltage of B should be
sufficient to give full scale deflection of the instrument. Use
any one of the potentiometers described above, following the
directions given for each instrument. After the connections
with the potentiometer have been properly made and its current
adjusted by balancing against the standard cell, throw the
selecting switch to the point marked "unknown." Set the
slider of the rheostat RS so that the voltmeter indicates about
one-tenth full scale deflection, and choose the largest decimal
ratio of the volt box giving a voltage within the range of the
potentiometer. Measure this voltage with the potentiometer.
In a similar manner check the voltmeter at 8 or 10 points dis-
tributed uniformly across the scale. Test the constancy of the
potentiometer current frequently by re-balancing against the
standard cell. Record voltmeter readings, potentiometer set-
tings, and volt box ratios. Note carefully the zero reading of
the voltmeter before beginning the test and again at the end,
after it has been deflected for some time, to see if the springs
show any elastic fatigue. With about two-thirds full scale
deflection, place the instrument in a vertical position to test the
accuracy with which the moving system is balanced. Bring
another instrument near this one, and see if there is any effect
from external magnetic fields. Tap the instrument gently with
the finger to see if the bearing friction is large. Does the pointer
swing past its final p/Dsition when a voltage is suddenly thrown
on?

Report. — 1. Obtain the differences between the readings of
the instrument and true voltages, and plot these corrections as
ordinates against readings of the instrument as abscissas. Draw
in straight lines connecting these points.

2. State your findings regarding the imperfections of the
instrument.

3. Would it indicate on alternating voltages?

CHAPTER V

MEASUREMENT OF CURRENT

66. Kelvin's Balance. — This is an instrument for the measure-
ment of current in which use is made, not of the action between
the magnetic field of a current and a permanent magnet, as in
the case of galvanometers and ammeters, but of the action
between the fields of two currents. It consists of six flat coils
placed horizontally, four of which are fixed while the other two,
mounted at the ends of a beam pivoted at the middle, are movable.
The general arrangement is shown in Fig. 36. The current to be
measured passes through all six coils in series, flowing in each in

Fig. 36. — Arrangement of coils in Kelvin's balance.

such a direction that A and C both urge E downward, while B and
D urge F upward. The force of attraction or repulsion between
two coils is proportional to the current in each. Accord-
ingly, when the coils are connected in series the force between
them is proportional to the square of the current. Thus, the
electrodynamic action between the fixed and the movable coils
is such as tC produce a torque on the movable ones in the counter
clockwise direction proportional to the square of the current.
This torque is counterbalanced by a weight which slides along a
graduated beam attached to the moving system. An index at
each end shows when a balance has been reached. Since the

MEASUREMENT OF CURRENT 71

torque due to the current is proportional to the square of the
current, and that due to the weight is proportional to the weight
and the length of the lever arm, we have, as the condition for
equilibrium,

KP = WL (1)

where W is the weight of the slider, L its distance from the zero
position, and K a constant depending upon the construction of
the instrument. Solving

W
P = ^L (2)

or _

I = const. \^L (3)

The constant is generally so given that one must use the
doubled square root of the length L, and, to facilitate observa-
tions, tables of these quantities have been prepared. For rough
work, however, a fixed scale is mounted directly behind and a
little above the movable one, from which the doubled square
root may, with fair approximation, be read directly. Since the
constant depends upon the weight of the slider, a means is
afforded for changing the range. Four weights are usually
supplied for which the constants are 0.025, 0.05, 0.1, and 0.2,
giving ranges of 1.25, 2.5, 5, and 10 amperes, respectively,
since the movable scale has 625 divisions, giving a doubled square
root of 50.

As with an ordinary balance, the beam must be in equilibrium
for no load, that is, no current flowing through the coils. If the
index at the end does jiot read zero, equilibrium may be obtained
by moving a small metal flag attached to the moving system so
as to throw more of its weight to one side or the other, as is
required. A special device mounted on the base and operated
by a handle below the case is provided for this purpose. The
movable system is carried by flexible ligaments made up of a
number of fine phosphor-bronze ribbons placed side by side.
As these are delicate and easily broken, an arrestment is provided
which is operated by a milled head at the bottom of the case.
Weights should never be changed without first raising the
arrestment. Since the balance must be in equilibrium for zero
current, no matter which weight is used, there must be a separate
counterpoise for each. These consist of brass cylinders, provided
with a pin, which are placed in a small horizontal trough at the
right-hand end of the moving system, with one end of the pin

ELECTRICITY AND MAGNETISM

passing through the hole in the bottom of the trough. Since the
direction of the torque is independent of the direction of the

Fig. 37. — Kelvin's balance.

current, the instrument may be used either on direct or alternat-
ing currents, indicating in the latter case, root mean square
values. Figure 37 shows the usual laboratory
form of the Kelvin's balance.

57. The Siemens Electrod3mamometer. —
This is another current measuring instru-
ment working on the principle of the electro-
dynamic action between two coils carrying
currents. The coils are rectangular in form
and placed perpendicular to one another, as
shown in Fig. 38. The movable coil, CF,
which is placed outside the fixed coil AB, is
carried by a fine point resting in a jewel and
the current is led to and from it by wires
dipping into mercury cups at a and h, situated
one above the other in the axis of rotation.
One end of a helical spring S is attached to
the moving coil, while the other is fastened
to a milled head D carrying an index read
from a fixed circular scale. When a current
flows through the two coils in series, the
movable one tends to set itself parallel to the fixed, but is
brought back to its zero position by turning the head D, thus

Fig. 38. — Ar-
rangement of coils
in Siemens electro-
dynamometer.

MEASUREMENT OF CURRENT

twisting the spring. The torque due to the current is propor-
tional to the square of the current since the coils are in series,
while that due to the spring, by Hooke's Law, is proportional to
the angle through which it is twisted. Accordingly, we have, as
the condition for equilibrium,

where A is a constant depending upon the size of the coils, number
of turns, stiffness of spring, etc, and </>, the angle through which the
spring is twisted. The range of the instrument is changed by
varying the number of turns in
one of the coils. The instru-
ment usually has two fixed coils
with separate binding posts on
the base. Since the magnetic
field of these coils is small, that
of the earth is appreciable in
comparison and may introduce
an error. For example, if the
earth's field is in the same direc-
tion as that of the fixed coil, the
instrument will read too high,
while if the earth's field is oppo-
site, it will read too low. This
error may be eliminated by re-
versing the currents and taking
the average. Since the direction
of rotation of the movable coil is independent of the direction of
the current, the instrument will indicate on alternating currents
as well as direct, giving in the latter case, root mean square
values. It may accordingly be calibrated on direct and used
on either direct or alternating currents. Figure 39 shows the
usual form of Siemens electrodynamometer.

58. Experiment 8. Calibration of an Electrodynamometer.^ —
In this experiment, an electrodynamometer is to be calibrated
in terms of a Kelvin balance, which is taken as the standard
instrument. Connect the instruments in series on a 20-volt

1 Jansky, Electrical Meters, chap. VIII.
Cabhart and Patterson, Electrical Measurements, chap. III.

Fig. 39. — Siemen's electrodyna-
mometer.

74 ELECTRICITY AND MAGNETISM

storage battery, including a variable rheostat and an ammeter
to observe roughly the currents used. Both instruments must
first be leveled and adjusted for zero on no current. Begin with
the lowest range of the Kelvin balance. For this use the carriage
alone and the smallest counter weight. When the limit of this
range has been reached, raise the arrestment, open the case, and
push the carriage moving mechanism a little to one side bringing
it forward enough for clearance. Place the first additional weight
upon the carriage, and the second counter-poise in the trough.
Whenever new weights are put in position, the zero must be
rechecked. Measure in this way the currents for ten points on
the electrodynamometer, taking them a little closer at the
lower end of the scale. Record electrodynamometer readings,
Kelvin balance readings, and number of counterpoise.

Report. — 1. Compute the current for each setting of the
instrument, also the constant A in equation (5).

2. Plot current as ordinates and settings as abscissas. What
is the shape of this curve?

3. What is meant by the root mean square value of an
alternating current?

4. Name some other electrical instruments operating on the
principle of the electrodynamometer.

69. Ammeters and Voltmeters.^ — An ammeter, as the name
implies is an instrument for measuring the current flowing in a
circuit; while a voltmeter, measures the difference of potential or

Line (V.M.) Load
(A.MO

Fig. 40. — Connections for ammeter and voltmeter.

electrical pressure existing between two points Since the former
indicates, at any instant, the rate of flow of electricity through a
conductor, it must be placed in series with the circuit, so as to be
traversed by the entire current; while the latter, being a pressure
gauge, is connected in parallel with the circuit, and carries a very
small current, which in general may be neglected. The regular
method of connecting these instruments is shown in Fig. 40.

1 Jansky, Electrical Meters, chap. III.
Karapetoff, Experimental Electrical Engineering, vol. I, chap. II.
Electrical Meterman's Handbook, chap. V.

MEASUREMENT OF CURRENT

While many different kinds of indicating instruments are in
use, each having its particular field of application, those generally
employed in direct current work are of the "moving coil" type,
and are the only ones which will be considered here. The
working parts of instruments of this class are the same in both volt-
meters and ammeters, the differences between them being only
in the method of connection. The instrument proper is, in reality,
a low sensibility, portable D'Arsonval galvanometer, consisting
of a coil of fine wire, well-balanced, and pivoted between the poles

^ ''■'>'>''

'■ --

-'^"-J^.T^'h^^^

^^^^^^^■^

ilHi .

Fig. 41. — Working parts of Weston ammeter.

of a strong, permanent horse-shoe magnet. The magnetic flux
through the coil is increased by placing within it a cylindrical iron
core, while the air gap is further reduced by pole tips shaped in
such a manner as to make the field as nearly radial as possible,
with respect to the axis of the coil. In this way, the torque acting
upon the coil, when traversed by a current, is independent of its
angular position, the condition necessary for equal scale divisions.
The current is led to and from the coil by spiral springs, which
furnish also the opposing torque. The current sensibility of puch
an instrument is such that a few thousandths of an ampere, or
less, will give a full scale deflection; and since the resistance of the
instrument is low, a few millivolts across its terminals will furnish

ELECTRICITY AND MAGNETISM

this current. Plgure 41 shows the construction of a Weston
ammeter.

When it is desired to construct an ammeter, the instrument
G is provided with a shunt, S, as shown in Fig. 42. The shunt,
which carries the current to be measured, has a resistance (always
low) such that it gives, when carrying the maximum current for
which it is designed, a fall of potential across its terminals equal
to that required for full scale deflection of the instrument. For
example, suppose 50 millivolts are required for full scale deflec-
tion, and an ammeter reading to 25 amperes is desired; the resis-
tance of the shunt must be

R = •^^= .002 ohms

By the law of shunts, the current through the instrument (neg-
lected in the above calculation) is proportional to that through
the shunt; and if the scale is divided into 25 equal parts, we have
an ammeter of the desired range.

KDi

-Tg Vww^

LINE

Fig. 42. — Internal connections
for ammeter.

Fig.

4.3. — Internal connections
for voltmeter.

The same instrument may be used as a voltmeter, if, instead
of the shunt, it is connected in series with a large resistance R,
Fig. 43, of such a value that, when the maximum voltage to be
measured is impressed across the outside terminals of G and R,
the drop across the instrument is that required for full scale deflec-
tion. For example, suppose the instrument, as above, requires 50
millivolts for full scale deflection, that it has a resistance of 10
ohms, and that it is desired to construct a voltmeter reading to
100 volts. By Ohm's law, R, is given by the following equation:

.050 ^ 10
99.95 R
Whence

R =

10 X 99.95
.05

19,990 ohms

MEASUREMENT OF CURRENT 77

Since the current through the instrument is proportional to the
external voltage impressed, if the scale is divided into 100 equal
parts, we have the voltmeter required. In some instruments,
e.g., Weston, especially for low ranges, the shunts and series
resistances, or multipliers, as they are generally called, are placed
within the case and cannot be seen; while in others, e.g., Siemens
and Halske, and R. W. Paul, they are mounted outside the case
and are detachable. The latter have the advantage of being
interchangeable, so that the same instrument, when provided
with a series of shunts and multipliers of appropriate values,
may serve either as a voltmeter or as an ammeter with any num-
ber of ranges for each.

60. Experiment 9. Electrical Adjustment of an Ammeter and a
Voltmeter. — It is the purpose of this exercise to illustrate the
fundamental principles of construction and operation of moving
coil ammeters and voltmeters. For this purpose, a Weston
switch-board type instrument, with transparent case, has
been provided with an ad justable external shunt and series
resistance. It is to be standardized and tested, first as an
ammeter, and then as a voltmeter. In order to accomplish
this, it is necessary to know three things concerning the instru-
ment: (1) Resistance; (2) current sensibility; (3) millivolts for
full scale deflection.

1. The resistance of the instrument may be obtained directly
by means of a Wheatstone bridge. Set the ratio coils Fig. 15
with 100 ohms in the right-hand bank and 10,000 in the left. Be
careful to connect the instrument so that the pointer moves for-
ward when operating the bridge.

2. To find the current sensibility of the instrument, which is
defined as the current for unit scale deflection, connect it in
series with an adjustable known resistance and a cell whose
E.M.F. has been determined. In all the tests to be carried out,
remember that the instrument is very sensitive, requiring but an
exceedingly small current for full scale deflection. Accordingly,
a resistance of at least 1,000 ohms should be included before the
circuit is closed. Determine the resistances corresponding to
five different indications of the instrument distributed uniformly
across the scale, and by Ohm's law, compute the current for
unit deflection. The E.M.F. of the cell may be obtained by
means of a low range voltmeter.

3. The voltage for full scale deflection is given at once by Ohm's

ELECTRICITY AND MAGNETISM

law as the product of the resistance, the current sensibility, and
the number of scale divisions.

Part I. Ammeter. — It is required to construct an ammeter of
range 0-5 amperes, from the instrument and adjustable shunt.
From Ohm's Law, find the resistance, which, when carrying 5
amperes, gives a potential drop across its terminals equal to the
voltage required for full scale deflection of the instrument.
Measure the total resistance of the adjustable shunt by means of
the bridge used above, correcting for the leads, and find what
length of wire is necessary for the required shunt resistance.

Fig. 44." — Connections for testing improvised ammeter.

Now connect the instrument, as shown in Fig. 44, where SA
is a standard ammeter and B, a storage battery of 6 volts, setting
the shunt at the computed value. Check your ammeter against
the standard ammeter at 8 or 10 points uniformly distributed
across the scale. Now compute, as above, the shunt resistance
required in order that your ammeter may have a range of 0-2.5
amperes, and test it in the same manner.

Part II. Voltmeter. — It is required to construct a voltmeter
of range 0-50 volts, from the instrument and an adjustable series
resistance used as a multiplier. From Ohm's law, compute the
resistance which, when placed in series with the instru-
ment, will give the potential drop across it necessary for
full scale deflection, when 50 volts are impressed across the
instrument and multiplier. Connect the apparatus, as shown
in Fig. 45, placing in M the computed resistance. 5 is a storage
battery of 50 volts, SV a standard voltmeter, and PD a potential
dividing rheostat of several hundred ohms, by means of which

MEASUREMENT OF CURRENT

any voltage between 0 and 50 may be impressed across the instru-
ments. Check your voltmeter against the standard at 8 or 10
points evenly distributed across the scale.

Report. — 1. Make a sketch of the instrument describing in
detail the essential working parts.

2. Outline the general principles involved in adapting it to
measure currents and potential differences.

3. Give in full your data and computations for shunts and
multiplying resistances.

€>

liii III

Fig. 45. — Connections for testing improvised voltmeter

4. Give data and curves for your ammeter and voltmeter
calibrations.

5. In calculating the resistance of the shunt, in Part I, the
current through the instrument was neglected. Compute the
error thus made.

61. Measurement of Current by the Potentiometer. — Since
the potentiometer measures potentials only, current measure-
ments made by it must necessarily be indirect. For this purpose,
use is made of a carefully standardized resistance capable of
carrying the current to be measured without appreciable heating.
The potentiometer measures the fall of potential across its termi-
nals produced by the current, which is then determined by Ohm's
law. If the resistance has some decimal value, the value of the
current will have the same significant figures as the potential
drop across it. Accordingly, if the potentiometer has been made
direct reading for voltage, it will indicate currents directly also.

ELECTRICITY AND MAGNETISM

Resistances for this purpose must be provided with two pairs of
binding posts, one for current and the other, for potential. The
potential leads are soldered securely to the posts between the
current terminals and the effective resistance is only that between
the points to which they are attached. Errors from imperfect
connections are thus eliminated. Such resistances should be
placed in an oil bath to keep the temperature constant. The
largest resistance giving, for the desired current, a potential
difference within the range of the potentiometer should be used.
62. Experiment 10. Calibration of an Ammeter by Potentio-
meter and Standard Resistance. ^ — Connect the apparatus, as
shown in Fig. 46. AM is the ammeter to be tested, B a storage
battery of 10 volts, S a rheostat for controlling the current, and

-nAAAAAAA^At

A.M

-f-

AVvVVVVWV\

Fig. 46. — Connections for standardizing an ammeter.

R a standard oil-cooled resistance provided with current and
potential terminals. The leads ah are to be connected to the
potentiometer. In connecting up the potentiometer and stand-
ardizing the current through it, follow the directions for the
particular type of instrument given in Chap. IV. After the
potentiometer has been adjusted, cause such a current to flow
in the ammeter circuit as will produce about one-tenth full scale
deflection and measure the fall of potential across R by means of
the potentiometer. The resistance R and the ammeter carry the
same current, since no current flows through a and b at the point
of balance. The current through the ammeter is equal to the

1 Jansky, Electrical Meters, chap. V.
Karapetoff, Experimental Engineering, vol. I, pp. 51-55.

MEASUREMENT OF CURRENT 81

reading of the potentiometer divided by R. Since R has a
decimal value, it is merely a question of properly pointing off this
indication. In a similar manner, check the ammeter at 8 or 10
points distributed uniformly across the scale. The balance
against the standard cell should frequently be tested and any
variations in the potentiometer current compensated.

Record ammeter readings, potentiometer settings, and the
value of R. Note carefully the zero reading of the ammeter
before beginning the test and again at the end, after the pointer
has been deflected for some time, to see if there is any elastic
fatigue in the springs. With about two-thirds full scale deflec-
tion, place the instrument in a vertical position to test the
accuracy with which the moving system is balanced. Bring
another instrument near this one to see if there is any effect due
to external magnetic fields. Tap the instrument gently with
the finger to see if the bearing friction is large. Does the pointer
swing past its final indication when a current is suddenly thrown
on? Record changes in reading in all of the above cases.

Report. — 1. Compute the differences between the readings of
the instrument and true amperes.

2. Plot these corrections as ordinates against ammeter
readings as abscissas. Draw straight lines connecting these
points.

3. State your findings regarding the imperfections of the
instrument.

4. Would it indicate on alternating currents?

CHAPTER VI
MEASUREMENT OF POWER

63. Wattmeters.^ — Whenever a current flows in a circuit,
there is a certain amount of energy consumed by the circuit, and
any instrument which measures the rate at which energy is
consumed is called a wattmeter, from the fact that electrical
power is generally measured in watts. Three kinds of watt-
meters are in common use; namely, indicating, recording, and
integrating. Instruments of the first kind show the power that
is being consumed at any instant; those of the second kind make
a permanent record on a revolving dial of the power consumption
during a given period of time ; while those of the third kind show
the total energy, that is, the integral of the power times the
time, delivered to a circuit during a definite period. Instruments
of the first kind only will be considered here, and of the various
types in use, only one will be discussed, namely, the
electrodynamometer type.

64. Use of an Electrodynamometer for the Measurement of
Power. — In case a steady current is flowing through a circuit,
the power is given by the product of the current and the fall of
potential across the circuit, or

Watts = Amperes X Volts
The watts may, therefore, be measured by simultaneously reading
an ammeter and a voltmeter. If, however, a single instrument
can be devised which will give indications proportional to both
current and voltage, it will automatically indicate their product,
and may be calibrated to read watts directly. In the discussion
of the electrodynamometer, it was pointed out that the torque is
proportional to the current in both the fixed and movable coils,
and, therefore, to their product. Accordingly, if one of the coils
can be made to function as an ammeter and the other as a volt-
meter, the instrument will be a wattmeter. For this purpose,
the fixed coil is made of a few turns of heavy wire and is connected

1 Jansky, Electrical Meters, chap. X.
Karapetoff, Experimental Engineering, vol. I, chap. IV.
Electrical Meterman's Handbook.

MEASUREMENT OF POWER

in series with the circuit like an ammeter, while the movable coil
is made of a great many turns of fine wire having a high resistance
and is connected across the circuit like a voltmeter and carries a
current proportional to the voltage. The torque is proportional,
therefore, to amperes times volts, hence, to watts. This is the
principle underlying the Weston Indicating wattmeter, the
connections for which are shown in Fig, 47. A and B are series
coils consisting of a few turns of heavy wire through which the
total current flows, while C is a voltage coil of many turns of fine
wire. It is connected across the load at the points H and K, and
is mounted so as to turn about an axis through its geometrical
center perpendicular to the plane of the paper. Attached to the
axle carrying this coil, is a pair of spiral springs, not shown in the

Fig. 47. — Schematic diagram for
Weston wattmeter.

Fig. 48. — Diagram showing com-
pensating and multiplying coils for
Weston wattmeter.

figure, whose restoring torque, as the coil is rotated, opposes that
due to the electrodynamic action of the currents. They serve
also as leads to and from the coil. The scale is so divided that
the instrument indicates watts directly.

The readings of such an instrument are subject to an error due
to the power consumed by the coils themselves. An inspection
of Fig. 47 shows that the current passing through the coils A
and B is the sum of the load current and that carried by the coil
C, hence the reading must be too large by the PR loss in this coil.
If it is attempted to overcome this by connecting the voltage

84 ELECTRICITY AND MAGNETISM

terminals on the "line" side of the current coils, the registered
voltage will be too large by the drop across the current coils thus
again making the reading too large. To overcome this difficulty,
a compensating coil M is provided as shown in Fig. 48, which is
usually placed inside A and B and so connected that its magnetic
effect weakens their fields, thus automatically correcting the
reading of the instrument. If the wattmeter is to be calibrated
by using separate sources of current and potential, this compensa-
tion is not necessary, and a separate binding post F is provided,
marked Ind. (Independent) on the instrument. This circuit
includes a resistance r equal to that of the compensating coil,
thus making the resistance between C and F equal to that
between C and L. The series resistance R is used as a multiply-
ing resistance in exactly the same manner as the multiplier in an
ordinary D.C. voltmeter. For example, if R is equal to the
resistance of the movable coil, the potential difference across it
will be equal to that across the coil, and if the instrument is
calibrated without R in circuit, when R is included, the readings
should be multiplied by the factor two.

65. Experiment 11. Calibration of a Wattmeter. — Wattmeters
are calibrated on direct currents and may be used on alternating
currents as well as direct. Separate sources of current and
electromotive force are generally used for purposes of calibration
since instruments of large capacity may then be standardized
with a comparatively small expenditure of power. Connect
the apparatus as shown in Fig. 49, where WM is the wattmeter
which is to be calibrated. B is ten-volt storage battery furnish-
ing the current which is controlled by the rheostat R and read
by the ammeter AM. C is another storage battery furnishing
the potential which is controlled by the voltage regulating rheo-
stat PS and read by the voltmeter VM. Since the field due to
the coils of the instrument is small, extraneous fields, such as
those of the earth or large currents, near-by instruments with
permanent magnets, etc., may cause errors as large as several
per cent. Hence it is necessary, when using this type of watt-
meter on direct currents, to reverse both potential and current
leads and average the two readings. Make two calibrations.
First, hold the current constant and vary the voltage so as to
check the instrument at eight or ten points uniformly distributed
across the scale. Next hold the voltage constant and vary the
current, checking approximately the same points as before.

MEASUREMENT OF POWER

Record volts, amperes, and indicated watts, both direct and
reversed, in all cases. With about two-thirds full scale deflec-
tion, bring an instrument with a permanent magnet near the
wattmeter and note the effect on the reading. Place the watt-
meter pointing in various directions and note any changes due
to the earth's magnetic field. Stand the instrument in a vertical
position and note any error due to imperfect balancing of the
moving system. Change the voltage terminal from the post F,
marked "Ind." to L and note the difference, which is the correc-
tion for internal energy consumption.

^ilil^lilili

-AAAAAWWVVWWV-

CI — 0.

066

L F G
W.M.

-oD

Fig. 49. — Connections for calibrating a wattmeter.

Report. — 1. Compute true watts from the average products
of amperes and volts.

2. Plot corrections as ordinates against wattmeter readings as
abscissas. Do the two curves (a) with current constant, and (6)
with voltage constant, coincide?

3. State your findings regarding internal energy consumption,
effects of extraneous magnetic fields, balancing of system, etc.

4. Why are the scale divisions in this wattmeter unequal and
those of the D.C. voltmeter and ammeter equal?

CHAPTER VII
MEASUREMENT OF CAPACITANCE

66. Condensers. — When a body is charged with a quantity of
electricity Q, the potential V which the body acquires is propor-
tional to Q. With a given charge, however, the potential
depends also upon certain conditions of the body such as size,
shape, surrounding medium, presence of other charges, etc. The
relation between charge and potential is given by the equation

Q = CV (1)

where C is a constant depending upon the conditions of the body,
and is called the "Capacitance" of the body. It is the ratio of
the charge to the potential and is numerically equal to the charge
when the potential is unity. The practical unit of capacitance is
the farad. A body is said to have a capacitance of one farad
when a charge of one coulomb raises its potential by one volt.
The farad is too large a unit for practical purposes, however, and
it is customary to take the millionth part of this, called the micro-
farad, as a working unit. Any device by which it is possible to
cause a large quantity of electricity to exist under a relatively
small potential is called a condenser. Such devices usually
consist of thin conducting plates, placed close together, but
insulated electrically by thin sheets of some good dielectric
material. If a positive charge is placed upon one plate and a
negative upon the other, the neutralizing effect of each on the
other, due to their close proximity, causes the potential difference
between them to be very much reduced over what it would have
been if they were far apart.

67. Grouping of Condensers. — Condensers, like resistances,
may be joined either in series or in parallel and used as a single
condenser. Figure 50 represents three condensers joined in
parallel. Let Ci, C2, C3 represent their individual capacitances,
Qh 92, qa their charges; and E, the difference of potential across
their terminals. Calling Q the total quantity of electricity stored
in the group, which would be obtained if they were discharged,
we have

Q = gi + 32 + gs (2)

MEASUREMENT OF CAPACITANCE

If C is the resultant capacitance of the group, we have, from
definition,

Q = CF = CiF + C^V + CaF (3)

C = Cx + C2 + Ca (4)

For condensers connected in parallel, the resultant capacitance
is the sum of the individual capacitances of the group. The
capacitance of n similar condensers thus joined is n times the
capacitance of a single condenser. Figure 51 represents three
condensers connected in series. As before, let Ci, Ca, Cz represent

H'hl^

i|i|h

Fig. 50. — Condensers
joined in parallel.

Fig. 51. — Condensers joined in series.

the values of the individual conaensers; gi, qi, qz their charges,
and Vi, V2, V3 the potential differences across each. It is evident
from the figure that

E = vi -j- V2 -^ vz (5)

Calling C the resultant capacitance of the group, and Q the total
charge, we have, from definition,

\j L/ 1 L- 2 ^-^ 3

(6)

A simple relation exists between these charges. We have tacitly
assumed that the condensers were uncharged before connection.
Suppose a unit charge passes from the battery to the outer coating
of Ci. A negative charge will then be induced on the inner coat-
ing and a positive unit charge repelled from it which will charge the
outer coating of C2 and induce a negative unit charge on its inner
coating and so on. The next unit charge from the battery will
do the same. It is evident then that the charge for each conden-
ser is the same, no matter what its capacitance and that the total
charge which may be obtained from the group on discharge is
the same as the charge from any one condenser. That is, the

ELECTRICITY AND MAGNETISM

positive charge on the outer coating of Ci neutralizes the charge
on the inner coating of C3 and similarly for the other condensers
of the group. Thus we have

Q = qi = q2 = qs (7)

and

(8)

C C 1 C2 v^a

For condensers joined in series, the reciprocal of the resultant
capacitance is the sum of the reciprocals of the individual capaci-
tances. The resultant capacitance of n similar condensers so

joined is - times the capacitance of a single condenser.

68. Standard Condensers. — Standard condensers are made of
sheets of tin foil separated by mica, alternate sheets of foil
being joined as shown in Fig. 50, and the whole finally imbed-
ded in solid paraffin. Figure 52 shows the connections for

EARTH — >— 1

!<!!
Ijii

iili

iiii

1!!
Ill
I'l'

iji j

Ill

<!'■

lllid)

05 .05 .2 .2 .5
CONDENSER

Fig. 52. — Connections for subdivided standard condenser.

one of the subdivided standard condensers used in the labora-
tory. One side of each of the sections shown by the dotted
lines is joined to a heavy bar marked "Earth" and the other
sides to one of the blocks. Another bar marked "Condenser''
is mounted opposite, and each bar is connected to a binding post.
When it is desired to use a certain capacity, e.g., 2 MF, place a
plug in the socket between C and the lower bar. If .7 MF is
desired, place another plug between E and this bar. Similarly
for the various other possible connections. When a section is
not in use, it should be short circuited by placing a plug in the
socket between the upper bar and the corresponding block. Care
should be taken never to place plugs at both ends of any block as

MEASUREMENT OF CAPACITANCE

that would short circuit the entire condenser, possibly injuring
apparatus to which it is connected.

Another method of assembUng subdivided standard condensers
is to join the units, not between the central lugs and one of the bus
bars as shown in Fig. 52, but to connect them between the lugs
as shown in Fig. 53. Thus between A and B there is .05 micro-

■05 46 .2 .2 .»

-^ '^

AHh

HhHh

MICRO-FARAD

Fig. 53. — Alternate method of connecting condenser units.

farads, between B and C, .05, etc. One more lug is required in
this case To connect all the units in parallel, plugs are inserted
in alternate holes on each side, but staggered. The method has
the advantage of permitting series grouping of the units, thus
giving a greater number of values of capacitance for a given
number of units. A subdivided standard condenser is shown in
Fig. 54.

Fig. 54. — Subdivided standard condenser.

69. Comparison of Condensers. — The capacitance of an
unknown condenser may be found by comparing it with a stan-
dard condenser. A ready means of doing this which gives results
sufficiently accurate for many purposes is to charge both the
unknown and the standard to the same potential difference and
discharge each in succession through a ballistic galvanometer.
The set-up for this purpose is shown in Fig. 9. Let Ci be the
unknown and C^ the standard condenser. First insert Ci, then
charge and discharge several times, taking the average deflection

ELECTRICITY AND MAGNETISM

which we will call (U. From the definition of capacitance we
have, as the charge in the condenser.

Qi = Ci7, (9)

and, since the deflection of a ballistic galvanometer is propor-
tional to the charge passed through it,

Qi = CiF = Kdr (10)

Now replace the unknown by the standard condenser and charge
and discharge as before. In a similar manner, we have

Q2 = C2F = Kd^ (11)

Dividing equation (10) by (11),

70. Bridge Method for Comparing Two Condensers. ^ — A more
accurate comparison of two condensers may be made by means

Fig. 55. — Bridge method for comparing two condensers.

of an arrangement similar to the Wheatstone bridge in which the
resistances of two of the arms are replaced by the two condensers
to be compared, and the current galvanometer is replaced by a
1 Carhart and Patterson, Electrical Measurements, pp. 213-220.
Smith, Electrical Measurements, art. IX.

MEASUREMENT OF CAPACITANCE 91

ballistic galvanometer. The connections are shown in Fig. 55.
Ci and d are two condensers and Ri and R2 two variable resis-
tance boxes. X is a charge and discharge key so connected that,
when the blade is pressed down, the battery B is connected to
the bridge, thus charging the condensers through Ri and R2
to the potential difference furnished by the battery. When
contact is made on the upper point, the battery is disconnected
and the condensers are discharged through the resistances. -No
matter what the values of Ri and R2 may be, the points P and Q
will come to the same final potentials on charge and again on
complete discharge, since, when no current is flowing through
the resistances there can be no fall of potential across them.
However, during the charging and discharging processes there
are currents through the resistances and, in general, there will be
a momentary difference of potential between P and Q causing a
deflection of the galvanometer in one direction on charge, and
in the opposite, on discharge. By properly adjusting Ri and
R2, it is possible to make the potentials at P and Q rise and fall
at the same rate which is the balance condition for the bridge,
from which the relation between the capacitances and resistances
may be deduced.

Let qi and 92 = instantaneous charges in Ci and d
Let z'l and 12 = instantaneous currents in Ri and R2

As in the ordinary Wheatstone bridge, we have

P.D. between A and P = P.D, between A and Q
P.D. between P and D = P.D. between Q and D

whence

and

Ci C2

(13)

Riii = Rill (14)

Differentiating (13) and remembering that i = -j^ we have
Idqi _ Idqi

92 ELECTRICITY AND MAGNETISM

Dividing (14) by (15)

RiCi = R2C2 (16)

It is to be noted that the ratio of the capacitances is the inverse
ratio of the resistances, whereas, in the bridge method for resis-
tances, it is the direct ratio.

71. Experiment 12. Comparison of Two Capacitances by the
Bridge Method. — Connect the apparatus as shown in Fig. 55,
using for B a battery of 40 volts. SK is a short circuiting key
to bring the galvanometer to rest after taking an observation.
Use for Ci a subdivided standard condenser and for Ci a fixed
condenser of about one-half micro-farad capacity. The problem
is to check the parts of Ci in terms of the whole. Set the plugs
of Ci so as to give the maximum available capacitance, and with
this as a standard, obtain several balances on C2, using different
values for Ri and R2 in each case. Now, taking this measured
value of C2 as a standard, determine the capacitance of each
part of Ci, making several independent balances for each. In
all cases use as large values for Ri and R2 as possible.

Report. — Tabulate your data in compact form. Your values
for the separate parts of Ci should add up to the total value indi-
cated on the top of the box.

72. Measurement of Small Capacitance by Commutator.^ —
The bridge method just described is not suited to the measure-
ment of small capacitances since the charging currents are so
minute that the potential drops through the resistances are
inappreciable. For the determination of a small capacitance,
such as that of an air condenser used in radio work or that between
the wires of a transmission line, a direct deflection method may be
used in which the condenser is rapidly charged to a known voltage
and then discharged through a standardized galvanometer by
means of a motor-driven commutator. If the interval between
discharges is small compared to the period of the galvanometer,
a steady deflection results which is proportional to the average
value of the current.

1 Fleming and Clinton, Proc. Phys. Soc. of London, vol. 18, 1901-03,
p. 386.

MEASUREMENT OF CAPACITANCE

Figure 56 shows the principle of the Fleming and Clinton com-
mutator, designed for this purpose, together with the wiring
diagram. Si and S2 are slip rings which revolve in a plane per-
pendicular to the paper while S3 is a series of posts alternately
connected to Si and S2. When brush 3 rests upon a segment
connected to Si, the condenser C is charged to the potential
difference of the battery B, and when 3 touches the succeeding
post, the condenser is discharged
through the galvanometer. Let n be
the number of discharges per second
and V the E.M.F. of the battery.
Then the current through the gal-
vanometer is

/ = nCV X 10-« = Fd (18)

Where C is the capacity of the con-
denser in microfarads, and F, the
figure of merit of the galvanometer,
i.e., the current for unit deflection.

In designing a commutator of this
type, special precautions must be
taken to secure good insulation be-
tween posts and sectors. They are
generally made with an air gap be-
tween posts since metal abraded by
friction otherwise embeds itself in a solid dielectric thus giving
a direct leakage path from the battery through the galvano-
meter. A speed counter, mounted on the shaft, indicates the
number of revolutions.

73. Experiment 13. Capacitance hy the Fleming and Clinton
Method. — Connect the apparatus as shown in Fig. 56 using for C
an air condenser of small capacitance. Drive the commutator
at speeds sufficient to give 50 to 100 discharges per second through
the galvanometer, and use for B a battery with voltage large
enough to produce a deflection of about 10 cms. Take special
precautions to insure good insulation between the galvanometer
and battery circuits. Use a number of different speeds and
voltages and determine the value of C by eq. (18). Determine
the figure of merit of the galvanometer by the method given
in Art. 23.

Fig. 56.

-Fleming and Clinton
commutator.

94 ELECTRICITY AND MAGNETISM

Report. — 1. Tabulate observations and results for the series of
measurements taken.

2. Check your results by measuring the dimensions of the
condenser, and computing its capacity from the formula for the
parallel plate condenser

C = .0885 X 10-« ^ microfarads (19)

where A is the area of one of the plates in square centimeters, d
the thickness of the dielectric, and k the dielectric constant
(K = 1 for air).

3. How do you account for the difference between the mea-
sured and computed values?

CHAPTER VIII
MAGNETISM

74. General Principles. — Magnetism is a universal property
of matter, since there is no substance which does not experience
a ponder-motive action when placed near the poles of a strong
magnet, though in many cases the effect is so weak that delicate
means are necessary for its detection. Substances may be
divided into two groups, in accordance with the manner in
which they behave when acted upon by a magnetic pole; those
which are attracted by the magnet are called paramagnetic, and
those repelled, diamagnetic. It is customary to add a third
group, including iron, nickel, and cobalt, which are characterized
by the fact that the ponder-motive action upon them is not
proportional to the strength of the attracting pole as in the case
of ordinary paramagnetic substances, but is much stronger,

75. Strength of Pole. — In the early study of magnets, it was
noticed that the magnetic property of a body is confined largely
to the areas about its ends and corners, and that opposite ends
behave differently toward other magnets. The term magnetic
pole was given to the regions where the property was most
pronounced and has been retained although it has been known
for a long time that magnetism is a volume and not a surface
phenomenon. A unit magnetic pole is defined as one which
repels an exactly similar pole at a distance of one centimeter in air
with a force of one dyne.

76. Strength of Field. — The space surrounding a magnetic
pole in which action upon another pole can be detected is called
a magnetic field, and is measured, at any particular point, by the
force in dynes with which a free unit positive pole is acted upon
when placed at that point. A field of unit strength or intensity is
one which will exert a force of one dyne upon a unit pole. Since
field strength is thus measured by force per unit pole, it is a
vector quantity; i.e., it possesses both magnitude and direction.
Both of these characteristics may be represented by imagining
lines drawn in space according to a definite convention; namely,

96 ELECTRICITY AND MAGNETISM

the magnitude of the field by drawing as many Unes per square
centimeter as the field has units of intensity, and the direction,
by making these lines coincide at every point with the direction
in which the unit measuring pole is urged. According to this
convention, if a sphere of unit radius is drawn with a pole of
strength m as a center, there must pass through each square
centimeter of its surface m lines, giving a total of 47rm lines from
the pole. From a unit pole there will emanate according to this
convention, Air lines of force.

77. Intensity of Magnetization. — ^Let us imagine an ideal
permanent bar magnet, of length L, and uniform cross section A,
magnetized uniformly and showing, therefore, pole-strength over
the ends only. That is to say, the magnetic lines all leave one
end, pass in regular curves through the outside space, and enter
the other end with no lines entering or leaving on the side, as in
any real magnet. Let the strength of pole be m. The pole

strength per unit area, -r, is defined as the intensity of magnetiza-
tion and is generally represented by the letter /.

78. Magnetic Moment. — Imagine the ideal magnet mentioned
above placed at right angles to a uniform magnetic field of
strength H. Equal and opposite forces of magnitude Hm will
act upon this magnet producing a couple of strength HmL.
If H is unity, the magnitude of the couple is mL, and this quan-
tity, which is exceedingly important in treating problems
involving magnets, is called the magnetic moment, and is designated
by the letter M. The moment of any magnet is, then, the torque
acting upon it when placed at right angles to a uniform field of
unit strength. Another definition of intensity of magnetization
in terms of magnetic moment may be obtained as follows:
Since the volume of the bar magnet is LA, we have

A~AL~~V ^ ^

Intensity of magnetization is thus defined as a volume rather
than a surface effect.

79. Magnetic Induction. — Let us imagine that, in an infinitely
long, uniform magnetic field of strength H, an iron bar is placed
with its axis parallel to the field. The bar becomes magnetized
to an intensity I and is equivalent to the ideal magnet considered
above. The number of magnetic lines through the space

MAGNETISM 97

occupied by the bar has been increased by the lines of magnetization
due to the bar. The total number of magnetic lines through the
bar, which is made up of the original hnes and the lines of mag-
netization, is called the magnetic flux, and is generally designated
by the Greek letter <l>. The number of lines per square centi-
meter through the bar is called the magnetic induction, and is
represented by the letter B. Thus

P T ^ ^- Total Flux <l>

B = Induction = ~^^^^^ = } (2)

The induction B is defined in the following manner: Imagine a
narrow crevasse cut through the middle of the bar at right angles
to its axis, and a unit positive pole placed within. The force in
dynes upon this pole measures B. The original field produces a
force of H dynes upon the pole, and since the iron is magnetized
to an intensity /, meaning / units of pole strength per unit area of
the crevasse, from each of which 4^ lines emanate, we
have, as the total lines per square centimeter through the gap
or the force in dynes acting upon the unit pole

B = H + 47r/ (3)

Lines of induction are continuous throughout the magnetic
circuit; that is, they never begin or end but form closed paths,
the parts in the air being called lines of force. If, instead of the
transverse crevasse we had bored a small hole through the bar
parallel to the lines of force and placed a unit magnetic pole within,
the force upon it would be the original strength of field H which
has produced the magnetic induction.

80. Permeability and Susceptibility. — For many purposes it
is convenient to define the magnetic quality of a given material
in terms of the relative increase in the number of lines or the
intensity of magnetization produced. For this purpose the terms
permeability and susceptibility are used. By permeability is
meant the ratio of the induction B to the field strength H, and is
represented by the Greek letter n. That is,

. = I (4)

where B is the induction produced in a given material when
acted upon by a field of strength H. When it is desired to express
the abihty of a material to acquire magnetism and to state its
condition in terms, not of the total induction, but of its own
magnetic lines alone, we use the term susceptibility. This is
7

98 ELECTRICITY AND MAGNETISM

defined as the ratio of the intensity of magnetization of the
specimen to the magnetizing field in which it is placed, and is
represented by the Greek letter k. That is,

A simple relation exists between these two quantities. Taking
the defining equation for induction

B = H + 47rl (6)

and dividing through by H, we have

I = 1 + 4t-^ (7)

/i = 1 + 47r/c (8)

81. Effects of the Ends of a Bar. — When a bar is magnetized
longitudinally by placing it in a magnetic field, the ends become
poles which act upon any other pole in the neighborhood, attract-
ing or repelling it according to the relative signs of the poles.
If the bar lies in an east-west position, magnetized with a north
pole at the west end, a unit north pole lying near the middle of the
bar, but outside it, would be urged from west to east or in a
direction opposite to that in which the bar is magnetized. If now
the unit pole is placed within the bar, the force is in the same
direction. Thus the effect of the poles is to produce a field within
the bar called a "demagnetizing field" which is opposite to the
direction of the field magnetizing it. This effect is greater the
shorter the bar is in comparison to its diameter. The actual
field producing magnetization is, accordingly, less than the field
before the bar was introduced. This phenomenon is allowed for
by computing the effective field H from the equation

H = H' - NI (9)

where H' is the original field and N a constant depending upon
the ratio of the length to the diameter of the bar, and is called
the " Demagnetizing Factor." Tables^ for A^ may be found in the
more advanced treatises on the subject. The same considera-
tions hold for solenoids, and hence it is necessary, when one
wishes a solenoid whose field may be computed readily from its
dimensions, to make it long in comparison to its diameter. If
one uses a ring solenoid, or a test specimen in the form of a ring,
1 Dtr Bois, The Magnetic Circuit, p. 41.

MAGNETISM 99

this correction is unnecessary since there are no free poles to
produce disturbing effects of this character.

82. The Magnetic Circuit. — In treating such phenomena as the
conduction of heat and the fiow of electricity, one makes use of a
general law in which the magnitude of the effect is given as the
ratio of a driving force divided by an opposition factor dependent
upon the properties of the medium in which the action takes
place. For example, the heat current Q, i.e., the quantity of heat
passing per unit time any cross section of a conductor of length L
and cross sectional area A, when the temperature at the ends are
ti and <2, is given by the expression

Q = ^^^ (10)

where t is a constant defining the ability of the medium to conduct
heat. T is called the specific thermal conductivity and is numeri-
cally equal to the quantity of heat passing through a centimeter
cube of the material, per unit time, when a difference of tempera-
ture of one degree is maintained across its faces. Similarly, the
electrical current flowing in the above conductor when its ends are
maintained at electrical potentials Vi and V2, is given by

A CA

where C = - is called the specific electrical conductivity of the

material and is numerically equal to the current flowing through a
centimeter cube when unit difference of potential is maintained
across its faces. Its reciprocal p is called the specific resistance,
and is the resistance of the centimeter cube. This equation is
called Ohm's law and is written

^ , Electromotive Force

Current = ^s — ^^

Resistance

In an analogous manner it is convenient, for purposes of calcula-
tion, to regard the region in which a magnetized state exists as
being the seat of a magnetic flow. The magnetic lines of induc-
tion are the stream lines along which the flow takes place, and
since magnetic hues are closed paths, the lines of flow are closed
circuits. Materials are then classified as good or bad magnetic
conductors according to the ease with which they are magnetized.

100 ELECTRICITY AND MAGNETISM

To make the analogy clear, consider a specimen of magnetic
material in the form of an anchor ring, wound uniformly with wire
through which a current is flowing, as shown in Fig. 57. We wish
to compute the total magnetic flux produced in the ring when a
given current is flowing.

Let N = total number of turns

L = mean length of magnetic lines
A = area of cross section of ring
B = magnetic induction in ring
n = permeability
/ = strength of current

As a direct consequence of the defim-
tion of the electromagnetic unit of
current, it is shown in elementary
textbooks that the work done in carry-

FiQ. 57. — The magnetic circuit. • •, j.- i j

mg a umt magnetic pole once around
a current of strength I E.M.U.'s, is 4tI ergs. The field strength
within the ring solenoid may be obtained from the fact that the
work done in taking a unit pole around this magnetic circuit is

Work = HL = ArNI (12)

since the pole is, in reality, carried N times around the current.
Whence

H = ^^ (13)

If / is expressed in amperes instead of electromagnetic units.

From the above definitions, the expression for the total flux is
obtained in the following manner:

^ = BA^,HA= '^^^ (15)

which may be written in the form

* = -^ (16)

The numerator of the right-hand member is of the nature of a
driving force, the denominator, an opposition factor depending
upon the medium, and their ratio, the effect produced. This

MAGNETISM 101

equation is called the "Law of the Magnetic Circuit" and is
written

-, .. T^i Magnetomotive Force

Magnetic Flux = —n—^ — i —

Reluctance

83. Magnetic flux, which represents the total number of
lines of induction, is analogous to flow of heat in calorimetery,
and to current in electricity. It forms a closed path which may
be spread out over a large area in some places and be concen-
trated within narrow limits in others. The unit of magnetic
flux is called the maxwell and is represented by one magnetic
line of induction through the total cross sectional area of the
magnetic circuit. Thus, if in a magnetic circuit, there are one
thousand lines, the flux is said to be one thousand maxwells.
In engineering practice, it is customary to define flux on the
basis of the E.M.F. induced in a conductor which cuts it.

Definition. — //, in a moving conductor, the induced E.M.F. is one
electromagnetic unit, the flux cut per second is one maxwell. *

84. The magnetic induction is defined as the total flux
divided by the area, and is, accordingly, the flux density. The
unit of magnetic induction is the gauss.

Definition. — Unit induction, or one gauss, exists in a magnetic
circuit in which the flux density is one maxwell per square centi-
meter. Thus

-. Maxwells

Gausses = ^ p^ — 17^-1 —

Square Centimeters

85. Magnetomotive force may be regared as the cause of
magnetic flux. It is analogous to electromotive force in the
electric circuit and is measured in a similar manner. Just as the
electromotive force of an electrical circuit is the work required
to carry unit electrical charge once around the circuit, so the
magneto-motive force in a magnetic circuit is the work required
to carry unit magnetic pole once around the circuit. The unit
of magnetomotive force is the gilbert.

Definition. — If the work required to carry a unit mxignetic pole
once around a magnetic circuit is one erg, the magnetomotive force
is one gilbert.

In case the magnetomotive force is produced by a current in a
closed solenoid, as in the above illustration, its value, as given

' Note. — The Units for the quantities involved in the magnetic circuit
here described were adopted by the International EUectrical Congress at
Paris, in 1900.

102 ELECTRICITY AND MAGNETISM

by equation (16) is .47rNI. The product NI is called the ampere
turn, and differs from magnetomotive force only by the constant
factor At = 1.26. Thus

M.M.F. in Gilberts = Air Ampere Turns.

Magnetomotive force, being thus measured in terms of work
per unit pole, is difference of magnetic potential. Accordingly,
if H is the average value of the magnetic field strength between
two equipotential surfaces, s cms. apart, having magnetic
potentials Mi and M^, respectively,

s As ^

where Ailf and As represent small differences in M and s, respec-
tively. Allowing the equipotential surfaces to approach indefi-
nitely close to one another, the limiting value of this ratio gives
the actual field strength at a given point. Thus

Magnetic field strength is the change in magnetic potential per
centimeter in the direction of H or the magnetic potential
gradient. The unit of magnetic field strength is called the
gilbert per centimeter.

86. Reluctance is the resistance a body offers to being mag-
netized and depends upon the constants of the circuit in a manner
similar to resistance in the electrical circuit. As seen from
eq. (16), it is directly proportional to the length and in-
versely proportional to the area and the permeability of the
medium. Permeability thus corresponds to specific conductivity,
and its reciprocal, corresponding to specific resistance or
resistivity, is often called "reluctivity." The unit of reluctance
is defined in terms of the law of the magnetic circuit and is
called the oersted.

Definition. — //, in a magnetic circuit, the flux is one maxwell
when the magnetomotive force is one gilbert, the reluctance is one
oersted.

Reluctances, hke resistances, may be joined in series or
parallel to form complex circuits, and laws similar to those for
resistances hold.

1. For reluctances joined in series, the total reluctance is the
sum of the individual reluctances.

MAGNETISM 103

2. For reluctances joined in parallel, the reciprocal of the
total reluctance is the sum of the reciprocals of the individual
reluctances.

87. Limitations. — While the idea of the magnetic circuit is an
extremely useful one for purposes of calculation, it must not be
regarded as a true physical concept, such as the electrical circuit,
but merely as an analogy serving a useful purpose. Among the
respects in which the analogy fails are the following:

1. There is no such thing as a magnetic substance in the sense
in which we have used it, and hence there can be no magnetic
flow.

2. When once the magnetic flux has been established, no
energy is required to maintain it, and there is nothing correspond-
ing to the PR consumption of energy in the electric circuit.

3. The reluctance of a circuit containing ferro-magnetic mate-
rial is not a constant for a given set of physical conditions but
varies with the flux, while the resistance of an electric circuit
is independent of the current flowing.

4. For ferro-magnetic materials, the reluctance is not a single
valued function of the flux, but depends upon the magneto-motive
forces to which they previously have been subjected. In other
words, there is no analogy, in the electric circuit, to Hysteresis.

88. Magnetization Curves. — Para-and diamagnetic substances
are characterized by the fact that, under a given set of physical
conditions, the permeability remains constant; that is, as the
magnetizing field is changed, the induction changes by propor-
tional amounts. This, however, is not true of ferro-magnetic
substances. If a piece of unmagnetized iron, for example, is
placed in a field which may be varied at will, it is found, starting
with H = 0 and gradually increasing it, that the induction B
increases slowly at first, remaining nearly proportional to the
field; then increases rapidly, for a certain interval of H, after
which a further increase produces only relatively small increments
in B. The curve showing the values of induction for different
magnetizing fields is called the "magnetization curve," and is
represented by OB of Fig. 58. The three parts of the curve,
differentiated by rather abrupt changes in slope, are accounted
for by assuming that, in the unmagnetized condition, the
magnetic axes of the molecular magnets are distributed
entirely at random, as many pointing in one direction as in any
other; and that the magnetic circuits, of which they form parts,

104

ELECTRICITY AND MAGNETISM

are small closed curves. Under the action of a weak magnetic
field, these molecular magnets are all sprung to a slight extent
from their initial positions, giving a resultant component in the
direction of the applied field, the amount of deformation being
proportional to the field. Thus the part of the curve near the
origin is obtained. With a further increase of field, some of
these local magnetic circuits are broken, and new alignments
formed; giving chains of molecules of considerable length. As
each local circuit breaks, becoming part of a chain, neighboring

Magnetization and hysteresis curves.

groups become unstable, break, and form other chains, thus
giving a sort of spontaneous magnetization, resulting in changes
in induction much greater than required for proportionality to
changes in field. Thus the steep part of the curve is given. As
the condition is approached in which all the local groups have
been broken up and the molecules placed in complete alignment,
the iron is said to become saturated, and further increases in field
produce only small changes in induction. So the upper part of
the curve which is nearly horizontal is obtained.

89. Hysteresis. — If, after the induction has been carried to
the point marked + Bmax on the curve of Fig. 58, the magnetizing
field is gradually reduced, the induction does not retrace the
magnetization curve, but takes on values, for a given field, greater
than those for the magnetization curve; and when H has been
reduced to zero, an amount of induction indicated by Br still
persists. If a reverse field is applied, the induction rapidly falls;
and when a certain value, —He, has been reached, the resultant
induction is zero; after this, a further negative increase in field to

MAGNETISM 105

— ^max gives a reversed value of induction — 5max equal in magni-
tude to +fimax- With a gradual increase in // to its original
positive value, B assumes values shown by the lower curve of the
figure, symmetrical with respect to the origin with the upper one
just described. This tendency of any material to persist in a
given magnetic state is known as "hysteresis," and the corre-
sponding curve is called the hysteresis curve. Bg is called the
retentivity, and He, the coercive field.

It may be shown that the area of the hysteresis loop is a
measure of the energy consumed by molecular friction in each
cubic centimeter of material when carried once through a mag-
netic cycle. For this purpose, let us refer to the ring specimen
described in Art. 75, and use the nomenclature there indicated.
The method of proof is based upon the fact that, as the current in
the magnetizing coil is changed, producing changes of flux in the
ring, a counter E.M.F. is induced, against which the magnetizing
ciirrent must flow. The electrical energy which thus disappears
is the energy consumed by hysteresis and reappears in the form of
heat within the ring. Let i represent the instantaneous magnetiz-
ing current and let dB and d<f> be the changes in induction and
flux, respectively, when a change di occurs in the magnetizing
current. If dt represents the time required for this change to
take place, the energy dw consumed during the change is given by

dw = eidt (19)

But

^-^ dt ~ ^^ dt

(20)

Therefore

dw = NAidB

(21)

Since

we have

XT' ^^

(22)

Substituting

HLAdB F„,^

dw = . = j^HdB

(23)

where V is

the volume of the ring. Summing

up for the complete

cycle, we 1

lave

V r V

1 dw =

W = JndB^-^-i.rea

of loop)

(24)

106 ELECTRICITY AND MAGNETISM

It is thus seen that the area of the loop divided by 47r gives the
energy lost per cycle per cubic centimeter of material. The
shape of the loop varies with the quality of the iron; hard steels
have both a high retentivity and coercive force; soft steels, a
high retentivity but a low coercive force ; while Swedish iron has
both a low retentivity and low coercive force. For a given speci-
men, the area of the loop depends upon the limits of induction.
Steinmez has made an exhaustive study of this relation and has
found that the energy lost is proportional to the 1.6 power of the
maximum induction. Expressed in symbols,

W = KB'-' (25)

K is called the Steinmetz coefficient.

90. Practical Methods.^ — For the measurement of magnetic
induction, there are three general methods, each of which possess
certain advantages as well as disadvantages. They may be
classified as follows:

1. The Traction Method.

2. The Magnetometer Method.

3. The BalUstic Method.

The first method consists in measuring the mechanical force
required to pull the magnetized specimen away from a massive
piece of iron. Since the specimen induces in the block at the point
of contact a pole of strength equal and opposite to its own, the
force required to separate them is proportional to the square of
the intensity of magnetization. In the second method, the
specimen is made in the form of a rod or elongated ellipsoid and
magnetized by being placed within a long solenoid. Its mag-
netic moment is determined by observing the deflection it
produces upon a small compass needle, called a magnetometer,
placed near it. From the magnetic moment, the intensity of
magnetization, and hence the induction, may be computed.
In the ballistic method, the specimen under test forms the whole
or part of a closed magnetic circuit, wound with suitable mag-
netizing coils, and also a secondary coil, connected to a ballistic
galvanometer. Any change in flux induces in the secondary a
quantity of electricity which is measured by the ballistic gal-
vanometer and from this quantity the change in flux is computed.
From the standpoint of accuracy and ease of performance the

1 EwiNG, Magnetic Induction in Iron, chap. II.
DuBois, The Magnetic Circuit, chap. XI.

MAGNETISM

107

ballistic method is much to be preferred and is the only one which
will be considered here.

91. Hopkinson's Bar and Yoke.^ — This is an application of the
ballistic method in which the samples to be tested are in the
form of rods, closely fitted into holes in a heavy yoke of soft iron.
The arrangement is shown in Fig. 59, where FF is the yoke and
CC the specimen under test. MM are the magnetizing coils
and F the secondary coil. The magnetic lines through the
specimen return, half through the upper and half through the
lower part of the yoke. Since the cross section of the yoke is
large in comparison with that of the specimen, its reluctance

©:

:::-

'.'.'. ¥.1..

Fig. 59. — Hopkinson's bar and yoke.

may be neglected without appreciable error and the entire
reluctance be considered as that part of the bar within the slot.
The rod consists of two parts joined at a point a little to the
right of F, one of which is clamped at C, while the other may be
drawn out by the ring at C. Springs are attached to the second-
ary coil F, generally called the "flip" coil, which runs between
guides. While the bar is being subjected to the desired mag-
netizing field, the part C is quickly withdrawn, releasing F which
is jerked suddenly out of the field, cutting the entire flux through
the specimen. The induction B, for a given value of //, is
computed in the following manner:

EwiNG, Magnetic Induction in Iron, p. 67-92.
Smith, Electrical Measurements, chap. X.

108 ELECTRICITY AND MAGNETISM

Let K = constant of the ballistic galvanometer

Uf = turns on flip coil
R = total resistance of secondary circuit
i = instantaneous current in secondary circuit
e = instantaneous E.M.F. induced in secondary circuit

df = deflection of galvanometer
<l> = total flux in specimen

As = area of specimen
Nm — magnetizing turns

Im = magnetizing current

Lft = length of rod (length of slot in yoke)

The quantity Q of electricity, expressed in coulombs, discharged
through the galvanometer is given by the expression

Q = Kdf = fidt (26)

But

Hence

_ e _ n/ dcf) _ rifAs dB

~~ R~ \mt Tt ~ Wr ~dt

(27)

Therefore

n,As r J. _ UfAsB

^ = ^^ ^^- (29)

The constant K of the ballistic galvanometer may be obtained by
means of the standard solenoid described in Art. 26. Substitut-
ing the value of K from Eq. 36.

The field strength to which the specimen is subjected is given by
the regular formula for the ring solenoid

rj _ 47riV ml m fn-,\

^ - ^olT (^^^

92. Experiment 14. Magnetization Curves by Hopkinson's
Bar and Yoke. — Connect the apparatus as shown in Fig. 60,
where C and Y are the bar and yoke, respectively. G' is a ballistic
galvanometer and DE a standard solenoid for calibrating it.
Since a considerable range of currents will be required, use two
ammeters, one of range 0-15 amperes and the other, a millam-
meter connected as shown, where Si is a knife switch which should
be left closed during all manipulations. Open Si when it is

MAGNETISM

109

desired to read the millammeter and then only when the 0-15
ammeter indicates a current less than the full scale reading of
the millammeter. S2 is also a knife switch. First compute, by
means of eq. (31), the upper and lower limits of current required
for field strengths ranging from 1 to 120 gilberts per centimeter.
Before proceeding to test a specimen, it must first be

:::

]_;

-0 0

iUJ

lua

fv ""On

■^— 1||||||| WWWVNA

Fig. 60. — Connections for Hopkinson's bar and yoke.

demagnetized. This is done by applying a magnetizing current,
somewhat greater than that required for the maximum test field,
and reducing it by small steps, reversing the commutator at
each step until a current barely readable on the millammeter has
been reached. The rate of commutation should not exceed 20
reversals per minute. In this way, the specimen is magnetized
first in one direction, and then in the other, each time to a less

110 ELECTRICITY AND MAGNETISM

extent, until finally all magnetism has disappeared. This
should be done with the flip coil out of position or with the second-
ary circuit broken, to avoid damaging the galvanometer. To
test for residual magnetism, flip the coil with no current in the
magnetizing coils. A deflection not exceeding a millimeter
should be obtained. Next, determine the constant of the gal-
vanometer. To do this, set the double pole double throw switch
so as to connect in circuit the primary of the standard solenoid,
and, with a steady current of about two amperes flowing, reverse
the commutator in such a direction as to cause the galvanometer
to swing to high numbers. Make several determinations in this
manner, using such values of primary current as will give deflec-
tions ranging from 2 to 14 centimeters. It is necessary here to
reverse the primary current, not simply to make or break it,
since that is the assumption on which the formula for the deter-
mination of the constant was derived. Use the average value
of the ratio of current to deflection in eq. (30).

Everything is now ready for the test proper. Set the double
pole double throw switch again so as to include the magnetizing
coils and the rheostat so as to include the maximum resistance.
Close the battery circuit and bring the current up to the
smallest value computed above. Flip the coil and note the
deflection of the galvanometer, which should swing in the same
direction as used when determining its constant. Obtain, in this
manner, about flfteen points on the magnetization curve, spaced
more closely together in the lower part of the field strength range,
where the curve rises steeply. Caution.— Points must be taken
always with increasing field strength. Do not allow the current
to rise too high and then decrease it. Obtain data for the magne-
tization curves for the samples of iron furnished. Check your
galvanometer constant before and after taking each set.

Report. — 1. Plot magnetization curves for the four samples
using B as ordinates and H as abscissas.

2. Calculate the permeability for each value of H and, on a
separate sheet, plot permeability as ordinates and field strength
as abscissas for each sample.

3. For the maximum field strength, compute the magneto-
motive force, total flux and reluctance of the magnetic circuit for
each sample, expressing each quantity in its proper units. What
is the relation between maxwells and gausses?

MAGNETISM

111

93. The Rowland Ring, i— In the bar and yoke method des-
cribed above, errors are introduced due to imperfect magnetic
contact between the ends of the rods and the rod and yoke. This
objection is overcome by making the specimen in the form of a
ring, either turned true in the lathe from a soUd block, or built up
of sheet stampings. The magnetizing coil is then wound uni-

E — 1||||| ^WWWW

Fig. 61. — Connections for Rowland Ring Method.

formly over the entire magnetic circuit with the secondary wound
over the primary. Since the turns on the inner side of the ring
are closer together than on the outer, tiie former part of the ring
will be subject to a greater mangetizing force than the latter, and
therefore, the thickness of the ring should be small compared to its

* EwiNG, Magnetic Induction in Iron, chap. III.
Smith, Electrical Measurements, chap. XII.
Rowland, Phil. Mag. vol. 46, 1873, p. 151.

112 ELECTRICITY AND MAGNETISM

diameter, the requisite area being obtained by increasing the
height. As the magnetic circuit cannot be broken, it is impossi-
ble to obtain any measurement of the magnetic state of a given
ring, so the method of observation is Hmited to measurements of
changes of magnetic state produced by definite changes in
magnetizing force.

A magnetization curve may be obtained by a series of reversals
carried out in the following manner : Suppose the iron to be in an
unmagnetized condition. Apply a weak magnetizing field. The
induction rises a small amount along the desired magnetization
curve. Reverse the magnetizing field. This causes the induc-
tion to change along the upper half of a small hysteresis cycle,
taking on a value the negative of what it had before the reversal
occurred. The change of induction, which is measured by the
ballistic galvanometer, is then twice the total induction existing
before the reversal. Now increase the field to a somewhat larger
value, thus carrying the induction to a higher point on the curve.
By again measuring the change of induction on reversal, twice the
new value of induction is obtained, and so on for a series of points
determining the entire magnetization curve. By this process,
the iron is taken around a series of successively larger and larger
hysteresis cycles, the apexes of whose corresponding curves lie
upon the desired magnetization curve. The induction B, for a
given value of H, is computed in the following manner:

Let K = constant of the ballistic galvanometer
Ns = turns on secondary coil
R = total resistance of secondary circuit
e = instantaneous E.M.F. in secondary circuit
i = instantaneous current in secondary circuit
ds = deflection of galvanometer
</) = total flux in ring
Ar = cross sectional area of ring
Nm = magnetizing current
Lr = mean circumference of ring

The quantity of electricity Q, expressed in coulombs, discharged
through the galvanometer is given by the expression

Q = Kds = fidt (32)

But

• = 1 = ^^^^ = ^sAr dB ,„„.

* R lO^dt lO^R dt ^ ^

MAGNETISM 113

Substituting

^^^ - IO^J_^f = lO^R- (34)

^ ^ 2iV^>- (36)

The constant K of the ballistic galvanometer may be obtained by
means of the standard solenoid method, as described in Art. 26;
Substituting the value of K from eq. (36), we have

„ AirNnA I ,

The field strength to which the specimen is subjected is given by
the formula

10L« ^^^>

94. Experiment 15. Magnetization Curves by the Rowland
Ring Method. — Connect the apparatus as shown in Fig. 61.
R is the ring specimen under test, and Nm and Ns the primary and
secondary windings, respectively. (? is a ballistic galvanometer,
and DE a standard solenoid for cahbrating it Since a consider-
able range of current will be required, use two ammeters, one of
range 0-15 amperes and the other a millammeter, connected as
shown, where Si is a knife switch which should be left closed
during all manipulations. Open Si when it is desired to read the
millammeter and then only when the 0-15 ammeter indicates a
current less than the full scale reading of the millammeter. First,
compute by means of eq. (37), the upper and lower fields from .5
to 100 gilberts per centimeter.

Before proceeding to test a specimen, it must first be demagne-
tized. This is done by applying a magnetizing current somewhat
greater than that required for the maximum test field, and reducing
it by small steps, reversing the commutator at each step until a
current barely readable on the millammeter has been reached.
The rate of commutation should not exceed 20 reversals per minute.
This should be done with the secondary circuit broken to avoid
damaging the galvanometer. Next determine the constant of the
galvanometer. To do this, set the double pole double throw
switch so as to connect in circuit the primary of the standard sole-
noid and with a steady current of about 2 amperes, reverse the
commutator in such a direction as to cause the galvanometer to
swing to high numbers. Make several determinations in this

114 ELECTRICITY AND MAGNETISM

manner, using such values of current as will give galvanometer
deflections ranging from 2 to 14 centimeters. It is necessary here
to reverse the primary current, not simply to make or break it,
since that is the assumption on which the formula for the galvano-
meter constant was derived. Use the average value of the ratio
of current to deflection in eq. (36).

. Everything is now ready for the test proper. Set the double
pole double throw switch again so as to include the primary on
the ring, and the rheostat R so as to include the maximum resis-
tance. Close the battery circuit and bring the current up to the
smallest value computed above. Bring the galvanometer to
rest, reverse the primary current, and note the galvanometer
deflection. Now bring the commutator back to its original posi-
tion, increase the current to a slightly greater value, and read the
galvanometer deflection again on reversal. Obtain, in this man-
ner, about 15 points on the magnetization curve, spaced more
closely together on the lower part of the field strength range
where the rise is rapid. Caution. — Succeeding points must
always be taken with increasing field strength. Do not allow
the current to rise too high and then decrease it. Obtain data
for the magnetization curves for two samples of iron. Check
your galvanometer constant before and after taking each set.

Report. — 1. Plot magnetization curves for the two samples,
using B as ordinates and H as abscissas.

2. Compute the permeability for each value of H, and, on a
separate sheet, plot permeabilities as ordinates and field strengths
as abscissas.

3. For the maximum field strength, compute the magneto-
motive force, total flux, and reluctance of the magnetic circuit for
each sample, expressing each quantity in its proper units. What
is the relation between maxwells and gausses?

95. Experiment 16. Hysteresis Curves by the Rowland Ring
Method.^ — Connect the apparatus as indicated in Fig. 61, and
observe the precautions regarding use of ammeters, switches,
rheostats, etc., indicated in Exp. 15. Instead of starting with
zero field and making changes of induction which are sym-
metrical with respect to the origin, as in the case of the magneti-
zation curve by reversals, start here with the maximum field and
make changes of induction by passing first to the retentivity

^ EwiNG, Magnetic Induction in Iron, chap. V.
Tayloh, Physical Review, vol. 23, p. 95.

MAGNETISM 115

point and then away from it. All measurements of induction are
accordingly to be made with respect to the retentivity point,
and we will, for the moment, regard this point as the origin from
which the upper half of the hysteresis curve is to be plotted.
The method will be made clear by reference to Fig. 58. Apply
first the maximum field, giving + ^max on the curve. Now
reduce the field to zero. The induction changes along the upper
part of the curve, and goes to the retentivity point, the actual
change being equal to Bi, Now apply the field — ^max- The
induction changes along the curve from Br to B^ax, the actual
change in induction being B2. Bi and B2 are thus located on the
curve with B^ as the origin. An intermediate point, such as
B3 may be obtained by applying again the field +//max and slowly
reducing to -hHs without breaking the magnetizing current.
If the magnetizing current is now broken, the induction again
returns to Br and the change, which is measured by the galva-
nometer deflection, is B3. This locates B3 with respect to Br.
The corresponding point, 5 4 may be obtained by applying the
field —/fa and observing the throw of the galvanometer. In a
similar manner, a series of points, corresponding to pairs of
positive and negative values of H, may be obtained and the
upper half of the curve plotted with respect to Br.

The actual manipulation of switches is as follows: Obtain the
constant of the galvanometer as explained in Exp. 14. With
the galvanometer circuit broken, set the rheostat to give the
maximum magnetizing current. Reverse this current several
times through the primary coil of the ring to remove the effects
of previous magnetization, and thus make sure that the iron will
follow the cycle desired. With maximum current flowing, close
the secondary circuit and bring the galvanometer to rest.
Break the primary circuit by the switch S2 and observe the throw
of the galvanometer which measures Bi. Bring the galvanometer
again to rest with the secondary circuit closed. Reverse the
commutator. Close S2 and note the deflection of the galva-
nometer which measures B2. Break the secondary circuit, reverse
the commutator, bringing the induction back to +i?max- Reduce
the current, without breaking the circuit, to give a value -j-Hs.
Close the secondary circuit and bring the galvanometer to rest.
Break the primary by means of «S2 and the throw of the galva-
nometer measures B3. Bring the galvanometer to rest, reverse
the commutator, and close ^2. The deflection of the galvanome-

116 ELECTRICITY AND MAGNETISM

ter measures B4, the induction corresponding to —Hz. The
other points on the curve are obtained in pairs in the same manner.
It is important to notice that before each pair of observations
is taken the induction must first be returned to +-Bn,ax, otherwise
a different cycle will be carried out for each pair. Obtain in this
way at least ten pairs of values for B, using field strengths rang-
ing from .5 to 30 gilberts per centimeter. It will assist the
calculation if deflections corresponding to positive and negative
fields are recorded in separate columns. Two samples are to be
tested.

The calculation of the values of B is carried out by the same
formula as used in Exp. 14 except here we wish the total change
in induction instead of half of it as was the case there. Accord-
ingly the limits of integration in eq. (33) are 0 and B instead of
-\-B and —B, giving as our final formula.

Before plotting the curve, the origin should be changed from Br
to 0. This is accomplished by adding Br to all values of induc-
tion corresponding to positive fields and subtracting all values of
induction corresponding to negative fields from Br. Br is
determined from the relation

Br = 5max — Bi = — — Bi (39)

The lower half of the curve, being symmetrical with the upper, is
plotted from these same data, merely changing the signs of all
values of B.

Report. — 1. Plot the hysteresis curves for the two samples of
iron, making the plots as large as convenient.

2. Measure the area of the curves by means of a planimeter, and
determine the energy loss per cycle per cubic centimeter. Since
it is not convenient to plot B and H to the same scale, if unit length
along the B axis represents b gausses, and unit length along the

H axis, h gilberts per centimeter, unit area will represent -j- ergs

per cc.

3. Compute the Steinmetz coefficient for each sample.

CHAPTER IX

SELF AND MUTUAL INDUCTANCE i

96. General Principles. — Whenever a change occurs in the
number of magnetic lines linking any electrical circuit, there is
induced within the circuit an electromotive force, which, if the
circuit is closed, will cause a current to flow. It makes no
difference by what means this change is produced; whether mag-
nets in the neighborhood are moved, currents in adjacent circuits
changed, or the current in the circuit itself varied, the nature of
the induced electromotive force is the same. The direction of
the induced electromotive force is given by a simple rule known as
Lenz's law, which may be stated as follows: Whenever a change
occurs in an electromagnetic system, the direction of the induced
electromotive force is such that the magnetic action of its current
opposes the change. For example, if the north pole of a magnet
is moved toward a closed helix, the induced current flows in such
a direction as to produce a north pole on the end toward the mag-
net, thus tending to repel it, and vice versa, when it is with-
drawn. The magnitude of this induced E.M.F. per turn is given
by the expression

where (j) is the total flux passing through the turn at any instant.
If the change of flux through the coil is produced, not by mov-
ing toward it a magnetic pole but by changing the current
in another coil placed near it, the phenomenon of the induced
E.M.F. is called mutual induction. The coil which is producing
the change of flux is called the primary and that in which the
E.M.F. is induced, the secondary. If the current in the primary
of two coaxial coils is rising, let us say in the clockwise direction,
on looking along the axis, an application of Lenz's law shows that
the current in the secondary is flowing counter-clockwise, while
if the current in the primary is decreasing, the secondary current

1 Duff, A Textbook of Physics, p. 445.

Reed and Guthe. College Physics, p. 365. Starling, Electricity and
Magnetism, chap. XI.

117

118 ELECTRICITY AND MAGNETISM

is in the same direction as the primary. Since the flux through
thje secondary at any instant is proportional to the current in the
primary, we may write for the total E.M.F. in the secondary

. = ^1 (2)

where i is the primary current and M a constant depending upon
the area of the two coils, their number of turns, distance apart,
the permeability of the medium surrounding them, etc. M is
called the coefficient of mutual inductance, the unit of which has
been named the henry.

Definition. — Two coils have one henry of mutual inductance, if,
when the primary current is changing at the rate of one ampere per
second, the induced E.M.F. in the secondary is one volt.

When the current through any coil is changing, there is a
change of flux, not only through any coil in the neighborhood, but
also through the coil itself, causing an induced E.M.F. within it.
This phenomenon is known as Self Induction. The direction of
this E.M.F., considering the coil to be its own secondary, is
determined by Lenz's law, as given above; i.e., when the current
is rising, the induced E.M.F. is in such a direction as to oppose
the current, and when the current is falling, it tends to maintain
it. The induced E.M.F. always opposes any change in the
current and is called a counter E.M.F. Since the flux through
the coil at any instant is proportional to the current, the induced
counter E.M.F. is given by

where i is the current at any instant and L a constant depending
upon the number of turns in the coil, its area, shape, permeability
of the surrounding medium, etc, L is called the coefficient of
self inductance and the unit is the henry.

Definition. — A coil has one henry of self inductance, if, when
the current through it is changing at the rate of one ampere per
second, the induced counter E.M.F. is one volt.

Since the henry is a relatively large unit, it is customary in
expressing the inductance of ordinary coils, to use a unit only
one-thousandth as large, called the millihenry. Variable stand-
ards of self and mutual inductance are made by mounting
two coils in such a way that their relative positions, and hence
their inductive interactions may be changed. If the coils are

SELF AND MUTUAL INDUCTANCE

119

connected in circuit separately, one being used as the primary and
the other as the secondary, a caUbration curve may be made
showing the mutual inductance between them for various posi-
tions. If, however, they are connected in series and used as a
single coil, a variable self inductance is obtained, since the resultant
self inductance of two coils, with mutual inductance between
them, is given by the formula

L = Li + L2 ± 2iW (4)

where Li and L2 are the separate coefficients of self inductance.
If the coils are mounted in such a manner that advantage may be
taken of both positive and
negative values of M, variable
self inductances of consider-
able range may be obtained.
Two forms of variable stand-
ard are in common use.
Figure 62 represents the
Ayerton and Perry variable
inductor which consists of
two coils mounted vertically
one of which is fixed and the
other movable. The coils are
wound on spherical surfaces,
and the inner one rotates
about a vertical axis. When
the planes of the coils are
parallel, the resultant self
inductance is a maximum or a
minimum,^according as the mutual is positive or negative. When
the coils stand at right angles to each other, the resultant self
inductance is the sum of the self inductances of the two coils,
since the mutual is zero for this position. For other positions of
the movable coil, intermediate values are obtained. The
relation between resultant self inductance and angular position
is nearly linear. Two pointers on the top read, one the angular
position of the coil in degrees, the other the self inductance in
millihenries. The coils are joined in series by a flexible conductor.
Separate binding posts for the coils are usually provided, and, when
used independently, the instrument serves as a variable standard
of mutual inductance also.

Fig. 62. — Ayerton and Perry variable
inductor.

120 ELECTRICITY AND MAGNETISM

The other instrument is known as the Brook's inductor and is
illustrated in Fig. 63. It consists of six coils mounted in pairs in
three hard rubber discs, placed one above the other in a hori-
zontal position. The upper and lower disks are fixed and the
middle one rotates between them. If the coils are joined in series
and connected so that their fields on one side are all directed
upward, and on the other side downward, the resultant self
inductance is a maximum; but if the middle disk is turned through
180°, the mutual inductance between the fixed and movable
coils will neutralize the self inductance and the resultant will be

Fig. 63, — Brook's variable inductor.

a minimum. By properly shaping the coils, an approximately
linear relation is obtained between angular position and
inductance. Separate binding posts enable the coils to be used
independently giving also a variable standard of mutual induc-
tance. The instrument is provided with two scales which read
respectively self- and mutual inductance in millihenries.

97. Comparison of Inductances.^ — Two coefficients of self
inductance may be compared by a bridge method in which the
two coils, whose inductances are to be compared, form two arms
of the ordinary Wheatstone bridge. Let Li and L^ of Fig. 64 be
two inductances having resistances Ri and R^, respectively,
and Rz and Ri be two non-inductive resistances, and let the
bridge be balanced for steady currents, as explained in Art. 31,
the condition for which is

R\Ri = R2R3

This condition signifies that when the currents, ^i and t2 are
constant, the potentials at C and D are equal, but less than the

1 Carhart and Patterson, Electrical Measurements, p. 255.
Smith, Electrical Measurements, p. 197-203.
Maxwell's, Elect, and Mag., vol. 2, p. 367.

SELF AND MUTUAL INDUCTANCE

121

potential at A. If the battery key Ki is opened, the current
ceases to flow and the potentials at C and D become equal to
that at A. When Ki is again closed, the potentials at C and D
on account of the counter E.M.F.'s of self induction in Li and L2
will not necessarily rise at the same rate, although they will
come to the same final values. Hence, there may be a short
interval of time during which a difference of potential exists
between C and D giving a deflection of the galvanometer if K2
is closed. By properly adjusting Li and L2 it is possible to cause

Fia. 64. — Bridge method for self-inductance.

the potentials at C and D to rise at the same rate when the bridge
is balanced for both steady and varying currents. The condi-
tions for such a balance is obtained in the ordinary way, except
that the equations must include terms representing the fall of
potential due to the counter E.M.F. Equating the difference
of potential at any instant between A and D to that between A
and C, also that between D and S, to that between C and »S, we
have

Riii + L

dii r> . .J dii

= /V2t2 ~r L12 j7

and

Jtzil = ^4**2

(5)

(6)

122 ELECTRtCtTY ANT) MAGNETISM

whence

dt "* dt

'■ wr. 1

«.^ = «-^ (7)

Eliminating ii and , , we have

RiRiii + RJji-jT = R%Rz,i\ + RaLz-jr. (8)

Since RiRi = RzRz, the condition for steady current balance, the
condition for the varying current or inductive balance is

L1R4 = L2R3 (9)

r = I' (i">

98. Experiment 17. Comparison of Two Coefficients of Self
Inductance by the Bridge Method. — Connect the apparatus as
shown in Fig. 64, where Li is the unknown inductance and L2
a variable standard. R3 and R4 may be two ordinary resistance
boxes with non-inductively wound coils connected with a slide
wire LM for accurate balancing. Ki and K2 should be two
ordinary press keys. First, using for B a battery of about two
volts, obtain a steady current balance by closing Ki first, and K2
after the current has had time to rise to its final value. Try to
keep Rz and R4 between one hundred and three hundred ohms.
For the inductive balance, use a battery of 20 volts. Close K2
first and then lightly tap Ki, never leaving it closed for more than
an instant, since the large currents would cause too great a heating
of the resistances. The motion of the galvanometer in this case
will be a sudden "kick" not a steady deflection. Adjust L2 until
this kick has disappeared. Read the value of L2 and compute the
value of Li from eq. (10). The unknown to be determined con-
sistsof a spool with two independent windings. Determine the
inductance of each separately, then join them in series, and
determine the resultant self-inductances with their mutual induc-
tances aiding and opposing, making in all four measurements.

Note. — It may happen that the balance point lies beyond the
range of the variable standard, making an inductive balance im-
possible. When this happens, the ratio Rz to R^ must be changed
so as to bring the balance point within the required range.
Since a steady current balance must always be obtained first,
this requires the insertion of a small non-inductive resistance in
series with either Ri or R2 as the case may demand. For ex-

SELF AND MUTUAL INDUCTANCE

123

ample, suppose the inductive kick of the galvanometer decreases
as L2 is increased to its maximum, but cannot be made zero or
reversed. The combination of the two balance conditions gives

L12 Ki R2
If, then, an appropriate resistance is connected in series with Ri
the new steady current balance condition will give a larger ratio
of JKs to Ri thus making the inductive balance possible. If, on the
other hand, L2 cannot be made small enough, the additional
resistance must be placed in series with R2.

Report. — 1. Tabulate your data for the determination of the
four inductances as indicated.

2. From the formula L = Li + L2 + 2M, compute M from
the cases where it is aiding and opposing the self inductance.
The agreement of these two values gives a check on the accuracy
of your work.

3. How are coils wound so as to be non-inductive?

Fig. 65. — Mutual inductance by Carey-Foster method.

99. Measurement of Mutual Inductances. ^ — The mutual
inductance of two coils may be measured in terms of capacity
and resistance by means of a method due to Carey-Foster, in
which the quantity of electricity induced in the secondary is
balanced against a known charge from a standard condenser.
The connections are shown in Fig. 65, where P and S are the

» Carey-Foster, Phil. Mag., vo. 23, p. 121.
Carhart and Patterson, Electrical Measurementa, p. 268.
Smith, Electrical Measurements, p. 217.

124 ELECTRICITY AND MAGNETISM

primary and secondary coils of the mutual inductance to be
measured, C a standard condenser, and G a ballistic galvanometer.
The primary circuit is represented by the path BPARi, while the
secondary is SR^DA including the galvanometer G. It will be
noted that the galvanometer is also included in circuit DCR\A
containing a standard condenser. When the primary circuit is
closed the galvanometer will be traversed by two distinct quanti-
ties of electricity: (1) The quantity induced in the secondary coil,
and (2) the charge entering the condenser, both of which may
easily be computed. If these two quantities are equal and pass
through the galvanometer in opposite directions, no deflection
will result, which is the balance condition sought.

The quantity Qi induced in the secondary coil is the time
integral of the secondary current, during the interval required for
the primary to rise from zero to its final value /.
That is,

^^ = /^-^^ = f /1>^ (12)

Mr MI

= rI ^' = ^ ^'')

where R is the effective resistance of the secondary circuit. The
quantity Q2 of electricity passing through the galvanometer to
charge the condenser is given by

.Q2 = CV = CRJ (14)

where V = RJ is the fall of potential across Ri which is charging
the condenser. Equating,

^= CRJ, (15)

M = CRiR. (16)

Since, at the point of balance, there is no current through the
galvanometer, and consequently no fall of potential across it,
the effective resistance R of the secondary circuit includes only
Ri and S. The final formula then becomes

M = CRr(R, + S) (17)

If C is expressed in farads, and the resistances iii ohms, M will be
given in henries.

100. Experiment 18. Mutual Inductance hy the Carey-Foster
Method. — Connect the apparatus as shown in Fig. 65, where PS

SELF AND MUTUAL INDUCTANCE 125

is a variable mutual inductance whose calibration curve is to be
obtained, C a subdivided standard condenser, G a ballistic gal-
vanometer of long period, and B a storage battery of 20 volts.
It is necessary that the four wires indicated at A should actually
meet at a common point, so a connector should be used. Since
a large voltage is connected directly across Ri there is danger of
burning it, so compute the minimum resistance which may be
used, allowing a maximum power consumption of 4 watts per
coil. To make sure that the discharges through the galva-
nometer oppose one another and are of the same order of
magnitude, try them first separately; that is, break the circuit at
C, make and break the primary circuit and note the direction of
the galvanometer deflection at the make, due to the induced
current in the secondary. Now close the circuit again at C,
breaking the secondary at R^, and note the deflection at make,
which is now due to the charge entering the condenser. If the
deflection is in the same direction as before, reverse the connec-
tions on either the primary or secondary coil. Close the circuit
at Ri and obtain abalance varying Ri, R2, and C. The resistance
of the secondary coil may be obtained by means of a post-office
box.

Report. — 1. Plot mutual inductance in millihenries, as
ordinates, and positions of coil as abscissas.

2. How would your results be affected if you had interchanged
primary and secondary coils? Explain.

CHAPTER X

ELEMENTARY TRANSIENT PHENOMENA^

101. Time Constant, Circuit Having Resistance and Induc-
tance.— When an E.M.F. is suddenly impressed on a circuit
containing resistance only, the current rises instantly to a definite
value determined by Ohm's law. If, however, the circuit con-
tains inductance as well as resistance, this is not the case, for
while the current is' being established, it produces within the coil
a magnetic flux which links the turns of the coil. Whenever a
change occurs in the flux through a coil there is induced within it
an E.M.F. in such a direction as to oppose the change which
produced it. From the definition of self inductance, the value

di
of this counter E.M.F. is L-r^ where L is the coefficient of self

inductance. It is thus seen that the impressed E.M.F. is opposed
R z, by two counter E.M.F.'s; one due to

— AVSAA/ nPPS^f^ — I the current flowing through the re-
sistance and the other due to the
rising current in the coil. Such a
circuit is represented in Fig. 66, where

E the resistance R and the inductance L

Fio. 66.— Circuit containing ^re shown Separately, although they
resistance and inductance. . . ., ^

may co-exist in the coil. Let the

value of the current, / seconds after closing the key, be i. Then

by Ohm's law, we have

E = Bi+4; (1)

This is a differential equation and can not be solved by the
ordinary rules of algebra. Dividing through by R and letting

E
I = nhe the final value of the current, we have

* Bedell and Crehore, Alternating Currents
Pierce, Electric Oscillations and Electric Waves.
Steinmetz, Transient Phenomena.

126

ELEMENTARY TRANSIENT PHENOMENA

127

Separating the variables, we obtain

di R _,,

J ; = Y dt

I — I L

Integration of both sides gives

-log {I -i) = -^t + C

(3)

(4)

where C is an arbitrary constant whose value may be obtained by
substituting corresponding known values for i and t. Counting
time from the instant the key is closed, when t = o, i = o, and
these quantities when substituted in eq. (4) give C = —log /.
Hence eq. (4) becomes, on replacing C by its value and rearranging,

log

(/ - i)

--h

Taking the antilogarithm of both sides,

I -i

= e

-?'

where e is the base of the Naperian logarithms,
have

i=l{l-e-r)

(5)

(6)
Solving for i, we

(7)

FiQ. 67.

Otitz ta <4 t

-Growth of current in a circuit containing resistance and inductance.

The graph of this equation for a series of values of L with constant
R and E is shown in Fig. 67. It is seen that when L = o the
last term of eq. (7) vanishes and the current rises immediately to
its final value ; but as L is made larger a longer time is required for
it to reach a given fraction of its final magnitude. It is obvious
that inductively wound coils might be classified according to the
time required for the current to reach a certain specified fraction

128

ELECTRICITY AND MAGNETISM

of its final values under a constant impressed E.M.F. The most
suitable fraction to choose is arrived at in the following way.

If, in eq. (7), t = „' there results

-^0-.^) =

,6327

-AAA/VWNA-

The quantity p is called the "Time Constant" for the coil and is

defined as the time required for the current to reach .632 of its
final value under the action of a constant E.M.F. The values
ti, h, ts, etc., in Fig. 67 represent the time constants for the various
values of L.

102. Circuit Having Resistance and Capacitance. — A case

quite similar to the one discussed
above is that in which an E.M.F. is
suddenly impressed upon a circuit
containing resistance and capacitance
in series. Such an arrangement is
shown in Fig. 68. As soon as the
key is closed, a current flows through
R and a charge begins to accumulate
in C. This charge at once produces
a counter E.M.F., which, added to
that due to the current through R, balances the impressed E.M.F.

Fig. 68. — Circuit containing
resistance and capacitance in
series.

The potential difference across the condenser is ^ or^ | idt

Accordingly, we may write

E = Ri +

^^J'

^ I idt.

(8)

It is more convenient to solve this equation in terms of the
instantaneous charge q in the condenser than of the current

through the resistance,
substitution in eq. (8),

Remembering that i = -r; vfe have, on

E =

^dt-^c

(9)

Multiplying through by C and putting CE = Q, the final charge
in the condenser, eq. (9) becomes, on separating the variables,

dq _ dt
W^ ~ RC

(10)

ELEMENTARY TRANSIENT PHENOMENA

129

Integrating both sides of cq. (10), we have

-log (Q-q) = -^^+K (11)

As before, K is an arbitrary constant of integration which may be
evaluated by subtistuting known values of q and i in eq. (11).
Counting time from the instant of closing the key, we have, when
t = 0, q = 0. Substituting in eq. (11)

K = -logQ
Replacing K by its value, and rearranging terms, eq. (11) becomes

1 (Q - q) t

,og i^SJ = - _ (12)

Taking the antilogarithm of both sides, we have

Q-q

= e

~RC

Solving for q, there results

q((i-

~RC

(13)

(14)

This equation is analagous to eq. (7) of the previous article and
its graph is shown in Fig. 69, for several values of R with constant

o tl «2 ^3 ti t

Fig. 69. — Growth of charge for a circuit containing resistance and capacitance.

E and C. li R = o, the condenser becomes charged instantly
to its final value Q, but when a series resistance is included, a
definite time is required for the condenser to become charged. Such
circuits may be classified according to the time required for the
charge to reach a specified fraction of its final value. As before
this fraction is arrived at by putting t = RC.

130 ELECTRICITY AND MAGNETISM

Eq. (14) then becomes

9 = (?(l -^)= .632Q.

The quantity RC is called the time constant for a circuit
containing resistance and capacitance, and is defined as the time
required for the charge to reach .632 of its final value. These
times are shown for the successive values of R by h, h, tz, etc., in
the figure. The time constant is an important concept in the
study of reactive circuits and will be referred to frequently in
this text in the discussions to follow.

103. Circuit Containing Resistance, Inductance and Capaci-
tance. Discharge of a Condenser. — To describe some of the phe-
nomena peculiar to a circuit containing resistance, inductance

and capacitance, it will be supposed

A A ^ * A ^..J^r.^^ c that the parts are connected in series

condenser has been charged by ap-
propriate means. Suppose further
that the key has been closed and

/JL

Fig. 70.— Circuit containing that it is discharging; also that the

resistance, inductance, and • j. j. j. • • j -i

capacitance. instantaneous current is i and the

charge in the condenser is q. Since
no external E.M.F. is acting, the sum of the differences of poten-
tial across the three elements of the circuit must be zero at all
times. Accordingly,

L^ +Ri-\-^\idt = 0. (14)

Differentiating and dividing through by L we have

dH R di 1 . , .

Since i = -jr' eq. (14) may also be written

d^q R dq 1 . .

dti+Ldt^LC^ = ^- ^^^^

eqs. (15) and (16) are sufficient, to completely describe a circuit

of this character. Since they are identical, only one of them,

e.g., (15), will be discussed.

This is a linear differential equation of the second order with
constant coefficients and may be solved in the following manner:
Let

i = ke""' (17)

ELEMENTARY TRANSIENT PHENOMENA 131

where k is an arbitrary constant depending upon the boundary
conditions and m, another constant, depending upon the coeffici-
ents of the original differential equation. Differentiating eq. (17)
twice and substituting in eq. (15), there results

m^^^m-^~ = 0. (18)

This equation gives the vdues that must be assigned to m in
order that eq. (17) may be the solution of eq. (15). Solving,

-RC ± VR^C^ - 4LC

^ = 2LC (1^)

It is thus seen that there are two values of m which will make eq.
(17) a solution of eq. (15). These give what are known as " par-
ticular solutions" and the "complete solution" is obtained by
adding them together. Accordingly,

rRC - Vr^C^ - 4LC-, rRC + Vfl'C* - 4LC-i

i = k^e L 2LC J _^ J^^ I 2LC J' (20)

The solution for q is identical except that different arbitrary
constants will appear. Call them ks and ki. It is to be noted
that the coefficient of t in the exponential term contains a radical,
the quantity under which may be positive, zero, or negative
according to the relative values of R, L, and C. The theory of
differential equations shows that the character of the solutions
under these circumstances is quite different, and that we have
three distinct cases to consider.

Case I. R^C^>4:LC. Non-oscillatory Discharge. — For simplic-
ity, let

_ 2LC _ 2LC

"■^ ~ RC- VR^C' - 4LC ^""^ ^' ~ RC + VR'C' - 4LC ^^^^
The solutions of eqs. (15) and (16) may then be written

i = kie '■i 4- k^e u (22)

q = k^e '•i + kiC u (23)

Ti and T2 are thus seen to be time constants and it is to be noted
that when both inductance and capacity are present, the circuit
possesses two time constants instead of one as in the cases
previously considered. The arbitrary constants fci, fcj, ks, kt
may be determined in the following way. If time is reckoned
from the instant the key is closed, then when

t = 0,i = 0,q = Q (24)

132

ELECTRICITY AND MAGNETISM

Substituting these values in eqs. (22) and (23) there results

0 = A;i + A;2 Q = A;3 + A;4 (25)

Differentiating eq, (23)

i = $= -%~'^ - h~^^ (26)

at Tl T2

Comparing coefficients in eqs, (22) and (26) we have

k^ = -^' and h = -^* (27)

Tl T2

Substituting the values of k^ and ki from eqs. (27) in (25) and
eUminating, the following values are obtained:

Q .. Q

ki =

T2 — Tl

Qti

Tl — T2

r% — Tl

Tl — T2

Substituting these values in eqs. (22) and (23) we have

(28)

I =

'^ — e

Tie

_ t '

'"2

_ t '
Tie ''•'■

(29)
(30)

T2 — Tl
Tl — T2

It is thus scon that the solutions are made up of two exponential
curves whose difference is to be taken.
In the case of the current, these curves
have the same initial ordinates but
approach the time axis at different
rates because of the different time
constants. The solution is shown
graphically in Fig, 71, where the
dotted curves are the separate ex-
ponentials and the full line represents

their difference. The current starts at zero, rises to a maximum

and then slowly dies away.

Case II. R-C^ = 4 LC. Critically Damped Discharge. — In

7?
this case the roots of eq. (18) are identical having the value — ^r

and the two terms of eq. (22) are the same. This equation
cannot be the complete solution for this case since it contains
but one arbitrary constant, whereas the complete solution
must have two, since the original differential equation is of the
second order.

-Aperiodic discharge
of a condenser.

ELEMENTARY TRANSIENT PHENOMENA 133

The theory of differential equations' shows that for this case
the sohitions of eqs. (15) and (16) are

i = kie 2L _|- k^te 2L (31)

Imposing the same boundary conditions as before, namely, when
t = 0, i = 0, and q = Q, we have

ki = 0 and ks = Q

Differentiating eq. (32)

Applying the first boundary condition to eq. (34) gives
, _ ksR _ QR

Comparison of coefficients in eqs. (31) and (33) gives

^ _ ^ ^ _ Q^- ^
"' ~ '21"' ~ 4L2- L

The complete solutions accordingly are

i = -yte~ 2l' (34)

These equations consist of the product of a straight line and an
exponential curve, and are similar to the corresponding ones for
Case I. If numerical values are substituted, it is found that they
rise to higher values and that they are more compressed along
the time axis. In fact, the theory shows that for this critical
case the discharge takes place in the shortest time possible.
Case III. R^C^<4LC. Oscillatory Discharge. — This is the
most interesting and important of the three cases. The quantity
under the radical sign of eq. (19) then becomes imaginary and the
two roots of eq. (18) are complex quantities. Call them

Ml = a -\- j/3 and niz = a — j^
where

' Murray. Differential Equations, p. 65.

134 ELECTRICITY AND MAGNETISM

Equation 20 may then be written

= e"'[A;i(cos fit + j sin fit) + fcg (cos fit - j sin fit)]
= e^iiki + ki) cos fit + {ki - ki) j sin fit]

Let

whence

A — jB
ki = ~— then ki -\- kz = A

k, = ^-^ti^ k^-k2 = -jB

i = e"\A cos fit + B sin fit]

By means of a well known formula of trigonometry this may be
written

i = ke"' sin {fit + 0) (36)

where

/- A

k = VZ2+52 and <^ = tan"! ;g

In a similar manner the solution of eq. (16) for this case may be
shown to be

q = k'e"' sin (fit + 0') (37)

The four arbitrary constants A;, A;', </>, </>' are real quantities and may
be determined by imposing the same boundary conditions as
used above. Substituting in eqs. (36) and (37) the values i = 0,
q = Q for ^ = 0 respectively they become

0 = k sin </) whence <^= 0 (38)

Q = k' sin «/.' k' = S-,

sm 4>'

Differentiating eq. (37) with respect to t

i = ^ = k'e"^ [a sin {fit + </>')+ i3 cos {fit + (j,')]

= fc'e«'[V«2 +/32 sin {fit + </>'+ tan-^^)] (39)

Using again the condition z = 0 for / = 0 we have

^ . fi V4LC - 722(72

tan <A = - ^ = ^e __

. ^, ^ Q _ Q _ ViLCQ

sin 0' . , V4LC - 722C2 \/4LC - 722(72
sm tan~^ — ^^^

ELEMENTARY TRANSIENT PHENOMENA

135

Comparing the coefficients of the sine terms in eqs. (39) and (36)
we have

The complete solutions may now be written

I =

\/4LC - R'^C^ '

sm-

2LC

9 =

V^LCQ
\/ALC - R^C'

fj t . \y/^LC - R^C^

Sin

2LC

t-\-

tan"

,^ ViLC - R^C^'

(40)

(41)

The current and charge are sine functions of the time and are
therefore oscillatory in character. The initial amplitude of the
oscillations is proportional to the charge given to the condenser

Fig. 72. — Damped sine wave.

and depends also upon the constants R, L, and C of the circuit.
Furthermore, the amplitude is multiplied by an exponential
factor which decreases with the time and the oscillations conse-
quently die out. An oscillation of this character is spoken of as a
"damped" sine wave. The graph for the current wave is shown
in Fig. 72. That for the charge is similar to it except that its
phase is ahead of the current by the angle whose tangent is given
by the last term in eq. (41). If R = 0, this angle is 90°.

The period T of the oscillation is obtained from eq. (40) by the
relation

^ V4LC - R^C^ ^ 27r
'^ 2LC T

136 ELECTRICITY AND MAGNETISM

whence

2LC

If R^C^ may be neglected in comparison to 4LC, this reduces
to the simple expression

T = 2tVLC (42)

104. Logarithmic Decrement. — The physical interpretation of
the phenomenon just described in mathematical terms is as
follows : When the condenser is given a charge, a definite amount
of energy, 3^C V^, is stored up in it. As it discharges and current
flows through the circuit, this energy is in part dissipated by the
resistance R and in part stored up in the electromagnetic field of
the inductance L. At the instant the potential difference across
the condenser is zero the energy which has not been dissipated as
heat is in the coil has the value }^LP. This energy, minus
that dissipated during the next quarter swing is returned to the
condenser charging it in the opposite direction and so on. If
the circuit were entirely free from resistance, the oscillations would
simply represent interchanges of energy between the condenser
and the coil at a frequency twice that of the circuit and would
continue indefinitely in much the same manner as a pendulum
suspended by frictionless bearings in a vacuum. It is obvious
that the greater the rate of energy dissipation, the smaller the
number of oscillations. The quantitative method of treating
the damping effect is as follows:
Write eq. (40) in the simplified form

i = /e~"' sin ut (43)

where

T ^ 2Q ^ R ^ \/4:LC^^R^^
V^C-R^C^"" 2L " 2LC
Let 1 1, h, h /n be the successive current maxima as indi-
cated in Fig. 72, let h, U, tz ^„ be the times at which they

occur, and let T be the period of oscillation. Since

sin cj^i = sin w/2 = =1

In = e-

t(.U + (n-l)T)

ELEMENTARY TRANSIENT PHENOMENA 137

The ratio of the first ampHtude to any succeeding one is

^^^a(n-l)T (44)

In particular, let n = 2. Then

It is easily seen that the ratio of any amplitude to the next one
succeeding it is constant and is equal to the value just given.
Taking the logarithm of both sides

log, ^ = aT = tR\y =8 (45)

The quantity 8 is called the "Logarithmic Decrement" and
is defined as the Naperian logarithm of the ratio of any amplitude
to the next one succeeding it in the same direction, and is given
in terms of the constants of the circuit by eq. (45). One
of the many applications that may be made of this quantity is
the determination of the number of oscillations that the circuit
will execute before the amplitude is reduced to an assigned frac-
tion of its initial value. For example, between /i and /„+i, there
are n oscillations.
Substituting in eq. (44)

^1 anT nS

Y = e = e

1 n + l

whence

lOge Y = ^5
i n+l

where

1, /l

w = ~ log. Y

8 in + l

•'"+1 ;c +K« noc,^T^r.A f».o«

■t n

106. Harmonic E.M.F. Acting on a Circtiit Containing Resis-
tance, Inductance and Capacitance. — The equation of E.M.F.'s
for this case is

LjA- Ri + ^jidt = E sin o/. (46)

The theory shows that when the second member of a differential
equation is different from zero, the complete solution is made up
of two parts : (a) the solution of the original differential equation
when the second member is put equal to zero, and (6) the par-

138 ELECTRICITY AND MAGNETISM

ticular integral. Part (a) has already been discussed and it was
found to represent a transient phenomenon which quickly
dies out. Part (6) corresponds to a ''forced" oscillation, and
represents a steady state. It is the part in which we are inter-
ested in problems of continuous alternating currents.

The student familiar with differential equations will remember
that equations of the form of (46) are best treated by means of
the differential operator "D." To apply this, first differentiate
eq. (46) with respect to t to remove the sign of integration

dH , R di , 1 . Eo) . , ._.

Introducing the operator D

(Z)2+^Z)+^)t= ^cosa,^ • (48)

The particular integral which we are seeking is then

i = 5 :r- ^^ COS 0}t (49)

The meaning of the "inverse" operator, as the quantity
immediately following the equality sign is called, is this: Find
a function of i such that when operated on by the coefficient of
i in eq. (48) it gives the right-hand member of that equation.
There is a weU known short method^ for treating the case of sines
or cosines such as eq. (49). It consists simply in expressing the
function of D as a function of D^ and replacing D^ by minus the
square of the coefficient of the independent variable. Accord-
ingly

1 Eo) , Eu}

I = p ^^j J- cos Oit — :j cos cot =

cos o}t =

Rw^ - Q,-L(^y

Eoi^RD -^ -Lco2)J

See Murray, Differential Equations, p. 77.

2 COS ut =

ELEMENTARY TRANSIENT PHENOMENA 139

Euiy;, — LCO^ j COS Oit

Eca^R sin ut

i22a,2 +Q- Lc^'Y «'«'+ Q - Lo:^)

ERsinoit , „ \Co} ^"/

+ E :, ; COS (at

combining into a single sine function

I = - sin

Ceo
o}t — tan~i

(50)

It is thus seen that the current is a sine function and has the
same frequency as the impressed E.M.F. In general, it is not in
phase with the E.M.F. but lags behind or leads according as

Leo is greater or less than 7;— • If Lw = tt", i.e., o) = /— — > the

Ow Oco -y/LC

current is in phase with the E.M.F. and in this case eq. (50)

becomes

. E . .
t = n sin col

which is identical with the current equation given directly by
Ohm's law for the case when no inductance or capacitance is
present. The maximum value of the current is obtained by
putting the sine function equal to unity: i.e.,

/= ^

V«^+(^"-(s)^

By analogy with Ohm's law the denominator is called the "Impe-
dance" of the circuit and the quantities Leo and 77- are called

Ceo

the inductive and capacitive "Reactances" respectively. Reac-
tance produces not only a phase angle between the current and
E.M.F. but also reduces the magnitude of the current.

106. Alternative Method. — For those unfamiliar with differ-
ential equations, an indirect method of obtaining the solution of
eq. (46) may be employed. Since an alternating E.M.F. is appUed
to the circuit, it is reasonable to suppose that the current will also
be alternating, that it will have the same frequency as the E.M.F.

140 ELECTRICITY AND MAGNETISM

and that it may not be in phase with the E.M.F. These assump-
tions are combined in the following expression

i = / sin (ut + <t>) (51)

where / and </> are arbitrary constants which are to be determined
by substituting eq. (51) in (46) and finding the values which must
be assigned to them in order that eq. (46) may be satisfied.

di I

-J. = loi cos (cat + 0) and J'idt = — cos {o>t + <t>)
at CO

Substituting these values, eq. (46) becomes

Lib) cos (ut -\- <f>) -\- RI sin (cot + <^) — ^^^ cos (cot -\- <j>) = E sin at

Since this equation holds for all values of t, we may write,
when

(ot ■}•(}> = 0, Lid — = —E sin </>

when at + (t> = ^^ RI = E COB 4>

Squaring and adding the above expressions,

Dividing one by the other

La - -^ La - j^

— tan 0 = 5 or </) = — tan-i y^

ih H

Substituting these values in eq. (51) we have

t = — j ~ sin

v«' + (^" - ly

La - ^

Ceo
at — tan~^

which is eq. (50) above.

107. Vector Diagrams. — In the discussion thus far, we have
spoken of alternating E.M.F.'s and alternating currents and have
used in each case the trigonometric expressions in discussing
them. For example the equations

e = E sin at

i = I sin (at — <i>)

ELEMENTARY TRANSIENT PHENOMENA

141

have been used to represent respectively an alternating E.M.F.

having a maximum value E and a frequency/ = ^' and an alter-
nating current of the same frequency with a maximum value I
lagging behind the E.M.F. by a phase angle <l>.

These may be regarded as being given by the projections on the
Y axis of the vectors OE and 01 respectively of Fig. 73 which
rotate with constant angular velocity in counter clockwise direc-
tion, the latter lagging behind the former by the angle <^. The
vectors OE and 01 represent the maximum values of the E.M.F.

Fig. 73. — Sine waves represented by rotating vectors.

and current. As a special case, consider that of an alternating
E.M.F. acting on a circuit having resistance and inductance. The
current is given by eq. (50) with C = oo , the condition for zero
capacitive reactance. Thus

I =

Vi^'+Z/'^co^

sin I Oil — tan~^

(52)

For the maximum value of the current, we have

From the form of the latter expression it is evident that E has
such a value that it may be given as the diagonal of a rectangle
whose sides are RI and Lw/, as shown in Fig. 74. The current /
is represented as a vector in phase with RI, since, from eq. (52),
the current lags behind the E.M.F. by an angle whose tangent is

L(i)

-p ' This is the angle <f> shown m the figure. If this figure is

rotated about the origin 0 with an angular velocity co in the posi-
tive direction, the projections of the vectors, E, RI, and Lul upon
the Y axis give the instantaneous values of the impressed E.M.F.

142

ELECTRICITY AND MAGNETISM

and the E.M.F. across the resistance and the inductance
respectively.

In a similar manner, a vector diagram may be constructed for
a circuit containing resistance, inductance and capacitance in
series. For this case the maximum current and phase angle are
given respectively by

I =

^m + (l. - ^)

0 = tan~^

Lfa) —
R

Lul >^

Lul

Fig. 74. — Vector diagram for resistance
and inductance.

Fig. 75. — Vector diagram for resis-
tance, inductance and capacitance.

The vectors are similar to those of the previous case except

for the additional vector, t^, which is shown drawn downward in

Ceo

Fig. 75, since in the equation it appears as a quantity subtracted

from Leo. In the figure, LcoZ is shown greater than 77- and the

Ceo

current lags behind the E.M.F. If Leo = jr~' the component of E

Ceo

perpendicular to I is zero and the current is in phase with the

E.M.F. On the other hand when 79- is greater than Leo, <^ is

Ceo

negative, and the current leads the E.M.F.

108. Electrical Resonance. — In discussing the discharge of a

condenser through a circuit containing resistance and inductance,

it was shown that when the resistance is less than a certain critical

ELEMENTARY TRANSIENT PHENOMENA

143

value, oscillations occur. If such a circuit is acted upon by an
alternating E.M.F. whose frequency is the same as the natural
frequency of the circuit, alternating currents of large amplitude
are set up in the inductance and condenser. This phenomenon is
spoken of as electrical resonance and is analogous to the motion of
a mechanical system possessing inertia and elasticity, when acted
upon by an alternating mechanical force having a frequency
corresponding to its own free period. Two distinct cases occur
depending upon whether the inductance and capacitance are in
series with the E.M.F. or are connected across it in parallel.
These are distinguished as "Series Resonance" and "Parallel
Resonance" respectively.

109. Series Resonance. — This case has been discussed above in
some detail. The instantaneous value of the current must
satisfy eq. (46) the solution of which is eq. (50). The amplitude
of the current is given by

I =

yi^' + (^- - ity

and it has already been pointed out this is a maximum when
1

Lo) = /^' which

Co?

is the condi-

tion for resonance. The cur-
rent is then given by E
divided by R as required by
Ohm's law. The resonance
condition depends upon the
relative values of L, C and co,

—nmnr-AmAr-i

Fio. 76. — Series resonance.

Fio. 77. — Effect of resistance on sharp-
ness of resonance.

and may be brought about by a suitable change of either one of
them, the other two being held constant. Bringing a circuit
into resonance is generally spoken of as "tuning" it.

The dependence of the current upon the constants of the

144

ELECTRICITY AND MAGNET I SiM

circuit may be illustrated by the curves shown in Fig, 77, where
the current aniphtudc is shown as a function of frequency for a
short range each side of resonance. The inductance and capaci-
tance are held constant and three different resistances are
indicated. A represents the current at resonance for a small
resistance and C that for a large. It is to be noted that the effect
of a change in resistance is much more marked at resonance than-
at a frequency somewhat removed. This is because at reson-
ance, resistance alone determines the current, while at low fre-
quencies, the capacity reactance j;r is an important term, but at

high frequencies, the inductive reactance Lw becomes effective in
reducing the current. It is to ])e noted also, that for low fre-
quencies the current leads the E.M.F., is in phase with it at
resonance and lags behind at high frequencies. When the resis-
tance is small, the rate of change of the phase angle in passing
through resonance is rapid.

110. Parallel Resonance.^ — When the E.M.F. is introduced in
the circuit in such a way that the inductance and condenser are in

parallel, the phenomena are strik-
ingly different from those of the
series arrangement just described.
The connections for this case are
shown in Fig. 78. Assuming that
the condenser is free from energy
absorption, the current through it
leads the E.M.F. by ninety degrees,
while that through the inductance
lags behind by an angle depending
upon R, L, and co. The current in
the main circuit is the vector sum
of these two and in determining it
the relative phases of the components must be taken into
account. Denoting the currents through the inductance and
condenser by h and Ic respectively, their amplitudes are obtained
from eq. (50) as follows

h = , , -- — Ic = ECoi (53)

Letting the vector OE of Fig. 79 represent the impressed E.M.F.,
1 Giro. 74. U. S. Bureau of Standards, p. 39.

L-VWVW^WH

Fig. 78. — Parallel resonance.

ELEMENTARY TRANSIENT PHENOMENA

145

the above currents are given by OIc and
OIl respectively, and the resultant cur-
rent 01, is the diagonal of the paral-
lelogram formed by them as sides. The
ampUtude of the resultant current, by
the law of cosines, is:

P = Ii} + Ic^ - 21 Jc cos yp (54)

The value of cos yp may be obtained by
remembering that the E.M.F. across
the coil is made up of two parts:
That across the resistance, RIl, and
that across the inductance Lw/l. The
former is in phase with II and the latter,
ninety degrees ahead of it. Accordingly

cos \p =

(55)

Substituting eqs. (53) and (55) in (54)
and combining we have

Fig. 79. — Vector diagram
for parallel resonance.

P = E^

C2aj2 +

2CcoLaj

R^ -\- L^u)^ R^ -h LW\

(56)

Multiplying numerator and denominator of the second term by
R^ -f L2a)2, eq. (56) may be written

= eJ(Co}

Lu)

R^ + L2,

"^ (/e^ TL2w2)"2 (57)

Equation (57) is the general expression for the current drawn
from the supply. The condition that this current should be in
phase with the driving E.M.F. is

P = h' - Ic'
Substituting the values from eqs. (53) and (56) there results

Co, =

«2 -j- U ««

(58)

The value of w obtained from this equation is not exactly that
corresponding to the natural period of the circuit but approxi-
mates it closely. If R is zero, it corresponds exactly. Introduc-

146

ELECTRICITY AND MAGNETISM

current with fre-
in the neighbor-

ing this condition in eq. (57) it is seen that the current, when in
phase with the E.M.F., is

^ ^ W+'IA? ^^^^

It is important to note that for small values of R, I is nearly-
proportional to R and that if R were zero, / would also be zero.
We thus have the extraordinary situation in which the larger
the resistance the larger the current. Figure 80 shows the varia-
tion of
quency

hood of resonance. It is
interesting to note that in
the case of series resonance
the individual voltages
across the condenser and
coil exceed the total volt-
age across the two com-
bined, while in parallel
resonance, the current in
each exceeds the two com-
bined. The series arrange-
ment gives a low imped-
ance at resonance, while
the parallel connection gives a high impedance at this point.
For this reason, the latter is frequently inserted in a circuit
when it is desired to suppress a particular frequency in a complex
wave.

111. Measurement of Inductance and Capacitance by Reso-
nance.— The phenomenon of electrical resonance furnishes a
convenient method for the determination of inductance and
capacitance, particularly when they are small. If two circuits,
adjusted to have the same natural periods are placed in inductive
relation, and one of them is caused to oscillate, the other will
oscillate also by resonance. It was shown on page 136 that the
period of an oscillating circuit is given by the expression

t = 2Try/LC.

Consequently, the condition for resonance is that the LC prod-
ucts for the two circuits must be the same or

Fig. 80. — Dependence of current on fre-
quency for parallel resonance.

iviCl — Ld2^2

(60)

ELEMENTARY TRANSIENT PHENOMENA

147

where the subscripts refer to the circuits 1 and 2 respectively.
If thrise of these quantities or one LC product and either L or C
are known, the fourth may be computed. In carrying out the
measurement it is more satisfactory to use a third circuit as a
source of oscillations, and then adjust both the standard and
unknown circuits to resonate to it. The inductance and capaci-
tance of the third circuit should be adjustable, but need not be
known. The three circuits are shown in Fig. 81. The source
circuit is energized by means of the battery A and an ordinary
buzzer B which serves as an interrupter. When the armature

r'lmm^s^ T-nmm^ r^imum^

^DOC^

-11

FiQ. 81. — Circuits arranged for electrical resonance.

of the buzzer closes the circuit, the battery current flows through
the coil L3 and stores up energy in the electromagnetic field link-
ing its windings. When the armature of the buzzer breaks the
battery circuit, this energy is transferred back and forth between
C3 and L3 until it has been dissipated. A group of damped oscil-
lations is thus established in this circuit for each vibration of the
buzzer armature. Similar oscillations but of weaker intensities
will be set up in circuits 1 and 2 if they are adjusted to resonate
to 3.

If small inductances and capacitances are used the frequency
of the oscillations thus produced will be above the audible range,
and special means for detecting them must be employed. A con-
venient method is to use a pair of head phones and a crystal
detector such as is commonly employed for the reception of radio
signals. Because of the rectifying action of the point-crystal
contact the high frequency alternating voltage across the con-

148 ELECTRICITY AND MAGNETISM

denser will produce a series of high frequency unidirectional
pulses in the phone circuit. Because of the distributed capaci-
tance of the phone windings, these are smoothed out into a single
pulse which causes a vibration of the diaphragm. The sound in
the phones then has the period of the buzzer armature. If
a sufficient amount of energy is available, it is best to disconnect
the right-hand phone lead shown in the figure, and use only a
single wire from the phone through the crystal to the oscillatory
circuit. This is particularly important when the condensers are
small since the capacity between phone leads may introduce a
very appreciable error.

112. Experiment 19. Measurement of Inductance and Capaci-
tance by Resonance. — Connect the apparatus as shown in Fig. 81,
using for Li a standard inductance variable by steps, and for Ci
a variable standard air condenser. L2 should be a single layer
coil of uniform windings whose dimensions may easily be meas-
ured, d should be an air condenser with plates easily accessible
for measurement. First obtain resonance in circuit 2 by varying
the frequency of the source. Next obtain resonance in circuit 1 and
compute the LC product. Measure the dimensions of C2 and
compute its capacity from eq. (19) given on page 94. (See also
the Appendix.) Determine Li from eq. (60). Check your result
by computing the inductance of Li from dimensions using the for-
mula given in the Appendix.

113. Effective Value of an Alternating Current. — If an alter-
nating current is passed through an ordinary D.C. ammeter, no
indication will be registered, since such an instrument indicates
average values, which in this case is zero. However, if an alter-
nating current is passed through a resistance, heat is liberated,
the energy of which is furnished by the current. The reason for
the difference in effect in these two cases is that the torque on the
moving coil of the instrument is proportional to the current and
therefore reverses sign with it, while the heating effect of a current
is proportional to its square and is therefore positive no matter
what its direction.

It is customary to define the Effective value of an alter-
nating current as the equivalent direct current which liberates the
same amount of heat in a given resistance per unit time. In
deducing the relation between the effective value of an alternating
current and its amplitude or maximum value, it is sufficient to
equate the heat, in joules, developed by each during the time T

=^'«X(^

ELEMENTARY TRANSIENT PHENOMENA 149

of one complete cycle. Accordingly let i = I sin (at be the alter-
nating current and /« its effective value. When flowing through
a resistance 72, the heat liberated by each is

H = IMT = fi^Rdt = T^R flin oitdt (61)

Jo Jo

COS 2u}t\ ,.

-2-r

- /2p/t _ sin2fa)f\-|^^ PRT

Therefore :

h = -4- = -707 7 (62)

The above process is seen to be equivalent to squaring the
instantaneous values of the current, taking the average value
of the squares, and then extracting the square root. The
effective value accordingly is often spoken of as the "Root Mean
Square" value. The same considerations hold for an alter-
nating E.M.F.

114. Power Consumed by a Circuit Traversed by an Alter-
nating Current. — ^Let us suppose that an alternating E.M.F. is
impressed upon a circuit which contains reactance as well as
resistance so that the current and E.M.F. are not in phase. It is
desired to find the power consumed by the circuit. Let the
E.M.F. and current be given respectively by the following
expressions

e = ^ sin o}t; i = I sin(a)< + 0) (63)

where 0 is the angle of lag or lead.

The energy dH consumed in the time dt is

dH = eidt = EI sin tat sin(ojf + <i))dt (64)

The energy H consumed per cycle is

H = EI \ sin co<(sin (at cos </> ± cos (at sin <i>)di

= EI \ cos </) I sin* (atdt ± sin </> I sin (at cos (atdl

^J rn cos2a,<\,, ' ^ T- ,
= EI I cos (f) \ I ^ n — )dt ± sm <|) I sm (at cos (atdt

150 ELECTRICITY AND MAGNETISM

Carrying out the integrations and substituting limits the last two
integrals vanish and we have

H = EI J cos <^

_ Energy per Cycle EI

Power '^—y— = ^71^ C08 *

It is thus seen that the power consumed is the product of the
effective E.M.F. and current multiplied by the cosine of the
phase angle. Cosine ^ is called the "Power Factor" and varies
from zero to unity.

CHAPTER XI

SOURCES OF E.M.F. AND DETECTING DEVICES FOR
BRIDGE METHODS

Before discussing the various bridges which are to be employed
in the measurement of inductance and capacitance, the student
should become familiar with some of the sources of alternating
E.M.F. and detecting devices that are available. Inasmuch as
the alternating currents for commercial purposes are of frequencies
too low to give a tone suitable for telephonic balances, special
generators have been devised, a few of which will now be described.

115. The Sechometer. — In Exps. 12 and 17 methods were
employed for comparisons of capacitance and inductance respect-
ively in which batteries were employed to energize the bridges
and the E.M.F. 's due to the reactances were made manifest during
the rise and fall of the bridge currents following the closing and
opening of the battery circuit key. It was then found that the
galvanometer deflected in one direction on closing and in the
opposite on opening this key. If some means were available by
which the galvanometer leads could be interchanged each time the
key is opened and closed, the deflections would always be in the
same direction and if the interval between successive operations
of the key were small compared to the period of the galvanometer,
a steady deflection would result whereby the sharpness of the
bridge balance would be greatly increased. The Sechometer
is a device which accomplishes this purpose and derives its name
from the "Secohm" by which our present unit of inductance, the
henry, formerly was known.

It consists essentially of two commutators mounted on the
same shaft which may be driven at any desired speed by a motor.
The segments are set in such a relation to each other that the
galvanometer leads are interchanged by one commutator each
time the polarity of the battery is reversed by the other. The
connections are shown in Fig. 82. The device must not be driven
at too high a speed since sufficient time must be allowed for the
establishment of a steady state at each reversal. High speeds

151

152

ELECTRICITY AND MAGNETISM

also develop heat at the brush contacts resulting in errors due to
thermal E.M.F.'s.

It is well to get an approximate bridge balance by manipulating

Fig. 82. — Sechometer connections to bridge.

the battery and galvanometer keys with the sechometer station-
ary and then use it merely to obtain the final setting. The

Fig. 83.— The Sechometer.

application of this instrument is equivalent to using a generator
giving a square wave form and can, therefore, be used only with
bridges which balance independent of the frequency. Figure 83

DETECTING DEVICES

153

shows the assembled instrument provided with a crank for hand
driving.

116. The Wire Interrupter. — The vibrating wire interrupter,
shown in Fig, 84, consists essentially of a piano wire stretched
between rigid supports A and B, the tension of which may be varied
by the screw S. Vibrations are maintained by means of an
electromagnet M, intermittently energized by a battery Bi.
The mercury cup contact C\ interrupts this current when the wire
is drawn up, and the device operates in a manner similar to the
ordinary buzzer. The battery Bt, which supplies current to the

C^ M IpC

Fig. 84. — Vibrating wire.

To Bridge

bridge, is connected through the contact C% and this circuit is
also closed and opened at the frequency maintained by the wire.

Frequencies ranging from 25 to 150 are easily secured. This
device is particularly well suited for use with the vibration
galvanometer, since it permits of sharp, easy tuning, and may
readily be adjusted to resonance with the galvanometer. Since
the vibration galvanometer responds only to the fundamental
and not the overtones, the fact that the interrupter gives a square
wave form results in no disadvantage and the combination may
accordingly be used on bridge circuits which do not balance
independent of the frequency and the same results obtained as
though a source giving a pure sine wave were employed. If a
suitable condenser K is shunted across the contact Ci to prevent
arcing, the device will operate continuously for hours with Uttle
or no attention.

117. The Motor Generator. — Another inexpensive source of
alternating current is the small motor generator set manufactured
by the Leeds and Northrup Company shown in Fig. 85. The
generator, which is shown at the right in the figure, is of the

154

ELECTRICITY AND MAGNETISM

FlQ. 85. Motor Keiiciiilnr vet.

inductor type and has stationary windings for both field and
armature circuits. Direct current is suppUcd to the field coil at
the base thus energizing the magnetic circuit which includes the
broad toothed wheel carried on the armature shaft of the motor.
The reluctance of this magnetic circuit depends upon the position

of the teeth with respect to
the pole pieces. When the
wheel is driven, the flux
through the magnetic circuit
fluctuates at a frequency
equal to the number of teeth
passing the pole tips per
second. An alternating
E.M.F. is thus induced in
the armature windings placed
near the pole tips.
By properly choosing the shape of the pole tips and teeth, it is
possible to obtain a wave form that is relatively free from harmon-
ics although it is not possible to eliminate them completely. By
the use of a suitable filter, good wave forms may be obtained.
By means of a specially designed mechanical governor, the speed
of the motor may be maintained constant to 3^^ per cent.

118. The Microphone Hummer. — If it is not necessary to
supply the bridge with a constant frequency, a simple microphone
hummer furnishes a convenient and
inexpensive source of alternating cur-
rent. Such a circuit is shown in Fig. 86.
It consists of a microphone transmitter
facing a telephone receiver. The trans-
mitter and receiver circuits are con-
nected in the ordinary way by a tele-
phone transformer. Any stray sound
will cause a variation in the microphone
current which produces a sound in the
receiver. This sound is "fed back" to
the microphone which again produces a
sound in the receiver and the action is
thus continuous, the energy being supplied by the battery. The
interval between the application of the sound to the microphone
and its return by the receiver after having operated the electrical
circuits depends to a large extent upon the time constant of the

Fig. 86. — Microphone
hummer.

DETECTING DEVICES

155

secondary or receiver circuit. If the resistance of this circuit is
not too large, it may be made oscillatory by the introduction of a
condenser as indicated in the diagram. By giving suitable values
to the capacitance of this condenser a large range of frequencies
may be obtained. Another transformer, the primary of which is
in series with the receiver, furnishes a means of making the A.C.
power thus generated available for a bridge circuit. A conven-
ient form of this device is manufactured by R. W. Paul of London
under the trade name "Kumagen," the appropriateness of which
is easily understood. The microphone, receiver, and trans-
formers are contained in a felt lined case which serves to deaden
the sound, A condenser is also furnished which is variable in
steps chosen so as to give a number of suitable frequencies.
119. The Audio-oscillator. — The frequency of the microphone
hummer, described above, is somewhat variable depending upon

Fig. 87. — Audio-oscillator.

the strength of the driving battery and the load upon
the secondary of the output transformer. An adaptation of the
underlying principle has been made by Campbell by which this
objection is overcome. It consists in operating the microphone
button, not by sound waves from a telephone receiver, but by
means of a tuning fork whose mechanical period coincides with

156

ELECTRICITY AND MAGNETISM

the period of the electrical circuit which it energizes. Several
different forms are on the market. Figure 87 shows an instru-
ment of this type known as the audio-oscillator, manufactured by
the General Radio Co., and Fig. 88 gives the wiring diagram.
The "field coil" which is connected directly across the battery
serves merely to magnetize the fork and armature core to a point
on the magnetization curve near the maximum permeability and
this increases the attractive forces of the poles. The battery also
sends current through the microphone and primary of the input
transformer. When the battery key is closed, the current through

O^'-O

T"""T'

■Vs

, ^AO/tAruoe coil.

^TTrroTV

'("rnrrmH-

OorooT

tJULflJJUUU

OJC/LLATOfi
ct) naatooe ftasj

xeifo i. otv

/^£OtuM nton

FiQ. 88. — Wiring diagram for audio-oscillator.

the primary of the input transformer induces an E.M.F. in the
secondary which starts oscillations in the resonating circuit
which includes, besides the condenser and primary of the output
transformer, the armature coil. This oscillating current changes
the attraction between the armature pole tips and the prongs of
the fork. Since the secondary circuit is tuned to the period of the
fork, the fork resonates to it, thus building up a vigorous
vibration. The microphone button, being in contact with the
fork, supplies a varying current of this same frequency to the
primary of the input transformer and energy from the battery is
thus furnished to maintain the oscillations, and carry the load put
upon the secondary of the output transformer.

Each transformer coil has a small air gap to prevent distortion,
but their magnetic circuits are sufficiently closed to prevent
disturbing stray fields. The oscillator is self starting and may
be placed at some distance and operated by a key near the bridge.
The coils are so wound that a 6-volt battery furnishes "ample

DETECTING DEVICES

157

power. The device is not designed to furnish more power than
that required by a single bridge circuit. If overloaded, the
microphone is likely to pack. It is carried by a stiff spring
mounted on one prong of the fork and its inertia is suflBcient to
insure response to vibrations of the fork.

120. The Vreeland Oscillator. — None of the sources thus far
described produce alternating E.M.F.'s of a purely sinusoidal
wave form. There are a number of important bridges which do

Fig. 89. — Wiring diagram for Vreeland oscillator.

not balance independent of the frequency and when a telephone
is used as the detecting device, complete silence can not be
obtained with impure wave forms. In such cases, when the
fundamental has been balanced out, the overtones are still heard
and materially mar the sharpness of setting which would other-
wise be possible.

The Vreeland Oscillator is one of the best sources of pure sine
waves available. It is, in reality, a mercury arc rectifier operated
backwards, the connections for which are shown diagrammatically
in Fig. 89. The essential part of the device is a large pear shaped
mercury arc tube with two anodes Ai and Az having a common

158 ELECTRICITY AND MAGNETISM

mercury cathode Ki. It is well known that the mercury arc will
operate only when the mercury electrode is negative. When used
as a rectifier, the condenser and deflecting coil are removed and
the source of alternating E.M.F. is connected to the terminals
G.D. When G is positive and D negative, current will flow from
Alio Ki through the battery B, which is here shown as the load,
to D, and when D is positive, the path is A2K1 MG, these furnish-
ing a current through B in the same direction as before. The
reactances Xi and X2 serve to smooth out the fluctuations through
the battery.

To understand its operation as an oscillator, let us suppose
that the source of ^.C. is removed and that the deflecting coil and
condenser are connected to G and D as shown in the figure. The
battery B now becomes the source of power. An arc is started
between the electrodes Ki and K2 by shaking the tube slightly,
thus causing the mercury pools to unite and break again. The
tube is quickly filled with ionized mercury vapor and the arc
spreads to the anodes Ai and A2. The switch S is then opened
thus stopping the arc to K2. If the impedance of the two paths
MD and MB are equal and the tube is symmetrical, the arc will
divide equally between the anodes ^1 and ^2 which are thus at the
same potential and there is no charge in the condenser. If,
however, some irregularity in the tube causes more current to flow
momentarily to the anode Ai it will be at a higher potential than
A 2 and a charging current will flow to the condenser through the
deflecting coil LL. This coil, which really consists of two parts,
one in front of the tube and the other behind it, is placed so that
its magnetic field is perpendicular to the flow through the arc.
If the polarity is so chosen that the charging current deflects the
arc stream so as to further increase the current to A 1 a very appre-
ciable charge may be given to the condenser. When the condenser
discharges, the deflecting action of the current which is now
reversed will cause more current to flow to the anode A2 thus
raising its potential above Ai and charging the condenser in the
opposite direction. The deflecting coil serves the double purpose
of furnishing a self inductance to form, with C, an oscillatory cir-
cuit, and to automatically deflect the arc streams from one anode
to the other to maintain the oscillations. The frequency is
given by the expression

= 1

2Tr\/LC

DETECTING DEVICES 159

where L is the inductance of the deflecting coil in henries, and C
the capacitance of the condenser in farads. It is found that such
a device, when properly designed, will oscillate at frequencies
ranging from 100 to 4,000 cycles per second.

Another coil, placed near the deflecting coils, serves as the
secondary of an air cored transformer to supply current to a
bridge. It is found that the frequency is but little affected by
changes in the load on the secondary. Because of the relatively
large coils, the instrument possesses an appreciable stray field
and must be placed at some distance from the bridge, to prevent
direct induction in the coils which are being studied.

121. The Electron Tube Oscillator. — One of the simplest and
most effective means of obtaining alternating voltage of any
desired frequency is that in which a three element electron tube
is used to maintain continuous oscillations in a resonance circuit.
The underlying principle is the amplifying action of the tube
which will be described in chap. XIV. It will be sufficient for
the present purpose to point out that the electron tube consists
of a highly evacuated glass container in which are placed a fila-
ment and a metal plate with a grid mounted between them. The
grid consists of a fairly coarse meshed structure of fine wires.
When the filament is heated to incandescence by an electric
current, it emits electrons which may be drawn to the plate by
a battery connected through an external circuit between the
plate and filament. The positive terminal of the battery must
be connected to the plate.

Inasmuch as the electrons, to reach the plate, must pass through
the meshes of the grid, the number arriving at the plate may be
controlled by giving suitable potentials to the grid, and may be
stopped entirely, if the grid is sufficiently negative. Inasmuch
as the energy required to maintain a given potential on the grid
is small, the device acts as an electrical throttle valve, whereby
the available energy of the plate circuit battery may readily
be controlled. Figure 120 of chap. XIV shows the relation which
exists between the plate current and grid volts. The time
required for an electron to traverse the distance from filament
to plate depends upon the potential of the plate but is of the
order of 10~* seconds. Changes in plate current accordingly
follow changes in grid volts with remarkable swiftness.

There are many different circuits in which an electron tube may
be used to generate sustained oscillations. Figure 90 shows one

160

ELECTRICITY AND MAGNETISM

of the simplest. F, G, and P are the filament, grid and plate
respectively. The filament is heated by the battery A, whose
current is controlled by the rheostat R. The battery B furnishes
the potential to draw the electrons from the filament to the
plate. The inductance L and the condenser C form the oscilla-
tory circuit. To understand the way in which oscillations are
sustained, let us suppose that, by closing the switch >S, the estab-
lishment of a current through the coil L has produced a transient

oscillation in the circuit LC.
This would quickly die out if
energy were not supplied to it
to compensate for the losses.
Suppose that the oscillatory
current through L is in the
direction of the arrow and is
rising. Due to the self in-
ductance L there will be an
E.M.F. in the coil in the
direction DE. This lowers
the potential of the grid with
respect to the filament which
thus decreases the plate cur-
rent, flowing through the
part Lp. This decrease in the

"an
A

Fig. 90. — Electron tube oscillator.

plate current induces in L
E.M.F. which tends to keep the oscillatory current flowing,
continuation of this reasoning throughout the changes occurring
during a complete cycle will show that the variations in plate
current always induce in Lp an E.M.F. tending to drive the
oscillatory current in L in the direction in which it happens
to be flowing at any instant. The oscillations would increase
indefinitely in amplitude were it not for the fact that the grid
volt-plate current characteristic of the tube becomes horizontal
at each end.

Frequencies ranging from 1 cycle to several millions per
second may be obtained. Alternating current power for bridge
work may be obtained either by placing another coil near L
which then serves as the secondary of an air core transformer or
by connecting the primary of a telephone transformer in the
plate circuit as shown in the figure. The latter is to be preferred
since variations in the load have a smaller disturbing effect upon

DETECTING DEVICES 161

the frequency than is the case with the former arrangement. The
wave form is not as free from harmonics as that obtained from a
Vreeland oscillator, and a filter must be used in cases where
extreme purity is essential.

DETECTING DEVICES

122. Telephone Receiver.^ — The telephone receiver is one of
the most generally useful of the various instruments for detecting
the balance condition in a bridge circuit actuated by alternating
currents. It consists essentially of a horseshoe magnet upon
which is wound a pair of coils carrying the
current to be detected, and a soft iron dia-
phragm mounted near the poles as shown in
Fig. 91. The current through the coils mag-
netizes the core which attracts the diaphragm
with a force proportional to the square of the
induction produced. The sensitivity of the
receiver is increased by using for the core,
not a piece of soft iron, but £t permanent
magnet. The way in which this is brought
about may be seen from the following con- ^°' j-ec^jver^^ °"^
sideration. Let Bo be the constant induction
through the gap due to the permanent magnet, and let the addi-
tional induction which is proportional to the current i in the
coils be kii. Then the total pull on the diaphragm is given by

Pull = k^B^ = kiiBo + kxiy = kiBo^ + 2kik2Boi + ki%2i^

The first term represents the pull due to the permanent magnet
alone, the second, that due to the current and magnet combined,
while the third is that due to the current alone. If it is desired to
have the motion of the diaphragm follow the variations in the
current so that its motions may reproduce for the human ear
the sound waves acting upon the diaphragm of a distant tele-
phone transmitter, then the receiver must be so designed that the
second term is large compared to the last which contains the
square of the current. The first term need not be considered
since it is independent of the current. The desired effect is
attained by making Bo large compared to kii. Since Bo enters
as a factor in the second term, making it large has the effect of

1 Mills, Radio Communication, p. 27.
11

162

ELECTRICITY AND MAGNETISM

increasing the motion of the diaphragm and hence of making the
receiver, to a certain extent, an amplifying device.

If the third term is not negUgible compared to the second,
then, although there is a repetition with amplification there is
also distortion since the pull which it defines is proportional to the
square of the current. The nature of this distortion can be
understood by supposing that the current is sinusoidal, e.g.,
i = I sin oit. The last term then becomes

kiki^P sin^ 0)1 = kikiH'^

1 — cos 2(at

Quartz
Fibre

Mirror

and it is seen that the distorting pull is made up of two parts:

koki^I^
A constant part — ^ — which need not be considered and a

pulsating part having twice the frequency of the phone current.
Since the diaphragm of the telephone receiver is an elastic

body it will have a frequency of its own and will accordingly
respond more vigorously to frequencies which
correspond to its natural period, and another
source of distortion is thus introduced. For
bridge work, however, this fact may be utilized
to increase the sensitivity by impressing upon
the bridge the frequency to which the tele-
phone resonates. Phones for this particular
purpose are constructed in such a way that
their resonance frequencies may be varied
over a considerable range.

123. Thermo -galvanometer. — The Duddell
Thermo-galvanometer is an adaptation of the
Boys' radio-micrometer for the purpose of
measuring and detecting small alternating
currents. The moving system, shown in Fig.
92, consists of a single turn of silver wire at

Fig. 92. — Duddell -i i ij. /• t.- i. • x- xi. i

thermo-galvanometer. t^e bottom of which IS a tmy thermocouple
of bismuth and antimony. The system is
suspended by means of a fine quartz fibre between the poles of a
strong horseshoe magnet and carries a small mirror by means of
which its deflections are read with a lamp and scale. Imme-
diately below the thermo-j unction is mounted a resistance unit
through which the current to be measured is passed. The heat
from this current is carried to the thermo-j unction by convection

i^t

Bi rns6

[-WVWv-

DETECTING DEVICES 163

and radiation and causes a current to flow through the low
resistance silver loop which is deflected by the electrodynamic
action of the field. Since the heating effect is proportional to
the square of the current while the thermal E.M.F., for small
temperature differences, is proportional to the temperature, the
indications of this instrument are roughly proportional to the
square of the current.

Several heating units are provided with each instrument and
range in value from 1 to 1,000 ohms according to the current
sensitivity desired. For low resistances, they are made of fine
wire bent back and forth but for the higher values, fine platinized
quartz fibres are used. With the latter, current sensitivities of
10~* amperes are obtained. This type of instrument may be
calibrated on direct currents and then used to measure alter-
nating currents. Since it is practically free from inductance,
the instrument may be used for the measurement of currents of
very high frequencies. Because of the low resistance of the
moving system it is critically damped electromagnetically, and is
usually designed so as to have a period of from three to four
seconds. Because of the delicacy of the quartz fibre suspension
and the light silver loop, it is not a robust instrument and must be
handled with caution. The heating elements are easily burned
out and should always be protected by a high resistance which
may be reduced to zero when it has been ascertained that safe
limits of current will not be exceeded. Sudden changes in tem-
perature cause the zero to drift and the instrument is usually
enclosed in a tight wooden case.

124. Vibration Galvanometer.^ — The vibration galvanometer
is one of the most useful instruments available for the detection of
minute alternating currents of commercial frequencies. To
secure suitable sensitivity, advantage is taken of the principle of
resonance. That is, the moving system is so adjusted mechan-
ically, that its natural period coincides with that of the alter-
nating current to be detected. Although the instrument shows
very little response to direct currents or to alternating currents
to which it is not tuned, nevertheless when resonance has been
secured, a very appreciable vibration results. The vibrations
are indicated by means of a small concave mirror carried on the
moving system which focuses the image of an incandescent fila-

* Laws, Electrical Measurements, p. 434.
Wenneb, Bull. U. S. Bureau of Standards, vol. 6, 1909-10, p. 347.

164

ELECTRICITY AND MAGNETISM

ment on a ground glass scale. When the system vibrates,
the image is drawn out into a broad band of light, while very
slight motions are detected by a diminution in the sharpness of
the line.

One of the chief reasons for the superiority of this instrument
is the fact that its response is selective. In many measurements
it is necessary to use a pure sine wave, a thing difficult to secure.
Since vibration galvanometers may be made with a selectivity

llimiiii

BRioer

6UIDE

Fig. 93. — Leeds and Northrup
vibration galvanometer.

Fig. 94. — Tuning mechanism for Leeds
and Northrup vibration galvanometer.

4,000

of that

so high that their response to the third harmonic is

to the fundamental and to the fifth, yTT^' impure waves may

be employed with very little if any inaccuracy introduced. In
fact an interrupter of the vibrating wire type described above,
giving a square wave form, may be employed. The current
sensitivity of the vibration galvanometer is about the same as
that of a good telephone receiver — i.e., 10~^ amperes.

Obviously the instrument may be of either the D' Arson val or
the Thomson type. In Fig. 93 is shown one of the former, or
moving coil instruments, while Fig. 94 shows how the moving

DETECTING DEVICES

165

system is tuned. The coil is held in position by a taut phosphor-
bronze ribbon, the effective length of which is varied by means of
the movable bridge carried on the upper screw. By sliding this
bridge up or down rough tuning is obtained while fine adjust-
ments are secured by slightly changing the tension of the sus-
pension by means of the lower screw and spring.

Figure 95 shows a Tinsley instrument of the moving magnet
type. The vibrating system consists of a small permanent
magnet mounted on a taut metallic ribbon behind which is held

Fig. 95. — Tinsley vibration galvanometer.

the fixed deflecting coil. Specially shaped pole pieces concen-
trate the field of the large horseshoe magnet on the moving
magnet. Since the period of the system is determined largely by
the strength of this external field, tuning is obtained by changing
this field. This is accomplished by moving the soft iron magnetic
shunt along the horseshoe magnet. The milled head shown at
the front of the base operates a worm gear which moves the shunt.
125. Alternating Current Galvanometer. — The alternating
current galvanometer is one of the most sensitive devices avail-
able for detecting the balance condition in a bridge supplied with
an alternating E.M.F. It is essentially a D'Arsonval galva-
nometer with the permanent magnet replaced by an electro-

166

ELECTRICITY AND MAGNETISM

A.C.
Supply

To Bridge

Fio. 96. — Alternating current
galvanometer.

magnet energized from the same A.C. source as that supplying

the bridge. It operates upon the electrodynamometer principle

and the direction of the torque
acting upon the moving coil is
independent of the polarity of
the supply. Its operation is
complicated by the fact that,
when connected to the bridge,
there are present in the coil
two currents, one due to the
unbalanced condition of the
bridge, and one induced by
the alternating flux of the
galvanometer field. Since the
former is small and disappears
at balance, the latter by far
overpowers it and must either

be eliminated or made ineffective.

It may be shown in the following manner that when the current

induced in the coil is 90° out of phase with the flux through it,

the torque is zero.
Let <f> = ^sino)^ = instantaneous flux through the coil and

i = /sin(wf ± d) instantaneous current in the coil.

The torque acting upon the coil in a given position at any

instant is then

T = K^ sin ut I sin {cot ± 6)

where Kisa, constant depending upon the geometry of the instru-
ment. The average value f taken over the time T of one complete
cycle is

T = — I sm Oil sm(co' ± d)dt
= I sin w<(sin o}t cos 6 + cos cat sin d)dt

i- Jo

;=— COS 6 I sin'^ (j}tdt + sin 6 I sin oit cos wtdt

J'*T \ — cos 2o)t , sin 6 sin^ (at\ ^
0 2 - 2 J,

cos 6

T [

The resultant torque on the coil is, accordingly, positive or

DETECTING DEVICES 167

negative depending upon the sign of 6 and is zero for Q = ±-

Since the E.M.F. induced in the coil is 90" out of phase with the
flux producing it, the condition stated above is equivalent to
saying that the induced current must be in phase with the E.M.F.
In bther words, the bridge circuit to which the coil is connected
must be nonreactive. In certain bridges such as those for com-
paring two condensers or two inductances this is obviously-
impossible. The required condition may, however, be met by
shunting across the coil an appropriate variable reactance, e.g.,
an inductance with a variable series resistance in the former case,
or a condenser and resistance in the latter.

When the galvanometer is first connected to the bridge, it will
be found that, due to the action just described, the coil assumes a
very rigid position, including either a maximum or a minimum
amount of the field flux. If the former position results, a leading
current through the coil is indicated, and a shunt with an induc-
tive reactance must be applied. For satisfactory operation, a
certain amount of stability is required to give a constant zero
position, so inductive reactance across the coil should predomi-
nate. Since the reactance of the bridge is an important factor in
determining the rest position of the coil, the galvanometer key
must remain closed, and the balance established by opening and
closing the supply circuit to the bridge.

The alternating current galvanometer has an important advan-
tage over detecting devices such as the telephone or vibration
galvanometer, in that it swings to the right or left according to
the phase of the current at the galvanometer corners of the bridge
while with the latter, no such effect is possible. Furthermore,
if a direct current is supplied to the field, it becomes an ordinary
D'Arsonval galvanometer and may be used to determine the
steady state balance. Its sensitivity may be made 100 times that
of the telephone or vibration galvanometer. It has, however,
one distinct disadvantage, in that the deflection depends not
only upon the field and current through the coil but also upon the
phase angle between them. It can not therefore be calibrated to
measure currents. Furthermore, zero deflection indicates either
no current, or current 90° out of phase with the field. A simple
test for the latter condition is to shift the phase of the field by
inserting a resistance in series with the field coil.

CHAPTER XII
ALTERNATING CURRENT BRIDGES

126. General Considerations. — In order to obtain the reactive
effect of an inductance or a capacitance it is necessary that the
current through it should be variable. In the early bridge meas-
urements for comparing inductances or capacitances and even
for determining an inductance in terms of a capacitance the vari-
able current was obtained simply by closing and opening the
battery circuit leaving the galvanometer permanently connected
to the bridge. The galvanometer employed was usually of the
long period ballistic type. This procedure is open to two objec-
tions. First, the sensitivity thus realizable is not great and
second it may lead to results which are appreciably different from
the effective values of the condensers or coils when employed,
as is usually the case, in circuits traversed by alternating
currents. For example, the effective value of the self in-
ductance of the primary of a transformer when an alternating
current is flowing through it, depends upon the load across
the secondary. If measured by the make and break method
with a ballistic galvanometer as the detecting device, the result
is the inductance of the primary alone independent of the effect
of the secondary.

The ballistic galvanometer is an integrating instrument, and a
zero deflection does not necessarily mean that no current has
passed through it, but that equal and opposite quantities have
traversed it. The bridge may have been out of balance each way
during the time the current through it was changing. It is,
accordingly, much better to use alternating currents through
the bridge and employ a detecting device such as the telephone or
vibration galvanometer, a zero indication of which indicates that
at no time is there a current through it, and that the bridge is
balanced at all times.

It will appear in the discussion which follows that, in order for a

168

ALTERNATING CURRENT BRIDGES 169

bridge with reactive members to be balanced at all times, there
are two conditions which must be satisfied. First, the bridge
must be balanced for direct currents, "steady state balance,"
and second, it must be balanced for alternating currents, "vari-
able state balance." These two balance conditions may be
interpreted in the equation for the bridge in a simple manner.
An expression is deduced, involving one current and its time
derivative. The "steady state balance" means that the coeffi-
cient of the current term is zero. The "variable state balance"
means that the coefficient of the term for the changing current,
i.e., the time derivative, is zero. This applies to any bridge
for which the balance condition may be reduced to an ex-
pression involving only one current and its first time deriva-
tive. Such a bridge balances independent of the frequency.
If a second time derivative is involved, as, for example, in
Trowbridge's bridge and the frequency bridge, the wave form
of the current must be assumed, and the bridge no longer
balances independent of the frequency. In some instances, the
student will find it advantageous to obtain the former by the
use of a battery and direct current galvanometer, and later
apply an alternator and detector for the variable state balance.
After becoming experienced in this type of work, however,
both balances may be obtained simultaneously by the use of
alternating currents.

127. Maxwell's Bridge.' — One of the simplest methods for
determining an inductance in terms of a capacitance or vice
versa is the method known as Maxwell's bridge. It consists of
an ordinary Wheatstone's bridge with three non-inductive
resistances Ri, R2, and R3, as shown in Fig. 97, while the fourth
arm contains the inductance L to be determined. Let the ohmic
resistance of this coil be Ri. To offset the reactance of the coil
L, a condenser C is placed across the opposite resistance Ri.
When an alternating E.M.F. is applied to the bridge, the
current in the upper half will lead the E.M.F., while that
in the lower half lags behind it. Accordingly, if an A.C.
galvanometer or other detecting device is connected ahead of
the inductance and behind the condenser, the arms of the bridge
may be so adjusted that the potential changes at D and E are

' Maxwell Electricity and Magnetism, vol. 2, p. 387.

170

ELECTRICITY AND MAGNETISM

not only equal but also in phase, and no indication of the instru-
ment will result.

The conditions necessary for balance may be obtained in the
following manner. Let the instantaneous currents through the
various elements be designated as in the figure. By equating

Fig. 97. — Maxwell's bridge.

the fall of potential from A to D to that from A to E, and the fall
from Z> to 5 to that from E to B and noting that 12 is made up
of i and I'l the following equations result.

ii = ii + i (1)

Riii = Rsiz (2)

Riii = Rdz -|" Li

Rii\ =

-^ I idt

(3)
(4)

We thus obtain four equations between the four currents. The
currents may therefore be eliminated and the relations between
L, C, and the R's obtained which are necessary for a balance.
Eliminating 12 between eqs. (1) and (3) there results

R,is-\-L^ = R2(ii-{-i)

(5)

ALTERNATING CURRENT BRIDGES 171

Differentiating eq. (4) with respect to t and solving for i, also
substituting the value of is from eq. (2) in eq. (5), we have

7? R
If the bridge has first been balanced for the steady state, p

= Ri, whence only the terms containing the derivative of ii
remain. The second condition for balance is obtained by equat-
ing the coefficients of the derivatives, whence

L = RiRzC (7)

While the theory of this bridge is simple, its application in the
laboratory is somewhat tedious in case both L and C are fixed.
For example, suppose a steady state balance has been obtained,
and it is attempted to satisfy eq. (7) by changing R^ or R3. The
steady state balance is immediately upset and must again be
obtained before the test for the new value of R2 or R3 can be made.
If L or C are continuously variable, eq. (7) may be satisfied with-
out disturbing the steady state balance, and it is in this case that
the bridge is particularly useful. An experienced observer
however, quickly learns to make both balances simultaneously.

128. Experiment 20. Maxwell's Bridge for Self Inductance. —
Make the connections as shown in Fig. 97 using for L a continu-
ously variable inductance and for C a subdivided condenser.
As a source of E.M.F., use an alternator giving a frequency from
500 to 1,000 cycles per second, and a pair of head phones as the
detector. It may be well to use a battery and ordinary galvanom-
eter to obtain the steady state balance. Connect double pole
double throw switches so that each source and detector may be
quickly exchanged. For the steady state balance, care must be
taken to close the battery key before the galvanometer key.
Obtain the variable state balance by changing L. If the balance
does not lie within the range of L, change either C or one of the
resistances of eq. (7). If the latter is done, a new steady state
balance must be obtained.

Report. — Plot a calibration curve of L as a function of its scale
readings. Define coefficient of self inductance. If a copper
disk were held near the coil so that its face is perpendicular to the
axis of the coil would the inductance as measured in this manner
be changed? Explain.

172

ELECTRICITY AND MAGNETISM

129. Anderson's Modification of Maxwell's Bridge.' — It

was pointed out above that the adjustments for balancing Max-
well's bridge are hkely to be tedious because each attempt to
obtain a variable state balance necessitates a redetermination of
the steady state balance. Anderson has suggested a simple
device by which the variable state balance may be obtained
without destroying that for steady states. The connections are
shown in Fig. 98. It will be noted that the condenser C, instead
of being connected to the point D has the resistance r, placed in

Fig. 98. — Anderson's modification of Maxwell's bridge.

series with it so that its time constant may be varied. Since, in
determining the steady state balance, the condenser produces no
effect, it may be left in circuit during that process, and the only
change introduced is placing r in series with the galvanometer.
This reduces slightly the sharpness of balance which is of little
consequence. The steady state balance may accordingly be
made once for all, and the variable state balance obtained by
adjusting r to the proper value.

The determination of the balance condition is somewhat more
complicated and is as follows: Let the instantaneous currents
through the various elements of the bridge be designated as
before. The points P and E are now the ones remaining at the
same potential. Accordingly,

ii = ii + i (1)

Phil. Mag., vol. 31, 1891, p. 329,

■M

idt + n

(2)

ALTERNATING CURRENT BRIDGES

173

^J^ =

= Rsis

dii

Rzii -\- ri = L~ + Rds

(3)
(4)

Combining eqs. (1) and (3) with eq. (4), there results

Riiii + i) + ri = ^— i + ^ 1 idt
Eliminating ii between eqs. (2) and (5) we have

Imposing now the condition for the steady state balance, it is
seen that the coefficients of the integrals are equal and eq. (6)
then becomes

(5)

idt (6)

R^r . jy . L

III tisL'

Rearranging and using

(7)

(8)

Fio. 99. — Stroude and Gates bridge.

Another change in the arrangement of this bridge has been
suggested by Stroude and Oates^ which is usually an advantage.
The general theory of bridges shows that it is always possible to
interchange the source of power and the detecting device. Figure
99 shows the connections when this has been done with a slight

» Phil. Mag., vol. 6, 1903, p. 707.

174 ELECTRICITY AND MAGNETISM

change in the arrangement which improves the ease of manipula-
tion. The principal advantage in this method lies in the fact
that r is now in series with the bridge and a correspondingly
higher E.M.F. may be used without injuring the resistances. An
increase in sensitivity is thus secured. The same balance condi-
tion, eq. (8), applies.

130. Experiment 21. Stroude and Oates Bridge for Self
Inductance. — Connect the apparatus as shown in Fig. 99. For
power supply and detector use either an audio frequency gen-
erator and phones or city A.C. supply and alternating current
galvanometer. The latter is particularly well adapted to this
bridge. Arrange double pole double throw switches so that a
direct current source and ordinary galvanometer may quickly be
substituted for making the steady state balance.

As an unknown inductance, use two coils mounted in a fixed
position close enough to one another so that mutual inductance
exists between them. Measure the inductance of each sepa-
rately, then connect them in series and measure the resultant
inductance with the connections direct and reversed, that is with
the mutual inductance first aiding and then opposing the self
inductances. Calling Li and L2 the self inductances of the
individual coils, and La and Lo the two together when aiding and
opposing respectively, the following equations hold

L„ = Li + L2 -h 2M (9)

Lo = Li -\- L2 - 2M

Report. — Check your results by solving eq. (9) for M. Give
a physical interpretation for eq. 9.

131. Trowbridge's Method for Self Inductance. — In Art. 141
there will be described a method by which an inductance may be
measured in terms of capacitance using the two reactances in
series in one arm of a bridge. While this arrangement admits of
an exceedingly sharp adjustment, the bridge may be balanced for
only one frequency for given values of L and C. In fact one of its
most useful applications is the determination of frequency using
reactances of known magnitudes. Trowbridge^ has shown that
if the reactances are shunted with properly chosen resistances the
balance condition may be made independent of the frequency
while sensitiveness of balance is very inappreciably sacrificed.
Such an arrangement is shown in Fig. 100.

' Phys. Rev., vol. 23, 1905, p. 475.

ALTERNATING CURRENT BRIDGES

175

Let the currents be designated as indicated in the figure. Then
putting, for the moment, R4 = 0, the following equations must
hold for the balance condition :

Rill = 112^2
Rsii = Ri\ + n'e

Rii = ToH + Li

cy'

*2 = ii -\-i 4

ti ^ lb 1 ^6

For simplicity let Ri = R2, then ii = iz

(1)
(2)

(3)

(4)

(5)
(6)

Fig. 100. — Trowbridge's method for self inductance.

Eliminating i^ between eqs. (2) and (6) we have

(R3 — r) t'l = Rix — rii

(7)

Again eliminating t'e between eqs. (2) and (4) and substituting
the value of if, in eq. (7)

dii

{Rz -r)ii = Ru -r[R,C^

«^l*]

(8)

176 ELECTRICITY AND MAGNETISM

Substituting the values of ii and i* in terms of iz obtained from
eqs. (3) and (5), namely,

To . . L dis

in eq. 8 there results

Collecting terms we have,

,R -{- To 1 . , [(R3 - r)L y , „ ^ 72 + ro

iiJ

[(i?3 - r)^" - ro]^3 + [^^^ - L + rRzC

-rr.c]f;^[iRz-Rf-fY^ = 0 (10)

Clearing of fractions,

[(Rz - r){R + To) - Rro)]iz -\- [(Rz - r - R)L

+ rRz{R +ro)a - /2rr„C] ^ + (^3 - R)rLC ^ = 0 (11)

Assuming now that iz is an alternating current of the form

iz = I sin o)/
and substituting this in eq. (11) there results

[(R2 - r)(R + To) - Rro] sin c^t + [(Rz - r - R) L -[-rRz
{R + ro) C - RrroC] 7w cosojf - {Rz - R) rLCIu^ sin wt = 0 (12)

In an ordinary measurement in which C is expressed in micro-
farads and L in millihenries the last term is of the order 10~'
and may be neglected without appreciable error. Since eq. (12)
holds for all values of t, we have
when t = 0,

{Rz - r - R) L + [rRz{R + n) - Rrro] C = 0

whence

_ [Rrro - {R + ro)rRz]C ,^^.

[rro{R - Rz) - RRzr]C
Rz - R -r

when

t = |, {Rz - r){R + ro) - Rro = Q (14)

ALTERNATING CURRENT BRIDGES 177

which is seen to be the condition for a steady state balance.
If Rz = R, the last term of eq. (12) vanishes, and the expression
given by eq. (13) is exact and reduces to

L = ^^ = RR,C = RK! (15)

The bridge, when used in this manner, is well adapted to the
standardization of a variable inductance such as Brooks inductom-
eter but is not well suited to cases in which the values of L
and C are fixed or variable by steps, since it is impossible to adjust
for the variable state balance without upsetting that for steady
states. The author has pointed out that this difficulty may be
avoided by use of the resistance R^ as shown in the figure. If
identical boxes are employed for r and Ri and a steady state
balance obtained with R4 set at a suitable value, then the variable
state balance may be obtained by shifting plugs from one box to
the other, keeping r -\- Ri constant. If an equal arm bridge is
used, this has the effect merely of adding to both the upper and
lower right hand arms of the bridge the value of Rt. Eq. (13)
then becomes

r [rro (R - Rs + Ra) - RrjR^ - R,)]C ,_,

^ = R,-R,-R-r (^^)

132. Experiment 22. — Trowbridge's Method for Self Inductance.
Connect the apparatus as shown in Fig. 100 using the telephone
and suitable oscillator for detector and energy source respectively.
As an unknown use a smoothly variable inductance, and set the
four resistances Ri, R2, R3, and R at suitable values, e.g., 500 ohms
each. C should be a subdivided standard condenser. Measure
the unknown for several settings and plot its calibration curve.
Replace the variable inductance by one of fixed value, and meas-
ure it, making use of the resistance Ri as explained above.

133. Heydweiller's^ Network for Mutual Inductance. — In
Exp. 18, a method due to Carey-Foster, was used for the measure-
ment of mutual inductnce in terms of capacitance. The essential
feature of this method consists in balancing the charge of a con-
denser against the quantity of electricity induced in the secondary
of a mutual inductance when a certain current change takes place
in the primary. This balance was effected by discharging the
two quantities involved in opposite directions through a long
period ballistic galvanometer. While this circuit is satisfactory

' Annalen der Physik., vol. 53, 1894, p. 499.
12

178

ELECTRICITY AND MAGNETISM

when used with the make and break method of excitation, it
can not be used with alternating currents since there is no way of
adjusting the time constant of the condenser circuit.

This defect was overcome by Heydweiller by the introduction
of the resistance S as shown in Fig. 101 and a very satisfactory
method for the measurement of mutual inductance was thus
obtained. The resistance P includes that of the secondary
coil whose self inductance is L. The conditions which must hold

Fig. 101. — Heydweiller' s network for mutual inductance.

for zero current through the galvanometer may be obtained
as follows. Designating the instantaneous currents through
the various resistances as indicated in the figure, we have

(1)
(2)

(3)

(4)

(5)
(6)

= Iz

= «3 +

Rii

= ^ 1 iidt + Sii

^'dt

+ Piz

From eqs. (2) and (4) we have

dis . ^. ,^/di
~dt
From eqs. (3) and (1)

L-+P. = M(^ + f)

'' = m "'

= ^7^ I '^sdt + ^ *3

Differentiating eq. (6) and substituting in eq. (5), there results,
on collecting terms,

{R + S)ldH
R J dt

(7)

ALTERNATING CURRENT BRIDGES 179

Since an alternating E.M.F. is applied to this circuit the current
ii is also alternating and may be represented by

di
iz = / sin oit; whence -j' = /w cos ut (8)

Substituting these values in eq. (7) we have

(R + S)-] , _ , . Tn M

^ ] /co COS a)f + [P - ^^] / sin a>< = 0 (9)

Since eq. (9) holds for all values of t, we have
when

o,t = 0,L - M^^^^ =OorM = L^^ (10)

when

co< =|P - ^ = 0 or M = PRC

It is thus seen that there are two conditions which must be
satisfied in order that there should be no deflection of the galva-
nometer, when an alternating E.M.F. is apphed. The second of
these is the same as for the original Carey-Foster circuit, which is
obtained by putting S = O. The impossibihty of satisfying the
first condition under these circumstances is obvious for then
M = L, an inflexible condition, difficult to satisfy. This circuit,
while not strictly a bridge, resembles one in that two balances
conditions are necessary.

134. Experiment 23. Heydweiller's Method for Mutual Induc-
tance.— Connect the apparatus as shown in Fig. 101. For
M use a pair of coils whose relative positions may be varied, and
for C, a subdivided standard condenser. The purpose of the
experiment is to determine ikf as a function of the setting of the
movable coil. As source and detector use either the wire inter-
rupter and vibration galvanometer, or an alternator and tele-
phone. From the known E.M.F. of the source, compute the
minimum value of R in order that the power consumption in it
should not exceed 4 watts per coil.

Report. — Plot M as a function of the scale readings of the
instrument. In computing M use the second of eq. (10). Check
the accuracy of your results by substituting in the first of these
equations and note the constancy of the values for L. In con-
necting up the circuit is there any choice as to which coil is
used as the primary? Does the mutual inductance of two coils
depend upon which is the primary? Explain.

180

ELECTRICITY AND MAGNETISM

136. Mutual Inductance by Heaviside's Bridge.' — If one of

the arms of a Wheatstone bridge is inductive while the other
three are non-inductive it is impossible to obtain a balance since
the E.M.F. across the inductive arm will have a component 90
deg. out of phase with the current through it, and the E.M.F.'s
at the galvanometer corners of the bridge can never be in phase.
It was pointed out by Hughes that if in series with the galva-
nometer there is connected the secondary of a variable mutual

Fig. 102. — Heaviside's bridge for mutual inductance.

inductance, the primary of which is included in the supply
circuit, an E.M.F. in quadrature with this current and hence
opposite in phase to the E.M.F. due to the self inductance of the
bridge coil may be obtained and a balance thus secured.

In discussing this circuit, Heaviside pointed out that a more
satisfactory arrangement results if the secondary of the mutual
inductance is introduced, not in the galvanometer circuit, but in
the arm of the bridge adjacent to that containing the inductance
under consideration. The E.M.F. thus induced in Li by mutual
inductance may be made to compensate the difference in the
E.M.F's of self inductance in Li and L2. Such an arrangement is
shown in Fig. 102. The balance condition is obtained as follows:

1 Phil. Mag., vol. 19, 1910, p. 497.
The Electrician, vol. 76, 188&-86, p. 489.

ALTERNATING CURRENT BRIDGES 181

Designating by i, ii, and t2 the instantaneous supply and bridge
currents respectively, the following equations result.

i = ii + ii (1)

Rill = RiH (3)

Eliminating i between eqs. (1) and (2), we have

Substituting in eq. (4) the value of i^ from eq. (3)

Imposing now the condition for steady state balance, the terms in
ii vanish, whence

whence

M(«3 + Ri) = URi - URi (7)

If an equal arm bridge is used, i.e., R% = R^

M = 1[l, - Lx] (9)

Campbell has suggested a simple modification of this bridge
whereby self inductances may easily be measured in terms of
mutual, provided a continuously variable standard of the latter
is available. This inductance is introduced at I, shown short
circuited by a link in the figure. A balance is first obtained with
the link inserted. Let Mi be the reading of the variable standard
for this setting. Introduce the unknown by removing the link
and balance again varying Rior R2 to compensate for the added
resistance of the unknown coil. Let 71^2 be the new reading of
the standard. Then, for an equal arm bridge,

Ml = 3^(L2 - Li)
M, = i^(L2-Li+L.)
whence

L. = 2(M, - Ml) (10)

where Lx is the unknown self inductance to be measured.

182 ELECTRICITY AND MAGNETISM

A further simplification results if L2 is a variable inductance,
for then the first balance may be obtained by making Mi zero and
adjusting L2 until it is equal to Li. When a second balance has
been obtained, Lx is simply twice the value of M. Neither Li
nor Li need be known. This method is particularly useful where
a number of inductances of the same order of magnitude are to
be measured.

136. Experiment 24. Heaviside's Bridge for Self Inductance. —
Connect the apparatus as shown in Fig. 102, using for M a
variable standard of mutual inductance. L2 should be a
continuously variable self inductance. As a detector, use
phones, vibration or A.C. galvanometer with appropriate source.
Measure a series of self inductances.

Report. — Is there any choice, in this bridge, as to which of the
two coils of the mutual inductance is used as the primary?
May the leads to the primary be interchanged at liberty? Could
a variable state balance be obtained if the unknown were intro-
duced in the arm Ril Explain.

137. Maxwell's Bridge for Mutual Inductance.^ — The simplest,
though not the most sensitive bridge for the measurement of
mutual inductance is one devised by Maxwell. The method
consists in obtaining the mutual inductance of a pair of coils in
terms of the self inductance of one of them. The connections
are shown in Fig. 103. In the discussion of the Heaviside
bridge. Fig. 102, it was pointed out that a balance could be
obtained by introducing in the coil Li by mutual inductance an
E.M.F. which would compensate for the difference in E.M.F.'s in
the coils Li and L2. It might equally well have been said that the
E.M.F. in the coil L2 balances the difference between the E.M.F.
in Li due to mutual and self inductance. If the relative values
of the currents in the primary and secondary of M are changed
these E.M.F.'s may be made equal without the use of the coil L2.
This is the method employed in the Maxwell bridge, and is accom-
plished by shunting the entire bridge by the resistance R.

Indicating the instantaneous currents as shown in the figure,
the equations for the balance condition are as follows:

i = ii + U + is (1)

«3.. = B,i,+Lf-Mf (2)

7^2*1 = R*i2 (3)

Ri, = (/?3 + R*)i. (4)

> Maxwell, Electricity and Magnetism, vol. 2, p. 365.

ALTERNATING CURRENT BRIDGES

183

Eliminating i between eqs. (1) and (2)
But, from eq, (3)

and, from eq. (4)

^3 + -^4 . ^3 + Ri Rz •
*3 = ri *2 = fi • r^ tl

R Ri

■A/VWWWWNA-

Fig. 103. — Maxwell's bridge for mutual inductance.

Substituting these values in eq. (5) there results

(5)

Riii + L

^1 _ If fl 4- -^ 4- ^^ + ^« ^1 ^1 = RsRdi

dt I ^ Ri R Rj dt Ri

(6)

Imposing the condition for a steady state balance, the terms in ii
vanish, and we have

Since

we have

Ri _ Ra
R2 Ri

R3 -\- Ri Ri -f" R2

184 ELECTRICITY AND MAGNETISM

and cq. (7) may be written

M[l + ^^^ + ^i^^]=L (8)

138. Experiment 25. Mutual Inductance in Terms of Self
Inductance by Maxwell's Bridge. — Connect the apparatus as
shown in Fig. 103 using for M several pairs of coils with fixed
mutual inductances. Operate the bridge either with phones,
vibration or A.C. galvanometer, and appropriate source of supply.
After the determination of M for each pair of coils, interchange
primary and secondary and check your result.

Report. — Is the balance as sharply defined as in some of the
bridges previously used? Explain. At the balance point, are
the currents in R and Ri in phase?

139. The Mutual Inductance Bridge. — Figure 104 represents
a bridge in which two coefficients of mutual inductance may
readily be compared provided one of them is a variable standard.
Designating the various parts of the bridge as indicated in the
figure, the balance conditions are as follows:

r> • IT <^^l 1 TIT ^*l r> • IT di2 , ,, di2 ,^v

Rsii - ^1 ^' = ^2*2 - M2 ^' (2)

Suppose that a variable state balance has been obtained with
the secondaries disconnected and the galvanometer joined
directly to the points A and B. This balance may be facilitated
by the introduction of a variable inductance in series with either
^1 or R2 as the case may require. Under these circumstances
eqs. (1) and (2) become the same as those for the simple induc-
tance bridge, namely

R^i -{- lJ^^ = R2i2 -h L2^ (3)

(4)
(5)

and

' Rail = ^4^2

whence

. fls .

'' = K."

and

di2 R3 dii

dr~Ri~dt

ALTERNATING CURRENT BRIDGES

185

Substituting in eq. (3) we have

r, . , J dii _ RiR-ii'

+ L:

R3 dii
Ridt

(7)

Imposing the steady state balance, we have

R\ _ R3

Ri Ri
and

Li _ Rs

Introduce now the secondary coils as shown in the figure
and obtain a balance by adjusting the variable standard. This
balance indicates that the E.M.F.'s in the secondary coils are
equal and opposite. Since no current flows in the secondary
coils, the currents for the primaries which are defined by eqs. (3)

dii

and (4) are unchanged. Accordingly, the values for 1*2 and

given in eq. (5) may be substituted in eq. (1). Subtracting eq.
(3) from eq. (1) then gives

Ml _ Rs .

w, - F4 ^^^

This bridge is distinguished from those previously studied in that
three balances are necessary. This may seem at first sight to
result in an unduly cumbersome method, but experience shows
that in reality it is a relatively simple bridge to operate.

Fig. 104. — Mutual inductance bridge.

140. Experiment 26. Comparison of Two MiUual Inductances.
Connect the apparatus as shown in Fig. 104, using a pair of
phones and a suitable source of alternating current. Obtain the

186

ELECTRICITY AND MAGNETISM

three balance conditions as described above, for several different
unknown pairs of coils. Check your results by interchanging
primaries and secondaries for each pair.

Report. — Explain why it is permissible to introduce an extra
inductance in series with R\ and R2. Could this be introduced
in Ri or R^'i

141. The Frequency Bridge. — In all of the bridges thus far
discussed, the balance condition has, in no case, contained a term

depending upon the fre-
quency. The physical sig-
nificance of this is that these
bridges balance independent
of the frequency and hence
the form of the impressed
wave is of little consequence.
A bridge will now be studied
which, for given values of L
and C, can be balanced for
only one definite frequency.
The connections are shown in
Fig. 105. Three of the arms
are non-reactive, while the
fourth contains an inductance and a condenser in series. Let
the instantaneous currents through the upper and lower arms be
ii and U respectively. The balance conditions are then

Riii = Rzii (1)

Fig. 105. — The frequency bridge.

Rd, + ^ i' + e

Eliminating ii, we have

/e4i2 + L-| + ^

R^ii

. , R2R3 .
lidt = p «2

(2)

(3)

Imposing the steady state balance condition, which may be
obtained by short circuiting C, in case a battery and ordinary
galvanometer are used,

Assuming that a pure sine wave of E.M.F. is applied to the bridge,
the current through it will also be a sine wave of the same fre-
quency as the source. Let the current 12 then be given by

12 = I sin ut

ALTERNATING CURRENT BRIDGES 187

Then

'dt

r I

= I(/} cos (1)1, and I i2dt = cos <at

J (^

Substituting these values in eq. (4), we have

LIci cos (at — y^ I cos (ot = 0 (5)

whence

''- Vlc

Since <o = 27rn, where n is the frequency,

When two of these quantities are known, the third may be
computed. One of the most useful applications of this bridge is
the measurement of frequency using a subdivided condenser and
a continuously variable inductance. In case a complex wave is
applied to the bridge, complete silence in the phones can not be
obtained for any value of the LC product. It will be observed
however, that the relative intensities of the fundamental and
overtones will be changed as L is varied and that for a certain
setting, the fundamental will disappear while the overtones
remain.

In Art. 110, it was pointed that a circuit connected for parallel
resonance, possesses a very large impedance for the particular
frequency to which it resonances. If now such a circuit is placed
in series with the source, and adjusted to resonate to the fre-
quency of the fundamental as above determined, this frequency
may be suppressed and that of the strongest overtone measured.
Introducing now another resonance circuit in series with the
source to suppress this overtone, the next stronger one may be
measured and so on. In this way a qualitative analysis of the
wave form of the source may be made.

142. Experiment 27. Bridge Method for Measuring Frequency.
Connect the apparatus as shown in Fig. 105, using for C a
subdivided mica condenser, and for L, a variable standard of
inductance. Determine the frequency of several sources of
alternating current, using the phones as a detector. Determine
first the fundamental and then place a filter circuit in series with
the source to suppress this frequency and measure the frequency
for the strongest harmonic. In computing the frequency by eq.

188

ELECTRICITY AND MAGNETISM

(G), the iiuluctancc and capacitance must be expressed in henries
and farads respectively.

Report. — Compute a constant for the right-hand side of eq, (6)
which, when divided by \^LC will give the frequency when L
is expressed in millihenries and C in microfarads. Compute the
inductance, which, when used with a capacitance of 1 microfarad
will balance the bridge for a frequency of 60 cycles.

143. Circuits of Variable Impedance. — In the bridges studied
thus far for the measurement of self and mutual inductance, the
assumption has tacitly been made that the only E.M.F.'s induced
in the coils are those due to the primary current. For example,

in the case of mutual inductance, a
varying current in the primary pro-
duces an E.M.F. in the secondary
proportional to the rate of change
of the primary current hence in
quadrature with the primary cur-
rent for the case of a sine wave.
For self inductance, the coil is its
own secondary, and the same con-
siderations hold as for two coils.
The direction of the induced E.M.F.
is counter to the driving E.M.F.
while the current is rising, and in
the same direction when it is fall-
ing. The power associated with the
induced E.M.F. at any instant is equal to the product of this
E.M.F. and the current. Energy accordingly is alternately stored
in the electromagnetic field of the coil and returned to the circuit.
The theory shows, in fact, that this occurs at a frequency twice
that of the driving E.M.F.

When the circuit is of such a nature that energy is consumed
by the coil or parts connected with it in some other manner than
by heat developed within the primary coil, the quadrature rela-
tionship is destroyed and the impedance of the coil is no longer
constant. Among the more important causes of such extraneous
energy consumption are hysteresis, eddy currents, and motion of
parts. The telephone receiver is an illustration of such a circuit.
For simplicity, consider a coil of wire C wound upon a bar magnet
M near one end of which is placed a flexible iron diaphragm D as
shown in Fig. 106. When an alternating current flows through

Fig. 106. — Simplified telephone
receiver.

ALTERNATING CURRENT BRIDGES 189

C, the residual magnetism is alternately increased and decreased
by the current and the diaphragm vibrates with the same fre-
quency as the source. In addition to the Joule heat, represented
by PR, developed in the coil, energy consumptions result from
the three causes enumerated above in the following manner.

1. Hysteresis. — The E.M.F. induced in the coil is proportional
to the rate of change of the magnetic flux through it. This flux
is produced by the magnetomotive force of the coil, the latter
being in phase with the current and proportional to it. Because
of the hysteretic lag of flux behind the magnetomotive force, the
E.M.F. is no longer in quadrature with the current but has a
component counter to the current which results in a continuous
energy consumption. The greater the area of the hysteresis
loop, the greater the lag of flux behind the current and hence the
larger the energy component of the induced E.M.F. The
hysteresis loss is proportional to the frequency.

2. Eddy Currents. — The magnet M may be regarded as a
secondary coil consisting of a single turn about its own axis
having a relatively large cross section and a low resistance.
The changing flux through this turn induces in it an E.M.F. in
quadrature with the flux and the resulting current is known as a
Foucault or eddy current. Because of the small self inductance
of this single turn, the eddy current is practically in phase with
the induced E.M.F. producing it. The eddy current may in
turn be considered as a primary which induces in the coil a
quadrature E.M.F. Except for the hysteresis lag,, this final
E.M.F. in C is counter to the current because of the double
quadrature relationship, and hence introduces a large energy
consumption. Viewed from the standpoint of Joule heat
developed in the core by the eddy current, this loss is
proportional to the square of the frequency, for the induced
E.M.F. producing the eddy current is proportional to the fre-
quency and the heating effect of a current is equal to the square
of the E.M.F. divided by the resistance.

3. Motion of the Diaphragm. — The effect of motion of the
diaphragm may be understood by the following considerations.
Suppose a sound wave strikes the diaphragm. The varying air
pressures cause it to vibrate and in so doing, the air gap between
it and the magnet is changed and hence the reluctance of the
magnetic circuit of the magnet. This introduces a change in
flux through the coil which induces an E.M.F. within it. In

190 ELECTRICITY AND MAGNETISM

fact this is the principle of the "magneto-phone" which is often
employed where accurate reproduction is more essential than
energy delivered. As regards the magnitude of the E.M.F.
induced in the coil and the phase relation between it and the
motion of the diaphragm, it makes no difference whether the
motion is produced by a sound wave or by a current through C
In the latter case, the energy of the wave must be supplied by
the current, and the law of conservation of energy requires that
the induced E.M.F. due to the motion of the diaphragm must
have a component counter to the current to account for this
consumption.

If an alternating current of intensity I is passed through the
coil, and the power delivered to the coil is measured by appropri-
ate means, it is found that this is much larger than would be
computed from I^R when R is determined by using direct cur-
rent. On the other hand we may define a resistance Re such that

Watts = PRe.

Re is called the "effective" resistance of the coil. It is the resis-
tance of a fictitious coil, free from hysteresis, eddy-current and
motional reactions, which consumes the same power with a given
current. Again the effective resistance may be written

Re = R -\- Rff -jr Re ~\~ Rm

where the last three terms represent the parts contributed by
hysteresis, eddy currents and diaphragm motion respectively,
and, it is customary to speak of the resistance due to hysteresis,
eddy currents, etc. In a similar manner, the E.M.F.'s induced
in the coil by hysteresis, eddy currents and motion will have
components in quadrature with the current, and will change the
apparent inductance of the coil, and it is customary, in an
analogous manner, to speak of the inductance due to hysteresis,
eddy currents, etc.

Kennelly and Pierce^ have made a detailed study of the
motional characteristics of telephone receivers and have shown
how their performance in practice may be predetermined from
simple measurements. The receiver to be studied was placed
in one of the arms of an inductance bridge and its effective resis-
tance and inductance measured for a wide range of frequencies,
first with the diaphragm clamped, and again when free to move.

iProc. Am. Acnd. of Sci., vol. 48, p. 131, 1912.

ALTERNATING CURRENT BRIDGES 191

The difference between the corresponding values for the same
frequency were called "motional resistance" and "motional
inductance" respectively. The latter when multiplied by the
frequency for which they were determined, gave the "motional
reactance " for that frequency. Interesting results were obtained
for frequencies near that corresponding to the natural period of
the diaphragm. For example, the curve showing the motional
resistance as a function of the frequency closely resembles, near
the resonance frequency, the curve in optics, showing the varia-
tion of the index of refraction with frequency in the neighborhood
of an absorption band, while that for motional reactance exhibits
a sharp minimum at this point.

144. Experiment 28. Motional Impedance of a Telephone
Receiver. — Connect the apparatus as shown in Fig. 64 substi-
tuting for Lx the receiver to be studied. Use an equal arm
bridge making Ri an Ri approximately equal to the direct current
resistance of the receiver. Energize the bridge with a Vreeland
oscillator which has previously been calibrated for frequency,
and use a pair of head phones as a detector. Place an elec-
trostatic voltmeter across the output coil of the oscillator and
maintain a constant voltage on the bridge throughout the
experiment. Determine roughly the natural period of the dia-
phragm of the receiver by varying the frequency of the oscillator
keeping the voltage approximately constant by noting at what
frequency the response is loudest. Introduce a small wedge be-
tween the diaphragm and the cap to prevent motion and measure
the resistance and inductance for a range of frequencies above
and below the resonance frequency. Remove the wedge and
repeat with the diaphragm moving.

Report. — Plot curves showing the variation of resistance and
reactance with frequency for both blocked and moving dia-
phragm. Subtract the former from the latter and thus obtain
the "motional" resistance and reactance and plot each as a
function of frequency.

145. Power Factor and Capacity of Condensers.' — In a perfect
condenser, that is, one without absorption or leakage, the phase
of the current is 90° ahead of the E.M.F. impressed across its
terminals. Although many condensers approximate the ideal, it
is only with well insulated air condensers that this condition may

» Grovek, Bull. U. S. Bureau of Standards, vol. 3, 1907, p. 371.
WiEN, Wiedemann's Annalen, vol. 44, 1891, p. 689.

192

ELECTRICITY AND MAGNETISM

be regarded as actually realized. In condensers having a dielec-
tric made of paper impregnated with parafine or beeswax there is
an appreciable component of the current in phase with the
E.M.F. In such a condenser there is a measurable amount of
energy absorption which appears as heat in the dielectric, and as
far as phase relations are concerned, it may be regarded as a
perfect condenser with a small ficticious resistance in series with
it. In Fig. 107a, let C represent the equivalent perfect condenser,
and p the fictitious series resistance. The vector diagram 107 6
represents the phase relations for such a circuit. OE is the

-*-j

/\AAAAA-»

a E

Fig. 107. — Phase diagram for a condenser.

nppressed E.M.F. and 01 the current which falls short of the
90'' vjead by the angle 6 which is designated as the phase differ-
ence of the condenser. <^ is the phase angle as ordinarily defined
and the power factor is then

P.F. = cos 0 = sin d.

It is obvious from the figure that

tan 6 = pCo) (1)

A simple bridge method has been devised by Wein by which
both the capacitance and the power factor of an imperfect con-
denser may be simultaneously measured provided there is avail-
able for comparison purposes another condenser which shows no
absorption. Such a bridge is illustrated in Fig. 108, where Ci is
the perfect condenser and d the one with ficticious resistance p
to be studied. In series with these condensers are placed the
small finely adjustable resistances ri and rz. The purpose of
these is to bring about equality of phase in the currents through
the upper and lower branches of the bridge. It is clear that,

ALTERNATING CURRENT BRIDGES

193

without them, if one of the condensers possesses an equivalent
resistance while the other does not, the potential differences
from D to B and E to B can not be equal and in phase at the same
time. Accordingly a perfect balance of the bridge can not be
obtained. If, however, a suitable resistance ri is introduced in
series with Ci of such a value that the time constant of the arm
DB equals that of EB, this difficulty is obviated. In practice
it is generally more convenient to introduce r2 also and take
account of it in deducing the balance conditions.

Fig. 108. — Bridge for measuring phase difference of a condenser.

The equations for balance may be derived in the following
manner, calling ii and ii the instantaneous currents in the upper
and lower arms respectively.

Riii = i22*2

kf'

idt + riii = — I i-4t + (p + r2)i2

Eliminating (2 we have

rj''"'' + --|5j

iidt + (p + r2)^ii
K2

Imposing the condition for a steady balance, namely

R2 r2-\- p
there results

C\ _ R2

C2 Ri
The phase difference 6 may be obtained as follows:

(2)
(3)

(4)

(5)
(6)

194 ELECTRICITY AND MAGNETISM

Combining eqs. (5) and (6) we have

?-' = ''^ (7)

Multiplying numerator and denominator on the left by w and
clearing of fractions, there results

Cicori = C2co(r2 + p) (8)

Referring to Fig. 1076 and solving eq. (8)

tan 6 = C^p = CiciTi — duVz (9)

146. Experiment 29. Measurement of Phase Difference and
Capacitance of a Condenser. — Connect the apparatus as shown in
Fig. 108. Ci is a standard mica condenser whose phase differ-
ence is regarded as negligible, and C2 is a telephone condenser
with parafRne paper dielectric whose phase difference and capaci-
tance are to be determined as a function of the frequency. The
resistances ri and r2 are small in value and should be joined by a
slide wire for fine adjustment. As a source of power use an
oscillator giving a pure wave form whose frequency may be
varied over a considerable range, such as the Vreeland, with
phones as detector. In obtaining a balance, set r2 = 0 and get
as good silence as possible. Introduce such a value of ri as makes
the best improvement, then change i^i or R2 and again adjust ri
and so on until complete silence is reached. Keep r2 as small as
possible. Make a series of balances using as wide a range of
frequencies as may be obtained from the oscillator.

Report. — Plot capacity and phase difference of the unknown
condenser as a function of the frequency. Show that in a
perfect condenser the current leads the E.M.F. by 90°. Define
Power Factor.

147. Resistance of Electrolytes. — The measurement of the
resistance of an electrolyte offers special difficulties not encoun-
tered in determining the resistance of metallic conductors.
This is due to the fact that current is carried through a solution
by virtue of the migration of ions, a double procession in opposite
directions. These are deposited on the electrodes, where sec-
ondary chemical reactions often take place. In general, the
deposits on the electrodes set up counter E.M.F. 's in the cell
which affect the measurements in the same manner as added
resistance. Obviously then an electrolytic resistance can not be
measured by a Wheatstone bridge employing direct currents.

ALTERNATING CURRENT BRIDGES

195

If an alternating current is used this effect is eliminated since
the counter E.M.F. is with the bridge current during one half of
the cycle and opposite to it during the other.

In case an electrolyte is measured in which a gas is formed at
one of the electrodes a further complication is introduced since
the cell behaves as though it contains capacitance. This
results from the fact that a gas layer separates the liquid from the
electrode thus forming a condenser. Since the gas layer is, in
general, very thin, a capacitance of considerable magnitude may

Fig. 109. — Bridge for electrolytic resistance.

result. The ceU then behaves Uke a condenser and resistance
in parallel, and it must be so regarded when connected in one
of the arms of a bridge. The resistance in the adjacent bridge
arm must also be shunted by a condenser else a balance can not
be obtained. Such a bridge is shown in Fig. 109, where Rt and
Ci represent the resistance and capacitance of the electrolytic cell,
and Rs and Ci its counterpart in the adjacent arm. Designating
the currents as indicated in the figure, we have

is + *5 (1)

ti =

ii = U + it
Rii\ = Riii
Rzii = Rd^

i'adt = Riiz

if,dt = Rdi

(2)
(3)

(4)

(5)
(6)

196

ELECTRICITY AND MAGNETISM

Eliminating i^ between eqs. (1) and (5) and h between eqs. (2)
and (6) there results

i, = u + c,/e3~! (7)

t2 = ii + CiRi

dt
dii
It

(8)

Substituting in eq. (8) the values of i^ and t4 from eqs. (3) and (4)
and eliminating ^l between the resulting equation and eq. (7)
we have

dis R2 R3. , r( T> Ridi^

''-^^'^'dt^R, R,

Ridt

(9)

Imposing now the condition for steady state balance, i.e.,

R\ _ R3
R-2 Ri
we have

Ci R2

(10)

C2 Ri

(11)

In carrying out measurements of the resistance of solutions,
the design of the electrolytic cell is a matter of considerable

Fig. 110. — Cell for measurement of electrolytic resistance.

importance. It has been found that different electrolytes require
different types of cells and even for the same electrolyte a given
cell is not always suited to wide ranges of concentration. For
example, polarization may occur in some cases unless the elec-
trodes are platinized, and in other cases platinized electrodes

ALTERNATING CURRENT BRIDGES' 197

appear to act calclytically and assist chemical action. Again
platinized electrodes may, because of their spongy nature, absorb
so much of the electrolyte as to cause errors in measurement when
used later with solutions of a different nature or concentration.

Figure 110 shows a cell designed by Dr. Washburn and manu-
factured by the Leeds and Northrup Co. The electrodes are of
platinum and are mounted by sealing their supporting wires into
tubular glass stems. These wires project through the seals and
connections with them are made by filling the stems with mercury.
Side tubes, above and below the electrodes respectively, are
attached for filling and washing out the cell. These tubes are
bent so as to form supports for holding the cell in a suitable bath
for maintaining a constant temperature.

148. Experiment 30. Resistance of Electrolytes. — Connect
the apparatus as shown in Fig. 109, placing the solution in a cell
specially designed for the purpose. Energize the bridge with the
Vreeland oscillator and detect the balance with a telephone
receiver. Determine the resistance of a series of solutions fur-
nished by the instructor. Measure the dimensions of the cell and
the distance between electrodes and compute the specific resistance
of each solution.

Report. — Explain why a bridge can not be balanced using direct
currents when it contains an electrolytic cell. What is the
essential difference between metallic and electrolytic conduction?

CHAPTER XIII
CONDUCTION OF ELECTRICITY THROUGH GASES^

149. Electrons. — When a high tension discharge passes between
electrodes sealed into a partially evacuated vessel, the gas
becomes luminous showing a series of highly colored glows
which are often very beautiful. If the pressure is sufficiently
reduced, a series of streams appears, proceeding in straight lines
from the cathode. These streams are known as "cathode rays,"
and are found to be independent of the position of the anode, and
often penetrate regions occupied by other glows in the tube.

The researches of modern physics have shown that these rays
are streams of discreet particles of negative electricity, called
"electrons." Their properties do not depend upon the material
of the electrodes nor the nature or presence of the gas through
which the discharge takes place. They may be produced from all
chemical substances, and consequently must play an important
part in the structure of matter. The velocities with which they
move through the tube vary from one-thirtieth to one-third that
of light. The ratio of the charge of an electron to its mass is con-
stant and is equal to 1.77 X 10^ electromagnetic units per gram.
The charge of an electron is 1.5 X 10"^" electromagnetic units and

the mass is about .. ^^^ that of the hydrogen atom. The radius

of an electron is estimated, at 1.9 X 10"^^ cms., which is about

-p. ^p.^ that of the atom. For many years the mass has been

regarded as purely electromagnetic in character; that is, while
exhibiting inertia, it shows no gravitational attraction in the sense
possessed by ordinary matter. Recently, however, certain
experimental and theoretical evidence has been produced which
makes it appear likely that this cannot be entirely the case.

1 Crowther, Ions, Electrons and Ionizing Radiations.
McClung, Conduction of Electricity through Gases and Radioactivity.
MiLLiKiN, The Electron.

Thomson, Discharge of Electricity through Gases.
TowNSEND, Electricity in Gases.

198

CONDUCTION THROUGH GASES 199

Many attempts have been made to discover evidence of
quantities of electricity smaller or larger than the electron, but
none smaller have ever been found. In fact, when quantities
comparable to the electron have been isolated, they have always
proved to be exact integral multiples of it. The evidence points
to the conclusion that electricity is atomic in structure and that
the smallest possible element is the electron, which thus con-
stitutes our natural unit of electricity. Electric currents through
conductors, as we know them in every day practice, are simply
streams of electrons through or between the atoms and molecules
making up the conducting body.

150. Conductivity of Gases. — A gas in its normal state is one
of the best insulators known. This may be shown by mounting
a gold leaf electroscope inside an inclosed space, and allowing
only a small rod carrying a polished knob, for the purpose of
charging, to project out. If the support carrying the electro-
scope is well insulated from the container, the electroscope will
remain charged for a long time, showing that the air or what-
ever gas surrounds the electroscope is a poor conductor of
electricity.

If, however, X-rays are allowed to shine through the enclosure,
or if a small quantity of some radio-active substance such as
thorium or radium is placed inside it, or again if the products of
combustion of a flame are drawn through it, it is then found that
the gold leaves collapse quite rapidly, indicating that the gas has
lost its insulating properties. That the leakage has taken place
through the air and not across the insulating support may be
shown by using a second chamber connected with the electrom-
eter enclosure by a glass tube, and introducing the X-rays, the
radio active substance or other agent into this, and then drawing
the air thus acted upon into the first chamber. The same effects
are observed. However, if glass wool is introduced in the con-
necting tube, or if the air is passed between two insulated plates
connected to a battery before entering the electrometer chamber,
it is found that its insulating properties are restored. Experi-
ments of this sort as well as many others of an entirely different
nature have shown that the conduction of electricity through
gases is due to carriers of electricity, and that the carriers aie of
two distinct types, positive and negative; the former are similar
to the carriers of electricity through solutions and are called
positive ions, while the latter are either negative ions or electrons.

200 ELECTRICITY AND MAGNETISM

161. Structiire of the Atom. — To explain the phenomena of
the conductivity of gases, it is necessary first to make a brief
statement concerning the structure of the atom. While our
knowledge is far from complete, it is well established that the
atom consists of a nucleus of positive electricity, about which
revolve in closed orbits, electrons, in much the same way that
the planets revolve about the sun, and that the relative dimen-
sions of electrons, nucleus and orbits are about the same as in
the solar system. The number of electrons present in a given
atom has been estimated in various ways, and while the results
are not entirely in agreement, it is probable that it is the same
as the atomic number, that is, its number in the list of elements
arranged in order of ascending atomic weights. The atomic
number, except for the case of hydrogen, is approximately half
its atomic weight. Since the atom as a whole is neutral, it is
necessary that the positive nucleus should have a charge equal
to ne, where e is the charge of the electron and n the number of
electrons. The shape of the orbits, the law of force between
nucleus and electron, and even the conditions of stability are
problems which have not yet been solved, but are now being
attacked from many angles.

When external agencies such as X-rays, ultra violet light, radia-
tions from radio active materials, etc., act upon a gas, it is found
that the atomic structure is broken up. One or more electrons
may be torn away from the system leaving it with an excess of
positive electricity. We thus have present in the gas positive
ions and negative electrons. The gas is then said to be ionized,
and the means by which this condition is brought about is
called the ** ionizing agent." If two electrodes are introduced,
and a difference of potential is maintained between them, the
electrons move to the positive electrode, and, entering it, pass on
through the external metallic circuit. The positive ions, on the
other hand, move to the negative electrode and receive electrons
from it, thus becoming again neutral molecules. Unless an
ionizing agent acts continuously, the current through the circuit
will persist only until the ions and electrons have been removed
from the gas.

152. The Ionization Current. — Suppose now that an ionizing
agent is acting continuously upon a gas in an ionization chamber,
as an arrangement such as that just described is called. At first
it might be supposed that if the agent acts long enough all of

CONDUCTION THROUGH OASES 201

the atoms would be ionized. This, however, is not the case; for,
due to their undirected heat motion, ions and electrons collide,
and recombine. When the rate of recombination is equal to
that of ionization, a steady state is reached where only a definite
fraction, usually a very small number, of the total number of
molecules are in the ionized state. If the difference of potential
between the plates is varied, and the current between them is
measured and plotted as a function of voltage, it is found that
the current increases with the voltage almost linearly at first, in
accordance with Ohm's law; but for higher voltages, the curve
is concave downward and when a certain voltage has been
reached, no further increase in current can be obtained, unless
the voltage is raised to very large values. The constancy of the
current is due to the fact that all of the ions and electrons pro-
duced are swept out by the field. This current is spoken of as
the "saturation current," from the similarity between the shape
of this curve and the magnetization curve for iron. The voltage
at which the horizontal part of the curve begins is called the
"saturation voltage."

If the distance between the electrodes is increased, it might, by
analogy with metallic conductors, be thought that the saturation
current would be reduced because of the increased path the ions
and electrons must travel. It is found, however, that the cur-
rent is actually increased. This is because there is a larger
number of gas molecules subjected to the action of the ionizing
agent, and hence more carriers are produced. Again, it is found
that if the pressure of the gas is increased, the ionization current
is increased. Both of these facts show that the saturation cur-
rent through a gas is proportional to the mass of the gas between
the electrodes.

163. Ionization by Collision. — If the voltage between the
plates of the ionization chamber is increased to sufficiently large
values, the saturation current does not remain constant indefi-
nitely, for fields may be reached at which the current again
begins to rise, slowly at first and then very rapidly, finally result-
ing in a disruptive spark accompanied by the passage of a current
of considerable magnitude. The field required for this increased
current depends upon the distance between electrodes, their size
and shape, and the nature and pressure of the gas. For air at
atmospheric pressure and spherical electrodes of moderate dimen-
sions, e.g., 1 cm. diameter, it is of the order of 10,000 volts per

202 ELECTRICITY AND MAGNETISM

centimeter. It diminishes, however, as the pressure is reduced,
and is most conveniently studied at pressures below 10 milli-
meters of mercury.

This increase in current is due to the fact that ions are produced
by collisions taking place between neutral molecules and ions as
well as electrons already existing in the gas. The mechanism
of this process is somewhat obscure, but it is clear that a definite
amount of energy is required to disrupt a neutral atom. The
kinetic energy of motion of the ions and electrons depends upon
how far they have moved under the accelerating field before
being stopped in the same way that the energy of motion of a
freely falling body depends upon the distance through which it
has fallen before being arrested. Thus, as the pressure of the
gas is reduced, the average length of free travel is greater and the
acquired energy available for ionizing purposes is increased.
The conductivity of a gas therefore increases as the pressure is
reduced. Since, however, the conductivity depends upon
carriers which come originally from neutral molecules, the con-
ductivity can not increase indefinitely with decrease of pressure,
for the effect of the decreased available supply will eventually be
felt. An optimum pressure therefore exists at which the
'increased range for acceleration is just balanced by the decreased
•supply of molecules. For air, this pressure is of the order of a
few tenths of a millimeter of mercury. A further decrease in the
pressure results in a rapid increase in the resistance of the gas.
If a perfect vacuum could be obtained, the free space between
electrodes would be a perfect insulator. While this is, of course,
impossible, it is, nevertheless, easy with modern methods of
evacuation to obtain pressures so low that no appreciable dis-
charge can be detected with the highest fields available in the
laboratory.

154. Experiment 31. Resistance of a Discharge Tube. — The
• apparatus consists essentially of a discharge tube, as shown in
Fig. Ill, about fifteen inches in length through the ends of
which are sealed wires attached to electrodes of relatively large
area. It is connected to a high vacuum pump by means of which
the pressure may be reduced to any desired value. A manometer
and McLeod gauge are also joined to the tube to measure the
pressure.

Connect a small high tension transformer across the tube to
supply the voltage for the discharge. Place an electrostatic

CONDUCTION THROUGH GASES

203

voltmeter across the tube and an A.C. milliameter in series with
it. The impressed voltage may be controlled by a series resis-
tance in the primary circuit. Starting at one atmosphere, reduce
the pressure until a current of 10 or 15 milliamperes is obtained
through the tube. Measure the required voltage. Take a
series of readings at various pressures measuring the voltage

To Pump and Gauge

High Tension
Transformer

Fig. Ill . — Resistance of discharge tube.

required to maintain a definite predetermined current. Com-
pute the resistance of the tube by Ohm's law. Repeat the experi-
ment using twice this current.

Report. — Plot a curve showing the resistance of the tube as
a function of pressure. Why must the current be held constant in
this experiment? Explain the operation of the McLeod gauge.

155. Phenomena of the Discharge Tube. — If electrodes are
mounted at the ends of a tube such as shown in Fig. 112, con-

204 ELECTRICITY AND MAGNETISM

taining air at ordinary pressures and a sufficiently high voltage
is impressed between them, the phenomenon first observed is the
ordinary spark similar to that between the electrodes of a static
machine. If, however, air is gradually removed, the sparks (
become less violent, and fine streamers of bluish color are \
observed. As the pressure is further reduced, these streamers
broaden out and fill the entire tube, and a pink color appears 1
at the anode. With further exhaustion, the pink color extends
some distance from the anode and dark spaces appear in the
region of the cathode. When the pressure has been reduced to

12 3 4 5

Fig. 112. — Luminous regions of discharge tube.

• about half a millimeter of mercury, the discharge assumes a very
characteristic appearance. Closely surrounding, but not quite
touching the cathode, is a thin layer of luminosity known as the
cathode glow. Next to this is a region, from which no light is
observed, called the Crooke's dark space, and beyond this is a
rather broad region of luminosity known as the negative glow.^
Following this is another non-luminous region, called the Faraday 4.
dark space. Between this dark space and the positive electrode
is a region called the positive column, which may be seen as a s
continuous band of light or, under certain conditions of current
and voltage, as a series of light and dark striae. The positive
column seems to be definitely associated with the anode, for if
the tube is increased in length or bent into a curve, the positive
column increases or bends with it, while the other parts of the
discharge remain fixed and are thus shown to be associated with
the cathode. These luminous regions are indicated in Fig. 112.
If the pressure is still further reduced, the striae of the positive
column become fewer in number and wider in extent and finally
disappear. The regions associated with the cathode also become
larger and, with the disappearance of the positive column, the
dark spaces fill nearly the entire tube. With sufficient ex-
haustion, the Crookej dark space completely fills the tube,
and the voltage required for a passage of current becomes

CONDUCTION THROUGH GASES 205

very high. At this stage, the walls of the tube fluoresce bril- ^
liantly with colors depending upon its chemical composition,
being bluish for soda, and bright green for German glass. If
the exhaustion is carried far enough, the tube becomes a non-
conductor of electricity.

156. Theory of the Discharge.' — Since no external ionizing
agent is acting, it is obvious that the discharge is maintained
by ions produced by collision, and the varied distributions of the
luminous regions indicate that the electric fields and the velocities
of the carriers can not be uniform throughout the tube. It has
not yet been definitely determined whether luminescence arises
from ionization of neutral molecules or whether it accompanies
the recombination of an ion and an electron to form a neutral
molecule. At the present time, the evidence seems to favor the
latter hypothesis. Another widely accepted view is that when a
molecule has been shaken up by collision with an ion or electron
to such an extent that its electronic orbits are badly distorted,
but not disrupted, light emission accompanies its return to the
equilibrium state. On the latter theory, luminous regions do not
necessarily coincide with regions of ionization. Some of the more
important {^enomena characterizing the several regions enu-
merated above are the following.

1. Cathode Glow. — The field strength in this region is large and
often the greater part of the entire potential difference occurs in
this limited space. The magnitude of the fall in potential depends
upon the nature of the gas and the material of the electrode,
ranging from 470 volts for water Vapor to 170 volts for argon with
platinum electrodes. If metals such as magnesium, sodium, or
potassium are used, much smaller values are obtained because
of the greater ease with which these substances emit electrons.
The large potential gradient here is caused by the accumulation
of positive ions in front of the cathode. Because of the greater
mobility of electrons, they rapidly move away from this region
thus leaving a preponderance of positive ions. The ionization
is caused by collision of the positive ions either with gas molecules
or the cathode itself.

2. (^T^/l/l^^t^ /^nr/p Spnnp.. — It was pointed out above that a
certain amount of energy is required to produce ionization. The
electrons from the cathode glow must move through a certain

• Cbowther, Ions, Electrons, and Ionizing Radiutions, chap. VI.
TowNSEND, Electricity in Gases, chap. XI.

206 ELECTRICITY AND MAGNETISM

difference of potential before they possess the requisite kinetic
energy for this purpose. The Crookes dark space represents this
distance for it is here that electrons, liberated in the cathode glow,
are acquiring the necessary energy of motion to produce the
ionization of the negative glow. It is, in general, a rough measure
of the mean free path of the electrons. No ionization .occurs in
this region and the current is carried almost exclusively by the
electrons.

3. Negative Glow. — The luminosity of this region is due to
ionization by electrons from the Crookes dark space. The positive
ions produced here move slowly out of the negative glow into the
Crookes dark space and their presence reduces the potential
gradient to such an extent that electrons, originating in the nega-
tive glow, do not gain sufficient speed to produce ionization; and
hence, after those entering from the Crookes dark space have been
stopped by the ionization process, no further ionization occurs.

4. Faraday Dark Syace. — The current in this region is due
largely to electrons which enter it from the negative glow.
Because of the accumulation of electrons in the negative glow, the
potential gradient through the Faraday dark space and even up to
the anode is quite large. The electrons are accordingly accel-
erated through this dark space and when they have attained
velocities sufficient for ionization, the positive column commences.

5. Positive Columri. — The potential gradient is practically
constant throughout this region and ionization by collision may
take place all the way, resulting in a uniform column of light.
Ordinarily, however, there are local accumulations of positive
ions, which result in a decrease in the potential gradient with a
consequent reduction in the acceleration of the electrons. There
are then regions in which the velocities are too small to produce
ionization and the striae commonly observed, result. Under
these circumstances, the positive column is, to a certain extent,
a repetition of the phenomena of the Crookes dark space, and the
negative glow.

157. Investigation of the Field Strength at Various Points
in the Discharge.^ — The potential at any point in a tube may be
determined by inserting an auxilUary electrode. A small plati-
num wire is most frequently used for this purpose. If the region
happens to be one of high potential, the wire will attract to it
positive ions until its potential is the same as that of its surround-

> Graham, Wied. Ann., vol. 64, 1898, p. 49.

CONDUCTION THROUGH GASES 207

ings, which is then indicated by an electrometer to which the wire
is attached. Accurate results can be obtained by this method
only when there is a plentiful supply of ions of both signs. For
example, suppose the wire is introduced near the anode, where
only electrons are present. The forces of the field will cause
electrons to strike the wire until it is so highly charged negatively
that no more can reach it because of repulsion, and the wire thus
has a negative potential considerably in excess of the region in
which it is placed. If positive ions also were present, they would
be drawn to the wire until its potential is the same as the
surrounding region.

If two test electrodes are used, the field strength at various
points through the discharge may be determined by measuring
the potential difference between them and dividing by their
distance apart. Except for regions close to the electrodes, where
only one type of ion is present, this method gives reliable results.
Because of the mechanical difficulty of moving a pair of test wires
through a tube with fixed electrodes, it is more convenient to use
a tube with fixed test wires tt and moveable electrodes as shown in
Fig. 113. The anode A and cathode C are held at a fixed distance
apart by means of a glass rod d with flexible leads connecting to
the seals through the tube. A small lug of iron / is acted upon by
a magnet so that the electrodes may be moved along the tube,
placing the test wires at any desired part of the discharge.

158. Experiment 32. Measurement of Field Strength in the
Discharge through Air. — Connect the apparatus as shown in Fig.

3^MOT^— ^

To Electrometer

iFly^mnm

To Pump and Gauge
Fio. 113. — Tube for measuring potential gradients.

113, using as a source of power either a battery of flash light cells
or a motor generator set giving an E.M.F. of about 1,000 volts.
Include a graphite resistance in series with the tube to prevent
arcing when the conductivity is high. Measure the difference
of potential between the test electrodes by means of an electrom-

208 ELECTRICITY AND MAGNETISM

eter which has been cheeked against a standard voltmeter.
Start the pump and note the character of the discharge from the
highest pressure at which a current can be maintained to the best
vacuum that the pump will give. An E.M.F. of 1,000 volts is
not in general sufficient to start the discharge although it will
maintain it, once it is going. To start it connect a small spark coil
across the tube with an air gap in series to prevent shorting the
generator or battery through the secondary of the coil. Deter-
mine the field strength at various points through the discharge
for two pressures (a) the highest at which a uniform discharge
can be maintained, (6) one at which the discharge has the
characteristic appearance shown in Fig. 112. Measure the pres-
sures by means of a McLeod gauge, and the voltage across the
tube by an electrostatic voltmeter.

Report. — Indicate by sketches the character of the discharge
for several different pressures. Plot field strength as a function
of distance from the cathode for the two cases studied. Plot
voltage as a function of distance from cathode. Obtain the
latter from the area under the field strength — distance curve.

159. Cathode Rays. — It was pointed out above that when the
pressure in a discharge tube has been reduced to a certain value,
e.g., a hundredth of a millimeter of mercury, the character of the
discharge is entirely changed from that represented by Fig. 112.
The positive column shrinks back and disappears entirely and
the Crooke's dark space occupies the entire volume of the tube.
The glass now shows a bright fluorescence, green or blue, depend-
ing upon its composition. This fluorescence is due to bombard-
ment by electrons shot out from the cathode or the region
immediately in front of it. They travel in straight lines perpen-
dicular to the cathode, and possess many interesting proper-
ties. For example, if they strike a piece of platinum foil, it may
be heated to incandescence by their bombardment, or if they
impinge upon substances such as willimite, calcium tungstate,
barium platino-cyanide, etc., they cause them to fluoresce
brilliantly. These streams of electrons are called cathode ray*

The fact that they possess a negative charge may be demon-
strated by placing two parallel plates within the tube between
which there exists a difference of potential. A stream of cathode
rays passed between them will be deflected away from the nega-
tively charged plate toward the positive. Again, if a magnetic
field is introduced across the tube, the stream will be deflected

CONDUCTION THROUGH GASES

209

at right angles both to their motion and to the field in the manner
required by the ordinary rules of electrodynamic action for
currents.

160. Velocity and Ratio of the Charge to the Mass of an
Electron.^ — The fact that an electron, when moving through a
magnetic field, is acted upon by a force at right angles both to its
motion and the direction of the field may be used to determine
the ratio of the charge to the mass of an electron and the velocity
with which it moves. Apparatus arranged for this purpose is
shown in Fig. 114, A vacuum chamber C is constructed from a
brass tube from which there projects a smaller tube A also of

To Pump

rmcrrT"

Fig. 114. — Apparatus for measuring

brass. The end of A is tapered and fitted to one end of a ground
glass joint. The other end of the glass tube is closed and carries
the cathode K. A piece of plate glass P, on the inner side of
which has been placed a thin coating of fluorescent material
such as calcium tungstate closes the vacuum chamber. The end
of the smaller tube A contains a brass plug through which has
been bored, with a jeweler's drill, a very fine hole.

When a suitable vacuum has been obtained, a discharge pro-
duced between A and K by a, static machine M causes a stream
of electrons to pass from Kto A, the individual electrons of which
move in straight lines normal to K. All but those lying in a

1 TowNSEND, Electricity in Gases, p. 453.
Crowther, Ions, Electrons and Ionizing Radiations, p. 92.
Duff, A Text Book of Physics, p. 492.

210 ELECTRICITY AND MAGNETISM

very narrow beam, defined by the hole through A, are stopped

but those passing through, enter the chamber C and produce on

P a bright fluorescent spot. If now the solenoid is energized,

the magnetic field causes a deflection of the beam and the spot

is moved a distance d perpendicular to the plane of the paper,

(shown in the plane of the paper in Fig. 114).

If the magnetic field is uniform, the path of an electron is

circular, since the force, in this case, is constant in magnitude,

and is always at right angles to the motion. The magnitude of

the force may be obtained as follows: Let I be the length of path

of an electron through the magnetic field. When it has traversed

the distance I, a quantity of electricity e has been transported

through this distance and may be replaced by a steady current of

g
strength i defined by i = t, where t is the time required for the

electron to travel the distance I. The theory of electrodynamics
gives, for the force acting on a conductor of length I, carrying a
current i, the expression

F = mi = Hel = Hev ,^.

where v is the velocity with which the electron moves.

Since the electron moves in a circle, whose radius we will call
R, the force given by eq. (1) must balance the centrifugal force.
Accordingly, we have

= -^ (2)

where m is the mass of the electron. The velocity v is acquired
while the electron moves through the difference of potential E
maintained between the anode and cathode by the static machine.
Since there is no potential difference between A and P it travels
this distance with constant velocity. The kinetic energy
acquired in moving from K to A is equal to the loss of potential
energy over this distance. From the law of conservation of
energy and the definition of potential difference, we have

Ee = >^mz;2 (3)

Eliminating successively v and — from eqs. (2) and (3) there

results

e 2E , 2E ,..

- = ^,^,and. = -^ (4)

CONDUCTION THROUGH GASES 211

The radius of curvature R is obtained from the sagitta formula

The magnetic field strength H is computed from the dimensions
of the solenoid and the current through it by the formula

„ 4nrNI

^ = ior

where N is the number of turns on the solenoid and L its length.
The accelerating potential E is measured by an electrostatic
voltmeter.

161. Experiment 33. Measurement of — and Velocity for

an Electron in a Cathode Ray. — Connect the apparatus as shown
in Fig. 114. Start the static machine and pump the vacuum
chamber until a green fluorescence is seen near the anode A.
Should this color appear at K the leads to the static machine
should be reversed. A bright spot will appear at P. Energize
the solenoid and determine the current required for a suitable
deflection d. In taking observations, reverse the solenoid current
and measure 2d. It will be found that by varying the vacuum,
different voltages may be maintained across the discharge while
the static machine is driven at a constant speed. With the
two halves of the solenoid as close together as possible, take a
series of observations using different accelerating voltages, and

deflecting fields and determine — and v. The fact that the parts

of the solenoid must be separated to permit the entrance of the
discharge tube introduces a non-uniformity in the field. To
determine this error, take a series of observations, keeping the
accelerating potential and the solenoid current constant and
increase the separation of the solenoid parts from the smallest

amount up to 10 cms., and plot the apparent values of — and v

as a function of the separation. The intercept of this curve,
when extrapolated to zero separation gives the correction to be

applied to the results obtained above. Since the value of —

is usually given in electromagnetic units per gram, it is necessary
to express E and H in eq. (4) in that system.
Report. — Plot the correction curve called for above and apply

to average values of — and v. Compute the velocity of an elec-

212 ELECTRICITY AND MAGNETISM

tron which has fallen through the following differences of poten-
tial using your value for — : 300, 3,000, 30,000 volts. Compute

the time required for an electron to move from X to A for some
one of the conditions actually used in this experiment. If the
charge on an electron is 4.77 X 10~^° electrostatic units, compute
the number of electrons passing per second across a plane in a
wire through which a current of one ampere is flowing.

162. Radio-active Substances.' — If the region surrounding any
radio-active substance such as uranium, radium, thorium, etc.,
is examined by appropriate means, it is found that these sub-
stances emit definite radiations which have very unusual prop-
erties. These radiations, for example, are able to darken a
photographic plate, to convert an insulating gas into a conductor,
and to cause a fluorescent screen to emit light. Moreover, they
are different from ordinary light in that they are able to pene-
trate many substances usually regarded as opaque. It has been
found that each radio-active substance is a definite chemical,
element and that its activity is due to a spontaneous decomposi-
tion or disintegration of its atoms. Furthermore, when certain
of the rays are emitted, there is a definite reduction in the atomic
weight of the substance, which naturally leads to the view that
the atoms of these substances are made up of complex systems
which have the same intrinsic character and differ from one
another only in their order of arrangement or degree of complex-
ity. Three distinct types of radiation have been found which
are designated as a, /3, and 7 rays.

163. The Alpha Rays. — These rays are distinguished from the
others by the fact that they are easily absorbed on passing
through gases or thin sheets of metal and that their action on a
photographic plate is weak. On the other hand, they are very
effective as a means for ionizing a gas, and they cause fluorescent
substances to emit light. If a screen upon which they are
acting is examined by a microscope, it is found that the illumina-
tion is not uniform but is made up of a large number of separate
flashes as though the screen were under bombardment. In
fact it has been found that a rays are discrete particles shot out

^ Crowther, Ions, Electrons and Ionizing Radiations, chap. XI.
McClung, Conduction of Electricity through Gases and Radioactivity.
Part II.

DtJFF, A Text Book of Physics, p. 502.

CONDUCTION THROUGH OASES 213

from radio-active substances and it is possible by suitable
experimental arrangements, to photograph their zig-zag courses
as they make their way through a gas, abruptly deflected by
some of the gas molecules, and stopped by others.

If a beam of a particles is shot at right angles to an electric
or a magnetic field, the path is curved in much the same manner
as the cathode ray stream described above, except that the
deflection is much smaller in magnitude due to their larger mass
and is in the opposite direction, indicating that they are posi-
tively charged. By making use of electric and magnetic deflec-

tions, the value — of the ratio of the charge to their mass and the
m

velocity with which they are emitted, have been measured. The

results show that — is the same for all a particles, no matter

what their source and is equal to 4,823 electromagnetic units per
gram, and that the velocities range from 1.5 X 10^ to 2.2 X 10^
cms. per sec.

The ratio of the charge to the mass for the hydrogen ion in
electrolysis is twice that for the a particle, and at first sight
it might be supposed that the latter is a hydrogen molecule
consisting of two atoms. However, it has been found that the
charge carried by the a particle is twice that of the hydrogen
ion, and hence its mass must be four times that of the hydrogen
atom. Since the particle is atomic in size and is of the same
order of magnitude as the atom of helium whose atomic weight
is 3.96, the most natural assumption is that it is an atom of helium
with twice the electric charge of the hydrogen ion. This hypo-
thesis is supported by the fact that both chemical and spectro-
scopic analyses show conclusively that helium is always present
where radio-active transformations are taking place.

164. The Beta Rays. — The /? rays are distinguished from a rays
in several important respects. In the first place, they have
a far greater penetrating power. While the a rays are completely
stopped by a sheet of aluminum foil }{o mm. in thickness, /3 rays
still produce noticeable effects after passing through sheets
100 times this thickness. Again, they are much more easily
deflected by a magnetic field. The deflection of the a rays is
appreciable only in the largest fields available, and even then
special methods have to be employed. The /3 particles, on the
other hand, travel in circles of large curvature when moving at

214 ELECTRICITY AND MAGNETISM

right angles to fields of ordinary magnitudes. The direction of
the deflection shows that th6y carry a negative charge, and all
the evidence indicates that they are identical with the cathode
rays of the ordinary discharge tube, i.e., electrons.

By subjecting /3 rays to the deflecting action of electric and

magnetic fields combined, the values of — and the velocities

with which they are emitted may be measured. It has been
found that while the former is the same as for the cathode-ray
particles, the velocities of emission are considerably higher than
those observed in discharge tubes, ranging from 6 X 10* to
2.8 X 10^° cms. per second. The latter is very close to the
velocity of light, 3 X 10^° cms. per second.

A careful study has been made by Kaufmann of the value — for

the particles as a function of velocity, and it was found that — is

not constant, but decreases as the speed increases. This can be
explained only by assuming that e decreases or that w increases
as the velocity becomes larger. The evidence furnished by other
lines of study indicates that the charge of the electron is one of the
fixed constants of nature, and therefore it is concluded that the
mass of the electron depends upon its velocity. Theoretical
considerations have shown that the apparent mass of an electron
is due wholly, or in part, to the motion of its electric charge. In
fact, for a number of years, the view was held that the mass of
the electron is entirely electromagnetic in character, but some
very recent work indicates that this can not be the case entirely.

165. The Gamma Rays. — The nature of the y rays is very
different from that of the a and j8 rays. They are distinguished
by the fact that they possess very much greater power of pene-
tration. In fact they may easily be detected after passing through
sieveral cms. of iron. Though subjected to the most powerful
electric and magnetic fields available, they show no deflection,
and can not therefore carry an electric charge. They cause a
fluorescent screen to emit light, and affect a photographic plate.
When passed through gases they produce ionization, and, in
fact, are usually detected by this action.

Searching investigations have shown that they are similar in
character to X-rays, that is, electromagnetic waves of very short
wave length. The similarity of the relation of /3 rays to cathode

CONDUCTION THROUGH GASES 215

rays and 7 rays to X-rays is very close. When the target of an
X-ray tube is struck by a rapidly moving electron, the electronic
orbits of one of the atoms of the former undergoes some sort of
rearrangement ; that is, they change over from one stable config-
uration to another possessing a different amount of potential
energy, and a train of X-rays is emitted. The emission of the X-ray
occurs as the result of suddenly stopping a high speed electron.
Similarly, when a radio-active substance emits a /3 particle, send-
ing it forth with a velocity comparable to that of light, a
rearrangement of the electronic orbits also occurs, which is accom-
panied by the emission of the 7 ray. The 7 rays thus accompany
the rapid acceleration of electrons. The fact that 7 rays are
always present when /3 rays are emitted supports this view.
Measurements have shown that the wave length of the 7 rays
is somewhat shorter than that of the most penetrating X-rays.

166. Radio-active Transformations. — Careful investigations
of the phenomena accompanying the emission of the rays just
described, show that radio-active substances are distinguished
from ordinary ones in that they are constantly undergoing changes
of character, never observed in ordinary materials. Each sub-
stance is entirely distinct from the other, and has its own charac-
teristic physical and chemical properties. However, instead of
enduring indefinitely as is the case with ordinary elements, such
as copper, iron, gold, etc., each radio active substance has a definite,
measurable period of existence, after which it disintegrates and
becomes a new chemical substance, and it is during these proc-
esses of transformation, that the emission of rays occurs.

All molecules are made up of atoms which consist of positive
nuclei with electrons rotating about them in closed orbits. The
electrons are held in their orbits by the electric attractions
existing between them and the nucleus while the atoms are held
together by the electric forces between their parts, or the magnetic
forces due to the circulating electrons. This complicated struc-
ture becomes unstable for some reason or another, and an a
or a /3 particle or both is emitted. After a rearrangement of the
remaining particles, a new state of stable equilibrium ensues,
giving a new substance of different physical and chemical pro-
perties. As an illustration, take the substance radium. Although
the individual molecules do not have the same periods of existence,
the life of an average molecule is 2,000 years. At the end of this
time, it emits an a particle, and the residue is called radium

216 ELECTRICITY AND MAGNETISM

emanation. The emanation persists for a period of 3.75 days
when it gives off another a particle, and becomes radium A. In
this form it lasts for 3 minutes, then again emits an a particle and
becomes radium B. This state persists for 26 minutes when it
gives off a /3 particle accompanied by a 7 ray and becomes radium
C, and so on. The entire series has been carefully worked out,
starting with uranium, going through ionium and the various
phases of radium, and thorium to those of actinium. The duration
of the different phases ranges from a few minutes to 10^° years.
Some of the transformations are apparently not accompanied by
the emission of any rays. These transformations are explained by
supposing that the ray is present but possesses such a low velocity
as to be unable to ionize a gas and is therefore not detected.

It is important to note that each time an a particle is emitted
the atomic weight decreases by 4, i.e., theatomic weight of helium.
Furthermore, the last radio active product, radium F, or polon-
ium, has an atomic weight equal to that of lead, and possesses
the properties of ordinary lead.

It is easy to conjecture that each of the chemical elements as
we know them, has been derived from one higher in the scale of
atomic weights by the emission of one or more a particles, and
that transformations are going on continuously but at a rate so
slow as to escape detection by methods at present available.

167. Experiment 34. Ionization by Radio-active Substances. —
The apparatus for this experiment is shown in Fig. 115. It
consists of an ionization chamber made entirely of metal. The
radio-active substance, in the form of a powder is spread over the
plate A which may be moved up or down. An insulated plate D
is connected to an electrometer E mounted in another chamber
and connected with the ionization chamber by a removable brass
tube. The electrometer is charged by means of a battery B of
small dry cells by pressing down the wire W which must be
insulated from the container. When the rays from the radio-active
substance pass up through the metal gauze G they ionize the air
between G and D. Either electrons or positive ions, depending
upon the sign of the charge on D and E are drawn toward D and
neutralize this charge. The deflections of the electrometer are
read by means of a long focus microscope provided with an eye
piece having a graduated scale. The time required for the gold
leaf of the electrometer to fall through one division is inversely
proportional to the ionization current.

CONDUCTION THROUGH GASES

217

It is necessary first to determine the rate of discharge of the
ionization chamber and electrometer due to leakage alone. For
this purpose, cover up the radio active substance by a close fitting
metallic plate P. Charge the electrometer by connecting it for
an instant to the battery by means of W, and note the time
required for the gold leaf to fall one division. Take several
readings and average. This leakage is due partly to imper-
fect insulation, and partly to y rays which penetrate the metal
cover.

Remove the shield P, charge the electrometer as before, and
with A near the bottom of the chamber, determine the rate of

-h

Fig. 115. — Apparatus for ionization studies.

discharge as above. The difference between these two rates is
a measure of the ionization produced by the ^ rays from the radio
active substance, provided the distance AC is greater than 10
centimeters, the range of the a particles. Take a series of
observations determining the rate of leak each time raising A }4.
cm. until a marked increase in the rate of leak is observed. This
indicates that the a rays have penetrated the space between the
gauze and D. Take readings each millimeter until the rate has
become nearly constant again. Continue until the plate A is
as high as it can be raised.

Report. — Plot inverse time of leakage against distance between
A and D. Draw horizontal lines representing the leakage cur-
rent, and that due to the currents produced by both /3 and a ray
ionization.

CHAPTER XIV
ELECTRON TUBES ^

During the last decade, the electron tube has had a develop-
ment little less than phenomenal. Because of the multiplicity
of its uses, e.g., as detector, amplifier, oscillator, modulator, etc.,
it finds many appHcations not only in the art of radio communi-
cation but also in engineering work and the general research
laboratory. No student of electrical engineering or physics can
afford to be unacquainted with this device, and it is the purpose
of the present chapter to set forth and illustrate the fundamental
principles upon which it operates.

168. Free Electrons. — It is customary to distinguish an insu-
lator from a conductor by saying that in the former, electrons
are held in their orbits about the positive nuclei with forces so
great that they can be dislodged, if at all, only by exceedingly
large fields while in the latter, the attracting forces are so weak
that they are easily torn from their positions of equilibrium and
move about through the body. The idea of such easily disrup-
table atoms is held by some to be inconsistent with the rigid
mechanical properties of metals, and the ease with which elec-
trons move through conductors is explained by saying that while
the forces holding atoms together are very large, nevertheless,
due to the closeness of approach during collisions, the nucleus of
one atom may attract an electron of another with so great a force
in the opposite direction that the electron is nearly in equilib-
rium, and a slight field may cause it to leave its original atomic
system and enter that of a neighboring one. The one from which
it escaped would be left with an excess positive charge and might,
in a similar manner, capture an electron from another atom.
According to this view, electrons move through a conductor, not
by zig-zag paths between molecules, but by passing through
the molecules, and forming distinct parts of the atomic structures

* Richardson, Emission of Electricity from Hot Bodies.
Van der Bijl, Thermionic Vacuum Tube.
MoRECROFT, Principles of Radio Communication.
Lauer and Brown, Radio Engineering Principles.

218

ELECTRON TUBES 219

on the way. Because of the many coUisions taking place in
consequence of thermal agitation, the large number of electrons
required to explain observed currents is easily accounted for.

169. Electron Emission. — From the fact that electrons move
thus freely from one part of a conductor to another, going either
between the molecules or through them, and pass readily out of
one conductor in to another in contact with it, it might be inferred
that they could also be drawn easily out of a conductor into a
vacuVous space. It is found, however, that this is not the case,
and tnat special means must be used to cause them to thus
emerge. For want of a better explanation, it has been assumed
that there exists at the surface of a conductor, a force which
tends to keep the electrons within the body. The exact nature
of this force is unknown, but recent developments regarding the
structure of the atom tend to support the view that such a force
really exists. If this is true, a certain amount of work must be
done on the electron to move it out of a body against this attract-
ing force.

One of the ways in which electrons may be dislodged from a
metal is by the application of electromagnetic radiation of very
short wave length. For example, if ultraviolet light falls upon
an insulated piece of zinc, it acquires a positive charge, or, if
originally charged to a negative potential, it loses this charge
and becomes positive. This is explained by saying that the
electrons within the atoms of the metal absorb energy from the
incident hght waves and are stimulated to vibrate with ampli-
tudes so great that they possess energy sufficient to overcome
the surface forces and escape into the surrounding space with
velocities which depend upon the energy of the light wave and
that lost in moving against the surface attraction. This is
known as the photo electric effect, and while it is most pronounced
in zinc it is found to exist to a greater or less extent in all metals.
Another way in which electrons may be dislodged from a body
is by bombardment with other electrons. Certain metals,
notably copper and nickel, when struck by electrons having
sufficient energy of motion may emit as many as twenty other
electrons for each one striking. This is known as secondary
emission, and has been made use of in a number of electron
devices.

The most effective way to get electrons out of a body is to
heat it. The explanation of this effect is as follows: Since the

220 ELECTRICITY AND MAGNETISM

body possesses temperature, its molecules must be in motion
and the average kinetic energy of the molecules is a measure of
the temperature. The electrons being free to move about within
the body must also possess undirected motion of thermal agita-
tion from impact with the molecules. In fact it is generally
supposed that they are in thermal equiUbrium with the molecules,
that is, the average value of their kinetic energies is the same
as that of the molecules. Since kinetic energy is 3-^ mv^, and the
mass of an electron is very much less than that of a molecule, it
follows that the velocities of the electrons must be many times
larger than those of the molecules. The temperature accord-
ingly need not be very high (dull red) before an appreciable
number of electrons will possess sufficient energy of motion to
overcome the surface force of attraction and escape into the
surrounding space. If the emitting body is insulated, it will
take on a positive charge because of the loss of electrons. If the
body is in a closed vessel so that the electrons can not move far
away, some of them will be drawn back into it, and an equilib-
rium condition will be established in which the number emitted
is equal to the number falling back. The number of electrons
emitted per unit time is given by the formula^

_ b
N = AVTe ^

where T is the absolute temperature of the body, e, the base of the
Naperian logarithms, and A and h are constants depending upon
the nature of the substance, its size, shape, and certain other
characteristics.

170. The Two-element Electron Tube. — This is a device in
which application of thermionic emission is made for the rectifica-
tion and control of currents. It consists of a filament F which
may be made of tungsten or of platinum coated with oxides of
barium, stromtium or calcium and a plate P mounted within an
enclosure which has been exhausted to a very high vacuum. The
filament may be heated to any desired temperature by a battery
A as shown in Fig. 116. Another battery B is connected between
the plate and the filament in such a way as to make P positive
with respect to F. On heating the filament, electrons pass out
into the enclosure, and, were it not for the battery B, would fill
the space with a definite density depending upon the temperature

» Richardson, Phil. Trans. (A) vol. 201, 1903, p. 543.

ELECTRON TUBES

221

Fig. 116. — Two element
electron tube.

of the filament, and an equilibrium state would be reached in
which the number returning to the filament is equal to that of
those leaving it. The battery B produces an electric field which

causes electrons to move from the

filament to the plate, which they

enter and pass through the external

metallic circuit and return to the

filament. They constitute an electric

current which, according to common

parlance, is said to flow into the plate

and out of the filament. The passage

of electrons through the vacuous space

between the electrodes is called the

"space current." The magnitude of

the space current is limited by two important considerations

which may be made clear by the following experiments.

171. Voltage Saturation. — Suppose that the temperature
of the filament is held constant at a value somewhat less than that
required for normal operation of the tube. Let the voltage of the
battery B be gradually increased from zero to some specified
value, and let the space current be measured and plotted as a

function of B. It will be
found that for small values
of plate potential the space
current gradually increases
as shown by the curve of
Fig. 117, marked Ti, but that
it soon stops -rising and re-
mains constant, no matter
how much the voltage is
increased. This limitation of
the current is due to the fact
that there is available at the
filament only a finite number
of electrons which depends
upon its temperature as
shown by eq. (1). When the voltage is sufficient to draw to
P aU the electrons which are emitted at a given temperature,
the maximum current for that temperature is reached, and no
increase in voltage, however great, can further increase the cur-
rent. If now, the experiment is repeated using a higher filament

Fig. 117.-

Plate Potential
-Voltage saturation curves.

222 ELECTRICITY AND MAGNETISM

temperature, the lower part of the curve will be the same as in
the previous case. When such a voltage is reached that the
available electrons are all drawn to the plate, the current again
becomes constant but this time at a higher value, as shown by
Ti of the figure. In this way a series of curves may be
obtained which are coincident at their lower extremities but
become horizontal at definite values of voltage for each filament
temperature.

The constant current which results when all the available
electrons are used is somewhat inappropriately called the "satura-
tion current" from the similarity of shape between these curves
and the magnetization curves of iron, in which the knee of the
curve is called the saturation point. The voltage required to
produce the saturation current at any temperature is called the
"saturation voltage" for that temperature. The saturation
current is thus a measure of the total electron emission at a given
temperature.

172. Space Charge. — From the experiment just described, it
might be inferred that the rate of electron emission is the only

limitation to the magnitude of
the space current and that if
filaments of sufficient areas were
provided, currents of any magni-
tude could be obtained. That
this is not the case is shown by
the following experiment.

Suppose now the voltage of
the battery B is held constant
and the space current measured
as the temperature of the fila-
ment is changed. Starting with

Filament Temperature ., ^, . i i •, -n i

the filament cold, it will be

Fig. 118. — Effect of space charge. fjj.ij.xi j. •

found that the space current is
zero, since no electrons are emitted. In fact for most filament
materials, there will be no space current, measurable with
ordinary instruments, until the filament is hot enough to show
a dull glow. When this point has been reached, it is found
that the space current rises rather rapidly with increasing fila-
ment temperature. This increase in space current does not
continue indefinitely for a temperature is soon reached above
which the space current remains constant as shown by the curve

ELECTRON TUBES 223

marked Vi of Fig. 118. Even though the temperature of the
filament is raised to the melting point, the space current remains
constant. If, however, the experiment is repeated using a larger
voltage for the battery B, it is found, on starting again with the
filament cold, that the relation between the space current and
filament temperature is the same as in the previous case for low
temperatures. At a certain temperature, however, the space
current again becomes constant but has a larger value than
before, and takes place at a higher filament temperature. This
is shown by the curve V2 of the figure. Repeating with a still
higher voltage, the curve Vz is obtained.

The limitation of the space current in this case is due to the
action of the electrons which constitute it. Consider an elec-
tron which has just emerged from the filament. If it were the
only electron between the filament and the plate, it would be
acted upon by an electric field which depends only upon the dif-
erence of potential between the filament and the plate. If how-
ever, there exists between this electron and the plate a second
electron, the force on the first electron will be less than in the pre-
vious case since it is repelled by the second electron. The fact
that the second electron may be in motion makes no difference so
long as its velocity is less than that of light. If, now, there is a
swarm of electrons of sufficient number between the filament
and plate, their repelling action on freshly emitted electrons will
just balance the attraction of the positively charged plate and there
is no tendency for them to move, until some of those near the plate
have entered it and thus reduced the number in the swarm.
Electrons from the filament will then enter the swarm keeping the
number between filament and plate constant, thus giving the steady
space current observed. If the plate voltage is raised, it will re-
quire a larger number of electrons in the space to neutralize this in-
creased potential gradient. Furthermore, electrons will be drawn
out of the swarm more rapidly thus requiring a larger number to
enter it to maintain equilibrium and the space current is
thereby increased.

From the explanation just given, it may be inferred that the
maximum space current which can be obtained for a given diff-
erence of potential between filament and plate depends upon the
shape, dimensions and spacing of the electrodes. For a tube
having a cylindrical plate of radius r, and a straight filament

224 ELECTRICITY AND MAGNETISM

placed along its axis, the current per unit length of filament is
given by the expression'

i = ^f^JlrL (2)

9 \m r

where V is the difference of potential between filaiiient and plate
and — is the ratio of the charge to the mass of the electron.

Substituting numerical values for -■> and expressing V/i, and r in
volts, amperes and centimeters this becomes

i = 14.65 X 10-«- (3)

The two element tube finds its chief supplication as a rectifier and
is often called the "Kenotron." Current can flow only when the
plate is positive with respect to the filament. When the plate is
negative, filament electrons are driven back into it as fast as
they are emitted, and so long as the plate is cold, none are emitted
there. Consequently there can be no reverse current. The tube
is thus a perfect rectifier for any voltage within the limits of the
mechanical and dielectric strength of the parts of which it is made.
It ceases to function as a rectifier however, if the plate becomes
too hot for it also then becomes a source of electrons. Even at
relatively low voltages, electrons acquire velocities of many thou-
sand miles per second in passing from filament to plate and thus
strike it possessing very appreciable amounts of kinetic energy
which is converted into heat by bombardment. In fact this
effect is made use of in heating the plates to ''outgas" them
during the evacuation process.

173. Experiment 35. Characteristics of the Two Element
Electron Tube. — Connect the apparatus as .shown in Fig. 116,
using for B a battery of flash light cells or a motor generator set
giving an E.M.F. of about 500 volts. The purpose of this
experiment is to obtain the two sets of characteristic curves for
the Kenetron rectifier illustrated in Figs. 117 and 118. Ascertain
from the instructor the normal filament current for the tube, and
using this and two smaller ones take plate potential-space current
characteristics for each, varying plate potentials from 0 to 500
volts. Determine also the filament current — space current
characteristics for three different values of plate potential.

1 Langmuir, Gen. Elec. Rev., 1915, p. 330.

ELECTRON TUBES 225

Report. — Plot the two sets of curves as indicated. Explain
what is meant by voltage saturation and space charge. From
formula 3 compute the dimensions of a tube that would carry one
quarter of an ampere with a difference of potential of 500 volts.

174. The Three -element Electron Tube. — In the discussion of
the two-element tube the dependence of space current upon fila-
ment temperature and plate potential was described, and it
was pointed out that its principal application is in the rectification
of high voltage alternating currents. By changing the tempera-
ture of the filament, thus regulating the supply of available
electrons it also serves as a means of controlling currents. In
this way, it acts as an electrical valve which may be opened or
closed to any desired fraction of its current carrying capacity.
Since, however, filament temperatures do not respond immedi-
ately to changes in heating current, this action is sluggish, and
it can not be used in this way to produce current variations that
are at all rapid.

It has been found that the space current may be controlled
with remarkable ease by the introduction between the filament
and plate of a third electrode in the form of a grid or mesh of fine
wires through which the electrons must pass on their way from
filament to plate. Such an arrangement is shown in Fig, 119.
If a difference of potential is established between the filament and
grid by means of the battery C, the grid tends to accelerate or
retard the electrons of the space current according as it is positive
or negative with respect to the filament. It thus counteracts or
increases the effect of the space charge. The operation of the
three-element tube may be best described by means of the curve
of Fig. 120, which shows the relation between the plate current
and the grid volts and is known as the "static characteristic."
If the grid is disconnected from the circuit, the tube behaves as
the ordinary two-element device in which the space current is
limited either by electron emission of the filament or by space
charge. Assuming there is available a sufficient supply of elec-
trons so that the space charge is the controlling factor, a negative
potential placed upon the grid adds to the retarding action of the
space charge, and the plate current is reduced, and may even
be made zero, if the grid is sufficiently negative. Again, if the
grid is positive, it neutralizes to a certain extent the effect of the
space charge, causing an increase of the space current. The space
current can not continue increasing indefinitely, for even though

226 ELECTRICITY AND MAGNETISM

the space charge were completely neutralized by positive charges
on the grid, the current would be limited by the electron supply at
the filament. This accounts for the horizontal part of the static
characteristic. If a higher voltage is applied to the plate, the
characteristic curve is not changed in shape, but is shifted toward
the left. This is because larger negative grid voltages are required
to reduce the space current to a given value.

This method of controlling the space current has a number of
advantageous features. In the first place, it requires the expendi-
ture of exceedingly small amounts of energy. If the grid is nega-
tive with respect to the filament, no electrons strike it and
consequently no current flows through the battery C, hence the
only energy drawn from it is that required to charge the con-
denser formed by the grid and filament, which is negligible in
most cases. If, however, the grid is positive with respect to the
filament, a few electrons strike it and a current is drawn from C
which then supplies energy to the tube. If, however, the grid
wires are very fine, this current may be made quite small even
though relatively large positive potentials are impressed on the
grid. The battery B may be one of high voltage and the space
current will therefore have large amounts of power associated
with it. Accordingly, by the expenditure of small amounts of
power in the grid circuit, large amounts of power in the plate
circuit may be controlled, and the device constitutes a relay
having a large energy ratio. '

In the second place, the response of the plate current to changes
in grid potential is exceedingly quick, almost instantaneous. If
the time required for an electron to travel from the filament to
the plate is computed by eq. (3) of chap. XIII, it is found that for
an ordinary tube with moderate plate voltages it is of the order
of one hundredth of a millionth of one second. This then is the
order of the time lag to be expected. For this reason it may be
regarded as a relay with no moving mechanical parts and is
therefore without inertia in its action.

Again, there exists for a considerable range, a linear relation
between grid potential and plate current so that the variations in
plate current are faithful reproductions of the changes in grid
potential and thus the device is a distortionless amplifier.

175. Experiment 36. Static Characteristics of a Three-element

Electron Tube. — Connect the apparatus as shown in Fig. 119.

^Use for the filament battery A, a set of storage cells, furnishing

ELECTRON TUBES

227

from 10 to 20 volts depending upon the size of the tube to be
tested. Ascertain from the instructor the normal heating cur-
rent for the filament, and be careful that this is not exceeded at
any time during the test. If the filament is of the oxide coated
type, it should be operated at a dull red heat, but if it is a tung-
sten wire, bring it up to about the same brightness as the ordinary
vacuum incandescent lamp. B may be a battery of flash light
cells giving 500 volts or a motor generator set. For C use a
battery of flash Ught cells giving about 60 volts. Bring the
filament up to normal temperature, and apply a plate voltage
of ^i normal. Apply a sufiicient negative voltage to the grid

Pig. 119. — Three element electron tube.

to reduce the plate current approximately to zero. Raise the
grid volts by steps to zero and positive values and note the
grid and plate currents for each setting. Repeat for several
values of plate voltage up to and including normal.

Report. — Describe the three element electron tube and outHne
its principal operation features. Plot the static characteristic for
the plate voltages studied, also the grid current as a function of
grid volts. Sometimes a negative grid current is obtained.
How can this be explained?

176. Amplification Factor. — The fact that the three-element
tube may be used as a relay has been referred to several times,
and it is necessary to define accurately what is meant by this
statement. By a relay, is meant any device by which a small
amount of energy may be used to turn on and off or control a
much larger source of energy. In the case of the electron tube,
the source of energy is the plate battery and the grid is the gate
by which it is controlled. Considering now the plate and grid

228

ELECTRICITY AND MAGNETISM

circuits, it is obvious that we may be interested in the relative
values of either the power, the currents, or the voltages existing
in these circuits, and that we may accordingly refer to either the
power amplification, the current amplification, or the voltage
amplification. The meaning of the first two of these expressions
is obvious; for example, by power amplification is meant the
ratio of the change in power drawn from the plate battery to
the change in power supplied to the grid, and a corresponding
meaning is given to current amplification.

However, in the ordinary use of the tube, the voltage of the
plate battery remains constant, and the meaning of the voltage
amphfication factor is not so evident. The significance of this

term can perhaps be un-
derstood by reference to a
series of static characteris-
tics as represented in Fig.
121, where the dependence
of space current upon grid
volts for a series of plate
potentials, at 50 volts in-
tervals, is shown. Sup-
pose, for example, the plate
voltage is 100 and the grid
volts zero. The space
current is then 10 milli-
amperes. It is desired to
increase the space current
to 20 milliamperes. This may be done either by raising the plate
voltage to 150 or the grid voltage to 5. Thus an increase of 5
volts on the grid produces the same change in the space current
as an increase of 50 volts on the plate. The voltage amplification
factor in this case is said to be 10, since one volt on the grid is
equivalent to 10 volts on the plate.

A working equation connecting these quantities may be
deduced as follows. It was shown in eq. (2) that for the two-
element tube, the plate current is proportional to the %
power of the plate voltage, i.e., Ip = aV^, where a is a constant.
Since a change in grid voltage is more effective by a certain
factor, which we will call k, in producing a change in plate current
than a change in the plate voltage, it follows that the plate cur-
rent in a given tube on which there is acting a plate voltage Ep

~~ Grid Volts

Fig. 120. — Characteristic for three element
electron tube.

ELECTRON TUBES

229

and a grid voltage Eg is just the same as though it were a two-
element tube with a plate voltage Ep + kEg. The expression
for the current then becomes

/p = a(Ep + kEg)^ (4)

Since eq. (2) refers to the case in which there is an abundance of
electrons at the filament and the current is limited only by the
space charge, eq. (4) holds only for the left-hand part of the
characteristic, i.e., up to the bend.

Referring to Fig. 121, it is seen that the static characteristics
all have a point of inflection, and that for a considerable portion
each side of this point, the curve is nearly a straight Hne. If the

— 25 20 10 5 0 5 10 15 20 23 •\-

Grid Volts
Fig. 121. — Dependence of static characteristics upon plate potential.

tube is used as a distortionless amplifier, it is necessary that
the range of applied grid volts should not appreciably exceed the
linear part. In this case, the simplified equation

Ij, = a{Ep + kEg) (5)

may be used in which a is the filament to plate conductance of the
tube, and is the slope of the linear part of the characteristic.
If Eg is sufficiently negative, the plate curreiit is zero. Calling
this value Ego we have

fc = - 1^ (6)

It is obvious that the amplification for a given tube depends
upon the spacing of the grid wires. If these wires are far apart,

230 ELECTRICITY AND MAGNETISM

a definite change of voltage is not as effective in controlling the
electron flow as though the meshes were smaller. As a matter of
fact, the amplification factor is inversely proportional to the dis-
tance between grid wires. Again, if the grid is close to the fila-
ment so that it acts upon the electrons before they have gained
appreciable speeds, it is more effective than if it is near the plate.
Thus, if it is desired to construct a tube with a large voltage
amplification factor it should have a grid with a fine mesh
mounted close to the filament. Tubes having amplification
factors as large as 100 have been constructed, but in actual prac-
tice factors from 10 to 20 are more common.

A simple method for obtaining the amplification factor of a
tube is to impress upon the plate a certain positive potential and
then apply to the grid a negative potential sufficient to reduce the
plate current to zero. The ratio of the plate and grid potentials
is then the amplification factor of the tube for this particular
plate voltage. It is found in practice that the amplification
factor of a tube is not constant but varies with the plate and grid
potentials used. This is due to the fact that the average distribu-
tion of electrons between the plate and filament changes with the
potentials on the grid and plate which in effect, changes their
relative positions. By taking, in this manner, measurements
over a series of values of plate voltage a fair idea of the behavior
of the tube may be obtained.

While the method just described yields results sufficiently
accurate for many purposes, it has nevertheless one serious error.
Unless the tube is very carefully designed, it does not have a
sharp "cut off." That is, the characteristic curve does not pro-
ceed straight down to the axis, but slopes off and approaches it
gradually. The actual negative grid potentials required to
reduce the plate current to zero are much larger than would be
obtained by continuing the straight portion of the characteristic
until it intercepts the horizontal axis. In actual use, this inter-
cept value is the on^ which is effective. A dynamic method in
which this error is' eliminated has been devised by Miller.'
His circuit is shown in Fig. 122. The tube is connected in the
ordinary way with a telephone receiver in the plate circuit,
and potentials supplied to the plate and grid by the batteries B
and C respectively. By properly adjusting the values of these
voltages, the tube may be set at any point on the characteristic

1 J. H. Miller, Proc. Inst, of Radio Engineers, vol. 12, 1918, p. 171.

ELECTRON TUBES

231

curve. Included in the grid and plate circuits are the resistances
Ri and R2 across which is connected the secondary of a telephone
transformer T. When an alternating current is supplied to the
primary of this transformer, small alternating voltages, i.e., the
resistance drops across Ri and R2, are introduced into the grid
and plate circuits respectively. It is obvious from the con-
nections that when the additional voltage on the plate is positive
that on the grid is negative and vice versa. By changing the
relative values of Ri and R2 the ratio of these voltages may be

H»i

WWWV^-

HllllHAVVVW

A ^

Fig. 122. — Dynamic method for amplification factor.

made to have any desired value. If it is such that the added
grid potential just balances that added to the plate, there will
be no change in the steady plate current and consequently no
sound in the phones. The amplification factor A: is then the
ratio of Rzto Ri. The advantage of this method is that it mea-
sures the amplification factor while the tube is operating in the
same manner as when actually used in practice. The dependence
of the amplification factor upon the plate and grid volts may thus
be easily and quickly obtained.

177. Experiment 37. Amplification Factor of a Three-element
Electron Tube. — Connect the apparatus as shown in Fig. 122, using
for P a pair of high resistance head receivers. The source B
should furnish a voltage equal to the maximum for which the tube

232 ELECTRICITY AND MAGNETISM

is designed, and if a power tube is under test, may be a high
vohagc generator. C should consist of a battery of small flash
light cells. The A. C. supply should have a frequency high
enough to give a good clear note in the phones, and the voltages
across Ri and R2 should be low enough so that the operating
point moves only a small amount along the static characteristic
curve. Make two tests. First hold the grid volts at some
predetermined value, and measure the amplification factor for a
series of plate voltages ranging from a small value up to the
maximum for which the tube is designed. Next hold plate
volts at normal value and measure the amplification factor for a
series of values of grid volts. Check the results of the first series
by the static method explained above. That is, for each different
plate voltage, finji the negative grid potential required to reduce
the plate current ^^ero.

Report. — Plot curves showing the dependence of the amplifica-
tion factor upon both the plate and grid potentials by the
dynamic method, and upon plate potentials for the static
method. How do you account for the differences between these
curves?

178. Internal Plate Resistance of a Three-element Electron
Tube. — Following the amplification factor, the next most
important characteristic of an electron tube from the standpoint
of operation is perhaps its internal impedance. It is a well known
principle of electrical practice that the impedance of a device
should equal that of the circuit on which it operates. Accord-
ingly, in designing a tube to operate on a particular circuit or
conversely in adjusting a circuit to fit the tube which is supplying
power to it, it is necessary to know the plate to filament impe-
dance of the tube. The mechanism by which the vacuous space
offers resistance may be understood by the following considera-
tion. When a current flows through a conductor, heat is
developed within it. This energy is furnished by the driving
electric field which urges the electrons along through the con-
ductor. Resistance, in this case, is due to a direct interference
with the motion of electrons. As a consequence of this view of
the nature of resistance, it might at first be thought that a perfect
vacuum would be a perfect conductor of electricity since there is
nothing to interfere with the free motion of electrons. That this
however, is not the case is at once evident when one remembers
that relatively large voltages are necessary to cause small currents

ELECTRON TUBES 233

to flow through the ordinary electron tubes, even when the condi-
tions are far removed from those of current saturation. More-
over, the fact that it is easy to heat the plate red hot by the passage
of current, indicates that it is accompanied by a consumption of
energy.

When an electron is emitted by the heated filament, it finds
itself in the electrostatic field existing between filament and
plate, and it is at once accelerated toward the plate. Since the
electron possesses mass, it necessarily gains kinetic energy as it
moves toward the plate. This energy is abstracted from the
electric field which accelerates it. When the electron strikes the
plate, it possesses a velocity of the order of several thousand miles
per second even under moderate potential differences At the
plate it is suddenly brought to rest and its kinetic energy of
motion is converted into heat energy of the molecules of the
plate. While the tube does not possess resistance in quite the
same way that an ordinary metallic conductor does, it, never-
theless, consumes energy when a current passes, and it is cus-
tomary to speak of its resistance and to define it on the basis of
the energy it consumes. Thus, if I is the current flowing through
the tube, and W the watts consumed by it, its resistance R is
defined to be such that

W = PR (7)

Since this is the same equation as holds for the power converted
into heat by the ordinary conductor, we may determine the
resistance of the tube by the voltage required to furnish a given
current through it. An application of Ohm's law to correspond-
ing values of plate volts and plate current as read from the static
characteristics shows that the resistance of a tube is not constant
but depends upon the values of both the plate and grid potentials,
and also upon the electron emission from the filament in case
saturation voltages are used. It is necessary therefore to define
the resistance of the tube for a particular point in the char-
acteristic curve. This is done by saying that the resistance of the
tube is the ratio of the change in plate volts to the change in plate
current produced by it, when this change is made vanishingly
small. That is

Thus the resistance is the reciprocal of the slope of the plate
potential, plate current characteristic. Since this curve is seldom

234

ELECTRICITY AND MAGNETISM

taken in practice, R may be obtained from the plate current-grid
potential characteristic by remembering that

Ej, = kE, , t (9)

whence
Therefore

dE„ = kdEa

R = k

dE,
dij,

(10)
(11)

The internal plate resistance is then the product of the amplifica-
tion factor and the reciprocal of the slope of the plate current-grid
potential characteristic.

While this method is satisfactory for many purposes, it is open
to the objection that it requires a determination of the amplifica-
tion factor k. A dynamic null method has been employed
by Ballantine^ in which the resistances may be measured

A.C.Supply
Fig. 123. — Connections for measuring resistance of tube.

directly. The connections for this circuit are shown in Fig. 123.
It will be noted that the arrangement is essentially a Wheatstone
bridge in which the plate to filament path through the tube is one
of the arms. Because of the battery B a steady current flows
through all four arms of the bridge and also through the phones.
The phones, however, respond only to the variable currents
' Ballantine, Proc. Inst. Radio Engineers, vol. 7, 1919, p. 129.

ELECTRON TUBES 235

furnished by the A.C. supply. The resistance thus measured will
be those defined by eq. (8),

179. Experiment 38. Plate-filamenl Resistance of an Electron
Tube. — Connect the apparatus as shown in Fig. 123. As a source
of alternating voltage use any oscillator giving a good clear note
furnishing an E.M.F. of about 10 volts. The resistance Rs
should be of the same order of magnitude as the tube under test,
i.e., several thousand ohms. Ascertain the normal plate voltage
for the tube, and make a series of measurements of internal
resistance varying the grid volts over a considerable range, both
positive and negative. Repeat using plate voltages three-quar-
ters, one-half and one-quarter normal. Disconnect the tube
from the bridge and determine its static characteristic for normal
plate voltage.

Report. — Plot internal resistance as a function of grid volts for
the four series of observations. Plot the static characteristic
and check your results by the first method described. The
amplification factor may be obtained by extending the straight
portion of the characteristic and taking its intercept on the
horizontal axis as the value of the grid volts necessary to reduce
the plate current to zero. See Art. 176.

180. The Tungar Rectifier. — In the case of tubes operated
on a pure electron discharge, it is possible, at best, to obtain
currents of but a fraction of an ampere, and these only by the
employment of several hundred volts. While such tubes are
satisfactory for the rectification of high voltage currents they
are, nevertheless, unsuitable for cases in which several amperes at
low voltage are required, as, for example, charging storage
batteries from ordinary city lighting circuits. For this purpose,
a satisfactory tube, known as the tungar rectifier has been
developed by the General Electric Co.^ It is a two-element
tube, the cathode of which is a heated tungsten filament in the
form of a helix, while the anode is a conical piece of tungsten
mounted about 3 mm. from the filament. Instead of a vacuum,
the tube contains pure argon at a pressure of 8 or 10 cms. of
mercury.

The purpose of the argon is to furnish positive ions which
neutralize the space charge encountered in pure electron tubes,
and thus to reduce by many fold the voltage required to maintain
the current. Furthermore the positive ions take part in trans-

1 Gen. Elec. Rev., vol. 19, No. 4, 1916, p. 197.

236 ELECTRICITY AND MAGNETISM

porting electricity between the electrodes and thus mateTially
increase the carrying capacity of the tube. In the early attempts
to utilize positive ions, it was found that many gases have
injurious effects. For example, in the presence of oxygen, the
electron emission of tungsten is cut down to a small fraction of
what it is in high vacuum. Again, many gases unite with the
heated filament forming compounds, which are highly volatile
at normal operating temperatures and thus cause it to disinteg-
rate. Furthermore, when a gas is present in only small amounts,
the mean free path of the positive ions may be so great that they
acquire velocities sufficient to chip off particles of the filament
softened by heating, and thus hasten its disintegration. By use
of an inert gas such as argon, the first two difficulties are over-
come and by shortening the mean free path by using relatively
high pressures, the speeds are so reduced by frequent collisions
that the disintegration by bombardment is insignificant.

In order to avoid the formation of volatile compounds it is
necessary that the argon be very pure, and in the early tubes
great pains were taken to secure this. It has been found possible
to mount within the tube, usually on one of the filament leads,
substances which react chemically with the impurities, which
thus keep the argon in a pure state. For the larger sized tubes,
a graphite anode mounted on a tungsten support is often
used, and the purifying agent may then be introduced in the
anode. As impurities are given off from the electrodes or interior
walls, the drop across the arc increases, liberating more heat at
the anode, which thus causes vapors to be given off by the purify-
ing agent and in this way the argon is maintained in a state of
high purity.

After the arc has once been started, the filament may be kept
heated by positive ion bombardment after the heating current
has been shut off. In this case, the arc confines itself to a very
limited portion of the filament. This spot wastes away more
rapidly than the rest of the filament and the life of the tube is
materially shortened when operated in this way. For the larger
sized tubes, i.e., those with a current capacity of 20 to 40 amperes,
a fine tungsten point is independently mounted close to the fila-
ment. This may be heated to a high temperature by using it
as anode with the filament as cathode. If the connections are
then shifted, this hot point may be used as the cathode against
the regular anode, its temperature being maintained by positive

ELECTRON TUBES

237

ion bombardment as just explained. The filament serves then
as a starting device only and the tube has an exceedingly long
life. Since relatively large amounts of power are consumed by
the filament current, it might be expected that the latter method
of operation would result in a material increase in efficiency.
This is not the case, since the voltage across the arc rises when
the filament current is cut off, and the resulting increase in energy
consumption in the arc itself practically balances the saving
effected in the filament. Commercial sets are usually made for
the purpose of charging automobile storage batteries with a
maximum E.M.F. of 60 volts directly from 110 volt alternating
current circuits. To avoid losses in controlling rheostats, a step
down transformer is mounted within the case to reduce the A.C.
voltage to the desired value before rectifying it. A separate low
voltage winding is included for heating the filament.

181. Experiment 39. Study of the Tungar Rectifier. — For
simplicity of operation, obtain the characteristic curves by the

Fig. 124. — Connections for tungar rectifier.

use of direct currents. Mount the tube in a special socket and
connect it in circuit as shown in Fig. 124. Ascertain the normal
heating current for the filament and be careful not to exceed this
value. With this arrangement four curves are to be taken: (a)
The volt-ampere characteristic for the arc; (6) the efficiency of the
rectifier with external filament heating current; using 30 volts
on the plate; (c) the efficiency of the rectifier with filament
heated by positive ion bombardment, 30 volts on plate, and (d)
same as (&) using 60 volts on plate. In all cases vary the arc

238

ELECTRICITY AND MAGNETISM

current by means of the rheostat R through as wide ranges as the
arc will permit. The power consumed by R is taken as the load
or useful output of the device. Next place the tube in the socket
of the regular rectifier set and make an efficiency run using the
110 volt A. C. circuit as a source of power. Measure the input
by means of a wattmeter and the output by the volt-ampere
product for the load rheostat R. Vary the load amperes through

-^mw

W.M.

l.C.Supp

Fig. 125. — Connections for rectifier mounted in commercial set.

as wide a range as possible. The connections for this test are
shown in Fig. 125. Before starting the test open up the housing
for the set and study carefully the internal connections.

Report. — Plot the volt-ampere characteristic for the tungar
rectifier, also the various efficiency curves as a function of the
load current. Is there any similarity between the volt-ampere
characteristics for the tungar rectifier and that of the ordinary
carbon arc.

CHAPTER XV

PHOTOMETERi AND OPTICAL PYROMETER

182. Intensity of Radiation. — The brightness of light, as
estimated by the eye, is not capable of precise measurement,
since it depends to a large extent upon the color of the light and
the sensitiveness of the eye which receives it. Accordingly, the
only consistent way in which intensity may be specified is in
terms of energy. Proceeding on this basis, the intensity of
waves, whether they are those of sound, light or of any other type,
is measured by the amount of energy passing per second through
a square centimeter of area at right angles to the direction of
propagation. If there is no loss in the medium, and if the
medium contributes nothing to the intensity, the same quantity
of energy will persist in a given wave no matter how far it travels,
or how the dimensions and form of the wave front may change as
it advances.

The variation of intensity with distance from the source
depends upon the shape of the wave front, or what amounts to
the same thing, the number of dimen-
sions in which the wave spreads out.
For example, if the wave front is
plane, as in the case of a sound wave
travelling along a speaking tube, or
the beam from a searchlight, the
wave front maintains a constant
area, and the intensity is independent
of the distance from the source.
Again, if a pebble is dropped in the
lake, waves travel outward in circles
and are propagated in two dimen-
sions. In this case the energy remains constant in a circle which
increases as the distance from the center and the intensity varies
inversely as the distance from the source. In the case of spherisal

1 Duff, Text Book of Physics, arts. 259, 637-639, 724.
Karapetoff, Experimental Electrical Engineering, arts. 205-211.
Nutting, Outlines of Applied Optics, p. 169.

239

Fig. 126. — Propagation of
spherical waves.

240 ELECTRICITY AND MAGNETISM

waves with which we are particularly concerned here, the energy
emitted per vibration of the source is confined within a spherical
shell whose thickness is that of one wave length, and this remains
constant as the wave advances. Let 0, Fig. 126, be a source
from which waves are sent out in all directions. Let S be the
strength of the source, i.e., the amount of energy emitted per
second. Also let di and d^ be the radii of a given wave at two
different distances from the source and let /i and 1 2 be the
corresponding intensities. Then

S = Aird^n, = 4rd2^l2 (1)

whence

k^% (^)

Thus the intensity varies inversely as the square of the distance
from the source.

183. The Photometer. — An instrument for the comparison
of two sources of light is called a photometer. While the eye is
unable to estimate absolute intensities at all accurately, it is,
nevertheless, quite sensitive to differences in illumination.

* i *

lliiliiliiliiliiliiliiliiliiliiliiliiliiliiliilnliiliiliiliilliliiliill
\< dri 4< ck >J

Fig. 127. — Principle of the photometer.

Accordingly, if light from two different sources is allowed to fall
upon a screen in such a way that the areas of the separate illumi-
nations are adjacent, equality in the two intensities may be
determined by the disappearance of the line of demarkation
between them. An instrument for this purpose may be
arranged as shown in Fig. 127 by mounting two lamps Li and L2,
which are to be compared, at the ends of a bench provided with a
scale along which runs a carriage supporting a screen of white
paper. The central portion of this screen is impregnated with
parafRne which renders it semitransparent. This spot appears
darker than its surroundings if viewed by reflected light, but it is
brighter in transmitted light. If, however, the intensity of
illumination is the same on both sides, the spot disappears since
the amounts transmitted in the two directions are equal.

PHOTOMETER AND OPTICAL PYROMETER

241

If (Si and S'i are the strengths of the two sources and di and dj
their respective distances from the screen, then by eq. (1) the
ilumi'nation on each side of the screen is given by

/ = -^1 = -A^ /ox

Si _ d^

S, ~ d,' W

If one of the sources, e.g., S2 is a standard lamp, ,Si may be
computed.

184. The Lummer-Brodhun Photometer. — A comparator con-
siderably more sensitive than the grease spot screen just described
has been developed by Lummer and Brodhun. The special

Fig. 128. — I-umnier-Brodhiin photometer.

feature of this instrument is the optical device for simultaneously
viewing the two sides of the comparison screen W, as shown in
Fig. 128. Light from each side is reflected by two mirrors or
prisms Mi and M2 so as to enter the optical system AB. This
consists of two totally internally reflecting prisms placed back
to back. The reflecting surface of one is plane, while that of the
other is spherical with a small portion ground flat. The flat
surface of the latter is placed in optical contact with the former.

242 ELECTRICITY AND MAGNETISM

Light entering either of these prisms and striking the contact
surface will be transmitted, but light striking any portion of the
reflecting surfaces backed by air will be totally internally
reflected. Light emerging from the prism B consists of two parts,
that from the contact portion of the two prisms and that from
the surrounding area. The former comes entirely from the left-
hand side of W while the latter is from the right-hand side. If a
telescope is placed at T and focused on the contact area of the
two prisms, the central portion appears brighter or darker than
the surroundings according as the illumination of the left- or the
right-hand side of W is more intense, but the entire field appears
uniformly illuminated when a balance is secured.

A convenient form of laboratory instrument is one in which a
single socket, to receive in succession the unknown and standard
lamps, is mounted at a fixed distance from the comparison box.
On the other side is a movable socket containing a small six volt
lamp for comparison purposes. The distance of this lamp from
the screen is read by an index registering on a fixed scale. A
slow motion device is also provided. The process consists then
in placing the unknown lamp in the fixed socket and obtaining
a balance by moving the comparison lamp to or from the screen
until the line of demarkation between the outer and central posi-
tions of the field of the telescope disappears. The lamp is then
replaced by the standard and a balance again obtained. The
screen should be reversed and readings taken in each position
and averaged to eliminate differences in reflecting power of the
two sides. The equation for computing the strength of the
unknown lamp may be derived as follows:

Let S, U, and C be the candle powers of the standard, the
unknown, and comparison lamps, respectively; letd, andd„ be the
distances of the comparison lamp from the screen when balanced
against the standard and unknown, and let D be the fixed dis-
tance of both standard and unknown from the screen. Then, for
the two balances, the following equations hold:
U ^D^ S ^D^

C ~ dj' C ds^

Dividing one equation by the other, we have

U d.2

^ = d? ^^^

Care must be taken to maintain the same voltage on the compari-
son lamp throughout the test.

PHOTOMETER AND OPTICAL PYROMETER 243

185. Experiment 40. Study of Incandescent Lamps. — The
purpose of this experiment is to determine, as a function of the
voltage upon which they are operated, the candlepower, wattage
consumption, watts per candlepower, and resistance of four
lamps differing as widely as possible in design. Each lamp,
including the comparison lamp, should be provided with a
voltmeter and a control rheostat. An ammeter should be placed
in series with the unknown. Use five different voltages between
90 and 130. Do not operate the lamps at the higher voltages
longer than is necessary for making the observations. The
standard lamp should be operated only at the voltage for which it
is rated. Make several settings for each observation using the
screen in both the direct and reversed positions.

Report. — Describe the Lummer-Brodhun photometer and plot
the four curves indicated for each lamp. Why does a tungsten
lamp reach full brilliancy more quickly after closing the switch
than a carbon? Why does the gas-filled lamp have a higher
efficiency than a vacuum lamp?

THE OPTICAL PYROMETER'

186. General Principles. — It is a matter of common experience
that when a body is heated to a high temperature it emits light
and also that the intensity of this, emitted radiation varies
rapidly with the temperature of the source. For example, a small
change in the voltage across an incandescent lamp produces a
relatively large change in the brightness of the filament. Meas-
urements show that a body at 1,500° C. emits more than one
hundred times as much as it does at 1,000° C, and if the tempera-
ture is raised to 2,000° C, the radiation is increased more than two
thousand fold. This fact is often made use of in the measure-
ment of temperatures, and pyrometers operating on this principle
have the marked advantage that it is not necessary to heat any
part of the measuring apparatus to the temperature of the body
being studied. This is particularly important for work above
1,600° C, for there is no substance which retains its temperature
measuring properties uniform when subjected to such extreme
heats. Again, the products of combustion in furnaces contami-
nate any pyrometric material introduced, thus necessitating
frequent recalibrations.

' LeChatelier and Burgess, Measurement of High Temperatures, pp.
237-243, 291-303, 325-327, 336-337.

Griffiths, Methods of Measuring Temperature, pp. 113-118.

244 ELECTRICITY AND MAGNETISM

The radiation method of measuring temperatures, however, is
complicated by the fact that incandescent bodies differ materially
as regards both the intensity and quality of the light which they
emit. For example, the radiation from iron or carbon is much
greater than that from such substances as magnesia or polished
platinum at the same temperature. If a pyrometer were cali-
brated by measuring the radiation from one substance and then
used to measure the temperature of another possessing different
radiating properties large errors would result in many cases.

This difference in radiating properties has led to the use of
"black bodies" as standard radiators and absorbers. A black
body is defined as one which absorbs all the radiation falling upon
it, and it therefore neither reflects nor transmits any radiation.
It also has the property, when heated, of emitting radiation whose
intensity is a function of temperature only and depends in no
way upon the physical constants of the material of which it is
made. Further, the intensity of the radiation from a black body
at a given temperature is greater than that from any other body
at the same temperature.

187. Black Body Furnace. — Experimentally, a black body is
very closely approximated by a hollow opaque inclosure with a
small opening. If the internal area of the inclosure is large
compared to the opening', radiation falling upon it enters the
inclosure and is reflected diffusely back and forth so many times,
that it is practically all absorbed before any can emerge. Again,
if the walls are heated uniformly to any temperature, the
radiation emerging from the opening has been reflected back and
forth so many times that it no longer has properties characteristic
of the material of the walls. Such a body is at the same time a
perfect absorber and a perfect emitter. The radiation from
a crack or other small opening in an ordinary furnace is nearly
black body radiation, so also is that from the inside of a narrow
wedge formed by folding a thin metallic ribbon into a very flat V.

A black body, satisfactory for experimental purposes, is made
by winding a porcelain tube with thin platinum foil through
which a heating current may be passed. The center of the tube
is closed by a porcelain disk and between this and the end,
through which observations are made, is arranged a series of dia-
phragms, also of porcelain, whose apertures increase in diameter
successively toward the end. These minimize the disturbing
effects of air currents and increase the number of internal reflec-

PHOTOMETER AND OPTICAL PYROMETER 245

tions which the radiation must make before it emerges. To
protect the internal tube from external disturbances and reduce
the heat losses to a minimum, it is surrounded by another tube
upon which is wound a second heating coil of some alloy such as
nichrome or therlo. Outside of this is a series of several addi-
tional tubes with air spaces between them, the outer one usually
being surrounded by powdered magnesia. By properly adjust-
ing the heating currents through the two coils, any desired
temperature up to 1,600° C. may be maintained with a high
degree of constancy. The temperature of the black body is usually
measured by a platinum, platinum-rhodium thermocouple, the
junction of which is supported by two small holes through the
central disk, with the insulated leads passing out through
the rear of the furnace.

188. Distribution of Energy in the Spectrum. — If one measures
the total energy emitted by a black body, he finds that it increases
rapidly as the temperature is raised. The law connecting black
body radiation with temperature was first stated by Stefan and
later deduced theoretically by Boltzmann. It is

E = ST^ (6)

where E is the total energy radiated, T the absolute temperature,
and S, a constant which is approximately 5,6 X 10~", ergs per
square centimeter per second. Although this law is rigidly true
only for a black body it is found to hold approximately for most
surfaces, the constant S being different for each.

If the radiation from a black body is separated out into a
spectrum and the energy associated with each wave length is
measured, it is found that not only is there a continuous change
in the amount of energy as we go from one wave length to another,
but also that the distribution of energy among the wave lengths
changes as we vary the temperature. Figure 129 gives the dis-
tribution of energy among the wave lengths for a series of
temperatures. It will be noted that as the temperature is raised,
the energy in each wave length increases but not in the same
proportion. Also that the wave length containing the maxinmm
energy decreases as the temperature is raised. This is in accord
with the common observation that, starting with low tempera-
tures, a body appears at first dull red, then yellowish or cherry
^ red, and finally becomes "white hot" as extreme temperatures
are reached. Wien has shown that the wave length for maximum

246

ELECTRICITY AND MAGNETISM

energy and the absolute temperature are connected by the
simple law

^ maxT' = const. (7)

He has also shown that the distribution of the energy among the
wave lengths at a given temperature, as illustrated by Fig. 129,
follows very closely the law

E\ = CiX-'^e -x3

(8)

120-

- CO

.001

.006

.002 .003 .004 .005

Wave Length in Millimeters
Fig. 129. — Energy distribution for a black body.

where Ex is the energy in the wave length interval X to X + d\;
e is the base of the Naperian logarithms; T the absolute tem-
perature, and Ci and C2 are constants. For other radiating
surfaces, it is found that E^ follows very closely the above law
but different constants must be used.

189. Application to Pyrometry. — It is obvious that any of the
three equations just given might be used to measure tem-
peratures. It is found, however, that eq. (8) is most suitable, and
when it is applied, only one wave length is used, or at least only
those lying within a very restricted range. This equation lends
itself more easily to calculation if it is put in the form :

log. Ex = k

(9)

PHOTOMETER AND OPTICAL P YROMETER

247

where

k = logeCi — 5 log X.
Let El and Ez be the energies for a particular wave length radi-
ated at the temperatures Ti and Ti, respectively. Substituting
these values in eq. (9) and subtracting, we have

i^„ El C2 / 1 1 \

(10)

If T2 is a standard temperature and Ti an unknown, then by
measuring Ei and E2 or their ratio, by appropriate means, Ti
may be computed. Solving eq. (10) for Ti and using common

logarithms,

Ti =

(11)

XTa

-f 2.308 logio§^

190. The Optical Pyrometer. — One of the most convenient
forms of the optical pyrometer is that devised by Holborn and

L|||||f_0^vAA/

Fig. 130. — Holborn and Kurlbaum optical pyrometer.

Kurlbaum. It consists of a telescope in the focal plane of which
is mounted a small six volt lamp with either a carbon or tungsten
filament, as shown in Fig. 130. When the telescope is focused
on the furnace and the filament is lighted, there is seen, on looking
into it, a field of uniform illumination with a fine Une extending
across it. If the filament is hotter than the furnace, it appears
as a bright line across a dark background; but if the furnace is
hotter, there is seen a dark line across a bright background. If
filament and furnace are at the same temperature, the line disap-
pears and the field is uniform throughout. The eye is very
sensitive to differences of brightness and a difference of two degrees
between furnace and filament may easily be detected. Current
for the filament is furnished by a storage battery, controlled by a

248 ELECTRICITY AND MAGNETISM

rheostat and measured by an ammeter. The indications of the
pyrometer are thus in terms of the filament current. If the
furnace is held in turn at a series of known temperatures and
the filament currents for balance obtained, a calibration curve
may be plotted showing temperature as a function of current.

A number of improvements in the original form of the Holborn-
Kurlbaum pyrometer have been made by Mendenhall.^ One of
them is a method by which such an instrument may be caUbrated
over a wide range using only one standard temperature. This is
accomplished by holding the temperature of the furnace constant
and rotating between it and the pyrometer a sectored disk which
allows only a known fraction of the energy to enter the telescope.
This is equivalent to reducing the temperature of the furnace.
Suppose the fraction of the energy transmitted is R. Then Ei
= RE 2. Substituting this value in eq. (11), we have, for the
apparent temperature of the furnace,

T, = -'-i ——1— (12)

^ ^^|,-+ 2.303 logio-^

By using a series of sectors, for example with R equal to ^^, 3^^, ^i,
etc., a series of apparent temperatures are obtained, and the
filament temperatures corresponding to each may be determined.
This gives a calibration for the instrument for ranges below the
standard temperature actually maintained in the furnace. The
necessary narrow wave length band is secured by mounting
behind the eyepiece a disk of red glass of special quality. The
instrument also may be used to measure temperatures above
that of the standard by using the sectored disk when taking
observations on the unknown temperature, thus reducing it to an
apparent lower temperature within the calibration range just
determined. For example, if an unknown temperature is
observed through a sector of transmission ratio R and is found to
be the same as the standard temperature T^ then the unknown
temperature is obtained from eq. (11) by putting E^ = REi which
gives

rp ^C2 1 (13)

^ X C2 + 2.303 logioR
XTa
In a similar manner a calibration curve may be computed for a
given sector extending the range of the instrument to any desired
1 Mendenhall, Phys. Rev., vol. 35, 1910, p. 74.

PHOTOMETER AND OPTICAL PYROMETER

249

value. For this purpose, it is only necessary to substitute for T2
in eq. (13) the value of temperature corresponding to each par-
ticular current read off from the original calibration curve. These
computed values of Ti plotted against the corresponding values
of filament current give the cahbration curve for the instrument
when used with the sectors to measure unknown temperatures.

It should be borne in mind that the Wien radiation law upon
which this method is based holds only for black body radiation,
and that the method of calibration just described makes use of a
black body as a source of radiation. If it should be used to
determine the temperature of some other body such as a heated
filament or strip of metal not within an enclosure, its indications
will be the temperature of a black body which would emit the
same amount of energy at the particular wave length used in the
calibration. Since no other body emits more energy at any
wave length than a black body at the same temperature, and
most substances emit less than a black body, the reading of the
optical pyrometer will, in general, be too low. The reading
obtained is called the "black body temperature." Mendenhall
and Forsythe^ have made an extended study of the differences
between the "black body" and "true" temperatures of a great
many substances with the result that the optical pyrometer
may now be very generally used to determine actual tempera-
tures. A few of their values for carbon and tungsten are given
below :

Black body
temperature

Corresponding true temperature

Degrees Centigrade

Tungsten, Degrees

Carbon, Degrees C,

1,000

1,068

1,012

1,200

1,273

1,222

1,400

1,486

1,430

1,600

1,700

1,638

1,800

1,910

1,847

2,000

2,126

2,056

2,200

2,345

2,400

2,565

2,600

2,783

2.700

2,890

1 Mendenhall and Forsythe, Astrophysical Jour., vol. 37, 1913, p. 389.

250 ELECTRICITY AND MAGNETISM

191. Experiment 41. — Connect the apparatus as shown in
Fig. 130. Ascertain the heating currents to be used through the
two windings of the furnace and take care that they are never
exceeded, particularly through the inner platinum winding.
Find out, also, the maximum current allowable for the filament
of the pyrometer lamp Fill the vessel containing the cold
junction of the thermocouple with cracked ice to maintain it at
0° C. Measure the E.M.F. of the thermocouple with a low resis-
tance potentiometer, special instructions for which are given in
chap. IV. A calibration curve is furnished with the thermo-
couple. Focus the eyepiece of the telescope on the lamp fila-
ment and as soon a the furnace is warm enough to permit
it, focus the telescope so that the inner circle of the furnace is
distinctly seen As the furnace heats up, determine its tem-
perature with the thermocouple and balance the pyrometer every
few minutes.

When a temperature of 1,200° C. has been reached, reduce the
heating current through both windings and allow the temperature
to rise slowly to about 1,300° C. and then hold the furnace con-
stant at this value. When holding the temperature constant, it
is best to leave the potentiometer setting fixed and keep the
galvanometer balanced by adjusting the heating current rheostat.
When a steady state has been secured, make several settings of
the pyrometer. Then introduce the 3^^ sector and with the motor
running, again make several settings on this apparent temperature.
Repeat, using the 3^^, Ko> Mo^ ^io, and H20 sectors. Check the
settings with no sector between each replacement to insure con-
stancy of conditions. Two observers are required for this
experiment, one to manipulate the pyrometer, and one to hold
the furnace temperature constant. Measure the temperature
of the filaments of several incandescent lamps of different types
and candle power using such a sector that the current through the
pyrometer lamp lies within the range covered by the calibration.

Report. — Compute the effective temperatures below the stand-
ard temperature secured by the various sectors by use of eq. (12).
Use for C2 the value 14,350 and for the wave length 0.658.
The standard temperature is that in degrees absolute at which
the furnace was held constant and is. obtained from the calibra-
tion curve for the thermocouple. The computed values of Ti
are also in degrees absolute. Plot temperatures below that of the
furnace.

PHOTOMETER AND OPTICAL PYROMETER 251

Plot calibration curves for values above that of the furnace for
the Ko> Ho J and H20 sectors, by use of eq. 13. To do this,
read from the first curve the values of T for a series of values of
filament currents. Substitute these values of T2 in eq. 13 using
for R the appropriate ratio. These values of T, when plotted
against the currents, give the calibration curve for a given
sector for the high range.

Read from these curves the black body temperatures of the
lamps measured and by use of the tables given above, determine
their true temperatures in degrees centigrade.

If the temperature of the sun is about 6,000° C, find the size
of sector opening necessary to measure it on the instrument used.

APPENDIX
CALCULATION OF INDUCTANCE AND CAPACITANCE

In designing electrical apparatus and in checking the results of
bridge measurements it is often advantageous to determine the
inductance of coils by calculation from their dimensions and
number of turns. In connection with its work in establishing
primary units of inductance, the United States Bureau of Stand-
ards made an exhaustive study of the formulas for this purpose,
and, besides extending those available at the time, developed a
number of new ones. A comprehensive collection of inductance
formulas, together with numerical examples, is given in the
Bulletin of the Bureau of Standards, vol. 8, 1912, pp. 1 to 237.
This publication is known also as Scientific Paper 169. In
another publication, "Radio Instruments and Measurements,"
Circular 74, there is given a series of simplified formulas which
yield results accurate to one-tenth of one per cent.

Three formulas, taken from Circular 74, are given below. They
apply to the coils most commonly used in every day practice.
Lengths and other dimensions are expressed in centimeters,
and the inductance calculated is given in microhenries. One
henry = 10^ millihenries = 10^ microhenries = 10* C.G.S.
electromagnetic units. It is assumed that the coil is placed in
air or other medium whose permeability is unity, and that no
iron is in the vicinity.

I. Single Layer Coil or Solenoid. — Nagaoka's Formula.

J 0.03948a2n2

L = ^ ^ K (l;

where n = number of turns of coil

a = radius of coil, i.e., axis to center of any wire
b = length of coil, i.e., number of turns times distance
between centers of adjacent turns.
K is a correction factor made necessary by the demagnetizing

action of the ends of the coil and is a function of -r • Its value

may be read from Table I. If the coil is very long compared to

252

INDUCTANCE AND CAPACITANCE

253

its diameter, K = 1. Formula (1) takes no account of the size
or shape of the cross-section of the wire and assumes that the
diameter of the wire is small compared to the dimensions of the
coil, and that the coil is compactly wound.

Table I. — Values of K for Use in Formula I

Diameter

Differ-

Diameter

Differ-

Diameter

Differ-

Length

ence

Length

ence

Length

ence

0.00

1.0000

-0.0209

2.00

0.5255

-0.0118

7.00

0.2584

-0.0047

.05

.9791

203

2.10

.5137

112

7.20

.2537

.10

.9588

197

2.20

. .5025

107

7.40

.2491

.15

.9391

190

2.30

.4918

102

7.60

.2448

.20

.9201

185

2.40

.4816

7.80

.2406

0.25

0.9016

-0.0178

2.50

0.4719

-0.0093

8.00

0.2366

-0.0094

.30

.8838

173

2.60

.4626

8.50

.2272

.35

.8665

167

2.70

.4537

9.00

.2185

.40

.8499

162

2.80

. 4452

9.50

.2106

.45

.8337

156

2.90

.4370

10.00

.2033

0.50

0.8181

-0.0150

3.00

0.4292

-0.0075

10.0

0.2033

-0.0133

.55

.8031

146

3.10

.4217

11.0

.1903

113

.60

.7885

140

3.20

.4145

12.0

.1790

.65

.7745

136

3.30

.4075

13.0

.1692

.70

.7609

131

3.40

.4008

14.0

. 1605

0.75

0.7478

-0.0127

3.50

0.3944

-0.0062

15.0

0.1527

-0.0070

.80

.7351

123

3.60

.3882

16.0

.1457

.85

.7228

118

3.70

.3822

17.0

.1394

.58

.90

.7110

115

3.80

.3764

18.0

.1336

.95

.6995

111

3.90

. 3708

19.0

.1284

1.00

0.6884

-0.0107

4.00

0.3654

-0.00.52

20.0

0.1236

-0.0085

1.05

.6777

104

4.10

. 3602

22.0

.1151

1. 10

.6673

100

4.20

. 3551

24.0

.1078

1.15

.6.573

4.30

.3502

26.0

. 1015

1.20

.6475

4.40

. 3455

28.0

.0959

1.25

0.6381

-0.0091

4.50

0.3409

-0.0045

30.0

0.0910

-0.0102

1.30

.6290

4.60

.3364

35 . 0

.0808

1.35

.6201

4.70

.3321

40.0

.0728

1.40

.6115

4.80

.3279

45.0

.0664

1.45

.6031

4.90

.3238

. 40

50.0

.0611

1.50

0.5950

-0.0079

5.pO

0.3198

-0.0076

60.0

0.0528

-0.0061

1.55

.5871

5.20

.3122

70.0

.0467

1.60

.5795

5.40

.3050

80.0

.0419

1.65

.5721

5.60

.2981

90.0 .

.0381

1.70

.5649

5.80

.2916

100.0

.03.50

1.75

0.5579

-0.0068

6.00

0.2854

-0.0059

1.80

.5511

6.20

.2795

1.85

.5444

6.40

.2739

1.90

.5379

6.60

. 2685

.52

1.95

.5316

6.80

.2633

254

ELECTRICITY AND MAGNETISM

II. Multiple Layer Coil. — For a long coil with few layers, the
inductance is given by

L^Ls- 5:5125W«(0.693 + B.) (2)

where Ls = inductance of mean single layer given by formula (1)
n = number of turns of the coil
a = radius of coil measured from the axis to the center

of cross-section of the winding
b = length of coil = distance between centers of turns

times number of turns in one layer
c = radial depth of winding = distance between centers

of two adjacent layers times the number of layers.

Bs = correction given in Table II in terms of the ratio - *

Table II. — Values of Bs

FOR Use in

Formula II

b/c

b/c Bs

b/c

0.0000

0.2844

0.3116

0.1202

0.2888

0.3131

0.1753

0.2927 ,

0.3145

0.2076

0.2961

0.3157

0.2292

15 0.2991

0.3169

0.2446 ! 16 0.3017

0.3180

0.2563 17

0.3041

0.3190

0.2656 18

0.3062

0.3200

0.2730 19

0.3082

0.3209

0.2792 20

0.3099

0.3218

III. Short Circular Coil with Rectangular Cross Section. — For
a coil having a shape such as shown in Fig. 131, the inductance is
given by a formula due to Stefan. It is deduced on the assump-
tion that the wire is rectangular in cross-section, and that the
insulating space between turns is negligible. Further, the axial
and radial dimensions of the winding are supposed to be small
compared to the mean radius of the coil.

Let a = the mean radius of the winding measured from the axis
to the center of the cross-section
b = the axial dimension of the cross-section
c = the radial dimension of the cross-section
d = -y/ft^ _|_ (.2 = the diagonal of the cross-section
n = number of turns of rectangular wire.

INDUCTANCE AND CAPACITAMCE

255

There are two cases depending upon the relative values of b
and c.

Case 1. 6>c.
L = 0.01257an'[2.303(l + ^, + .Qiog,."^ - „ +

16^2^'J (^)

Case 2. 6<c.

L = 0.01257an' [2.303(1 +3|l+g4)log,

y^-^it^'^'] (^)

U— 6-

Fig. 131. — Multiple layer coil with winding of rectangular cross section.

The constants yi, yz, and 2/3 depend upon relative values of
b and c, and are given in Table III. The ratio of these quantities
is always to be taken so as to give a proper fraction; i.e., in for-
mula (3), use c/b, and in formula (4), use b/c. In eq. (3), yi is the
same function of c/b that it is of b/c in eq. (4).

256 ELECTRICITY AND MAGNETISM

Table III. — Constants Used in Formulas (3) and (4)

b/c or c/b

Differ-
ence

c/b

Differ-
ence

b/c

J/3

Differ-
ence

0.5000

0.0253

0.125

0.002

0.597

0.002

0.025

.5253

237

.05

.5490

434

0.05

.127

0.05

.599

.10

.5924

386

.10

.132

.10

.602

0.15

0.6310

0.0342

0.15

0.142

0.013

0.15

0.608

0.007

.20

.6652

301

.20

.155

.20

.615

.25

.6953

266

.25

.171

.25

.624

.30

.7217

230

.30

.192

.30

.633

0.35

0.7447

0.0198

0.35

0.215

0.027

0.35

0.643

0.011

.40

.7645

171

.40

.242

.40

.654

.45

.7816

144

.45

.273

.45

.665

.50

.7960

121

.50

.307

.50

.677

0.55

0.8081

0.0101

0.55

0.344

0.040

0.55

0.690

0 012

.60

.8182

.60

.384

.60

.702

.65

.8265

.65

.427

.65

.715

.70

.8331

.70

.474

.70

.729

0.76

0.8383

0.0039

0.75

0.523

0.053

0.75

0.742

0.014

.80

.8422

.80

.576

.80

.756

.86

.8451

.85

.632

.85

.771

.90

.8470

.90

.690

.90

.786

0.95

0.8480

0.0003

0.95

0.752

0.064

0.95

0.801

0.015

1.00

.8483

1.00

.816

1.00

.816

IV. Coil of Round Wire Wound in a Channel of Rectangular
Cross-section. If the insulation is not too thick, eqs. (3) and (4)
give a very close approximation for the cape in which ordinary-
round wire is used. When the percentage of the cross-section
occupied by the insulating space is large, the following correction
must be added to these formulas.

AL = 0.01257an2 [2.303 logio^ + 0.155]

where D = distance between centers of adjacent wires
do = diameter of the bare wire.

(5)

CALCULATION OF CAPACITANCE

The following formulas' may be used to calculate the capac-
itance of condensers of the common forms. The dimensions of
the condensers are measured in centimeters, and the capacitance
is given in micro-microfarads. In these formulae, no correction

1 Cir. 74, U. S. Bureau of Standards, p. 235.

INDUCTANCE AND CAPACITANCE 257

is made for the curving of the electrostatic field at the edges of
plates, etc., and it is assumed that the distance between plates is
small compared to their linear dimensions.

V. Parallel Plate Condenser

C = 0.0885x|, (6)

where S = surface area of one plate
T = thickness of dielectric

K = dielectric constant (K = 1 for air, and for most sub-
stances, lies between 1 and 10).

If, instead of a single pair of plates, there are N similar plates
with dielectric between them alternate plates being connected
in parallel,

C = 0.0885X^^ ~ ^^^ (7)

VI. Variable Condenser with Semi-circular Plates

C = O.mOK^^-'^^p'-''"^ (8)

where N = total number of plates

ri = outside radius of the plates
r2 = inside radius of the plates
T — thickness of dielectric
K = dielectric constant

This formula gives the maximum capacitance, i.e., when the
movable plates are entirely within the spaces between the fixed
plates. As the movable plates are rotated out, the capacitance
decreases in direct proportion to the angle through which they
are turned.

VII. Isolated Disk of Negligible Thickness

C = 0.354d (9)

where d = diameter of the disk

VIII. Isolated Sphere

C = 0.556d (10)

where d = diameter of the sphere

IX. Two Concentric Spheres

riTi

c = i.n2K-^^^^ (11)

ri - rj

258 ELECTRICITY AND MAGNETISM

where ri = inner radius of outside sphere
ro = outer radius of inner sphere
K = dielectric constant of material between spheres.

X. Two Coaxial Cylinders

C = 0.2416K — .,_,

1 ^1 (12)

where I = length of each cylinder

ri = inner radius of outer cylinder
r<i = outer radius of inner cylinder
K = dielectric constant of material between cylinders.

INDEX

Alpha rays, 212

Alternating current galvanometer,

165
Ammeter, 74
adjustment of, 76
calibration of, 80
Ampere turn, definition of, 102
Amplification factor of electron tube,
227
dynamic method for, 231
Anderson modification of Max-
well's bridge, 129
Atom, structure of, 200
Audio-oscillator, 155

Ballantine, dynamic method for
resistance of electron tube, 234
Battery test, 53
Beta rays, 213
Black body, 244
temperatures corrected to true,
249

Campbell, measurement of induc-
tance, 181
Carey-Foster bridge for resistance,
42
method for mutual inductance,
124
Cathode glow, 205

rays, 208
Checking devices for ballistic gal-
vanometer, 33
Comparison of cells, 63
Condensers, capacitance of, 86
comparison of, 89
grouping of, 87

measurement by Fleming and
Clinton commutator, 93

Crooke's dark space, 205
Current, measured by electrody-
namometer, 72

Kelvin balance, 70

potentiometer, 79

Damped sine wave, 135

Decrement, logarithmic, 136

Demagnetizing factor, 96

Discharge of condenser, aperiodic
discharge, 131
critically damped discharge, 132
oscillatory discharge, 133
through gases, theory of, 205

Duddell thermo-galvanometer, 162

Effective value of an alternating

current, 148
Electrodynamometer, Siemens, 72
Electrolytes, resistance of, 194
Electrons, 198
Electron tubes, 218

amplification factor of, 227

as oscillator, 159

characteristics for, 228

impedance of, 234

Faraday dark space, 206

Fleming and Clinton commutator,
92

Fluxmeter, Grassot, 31

Forsythe and Mendenhall, correc-
tions for black body tempera-
tures, 249

Frequency bridge, 186

Galvanometer, description of, 1 /
ballistic galvanometer, theory of,
25

269

260

INDEX

Galvanometer, constant of, 24
current galvanometer, 19
D'Arsonval galvanometer, 18
figure of merit, 22
Thomson galvanometer, 17

Gamma rays, 214

Gauss, definition of, 101

Gilbert, definition of, 101

Graham, potential gradient in dis-
charge tubes, 206

Grover, phase angle of condensers,
191

Heaviside's bridge for inductance,

180
Heydweiller's network for mutual

inductance, 177
Holborn and Kurlbaum's optical

pyrometer, 247
Hopkinson's bar and yoke, 107
Hysteresis, 104

measurement of, 114

Impedance, 139
Inductance, 117

calculation of, 252

coefficients of, 118

comparisons of, 120

standards of, 119
Induction, magnetic, 96
Insulation resistance, measurement

of, 46
Intensity of magnetization, 96
Internal resistance of cells, 50

condenser and ballistic galva-
nometer method for, 52

voltmeter ammeter method for, 51
Ionization, theory of, 151

Kaufmann, variation of — with

velocity, 214
Kelvin current balance, 70

Kelvin current balance, 70 double
bridge for resistance measure-
ment, 40
galvanometer, 17

Kennelly and Pierce, motional im-
pedance, 190

Kenotron, 224

Keys, 2

Kumagen, 155

Kurlbaum, optical pyrometer, 247

Lummer-Brodhun photometer, 241
M

Magnetic circuit, 99
shields, 18

Magnetism, general principles, 94

Magnetization curves, 103

Maxwell, definition of, 101

Maxwell's bridge for mutual induc-
tance, 169

Mendenhall, use of optical pyrom-
eter, 248
correction for black body tem-
peratures, 249

Microphone hummer, 154

Miller, amplification factor of elec-
tron tubes, 230

Motional impedance, 191

Motor generator, 153

Multipliers for voltmeter, 77

Mutual inductance bridge, 184

Nagaoka's inductance formula, 252
Negative glow, 206
Notebooks, 6

Oersted, definition of, 102
Ohm's law, 35

Permeability, magnetic, 97
Phase angle of condensers, 191
Photo-electric effect, 219
Photometer, 240

INDEX

261

Pierce and Kennelly, motional im-
pedance, 190
Pohl's commutator, 2
Polarization of cell, 54
Positive column, 206
Post-office box, 38
Potentiometer, description of, 55

Leeds and Northrup, 58

Tinsley, 62

Wolff, 60
Power, measurement of, 82

factor, definition of, 150

of condensers, 191
Pyrometer, optical, 243

Radiation, intensity of, 239
Radioactive substances, 212
Reactance, 139
Resistance, specific, 35

measurement of high resistance,
46
of low resistance, 40
temperature coefficient of, 36
Resistances for current measure-
ments, 80
Resonance, electrical, 142
parallel resonance, 144
series resonance, 143
Rheostats, 3

Root mean square value of an alter-
nating current, 149
Rowland ring. 111

Saturation current, 201
Sechometer, 151

Siemen's electrodynamometer, 72
Sine wave, vector representation of,

141
Space charge, 222

current, 221
Specific resistance, 35
Spectrum, distribution of energy in,

245
Standard cell, E.M.F. of, 11

temperature coefficient of, 12

Stefan Boltzman law, 245
Steinmetz coefficient, 106
Stroude and Gate's bridge, 173
Susceptibility, 97
Switchboard, 5
Switches, 2

Telephone receiver, 161
Temperature coefficient of resis-
tance, 36

standard cell, 12
Thermo-galvanometer, Duddell, 162
Time constant of circuit containing
resistance and inductance, 126

and capacitance, 128

inductance and capacitance, 131
Tinsley potentiometer, 62
Trowbridge's bridge, 174
Tungar rectifier, 235

Units, systems of, 7
electromagnetic, 8
electrostatic, 8
practical, 9

rationalized practical, 13
ratios of, 12

Variable impedance circuit, 188
Vibration galvanometer, 163
Volt box, 67
Voltmeter, adjustment of, 74

calibration of, 68
Vreeland oscillator, 157

Wattmeters, description of, 82

compensation of, 83

calibration of, 84
Wein, phase angle of condensers, 191
Wein's law, 246
Weston instruments, 75

standard cell, 64
Wheatstone bridge, 36
Wire interrupter, 153
Wolff potentiometer, 60

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