TERRESTRIAL MAGNETISM
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STAT
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STAT
TERRESTRIAL MAGNETISM[
Zemn2y
i LTorreatrial B. M. Yanovskiy
Magnetis ,
1953, Moscow,
Pages 3591
TABLE OF CONTENTS
1
Foreword
Introduction
4
Section 1. Brief historical information
Section 2. Basic laws of a stationary magnetic field
20
Section 3. Magnetic field of a closed linear circuit
24
Section 4. Magnetic potential of elemental circuit
26
Section 5. Magnetic field of a linear circular circuit
28
Section 6. The effect of finite dimensions of the Grose
section of the circuit
33
Section 7. Magnetic field of Helmholtz rings
36
Section 8. Magnetic field of a cylindrical solenoid
41
Section 9. Magnetic field of a multilayer solenoid (coil)
45
Section 10.
Magnetization of rocks
47
Section 11.
Magnetic potential of a magnetized body
51
Section 12.
Magnetic potential of a uniformly magnetized
sphare
53
Section 13.
Potential of a uniformly magnetized cylinder
55
Section 14.
Magnetic potential of an ellipsoid
55
Section 15.
field
Magnetization of rocks in a uniform magnetic
58
Section 16.
Determination of demagnetization coefficients
64
Section 17.
Effect of the shape of the sample on the mag
nitude of magnetic moment and magnetization
67
Section 18.
The permanent magnet and its properties
70
Section 19.
Magnetic materials and alloys
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Part One
The Permanent Magnetic Field of the Earth
90
Chapter I.
Description of the Magnetic Field of the Earth
90
Section 1.
The elements of terrestrial magnetism
90
Section 2.
Graphic representation of the earth's magnetic
field. Magnetic maps
93
Section 3. Methods of.nvestigating the magnetic field of
the earth. Magnetic surveyss magnetic observatories 99
Chapter II. Analytic Representation of the Earth's Mag ?
netic Field 106
Section 1. The earth's magnetic field as the field of a
uniformly magnetized sphere 106
Section 2. Expansion of earth's magnetic potential to a
series. The theory of Gauss 116
Section 3. Physical meaning of the terms of Gaussian seriesl2l4
Section 4. Separation of the earth's magnetic field into
"internal" and "e:cternaln 131
Section 5.
Vortical magnetic field
136
Chapter III.
Structure and Physical Theories of the
Origin of the Earth's Magnetic Field 138
Section 1.
Structure of the earth's magnetic field
138
Section 2.
Continental or residual field ahd its theory
110
Section 3.
Magnetic anomalies
152
Section 4.
Hypotheses of the origin of earth's magnetic
field
159
Chapter IV.
Secular Variations
174
Section 1.
Phenomena associated with,secular variations
174
Section 2.
Theory of secular variations
181
Part Two
Variable Magnetic Field of the Earth 187
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Chapter V. Classification of Variations of the Earth's
Magnetic Field and Methods for their Investigation
187
Section 1. General information and classification of
magnetic variations
187
Section 2. Methods for differentiation of daily magnetic
variations (statistical processing)
189
Section 3. The method of spheric and harmonic analysis of
variations
193
Section 4.
Magnetic activity
200
Chapter VI.
Magnetic Variations and Aurora Polaris
210
Section 1.
Solar diurnal variations
210
Section 2.
Lunar diurnal variations
224
Section 3.
Magnetic disturbances
227
Section 4.
Variations of high latitudes
236
Section 5.
The aurora
242
1.
Forms of aurora polaris
242
2.
Direction of rays of aurora polaris
244
3.
Height of aurora polaris
245
4.
Geographical distribution of aurora polaris
248
5.
Diurnal distribution of aurora polaris
249
6.
Spectrum of aurora polaris
250
7.
Connection between aurora polaris and magnetic
and solar activities
251
Chatper VII. Theory of Magnetic Variations and Aurora Polaris 253
Section 1. The ionosphere and its properties 253
1.
Propagation of radio waves within the ionosphere
253
2.
Methods of investigation of the ionosphere
258
3.
Measurement of the altitude of the reflecting
layer
260
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4?
Composition of the ionosphere and formation of
264
ionized layers
270
5.
Conductivity of the ionosphere
272
6.
Sporadic layer
272
7.
Tidal phenomena in the ionosphere
274
Section 2.
Theory of daily variations
275
1.
Theory of atmospheric dynamo
278
Section 4. Movement of changes within the magnetic field of
2. Theory of drifting current
Section 3. Fundamental propositions of the theory of aurora
polaris and magnetic disturbances
a dipole
Section 5. The theory of Chapman and Ferraro
section 6. The theory of A1'fven Ltransliterate./
Part Three
Practical Utilization of the Phenomena of Terrestrial
Magnetism
Chapter VIII. Magnetic Prospecting
Section 1. The magnetometric prospecting method and its
development
Section 2. Magnetic properties of rocks and minerals
Section 3. Residual magnetization of rock formations and
the causes of its occurrence
Section 4. Effect of variations on the state of magnetiza
tion of rocks
Section 5. Methods of investigation of the magnetic
properties of rocks
1. Ballistic method
2. The magnetoneetric method
3, The magnetometer of B., M. yanovskiy and Ye. T.
Chernyshev
285
287
298
303
308
308
'308
313
329
334
337
340
342
352
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4. The method of attraction and repulsion in a
nonuniform magnetic field 355
5. Cylinder method 360
6. Astatic magnetometer of S. Sh. Dolgin4v 362
7. The method of T. N. Roza 365
8. Constant magnetometer for measuring the magnetic
properties of.rocks and materials of small volume 367
Section 6. Procedure of conducting magnetometric operations 368
Section 7. Aerial magnetic surveying 371
Section 8. The main problem of magnetometry. Magnetic
fields of regularly shaped bodies 375
1. Sphere 377
2. Cylinder with small diametertolength ratio,
magnetized in the direction of its geometric axis 380
3. Ellipsoid 384
4. Infinitely long circular cylinder magnetized per
pendicularly to the axis 386
5. Elliptical cylinder of infinite length magnetized
along the major axis of the ellipse 389
6. A thin and infinitely long plate, magnetized in
width 390
7. Infinitely long rectangular prism magnetized
vertically. Vertical stratum 393
8. A thin, infinitely long plate. Thin horizontal
stratum 395
9. Horizontal, semiinfinite stratum 397
10. Vertically magnetized sloping stratum of infinite
length 398
11. The method of cards for solving the main problem
of magnetometry 400
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12.
Experimental method of solving the main problem
405
Section 9.
The possibility of utilizing the plane problem
407
Section 10.
The inverse problem of magnetometry
411
1.
Determination of the shape of a rock formation
411
2.
Multiplevalued nature of the inverse problem
of magnetometey
415
3. Analytical methods of solving the inverse
problem
417
Section 11.
Procedures of interpreting magnetic anomalies
437
Chapter IX.r;4Shipi's:Magnetic Compass Deviation
445
Section 1.
The basic equations of a ship's magnetic field
445
Section 2.
Transformation of Poisson's equations
447
Section 3.
Determination of deviation
449
Section 4.
Elimination of deviation
450
Section 5.
Deflector magnetometer of Dekolong
453
Part Four
Methods and Apparatus for the Measurement of the Elements of
Terrestrial Magnetism
457
Chapter X. Measurement of the Elements of Terrestrial
Magnetism
457
Section 1. Classification of measurement methods and of
measuring apparatus
457
Section 2.
The principle of measuring declination
460
Section 3.
Astronomical observations during magnetic
surveys
467
Section 4.
Measurement of dip with a needle inclinator
480
Section5.
Measurement of dip by means of an induction
inclinator
486
Section 6. Measurement of dip by the method of induction
in soft iron
498
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Section 7. Measurement of the, horizontal component. The
absolute method of Gauss
503
Section 8. Sequence of observations of the absolute method
516
Section 9. Determination of the constant coefficients
525
Section 10. Errors in determination of H caused by inci
dental errors of direct observations
532
Section 11. Systemic errors caused by imperfections of
instruments
536
Section 12. Absolute electric method of determining the
horizontal component
544
Section 13. Absolute magnetic theodolite of VNIIM
547
Section 14. The relative method of Gauss for determination
of the horizontal component
550
Section 15. Measurement of the horizontal component by the
method of deflections (abridged method of Gauss)
552
Section 16.
The relative electric method of measurement of Ii 554
Section 17.
The"combind'magnetic theodolite
557
Section 18.
Measurement of the horizontal component by means
of the quartz magnetometer
558
Section 19.
Double compass for measuring H
562
Section 20.
Measurement of vertical component by the
electric method
566
Chapter XI. Methods of Measurement used during Investiga
tion of Magnetic Anomalies
567
Section 1.
Introduction
567
Section 2.
M1 field magnetometer
568
Section 3.
M2 magnetic balance
575
Section 4.
The aeromagnetometer of A. A. Logachev
585
Section 5.
Magnetically saturated sondes
588
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Chapter XII. Variation Instruments
Section 1. General theory of variation instruments
Section 2. Variometers of horizontal component and declina
tion (unifilar)
magnetic balance
Section 7. Methods of determining scale division of vari
 ometers
601
601
605
Section 3. The influence of temperature on the unifilar
readings. Methods of compensation
610
Section 4. Unifilar as a northward and eastward component
617
variometer
Section 5. Vertical component variometer based on mag
619
netic balance
Section 6. Effect of temperature on the readings of the
623
by B. M. Yanovakiy
630
Section 9.
Construction of variation instruments
633
Section 10.
Magnetographa of B. 3?e. Bryunelli
638
Section 11.
High speed magnetographs of La Kur 2trana
Section 8. Variometer of the vertical component desighed
640
literates
641
Section 12. Station of A. G. Kalaahnikov
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The present edition of the Textbook of Terrestrial Magne
tism differs substantially from the first edition which was pub
lished more than 10 years ago. Those differences are due on one
hand to advances in our knowledge of terrestrial magnetism and on
the other hand to the intended purpose of the new edition. While
the first edition was designed to be a textbook for use at hydro
graphic institutes and faculties and had as its principal aim
familiarization of students with magnetic measurement procedures
as applied in conjunction with determination of the spatial distri
bution of the earth's magnetic field. The new edition is a text
for university students of the departments of physics and geology
who are specializing in geophysics. Hence the scope and contents
of the book are entirely different from the first edition since
it includes all the basiczpuablems of theoretical as well as of
practical nature relating to terrestrial magnetism phenomena.
The text is arranged in accordance with the schedule of
lectures given at the department of physics of the Leningrad Order
of Lenin, State University imeni A. A Zhdanov,,for students speciali
zing in geophysics. However, some problems are treated in a somewhat
wider scope, in order that those who study terrestrial magnetism
may be in a position to understand more thoroughly and clearly
phenomena which due to a lack of time cannot be dealt with in detail
in the lectures.
The course is divided. into 4 parts, each of which includes
a definite range of problems of an individual nature. Presentation
of the material as a whole is preceded by an introduction of considerable
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length describing some specific problems of magnetism not usually in
cluded in general texts of physics, but which are of great importance
to the mastery of many aspects of a course in terrestrial magnetism.
These problems include: calculation of the magnetic fields of var
ious coils, magnetization of ferromagnetic bodies, magnetic materials
and several others.
The first part constitutes a study of the permanent portion
of the earth's magnetic field and its secular variations. Attention
is focused primarily on the physical aspects of the phenomena and on
analytical investigation methods which enable ascertainment of the
structure of the magnetic field, while questions of a theoretical
nature are considered only to the extent necessary to form a general
conception of the situation that prevails in this field. This is due
to the fact that as yet there is no fully evolved theory, nor even an
adequate outline of the theory of terrestrial magnetism. There are
only a number of hypotheses none of which can claim any degree of
reliability.
The second part of the course relates to the variable portion
of the earth's magnetic field, magnetic variations and the phenomena
associated therewith, such as the aurora polaris and ionospheric
phenomena. Although the theories of these phenomena cannot be.con
sidered as being fully developed at the present time, their. foundations,
in contrast with the theory of the permanent magnetic field, are more
or less clear and can be accepted as objective facts. Therefore, more
space is allocated to presentation of the existing theories of
magnetic variations and aurora polaris.
The third part describes the practical applications of
terrestrial magnetism phenomena, namely magnetic prospecting for
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useful minerals, and compass deviation. These are independent
problems and are treated in separate courses. However, their in
clusion in a general text of terrestrial magnetism is necessary
to the extent that they follow directly from the study of ter
rostrial magnetism phenomena and are integrally associated with
the latter. At the present time it is impossible to visualize the
study of terrestrial magnetism divorced from its practical applications.
Since the present text is one of a purely physical nature wherein
phenomena are considered from the standpoint of physical laws, the
chapter on magnetic prospecting also is physical and thus no con
sideration is given to problems of a geological nature connected
with the interpretation of any given deposits.
The fourth part describes problems of procedure, including
the theory and practice of geomagnetic determinations in the course
of field work as well as of observatory observations. Many of the
problems in their part are considered on a somewhat wider scope than
is required by the curricula, but we believe such an elaboration is
fully justified because until the present there has been no manual
or monograph in which these problems would be presented in an
orderly and systematic fashion, although the need of such a manual
is very apparent.
I consider it my duty to express profound gratitude to'the
associates of the Institute of Terrestrial Magnetism K. V. Pushkov,
Professor Yu. D. Kalinin, X. P. Ben'kova, V. P. Orlov, S. I. Isayev,
S. M. Manourov, and others who contributed to the editing of the
manuscript and have made many, very valuable suggestions.
F
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Section 1. Brief Historical Review
Together with astronomy and geodesy, the science of terrestrial
magnetism has one of the oldest histories in the development of science.
The Chinese were the first in discovering the phenomenon of magnetism
and the possibility of its practical utilization. They have the pri
ority in the discovery of the basic magnetic properties and in the in
vention of the compass. According to Chinese chronicles the phenomenon
of magnetic polarity and the use of the compass were known in.China
more than 1,000 years prior to the beginning of the new era.
Thus the Chinese historian SuMaTzyan, who lived in the first
century B. C., in relating the event of the reception in the year 1100
B. C. of Vietnamese envoys by the Chinese emperor CheyKun, writes:
"CheyKun presented them with 5 travel chariots so designed that they
always indicated the direction of south. The envoys of YueChen (the
ruler of Vietnam) departed in these chariots, reached the seacoast,
passed Fu Nan and LinI, and one year later arrived in their homeland.
The chariots which indicated the south were always in the lead in
order to guide those that followed and to show the direction of the 4
cardinal points."
The first written record of the property of magnetic polarity
is found in a Chinese dictionary compiled aWmut 121 A. D. while a re
port of the use of the compass also is found in a later work entitled
"Treatise on Vehicles and Clothing" which, according to Chinese chronicles
was written ih'tthe fifth century A. D.
In this source it is stated that after the secret of making
chariots which indicate the south had been lost in the firstcentury
B. C., the scholar MaNuin again invented such chariots in 226 A. D.
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"On these chariots was a wooden figurine, the outstretched
aril of which pointed to the south. No matter how the chakiot was
turned or its direction reversed, this figurine continued to indi
Cate the south" (Figure 1).
Furthermore, at that time the Chinese already knew that a
needle which had been rubbed with a lodestone points,,:saa exactly
to the south, but is deflected slightly to the west, which is stated
in a number of instances in various works of,Chinese authors which
were written in the tenth and eleventh century A. D.
In Europe the first information concerning magnetism came
to light several centuries B. C. References to a "mysterious stone"
which possesses the remarkable property of attracting iron are found
in the writings of several Greek authors. At first it was referred
to as "Hercules stone" "Lydian stone," "Siderite" and also simply as
"stone." Later these terms were replaced by "magnet."
The basic properties of magnetic attraction were wellknown
to the Greeks back in?the VII century B. C. Thus, references to
this fact are found in the writings of Thales, who lived about 640
546 B. C. However, the phenomenon of polarity became known in Europe
only in the twelfth century. The use of a compass by Europeans is
first mentioned in the work of the English monk Alexander Neckhaa
written in the twelfth century: "Seafarers while sailing, when
they cannot orient themselves by the sun because of cloudy weather
or when the world is plunged into the darkness of night and when
they do not know in which direction to sail, make use of a freely
turning magnetic needle one end of which points to the north."
All this convinces us that atthe beginning of our era the
Chinese had knowledge of the properties ofmagnets and of the magnetic
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field of the earth which far exceeded that of the Europeans of the
same period.
Yet, the present day bourgeois scientists of Europe and
America in writings on the history of magnetism make every effort
to minimize the role and priority of the Chinese in the discovery
of terrestrial magnetism, by arguing, for example, that the Chinese
could not possibly have invented the compass before the Europeans
since they were not at that time a maritime nation and had no ships.
Such reasoning is found, for example, in the "Encyclopedia Britan
nica."
The earliest European treatise concerning the magnet and its
properties is the letter of Peter Peregrine to a certain Siger, dated
12 August 1269. In this writing Peregrine describes all the pro
perties of magnet known in his days and names for the first time
the poles of the magnet. The end of the needle pointing to the
north Peregrine proposed to name the north pole, and the opposite
end the south pole, But the prinlipal achievement of Peregrine
is the improvement of the compass, which at that time was a fairly
primitive instrument in the form of a magnet floating in a vessel
filled with water and having neither a pointer for taking readings
nor a card. Peregrine combined the compass with a marine astrolobe,
providing it, with a graduated scale and a base line which enabled
sailors not only to direct the ship, but also to determine the azi
muth of heavenly bodies. At first Peregrine made use of a floating
compass, but later on he adopted a compass turning on a vertical
pivot. Figure 2 shows the 2 types of compasses used by Peregrine.
The contents of Peregrine's letter did not become widely known
until the sixteenth century and therefore his discoveries were not
put to extensive use.
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Invention of the mariner's compass (13021318) in its
present form has been attributed to the Italian Flavio Gioia
who lived in the early nineteenth century. Gioia set the compass
needle on a pointed pivot as is done in modern compasses and pro
vided it with a paper disc (card) divided in 32 sections known as
the Rose of Winds or compass points.
The next step forward in the development of the science of
terrestrial magnetism was the discovery of magnetic declinations
by Columbus during his journey from Europe to America. Strictly
speaking, it is the starting point of the science of terrestrial
magnetism. Before the discovery of Columbus, i.e., up to the fif
teenth century, it was believed that the magnetic needle pointed
exactly to the north and this was thought to be duo to the attrac
tion exercised by the north star on the magnetic needle. Only
after the first voyage of Columbus to the New World was it known
that the magnetic needle changes its direction on passing f.rorz one
locale to another.
Several days after Columbus had sailed from Europe, on the
13 September 1492, it was noted to the utter amazement of the
mariners, that the magnetic needle had changed its direction having
become deflected to the NW. On the following morning the change
occurred again in the same direction and to the same extent. on 17
September the navigator, having determined the azimuth of the sun,
found that during the 4 days the needle had undergone a change in
direction amounting to an entire division of the compass and scale.
To reassure his crew Columbus had to resort to deceit and altered
the compass scale, explaining that it was not the needle that had
changed its direction, but the north star which had altered its position.
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Readings taken after thelaeirivsl in the New World, showed
that the needle was again Pointing exactly to the north.
Thus by the end of the fifteenth century the Europeans had
knowledge of 2 facts, namely: (1) the needle is deflected from
the true meridian, and (2) that the magnitude of the deflection
changes from one locale to another. These facts prompted the under
taking of measurement of the elements of terrestrial magnetism,
and consequently constituted the beginning of the science of
terrestrial magnetism.
The discovery of magnetic declination and inclination ap
parently was made by George Hartmann of Nuerenborg, a master
craftsman who made compasses and sun dials. He was the first to
determine the declination at Rome about 1510, but did not report
this discovery until 1544, when he mentioned it in a letter to Count
Albert of Prussia.
In the same letter he also stated that "the magnet is not
only deflected to the east by about 9?, as I have already reported,
but also dips downward. This can be shown in the following manner.
I made a needle about as long as a finger, which was mounted exactly
horizontally on a pointed pivot, but as soon as I?touched one of
its ends with a magnet the needle could no longer remain horizontal
but was deflected downward by an angle of
The first extensive series of carefully conducted observa
tions of declination at sea was carried out by Jean do Castro in
1538 during his journey from Europe to the East Indies.
The first work on magnetic declination was published by Bur
roughs in England in 1585. By the on. of the sixteenth century
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declination measurements bad been taken at almost a hundred differ
ent locations at various parts of the globe, including Russia. Thus,
in 15561557 declination readings were taken at the estuary of the
Pechora river, on Novaya Zemlya, Vaygach Island and at the town of
Kholmogorsk, and in 1580 at the towns of Astrakhan and Derbent.
Hartmann's letter had not been published and possibly was
not known to his contemporaries. In 1581 the English navigator
s.
hydrographer Norman published the results of his measurements and
was the first to express the idea that the cause of the needle's as
suming a certain direction is locatedwithin the earth. Norman also
showed experimentally that within the terrestrial field the magnet
undergoes only a rotary movement. He did this by placing the magnetic
needle in a vessel containing water and balanced it so that it could
move freely over the surface or within the water.
The next noteworthy step in the development of the science
of terrestrial magnetism was the publication in 1600 of Gilbert's
book in Latin "Magnets, Magnetic bodies, and the Great3$agnet  the
Earth."
In this work is stated for the first time a theoretical idea
of the causes of terrestrial magnetism, which has retained its Sig
nificance up to present time. Gilbert expressed the opinion that
the earth is a magnet, the poles of which coincide with the geo
graphical poles, and he substantiated his assertion by experiments
with a magnetized s\hsre. As the principal argument in support of
his theory Gilbert cited the fact that the dip of the magnetic needle
was found to be about the sane when using a small model of the ter
restrial globe made from a natural magnet, as in the case of the earth.
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To explain the phenomenon of "declination" which is in conflict
with his theory Gilbert postulated that the continents had mag
netic properties which deflect the needle. The significance of
Gilbert's theory was the fact that he had definitely established
the relationship between the magnetic field of the earth and ter
restrial globe, by pointing out that the cause ofterrestrial mag
netism must be sought not outside but within the earth.
Up to the end of the eighteenth century all the observations
of terrestrial magnetism were limited to measurements of declina
tion and dip, since no methods were available enabling determination
of the magnitude of magnetic force. Only in 1785, when Coulomb
found a method for measuring torque, was it possible to work out a
method for measuring the intensity of a magnetic field. The first
such method was proposed by Coulomb himself and it found immediate
extensive application during various expeditions in taking magnetic
measurements.
The method of Coulomb consisted of determination of the
period of oscillation of a pendulum and therefore gave only rela
tive values of the field intensity. Moreover, it involved the flow,
that the period of oscillation depended not only on intensity but
also on magnetic moment, which could very and thus alter the period
of oscillation.
In 1839 the classical works of Gauss; written in Latin and
entitled "Intensity of Terrestrial Yarnetic Force Reduced to an,Ab
solute Sclae" appeared. In it Gauss provided a theoretical founda
tion for the method of measuring the horizontal component on an
absolute scale, a method which has remained up to now unique for
this purpose, and at the same time he also provided an experimental
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technique which increased the accuracy of the measurements. Al
most simultaneously with the latter Gauss published another work
entitled "General Theory of Terrestrial Magnetism" in which with
out relying on any hypotheses he formulated the problem of inter
pretation of the terrestrial magnetic field in an entirely differ
ent manner. on the basis of the single assumption that the cause
of terrestrial magnetism is within the earth, Gauss was able to
formulate the magnetic potential at any point on the surface of the
terrestrial globe as a function of the coordinates of latitude and
longitude, expanded to an infinite series of spherical functions.
By using a finite number of terms of this series it is possible to
determine the coefficients of this series from observational data
and thus calculate theoretically the potential at any point of the
earth's surface.
Prior to 1634 all investigators assumed that magnetic de
clination varies only from one locale to another and that at a
definite point it remains constant. In 1634 Henry Hellibrand found
the declination at London to be +406', whereas Burrough and Norman
in 1580 had obtained the value +11015'. This fact showed that over
34 years the magnetic declination had undergone such substantial
hanges that they could not be attributed to observational errors
and therefore the fact of a gradual change of this element with
time had to be recognized, and which subsequently was designated
as "secular variation."
This was the manner in which the secular vaiations of de
clination were discovered. Diurnal varis.tions were first discovered
in 1682 by Guy Tachard, who on observing the declination of the town
of Luvo in Siam during 3 consecutive days found that it varies in a
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different canner from day to day. Since the observations were
probably made at different times of the day, the changes observed
were unquestionably diurnal variations.
A more thorough proof of this phenomenon was provided by
the London clockmaker Graham, who in 1722 made hundreds of obser
vations of the magnetic needle during a single day and also found
the occurrence of these same variations. The observations of
Graham were confirmed during the same year by Professor Celsius
at Upsala (Sweden), after which the diurnal variations became a
recognized fact.
Later observations revealed the existence of variations in
dip and at the end of the eighteenth century, after methods had
been worked out for measuring the horizontal component, variations
in this element also were found to occur. This gave rise to an in
vestigation of these variations by means of regular, continuous ob
servations at special magnetic observatories which were established
in Russia and in Western Europe in the second decade of the nine
teenth century.
The first Russian to devote serious attention to terrestrial
magnetism phenomena was the brilliant scientist M V. Lononosov,
who as early as1759 provided the solution for a number of problems
relating to terrestrial maaetism in his work "Discourse on the
Great Accuracy of Maritime Navigation."
In this work Loiaoetosov gave very valuable suggestions concern
ing compass design, ensuring more accurate readings. Like a true
scientist Lomonosov also considered the causes which bring about
the definite orientation of a magnetized needle in space. Prior to
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Lomonosov the prevailing notion of the magnetic field of the earth
was that it was similar to the field of a single magnet having 2
poles. Lomonosov was the first to consider the structure of the
terrestrial globe as a body consisting of minute, differently magne
tized particles which in combination form a nonuniformly magnetized
sphere, and which in turn causes unequal declination at different
parts of the globe. Thus he anticipated Gauss' idea of a random
magnetization of the terrestrial globe.
On the basis of such an assumption Lomonosov considered it
unfeasible to base any mathematical theory of terrestrial magnetism
upon a small number of observations and indicated the following as
the manner in which its true theory should be evolved: "the best
method for determining the truth is to derive a theory from observa
tions, and to correct observations by means of the theory." This
propositioh*of Lomonosov applies not only to the theory of terrestrial
Magnetism but is a prerequisite in the development of the theory of
any natural and social phenomenon.
To meet these requirements he recommended the establishment
of permanent observation points (observatories) on land and systematic
observations aboard 'ships at sea. As stated above, however, this
idea of Lomonosov was put into effect only some 60 years later.
Lomonosov also was the first'to suggest that variations in
declination with time are due to an external magnetic field not as
sociated with the magnetic field of the earth, although he gave an
incorrect interpretation of its origin.
Louonosov's ideas, which were many years ahead of *the contem
porary knowledge of terrestrial magnetism, unfortunately were not
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in Russia at that time. The work of Loronosov was forgotten and re
enlarged upon due to the attitude concerning science which prevailed
rained unknown for more than a century.
During the first half of the nineteenth century professor I.
M. Simonov (17941855) and Academician A. Ya. Kupfer (17991865)
made classical contributions to science which revived the basic
ideas of Lomonosov although the authors themselves did not suspect
that these ideas already had been expressed by Lomonosov.
Even before the publication of Gauss' treatise I. M.
Simonov, a professor at the Kazan University, published in 1935 in
the "Uchenyyo Zapiski" [Scientific Records] of the University a
paper entitled "Experiments on Mathematical Theory of Terrestrial
Magnetism."
In this paper I. M. Simonov showed that the Magnetic field
of the earth induced by the cumulative action of its magnetic
particles is equal to that of a dipole field, assuming that the
particles are uniformly distributed. The dipole potential as a
function of latitude and longitude was found to be identical with
the first term of the expansion of the potential as derived by Gauss.
The contributions of I. M Simonov and Gauss are the founda
tion of modern concepts of the magnetic field of the earth and
their publication may be regarded as the beginning of the current
phase of development of the science of terrestrial Magnetism..
This phase consists of rapid gathering and organization of materials
on the distribution of elements of terrestrial magnetism over the
surface of the earth, the evolution of number of hypotheses con
cerning the origin of the magnetic field of the earth, and finally
in recent times widespread practical utilization is being made of
magnetic observations.
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In addition to this work I. M. Simonov discovered the
periodic nature of variations in declination. In his "Notes
and Recollections Concerning a Journey to England, France, Belgium
and German in 1842," published at Kazan in 1844, Simonov states
that he was able to ascertain the following 3 periods of declina
tion: "the first period lasts one year and depends upon the move
ment of the earth around the sun; the second, which is'due to the
sun's rotation about its axis, lasts for about 27 days; the third
period is diurnal, and is determined by the position of the sun in
relation to the horizon. All these periods are confirmedby obser
vations."
In 1925 the work of Academician A. Ya. Kupfer was published
in which a fact entirely novel at that time, namely the simultaneous
occurrence of magnetic storms at Paris and Kazan, separated by a
distance of 430 of longitude, was established. This work was the
result of observations of variations in declination carried out
by Kupfer and Simonov at Kazan, and provided the stimulus for the
organization of systematic observations of variations in declina
tion at different points of the terrestrial globe.
The work of Simonov and Kupfer constituted the beginning
of detailed studies of magnetic variations and determination of
their causes.
It is of interest to note that as early as the eighteenth
century Russian inhabitants of the Kola peninsula had independently
discovered, another fact relating to variations in declination,
namely the occurrence of magnetic storms during aurora borealis.
A. Ya. Kupfer is credited with organizing systematic observation of
variations at a number of observatories which were established through
his efforts.
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In 1929 he organized a magnetic observatory at Petersburg,
where beginning an October 1829, observations of declination were
made every hour. _Due to the efforts of A. Ys. Kupfer observatories
were built in 1832 at the Siberian Mining Works of Norchinsk,
Barnaul and Kolyvan, with funds supplied by the Department of Mines
and in 1836 an observatory was established at Yekaterinburg (Sverd
lovsk) at the center of the Ural mining district.
It should be mentioned that of all the observatories
organized by Kupfer only the one at Sverdlovsk has been in contin
uous operation up to the present time.
The observatories at Nerchinsk, Barnaul and Kolyvan were
discontinued in the early sixties when the mining industry of
Siberia and the Urals, which had been based on forced labor, de
clined due to the abolition of serfdom. "But the same system of
serfdom which helped the Urals to rise to such a high level during
the epoch of incipient development of European capitalism also was
the cause of its downfall during the height of the capitalistic era."
(Lenin, complete works, Volume 3, 4th.edition, page 424.)
The Petersbuu4 observatory was not in operation between 1852
A. Yap Kupfer was the first to note changes in the moment
of magnetic needles with changes in temperature and to determine the
law goferning those changes, inabling temperature corrections in
deterainination of the horizontal component by the Gauss method and
thus 'increasing the. accuracy of these digterminations.
In a historical article published by?the in Geophysical
Observatory, Academician M. A: Rykachev wrote, concerningthe work
 16,
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conducted during the thirties and forties of the past century in
the field of terrestrial magnetism: "In no country did the dis
coveries of Gauss produce a greater effect than in Russia, where
at that time Kupfer was undertaking the establishment of a system
of magnetic and meteorological observations."
During the seventies of the past century I. N. Smirnov, a
docent at the University of Kazan, conducted an important magnetic
survey, and work was resumed at the Petersburg magnetic observatory,
which was later transferred to the town of Pavlovsk.
From 1871 to 1878 1. N. Sairnov, on his own initiative,
carried out a magnetic survey of almost the entire territory of
European Russia, which included determinations at 281 points, and
it was only due to his untimely death that he was prevented from
extending this survey over all of Russia. This survey revealed the
existence of a large magnetic anomaly in the region of'Kursk and
supplied valuable information concerning the magnetic field of the
territory of Russia. Its results provided the basic data for the
first magnetic maps of European Russia, which were drawn up by
A. Tillo in 1881 and 1885.
The work started by I. N. Smirnov was continued in the
late nineties by P. T. Passal'skiy, a docent at the Odessa VAIver
sit[, who in 1898 and 1900 made a detailed survey of the Crimean
peninsula and the adjoining areas, including observations at more
than 200 points.
At the initiative of Academician G. 1. Vil'd, observations
of variations were resumed in 1870 at Petersburg, and in 1878, they
were continued at Pavlovsk, where a first rate magnetic observatory
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had been established and which has served as a modelfor all such
,observatories throughout the world. Vil'd is accredited with having
developed and built a series of new instrument# for observatory re
cordings. He built 3 types of absolute magnetic theodolites, a
decimator and induction inclinator and designed a series of vari
ational instruments. All this equipment was in use at the Pavlovsk
observatory up to the time of its evacuation on 1941. The same
instruments also were used at the Sverdlovsk, Irkutsk and Tbilisi
observatories.
Until the end of the nineteenth century all investigations
of the magnetic field of the earth were conducted by independent
institutions and by individual scientists, with no consolidating
or coordinating agency. At the beginning of the present century it
became evident that there exists a correlation between the magnetic
field of the earth and its geological structure, and the undertaking
of a magnetic survey throughout Russia in accornce with a single
plan and under the general direction of the Academy, wasoposed
by the St. Petersburg Academy of Sciences, the costs being defrayed
from a special fund allocated by the State. Treasury. The initiator
of this undertaking was Academician M. A. Rykachev, director of
the Main Geophysical Observatory, who in advocating the survey in
an address delivered on 31 December 1901 at the Twelfth Congress
of Russian Naturalists and Doctors stated: "The general distribution
of magnetism over the surface of the terrestrial globe is such that
the entire globe can be regarded as a magnet, the magnetic axis of
which forms an angledthe axis of rotation of the earth. How
ever, over extensive regions and at small individual areas there are
sizeable deviations from this symmetrical distribution. The study
of these deviations shows that such 4;e 1 anomalies are correlated
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with the geological structure of the given area. From this it
is apparent that a study of the magnetic characteristics of the
earth is of great importance to the advancement of science."
The magnetic survey undertaken by the St. Petersburg
Academy of Sciences was initiated in 1910 and continued until
1914 when it was discontinued due to the wartime conditions, and
was resumed only after the October Revolution.
twentieth century include the work of the reknowned Russian
41
physicist N. A. UUov, professor it the Moscow University entitled:
"Geometric Expression of Gaussian Potential as a Means of Determin
ing the Laws of Terrestrial Magnetism," which contained the first
interpretation of the constant terms of the Gauss analysis, the
physical meaning of which until that tivie had remained unknown.
Following the Great October Revolution, when extensive
possibilities were provided for rapid and effective utilization
of all natural phenomena, a vigorous development took place in the
study of terrestrial magnetism.
At the initiative of V. I. Lenin, a thorough investigation
of the Kursk magnetic anomalies was started during the first years
of the Soviet regime, which led to the discovery of extensive iron
ore deposits.
In 1930 a general magnetic survey was initiated, which over
a period of 10 years covered 26,000 magnetic points throughout the
USSR, and which has enabled the construction of magnetic maps of the
USSR with an accuracy not thought to be possible prior to that time.
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In 1924 the first polar observatory in the world was estab
lished at Matochkin Shar strait on Novaya Zemlya. During the fol
lowing years similar observatories were established at a number of
points extending from Franz Joseph Land to Bearing Strait.
In 1932 the Institute of Terrestrial Magnetism was founded
which has grown to a large scientific establishment of the USSR.
The number of magnetic observatories directed by this institute
increased several fold after the revolution.
Under the Soviet regime, Soviet scientists have worked out
new investigation methods, gathered and consolidated vast amounts
of data concerning the distribution and variations of the magnetic
field over the surface of the earth and finally, a new branch of
terrestrial magnetism of an applied nature has been developed,
namely, magnetic prospecting, which through the work of Soviet
scientists has received both theoretical basis and widespread prac
tical application.
Section 2. Basic Laws of a Stationary Magnetic Field 
The magnetic field of the earth can be regarded as stationary
since its changes with tirosconstitute only a small portion of the
total field. It is sufficient to note that the amplitude of static
daily variations does not exceed several tens of gammas. Moreover
the frequency of variations ranges between 104 and 1Q~1 hertz, so
that they have very little effect on the magnitude of,an inductive
electric field in a study of many phenomena of terrestrial magnetism.
Therefore in most studies of terrestrial magnetism the laws of a
stationary field must be used, which are particular instances of the
general laws of an electromagnetic field expressed by Maxwell's equations.
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Maxwell's equations of a stationary magnetic field are of
the form
rot H  4irj (0.la)
div H = 0, (0.1b)
where H is the intensity of the magnetic field and j the current density.
The first equation provides a correlation between the intensity
of magnetic field and current density at a given point, while the
second expresses the continuity characteristic of the magnetic field.
Since the vector H has no origins, it can be made equal to the rotor
of another vector A, i,o.,
H = rot A. (0.2)
Thus equation (0.1a) assumes the form:
rot rot A a 4xj
or, replacing rot rot A by its equivalent expression we obtain:
grad div A // A a 4xJ,
where 61, is a Laplace operator.
If vector A is made to conform to the condition
div A = 0
we obtain the equation which vector A bust satisfy:
& A a  47Q.
(0.3)
Vector A is designated as the vectorpotential, and if it is
known the vector H nay be determined. The necessity of introducing
the new function of vectorpotential is due to the fact that equation
(0.1a), which correlates H and J, cannot beresolved directly, whereas.
equation (0.3) of the vectorpotential can be solved by mathematical
physics equations.
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The solution of equation (0.3) is 'of the form:
r dv,
where r is the distance of the volume clement dvthrough which
current of density j flows from the point at which the vector
potential is considered.
From this equation by means of rotor operations (differentiation)
for the coordinates of point P at which the vector A is being considered,
we obtain:
H rote A rote, dv rotpjdv ' 5 (i grade jdv.
Since the value of vector j does not depend on the point P,
it follows that
rot'pJ = 0.
1
gdp r
r
H  ( r] dv.
r3
This expression represents the law of BiotSavar in its
integral form.
(0.4)
From equation (0.1a) integration of both parts over a surface
S we yield
(rot HdS)  4n ~(jdS),
S
or, on applying the theorem of Stokes:
%(Shc) : 4x1, (0.5)
where I is the intensity of the current flowing across the surface,
and the integration gust be carried out according to the contour of
the surface.
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lquations (0.4) and (0.5) show that a constant magnetic
field within a medium of permeability equal to unity, may exist
only if electric conductivity currents, or equivalent convection
currents are present in the latter, with current density
9 = evn,
where e is the charge of a particle (electron, ion), v the velocity
of its movement and n the number of particles per unit volume.
The magnetic field within the portion of the medium con
taining no current must satisfy the equations:
rot H = 0,
(0.6a)
dives=0..
(0.6b)
In this case vector H may be represented by the gradient of
a scalar function , since rot grad 0, and the first equation
is satisfied. Taking
H = grad (x, y, z)
(0.7)
and taking into account equation (O.6b) we obtain:
div grad,, A0,
(0.8)
i.e., the function `.~ which is termed the magnetic potential,
satisfies the Laplace equation, and plotting of the function requires
solution of this equation. This solution is possible if the boundary
conditions are known, i.e., the distribution ofr or of its derivative
over the normal to a surface.
In an investigation of phenomena connected with the movement
of charges within 'a magnetic field the following equation, the
equation of Lorenz, mast be added to the above equations, which fully
define the status of a magnetic field:
F = eE 4 c [v, H],
(0,q)
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where F is the force acting on a charge e, moving at a velocity v
within the electric and magnetic fields, c is the velocity of
light; and e and E are exposed in absolute electrostatic units and
H in absolute electromagnetic units.
Section 3. Magnetic Field of a Closed Linear Circuit
In considering many problems of the theory of terrestrial
magnetic field the magnetic field of an elemental magnet (dipole)
or of its equivalent elemental circuit current are encounteredthere
fore, it is most important to know the laws which govern the magnetic
field of such models and how they are derived from the general equa
tions of a magnetic field.
Let us consider first the magnetic field of a linear circuit
current of any shape. A linear circuit is taken to mean a closed
conductor the cross sectional area of which is infinitely small, and
the intensity of the current flowing through this closed circuit has
a finite value 1. The field of such a circuit is determined by the
law of BiotSavar (0.4) which in this instance has the form:
H I d (dl, ]
r
because jdv = Idl, where dl is the element of the length of the
circuit. The component of vector H on the xaxis is:
Hx = I
rz
r3
dy  ry dz).
r3
(0.10)
Denoting the coordinates of point P, at which the vector is
applied, bys'l~ay~ yl, z1, and the coordinates of element dl by x, Y, z, then
r =yl  y,
y
rz=x1Z.
We introduce the auxiliary vector L, the components of
Lx = 0, Ly = 3.
r
(0.11)
(0.12)
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lrom the above correlations it is apparent that the direction
of vector L is fully determined by the coordinates of point P and
by element dl.
Here, formula (0.10) gray be written in the form:
Hx = I j (L, dl),
or, on applying the theorem of Stokes on the conversion of a contour
integral to a surface integral:
Hx = I [Prot LdS), (0.13)
wherein the integration must. be extended over the entire surface in
volved in circuit; the shape and dimensions of the surface being ir
relevant. On the other hand, the direction of the normal to the sur?
face element dS depends upon the direction of circuit eleesent dl,
which coincides with the direction of the current.
According to the formula we have for the scalar product:
(rot LdS) = rota LdSx + roty LdSy + rots LdSZ.
Substituting the rotor components according to vector analysis
formulas, and the components of the surface element by the correspond
ing cosines between the normal and the coordinate axes we obtain:
(rot LdS) [( La  bLY) cos (n, x) + (  Ls) cos (a, y) +
y Z A z o x
(i ) cos (n, z) ]dS. (0.14)
~x y
Further, using the correlations (0.11) and (0.12) to determine
the derivatives Lz and =y and substituting then in equation
y z
(0.14) we obtain:
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The cosines of the angles between the normal nto the surface
element dS and the coordinate axes are derivatives of the correspond
ing coordinates of the normal, and the above expression assumes the
~C x L .16t)
Substituting derivatives for hgpproduct of the rotor and the
surface element dS in equation (0.13), we have:
The components
By and Hz are determined analogously:
:Ir
whence,
/ o w ?
?( 11 r `)
expression cos(n, r) is in fact an element of the
solid angle d ? ? , at which the element dS is seen from point P, thus
where k is the solid angle at which the circuit is seen from point P,
and therefore the quantity may be designated as the magnetic
potential of the closed circuit, taking
Section 4. Magnetic Potential of AU Elemental Circuit
If the closed linear circuit constitutes an elemental circuit
the area of which is infinitely small, then in accordance with formula
(0.18) its potential
is given by the equation
U
or, in a vectorial form:
I,
( y i
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The potential produced by a magnetic dipole is of exactly
the save form, i.e., that of 2 point g tefic charges of different
sign located very close to each other, because by denoting the mag
nitude of the magnetic charge by m and the distance between the
charges by dl, we readily find by applying Coulomb's law, that
/(/ 17 m f E )~) _ }arm y ';,~ ( l ~/l
.fr Y ~
The product mdl is termed the magnetic moment u, which is a
vector coinciding in direction with dl and equal in magnitude to the
product of mass m by the distance between the charges, i.e.,
isdl = A .
Comparison of expressions (0.17) and (0.18) shows that they
are equated at:
IdS = mdl, (0.19)
i.e., on substituting the elemental current with a magnetic dipole
the magnetic moment of which is equal to the product IdS. Therefore,
by analogy, the quantity IdS acquires the designation of magnetic no
ment of elemental current. Thus it may be stated that the designation
of magnetic moment of elemental current is given to a vector which in
magnitude is proportional to the product of current intensity and
area of.circuit current, and coincides in direction with the normal
to the circuit area dS, i.e.,
P = IdS.
The orientation of the direction remains arbitrary. It has
been agreed to consider as positive that direction of the al"f
which coincides with the direction of advance of a corkscrew which
is turned in the direction of the current.
Thus:the potential of a magneticfield created by an elemental
circuit, and consequently also the magnitude of field strength, are.
proportional to the magnitude of the magnetic moment of the circuit, i.e.,
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(/1 ,)
I,.
Tt.
where n is a unit vector coinciding in direction with the direction
of the magnetic moment.
Therefore the notion of a magnetic moment plays the same part
in the treatment of the magnetic field of an electric current, as is
played by the notion of the magnetic charge in the treatment of the
field of permanent magnets. Applying this motion to a circuit of
finite dimensions, it can be shown that the intensity of the magnetic
field of a linear circuit also is proportional to the product of
current intensity and circuit area which is known as the magnetic
moment of the circuit.
In calculating the magnetic potentials of current circuits
formulas (0.17) and (0.18) make it possible to replace elemental
curront,j with magnetic dipoles.
Section 5. ? gnetic Field of a Circular Linear Current
To find the magnetic potential of a circular linear current
of radius R it is necessary to calculate the solid angle It as a
function of the coordinates of point P. Taking the axis of the cir
cular circuit OX as the basic axis of polar coordinates, then because
of the symmetry of the magnetic field in relation to this axis the
magnetic potential will depend only on the latitude 8 and the
polar distance r of point P from the origin of the coordinates 0
(Figure 3), i.e,
From the theory of spherical functions it is known that any
function of r and 0 which satisfies the Laplace equation can be ex
panded to a series in powers of r according to one of the following formulas:
cri 00
,il
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where pa(cos 0) are Legendre polynomials, and An and B. are constant
coefficients which are independent of the coordinates of ,point P.
The Legendre polynomials are algebraic functions of
power n and are the coefficients of A in the expansion to
the expression
27 cos 0))
Q()
7 4
0
consequently,
a series of
I
of all the properties of Legendre polynomials the following,
which are necessary for subsequent deviations, are>>eilioned:
(1) If the argument of the polynomial cos 0 changes its sign
the polynomials of even power remain unchanged, and the polynomials
of odd power change their sign;
(2) The cos 0 derivative of the Legendre polynomial is ex
pressed by the formula:
which may be obtained directly by differentiating expression ((p.22);
(3) At cos 0 = 1 all the polynomials are converted to unity,
i.e., Pn(l) 1;
(4) At cos 0 = 0 the odd polynomials become zero, and the even:
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Following these remarks we turn to the derivation of the ex
pression for the solid angle .~:j , i.e., to the determination of co
efficimnts An and Bn of equation (0.21).
Expausi4nof Z_ for some particular instance and comparison
A
of the coefficients of this expansion with the coefficients A. and B.
of the same powers, r:' or, are sufficient for determination of the
coefficients An and B. Taking the point P1 on the axis of coordinates,
the solid angle to its circular circuit can be readily determined.
Indeed, on circumscribing a spherical surface of radius
plc : P from point P1 we obtain
1 'j f
where r,~ is the angle OP,
  el y1 I(/.,, p",
C, and cos (n, p) has a value
of either plus
one, or minus one, depending upon the direction of the current in the
circuit. Assuming that the current is directed clockwise, on looking
at it from the origin of the coordinates, then
cos (n,p) = + 1,
where ` ` = / ? C' (Figure 3).
Assuming po and removing po outside the radical sign,
we have:
The expression within the second pair of parentheses may
written in the form of a Legendre polynomial series, so that:
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to the left of the origin of coordinates,
For points located
the expression for the potential is:
for which 6 ,
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f /  C ~`  'emu f r?
in the expression p1,
Substituting this value of cos
r (, 7
and after sirple transpositions, we .obtain
Y
The expression within the summation sign in brackets IG
1 according to
i
I
the
For points located on the axis of circuit 0 = 0, an
therefore the expression for the solid angle (0.21) beCO 0S one
of the following:
where po
circuit.
the above equations with the expression
Equating the first of
(0.25) we obtain: , the potential of a circular circuit at any
ConsequeAtly
and a has
point of its extent, under the conditions r ( Po
the f orn : .
e I, r fJ 1 ~"' L?
a
the polynou
ties of I$g.ndrUo polynotiaia I*)
a
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The components of the intensity of the aagtygtic field along
axis x and the axis y are determined by the rolationships
rte k /J i
ti
where P'n (cos 9) and P'n (cos 9 )
and cos A .
Furthermore, on substituting the expression in brackets ac
cording to formula (0.23) We obtain the following expression for
Hx:
Taking into account the formulas for Legendre polynomials and
substituting sin+ and cosy\' with their
A f fw '" ~~ x / ~~
!r J r 1 I `!
~y ' , F ! t Y 1 y/ 1
' / jJG~/,, 4 1
For points located on the y axis, where 9 = 900 and r = y, the
formulas for HX and Hy, which are accurate up to terms of fourth order,
assume the fora:
values at xo = 00, and pot we
l
denote the derivatives of cos 9
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fi
61
At the origin of coordinates, the point 0:
which result is known from elementary physics.
Since the magnitude of the field intensity does not depend
upon the selection of the coordinates, for practical purposes the
most convenient formulas are (0.29a) and (0.30a) in which the ori
gin of the coordinates coincides with a projection of the point
under consideration on the axis of the circular circuit.
Section 6. The Effect of Finite Dimensions of the Circuit Cross
Section
In the formulas for magnetic potential and field intensity
derived in the presiding section it was assumed that the current
flows over an infinitely thin (linear) wire. Actually the cross
section of the wire always has finite dimensions and therefore it
is necessary to ascertain the extent of the effect of these dimen
sions and the degree to which formulas (0.29a) and (0.30a) must be
altered in order that field intensity values calculated with their
aid coincide with the observed values.
Assuming that the cross section of the circular circuit has
the'slope of a rectangle with sides 2a and 2b, and that a current
of uniform density j (Figure 4) flows through this circuit.
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Dividing the area of rectangle S into an infinite number of
elemental areas dS, which may be regarded as cross sections of linear
circuits, and the circuit itself as regarded as a combination of an'
infinitely large number of linear circuits, then the magnetic poten
tial U produced by the entire circuit at point P is the sum of the
magnetic potentials produced by an infinitely large number of linear
circuits; or in the integral form:
U _ dU.
where dU is the potential produced by the linear circuit and equals:
dU=  ft dI.
where ii. is the solid angle at which one of the linear circuits Q is
seen from point P, and dl JdS = Iw/S dS. Thus
U =  w/S ICLdS,
where w is the number of windings in one of the circuits, and the
integration is taken over the entire area of cross section S.
The product I! may be regarded as the magnetic potential U'
produced by a linear circuit with current of intensity I, and there
fore we may state;
U : w/S 3 U'dS. (0.31)
But as we have seen, U' is a function of the radius of the
linear circuit R and of its distance x" from the origin of the co
ordinates. Taking one of the circuits with coordinates R0 and x0 as
being the initial, circuit, and taking the cross section of this circuit
at the center of the rectangle 01, if the transverse dimensions za and
zb of the circuit are small in comparison with the coordinates Ro and Xo,
the potential can be expanded to a series according to the formula of
Taylor:
U' .. Uo . f5/,fix (h"'io)x+ Up/) R(RrRO3 11*l/2wl uo/tog' XRRo)I X1Y
+ ...
3 Uo/x 3 R (xxo) (RR6)
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wherein Uo is the potential produced by the initial circuit which has
Substituting this value in formula (0.31), substituting
dS = dxdR, and taking into account that all the derivatives will be
constant upon integration, we obtain:
+a +b
U = Uo + w/S . n Uo/. x (xxo)dxdR + w/S . ' Uo/ R
a b
(RRo)dxdR + ...
Ha 4b
a b
The potential of the initial circuit Uo is expressed in the
form of a series (0.26) which may be written as follows:
Uo = Ul + U2 + U3 + ...
Fa +b +a +b
U  Ul + w/S . Ul/~} x i (xxo)dxdR + VS . ~` Ul/R a ib
b
+a 4b
(RRo)dxdR + ... +?U2 + w/S . ' U2/'.x (xxo)dxdR +
a b
+a +b
w/S . ?' U2/  R i (RRo)dxdR
a b
It is readily apparent that the integrals in which the dif
ferences xx. and RRo appear at odd powers become zero, while those
of even powers assume the values:
Analogous expressions are obtained for components of the
(0.32)
magnetic field intensity, in which.412 is derived for the corresponding
coordinates, instead of Ul
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lei
u y ((t
the first 2 terms, which are of the form
Applying this result to formula (0.29) for the component of
field intensity produced on the x axis by a circular circuit having
a rectangular cross section, and limiting the corrections made in
we obtain:
(0.33)
Section 7. Magnetic Field of Helmholtz Rings
f
t l !' ~ J
!
1 7 /J 11 1, 6
Two circular circuits of equal diameter placed parallel to
each other at a distance equal to their radius R, with their centers
on a common axis cc', are called Helmholtz rings.
The peculiarity of these rings is the uniformity of the mag
netic field at the center between the rings, and because of this as
a source of a uniform magnetic field.
To determine the intensity of the magnetic field14tdthes=.nringa
formulas (0.27) and (0.28) will be used, taking the origin of the co
ordinates at the center of the rings, and arbitrarily taking the
distance between the rings as equal to 2d (Figure 5).
71 Sind r, 8 and po are the sane for both circuits, angle1 differs
U
by 180 it follos that
/11 0 d Because of he 7oPerti0f q~ Gc t 1i ~') 17 
Y
~J Z J / ~ 4~'y ?~ L I/
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~~ _ ()J+
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Because of this all terms containing even powers of the
iegendre polynomial derivatives are cancelled and the odd powers
are doubled, consequently
00 2n1
Hx = 4iIw singf~/po 5 (`p ) P2n2(cos "P'1n1(cos
1 n = 1
Retaining only the terms of the fourth order we obtain
Hx=4nlw sin2~ [1+r2 p'3(cos ~)P2(cos 9) 4
0 p'5(cos 4)P4(cos A)] (0.34)
Po p po
In an analogous manner we obtain
( Ph
6 s s (Uria~
66~
(0.35)
Selecting angle in a manner so that the second order term
in equation (0.34) becomes zero, this requires that P'3 (cos i) = 0,
Cos
cost = d2/d2 + R2 , it follows that d = R/2.
Consequently, when the circuits are separated by a distance
equal to their radius, the term of the second order, on expansion
of H to a series, becomes equal to zero. Therefore for points located
at a distance r from the center, which is small in comparison with half
the distance between the circuits, the field intensity component on the
axis CC' will be almost the same, i.e., the magnetic field in the
central portion of the circuits may be considered as uniform and ac
curate up to the terms of the fourth order.
or 5/2 cos2  1/2  0, hence
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The advantage of this system over a solenoid, in the center
of which the field also is uniform, is the accessibility of this
field to the observer. The space within which the uniform field is
formed is entirely free of any apparatus and makes it possible to
place any specimens of materials or instruments inside the field, pro
vided their overall dimensions do not exceed those of the coil.
The disadvantage of Helmholtz rings with respect to a solenoid
is the fact that strong fields cannot be produced.
The intensity of the magnetic field produced by Helmholtz
rings usually does not exceed several tens of oersteds, whereas a
solenoid may produce a uniform field of the order of 1,000 Oe.
Substituting the values of p, sin ? and cos } , expressed
by means of R, for H and H in formulas (0.34) and (0.35) we obtain:
1.
_ .,"6 ' (0.36)
f,
Since,
_..( (0.37)
at cos 0 = x/r and r2 a x2 + y2, upon substituting these values in
the above expressions and bearing in mind that la is equal to 0.1 of
the absolute electromagnetic unit, we obtain
"i" r.,
7
(0.38)
1 7
s t, (0.39)
where x and y are the coordinates of point P, taken from the center of the
rings.
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To give an idea of the degree of uniformity of the magnetic
field of Helmholtz rings, tables 1 and 2 are included, showing
relative values of both components at different points in the space
within the rings, i.e., for different values of the coordinates x
and y. The value at the center of the field is taken as the unit
of field intensity and x and y are expressed as the length of the
radius of the rings.
VALUES OF H
0
0.05
0.10
0.15
0.20
1.00000
0.999997
0.999957
0.999781
0.999309
0.999993
1.000012
1.000036
0.999969
0.999547
0.999885
0.999968
1.000187
1.000444
1.000576
0.998157
0.998500
0.999496
1.001049
1.002995
VALUES OF Hy
These results refer to Helmholtz rings consisting of 2 linear
circular currents. In practice they usually consist of 2 coils of
rectangular cross section placed so that the central windings f*m
Helmholtz rings. The formulas derived in the preceding sections for
a circular circuit having a cross section with finite dimensions may be
used for making corrections for the finite dimensions of the cross section
of the coil.
0.15
0.20
0
0
0
0.05
0
+ 0.4
106
 5.8 ?
106
 24.8 '
106
63.3 ?
106
0.10
0
+ 9.3
10
+ 5.8 '
106
 2.4 ?
10i6
92.2 ?
106
Oak~
0
?35.6 ?
10 6
+ 51.8 ?
106
+ 29.2 ?
106
51 8 ?
106
0.20
+149.8 '
106
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Since in the Helmholtz rings the origin of the coordinates
is taken at their center,?in making corrections it is necessary
to use formulas of the general form (0.29), (0.30) and to determine
the appropriate correction for each of the terms.
For the first and third terms the corrections were given in
equation (0.33). The correction for the second term contains the
factor X0, and since the Helmholtz rings comprise 2 circuits, lo
cated at the sane distance X 0 from the origin of the coordinates, it
follows that for one circuit this correction has the factor Xo and
for the other Xo, and as a sum they are cancelled.
In applying the e*pression (0.33) to the Helmholtz rings the
condition Xo = R/2 must be assumed.Under the condition4equation (0.33)
becomes:
HX = 323TIw/5 5R [lb2/15R216/375 (36a631b2)r2/R4 (3 cos26l)],
or, on substituting x and y for r and 9, accurate to terms of the second
order:
H 32nIw/5 5R[lb2/15R216/375R`(2x2y2)(36a231b2)1.
If the dimensions of the cross section are selected so that
36a231b2=0, that is a/b =`/36
the corrections for the third term also become zero. Corrections of
the subsequent terms may be disregarded since they contain Rs and Rg in
the denominator.
Thus the component of field intensity on tbe.x axis, accurate
to terms of the fourth order, assumes the form:
Y 4
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Corrections of the y axis component may be disregarded
because they contain R at a power higher than 5 in the denominator.
of a Cylindrical Solenoid
Section S. MOagnetia Field
A singlelayer solenoid may be considered as a cylinder over
the surface of which a circular current flows perpendicularly to
the turns of wire which form the cylinder, assuming that the surface
density of the current remains constant. Taking the length L of the
solenoid and the number of turns w, the density of this current will be:
j  Iw/L,
where I is the intensity of the current in the winding.
The field of this solenoid is equivalent to the field of an
infinite number of linear circular currents each of which has a cur
rent intensity
dI = jdl = Iwdl/L,
where dl is the solenoid length.
Therefore the potential U of such current is equal to the sum
of potentials produced by each of then and is defined by the expres
sion (0.26), in which it is necessary only to substitute the former
for current intensity, i.e.,
U = Iw/L dl.
Taking theorigin of coordinates at some point 0 on the axi's of
the solenoid (Figure 6), the point P, at which the potential is being
considered, is at distance r fromthe origin of the coordinates, r
being less than the radius of the cylinder, and Q as one of the linear
circuits of the solenoid, defined by the coordinates Po and 1f), then
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wherein the integration must be carried out for the entire length L
In the integration r and 0 remain constant. Moreover, the
first term of the series j(lcost) also is constant and is of no
significance in determination of the components of field intensity.
Eliminating the latter and transposing the constant factors outside
the /sign of integration we obtain:
77
where the der0ative of PP(cos ) with respect to cos ' is designated
by P'n(cos The triangle OCB, in which OC is denoted by.L , yields:
Substituting the derived values for Po and dl in the function
within the integration sign, we obtain:
(0.41)
where 1 and 1 are the an&les at which the radii of the ends of the
solenoid are seen from point:0. From the statement:
i)
r (
(0.42)
(0.43)
The coefficients An are constant at the given dimensions of the
solenoid and vary changes in.the origin of coordinates. The series
(0.43) is absolutely converging only if r < a.
The components a and H of the intensity of the magnetic field
x y
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of the solenoid are determined by the formulas:
a~ eye
Substituting tie values of U from the expression (0.41) we
t/
r?
Replacing P'n (cos 0)
obtain for Hx the expression:
11
/r
ff
sin20 with value from equation (0.23), we
Since the origin of coordinates can be taken at any point. provided r
is less than the radius of the solenoid a, it is most convenient to
place it at the intersection of the axis of the solenoid with a per
pendicular from point P (Figure 9).
In this case 0 : 90, cos 0 = 0 and r = y. But at cos 0 = 0
all the polynomials P. having odd powers become zero, while the even
powers become constant numbers. The derivatives of P',(0) polyno
mials will be zero at even powers and
constants at odd powers, hence
( r r) ,
F;r
Substituting the value_of P2.(0) obtained from equation (0.24), we
~ r
(a. %r)
These formulas hold for every point withinttbirradius of the
solenoid. Thus the coefficients A2n and A2n4l arefunctions of the
distance of the origin of coordinates from the center of the solenoid.
 43,
have:
J )
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1
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For points located on the solenoid axis for which y = 0,
the formulas assume the form:
1
HY  2nIw/L Al; Hy s 0.
To determine the An coeff#leisntsi3it is necessary only to
insert the values of the Legendre polynomial derivatives in ex
pression (0.42) and to integrate, which gives:
It is seen from Figure 6 that the cosines and sines of the angles
are defined by the formulas:
/v T('/ C'
(0.46)
where 1 is half the length of the solenoid, and x the distance from
the, center of the solenoid, to point p.
Consequently, for points located on the solenoid axis:
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For the central point, where x = 0,
2 2
H 4nIw/L ? i/~{a + 1
(0.48)
and at 1, such greater than a:
H = 4%Iw/L [i_1/2(a/1)2+3/8(a/1)4?f ... ].
0
For the point located at the end of the solenoid, where x=l:
H1: 21rIw/L ? 21/11 a2 + 412
H1= 29Iw/L [11/2(2/21) 2 + 3/8(a/21) 4 ...],
i.e., the intensity of the field at the end of the solenoid is almost
onehalf the intensity at the center.
The nature of variations of field intensity inside the solenoid,
along its axis, is indicated in Table 3.
Distance from center in parts
of the solenoid length 0 0.01 0.02 0.05 0.1 0.5
Intensity of the field 1 0.99995 0.99981 0.99875 0.99503 0.98058
Section 9. magnetic Fiild of a Adultilayer Solenoid (Coil)
The magnetic field'of a multilayer solenoid, consisting of w2
layers with w1 turnsin each layer, of total lengthL, and having a
depth of layers D, is determined analogously to the case of a single
layer solenoid, by integration of the potential of an infinitely thin
wire, the current intensity in which is
dI = IW1w h/LD dLdD, _
A
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Taking 01M = a and 001 = x, (Figure 7) we have:
(L.,
Substituting Po, dD and dL with their values, we obtain:
^~' / l !
where a is the internal radius of the solenoid and b the external radius.
It must be remembered that the series converge on the condition
that point P. for which potential is sought, is located at a distance
from the solenoid axis, less than the internal radius of the solenoid
a. Inserting the former conditional values (0.42), we obtain as the
first integral: 
I'
A, b
Since the integral
a
Anda does not depend on the coordinates
of point p, on denoting this integral by Bn we
obtain the following
expressions for the components of the intensity of the magnetic field,
which are analogous to the expression (0.44) for a singlelayer
solenoid:
r:
6'. f
(0.49)
(0.50)
The coefficients BU may be found by simple integration of ex
pressions (0.42) in which sin f` and cos are replaced by their values
determined from the equation (0.46), i.
KAI
4z x IJ c
it is appa``snt the determination o the in egrais reduces it
self to an integration of irrational functions of the form;
'L ,!.
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where f(x, y) is a rational function of the variables x and y. Such
functions are integrated by substitution of ) a2 + c2 = za, where z
is a new variable.
easily integrated.
In accordance with expression (0.49) the field intensity of the axis of
the solenoid, where y = 0 is:
n n
where I is the current intensify amperes and Bl is the function
defined by equation (0.51).
The formulas derived above establish the correlation between
the intensity of the magnetic field and the dimension of the coils,
their shape and the current intensity, and consequently. enable deter
urination of the intensity of the magnetic field from measurement of
the geometric dimensions of the coil and current intensity.
Section 10. Magnetization of Geological Rock
All rocks possess magnetic properties by which they may be
classified according to 3 types: diamagnetic, paramagnetic and
ferromagnetic.
The rational functions which remain after the substitution are
Thus for example, for B1 the substitution yields:
Among the elements of thu periodic table of D. I. Msndeleyev
the ferromagnetic elements include iron, nickel, cobalt and gadolin
. The other elements are either diamagnetic or paramagnetic.
L y
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The ferromagnetic rocks of greatest interest and importance
since the local characteristics of the magnetic field of the earth
are associated with the preseii!e of ferromagnetic rocks within the
earth's crust.
Ferromagnetism, according to the current concepts developed
by the Soviet physicists Ya. G. Dorfman, Ya. I. Frankel, N. S. Aku
lov, S. V. Vonsovskiy, Ye. I.Kondorskiy and others, is brought
about by orientation of the natural magnetic moments u (spins) of
electrons in a single direction. However, such an orientation does
not occur throughout the entire volume of a given body but only in
9 3
a small portion of the order of 10 cm , called the region of spon
taneous magnetization, or domain. Each doman is oriented in such a
manner that in combination the total magnetization is equal to zero
and in its natural state the rock is "neutral", i.e., is'"bot magnetized.
Under the influence of a magnetic field a redistribution of the
dimensions of the individual domains and also in the direction of
their magnetization occurs and the rock becomes magnetized in a single
direction parallel to the field.
Without considering the process of this redistribution, which
explains the phenomenon of magnetization of ferromagnetic materials,
only the formal aspects of this problem which enable determination of
the quantitative correlations between the magnetizing field and the
magnetization of the rock will be discussed.
As previously stated, the theory of ferromagnetism is based
on the experimental fact of the presence in the electron of a natural
magnetic moment u which, as does the charge of the electron, consti
tutes elemental magnetic moment.
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The geometric sun of all the elemental moments within a
given rock is called the magnetic moment Y of the rock and char
acterizesthe degree of magnetization of the rock. Considerably
separate small portions of the rock volume, in every such part dv
the geometric sum of elemental magnetic moments, with an equal
value dv, can differ in, magnitude as well as in direction. indi
cating the magnetic moment of this volume by dM, the ratio of the
latter to the volume dv is called the intensity of magnetization
or simply the magnetization of the rock, and is denoted by the
letter J, i.e.,
J = dM/dv. (0.52)
As is apparent, magnetization is a vector which coincides
in direction with the vector dM and characterizes the magnetic state
of,the rock ataall of its points.
In accordance with equation (0.52) the magnetic moment M of
a rock is expressed:
Y' fJdv,
where the integration extends to the entire volume of the rock.
Magnetization may be distributed within a rock according to
any lain, A=vtded the condition that the flow of vector J through
its entire surface is equal to zero is satisfied, i.e.,
(J, ds) = 0. (0.53)
If the magnetization J is identical in magnitude and direction
at all points of a rock, the magnetization of the rock is termed uni
form. It is clear that in the case of a uniformly nagnitized rock
its magnetic moment is equal to the produce of the magnetization and
the volume, i.e.,
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in the CoSJ~ system the unit of mag4tization is the magneti
zation of a body of 1 em3 volume having a magnetic moment equal to
unity. The unit of magnetization has no specific name.
As stated above, under the influence of a magnetic field the
rock acquires a magnetic moment, i.e., its magnetization J becomes
different from zero. With increasing intensity of the field H the
magnitude of J also increases, but only to a certain magnitude which
is called saturation and is denoted by the symbol J . Consequently,
each value of H at any point of the rock has a corresponding value J.
The ratio of the quantity J to the quantity H is called mag
netic susceptibility and is denoted by the letter N, i.e.,
N = J/H.
In paramagnetic and diamagnetic rocks the quantity N does not
depend upon J and is constant. It characterizes the magnetic pro
perties of these rocks and is called the parameter of the rock.
In the case of ferromagnetic substances the susceptibility N
is a complex function not only of J but also of the previous history
of the magnetic state of the ferromagnetic substance, therefore its
magnetic properties of ferromagnetic substances only on the condition
that the magnitude of J to which it relates and. its magnetic state prior
to magnetization is specified.
The magnitude of specific susceptibility X, i.e., of the sus
ceptibility relating to a unit of dep4ity, ordinarily is used in
measuring X for the characterization of pars and diamagnetic substances.
X  x/d,
where d is the density of the substance.
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Section 11. Magnetic Potential of a Magnetized Body
Since every ferromagnetic substance consists of domains of
small dimensions the volumes of which may be considered infinitely
small and may be regarded as elemental magnets, i.e., as dipoles,
the magnetic potential dU of each of which is expressed as:
dU = (d ?* , r)/r3
where d f is the magnetic moment of the domain.
Since df J dv where dv is the volume, it follows that
DU (J, r)/r3 dv,
dU = (J grad 1/r) dv.
Consequently, the potential U of the entire body at point P
Applying the familiar vector analysis formula for the diver
genco of the vectorscalar product to the expression within the inte
gration, we obtain:
U div(J/r) dv  f div J/r dv.
The first integral is converted to a surface integral according
to the formula of OrtrogradskiyGauss, and therefore:
(Figure 8) is:
U =  j (J grad 1/r) dv (0.54)
the integration being carried out for the entire volume of the given
body, and the gradient taken for coordinates of point P.
If the gradient of function 1/r is formulated in coordinates
of a point Q, it is known t4 t
grad, 1/r = grad 1/r,
P
and the preceding expression becomes:
U = ( (J gradQ 1/r) dv.
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I
U = Jds/r  ) div J/r dv.
S v
(0.55)
The first integral applies to the entire surface of the mag
netized body, and the second to its entire volume.
The derived expression is entirely analogous to the expression
of the potential of electric charges distributed on the surface with
a density C and within the body with a density p, assuming that ficti
tious magnetic charges are distributed on the surface of the body with a
density
= Ju
and within the body, with a volume density
p = div J
(0.56)
(0.57)
Expression (0.55) holds for any point of space, both within and outside
the body.
Assuming vector J is constant in equation (0.54), i.e.,
assuming the body is uniformly magnetized, this equation is converted
as follows:
U : (JJ gradp 1/r dv).
Since the gradient operation is carried out with respect to
the coordinates of point P, and integration is carried out with
respect to the4coordinates of point Q, the sequence of the gradient
and integration operations may be reversed, and we have:
U : (J grad ( dv/r). (0.5$)
Introducing the expression
V : J dr/r,
we obtain a simple expression for the magnetic potential:
91
U . (J grad V), (0.59)
where the quantity "V is proportional to the gravitational potential by
the magnetized body on the assumption that the density throughout the
body is equal to unity.
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Thus, with the opposite sign, the magnetic potential of a
uniformly magnetized body constitutes the scalar product of mag
netization J and the gradient of the potential of the gravitational
mass of the magnetized body, assuming its density is equal to unity.
The equation (0.5$) is called the', theorem of Poisson, and it enables
determination of the magnetic potential of uniformly magnetized
bodies having a constant density, provided the gravitational potential
of a body of the same shape and dimensions is known.
Another oxpression for a uniformly magnetized body is derived
from equation (0.55).
(0.60)
To determine the magnetic potential by means of this formula
it is necessary to know the surface distribution of the normal compo
nent of the vector of magnetization. The shape of the magnetic body
determines which of the 2 formulas (0.59) and (0.60) should be used
in calculating the magnetic potential. For certain shapes, such as
a hares and ellipsoids, it is more convenient to use formula (0.59).
Since div J = 0, we have:
U = 1 Jn/r dS.
Because for other shapes such as prisms or cylinders the gravitational
potentials are known, equation (0.60) is indicated.
In the following sectionthe field of a uniformly magnetized
sphere, cylinder and ellipsoid, will be considered as examples of
great importance in terrestrial magnetism using the formulas derived
above.
:Section 12. I gnetie Potential of a Uniformly liagne.tized Sphere
The gravitational potential of a sphere V at an external point
locatedd at distance R from the center of the sphere, is:
V : v/R.
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Therefore the magnetic potential at the same point will be:
U  v (JR)/R3 or U  (MR)/R3
i.e., the magnetic potential of a uniformly magnetized sphere with
in its external space is equivalent to the potential of a dipole.
The potential inside the sphere at a distance R1 from its
center is determined by dividing the sphere into 2 hemispherical
surfaces of radius R1.
The magnetic potential U at a point located on this spherical
surface will be the sum of potential Ul produced by a sphere of
radius R1, and the potenthl U2 of the spherical layer.
In view of the above the first mentioned potential is expressed
by the equation
U1 a r/3 A R13/R13(JR1) = 4/3 n (JR1)
Gravitational potential within the spherical layer is a constant
quantity and its gradient is equal to zero, hence:
U2 = 0
Consequently the potential inside the sphere is:
U = Ul = 4/3 x (JRl),
and the intensity of the magnetic field inside the sphere is:
H  grad U = 4/3 x J
(0.61)
Thus, H is proportional to the magnetization J and has a direction
opposite to that of J.
The proportionality coefficient
N=4/3a
is called the coefficient of demagnetization.
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Section 13. Potential of a Uniforaly lfagsetized linder
Assuming that a cylinder can be uniformly magnetized along.
its axis, then the normal component Jn of the vector of magnetiza
tion j must be the same at both end surfaces and equal to the vector
J. Therefore to determine the magnetic potential it is more conven
ient to use equation (0.60) which for the external point P is:
U = J j dS/r1 ?. J dS/r2,
where the first integral extends to one end of the surface,the second
to the other.
In the general case, i.e., for any point in the space, the
integrals cannot be resolved in the simplest functions and therefore
discussion will be limited to the points located on the axis of the
cylinder. Indicating the radius of the cylinder by a, its length
by the distance of point P from the nearest surface S1 by R, and
the distance of the surface element dS from its center 0 by P, the
potential of the first end surface s is :
20 a ......~rte
U1: J 3 A pdedp/1 R2 + p2 = 2n J( / R2 ?4. &2 R) .
0 0
The potential of the second end surface is indicated by a similar
expression in which R +x appears in lieu of R, Hence the external
potential is:
U = 2rJ [ R2 + a2. j (H +1 )1 + a'' + Z).
For the point P (internal point) the potential is:
(0.62)
U : 2itJ [' .'. ?I (1 + R)2 + a2 ?. 2R). (0.63)
Section 14. lagnetic Potential of an Ellipsoid
The magnetic potential. of an ellipsoid is determined in ac
cordance with the theorem of Poisson. Since'rits gravitational
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potential is known and is expressed for a point p located outside
of the ellipsoid having coordinates x, y, z by the formula
where a, b, c are the halfaxes of the ollipsoid,~ (o)
'(b2 +0)(c2 +Q), and n is the root of the equation:
x2/a2 + n + y2/b2 + n + Z + n = 1.
(0.64)
First, the magnetic potential on the surface of the ellipsoid
is determined. in this case the root of equation (0.64) is n : 0 and
therefore the gravitation potential is
the form: T
This expression may be written in
Y 1 4 _J
where L, M, N and are constants and are expressed by definite
elliptic integrals
tS
/ ,? i~ J~rt .?` ,. C7 ~'! r ! r c. jr (71 (~~ 1 1 0, E:
Q \ ii
r
consequently the magnetic potential on the surface of the ellipsoid
.is of the form:
U : JXLX + Jy + JZWZ' (0.66)
and the products Lx, L1 and N. represent the corponents of the gravita
tional attraction force of the ellipsoid.
The magnetic potential of the ellipsoid at an external point
is, according to the 'theorem of Poisson,
U Jxfx + Jyty + JZfZ,
(a2+ 0)
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where fx, fy, fZ also are the components of the gravitational potential
at the external point P. To determine these forces an ellipsoid co
focal with the given ellipsoid is constructed through point P so that
their axes coincide. Then acending to the theorem of Maclaurin,
assuming both ellipsoids are of the same density the attraction of
such an. ellipsoid is greater by as many times, as its volume is
greater than that of the given ellipsoid, i.e.,
f'x/fx = f'y/fy = f'z/fz = alblcl/abc
where aI b1 c1 are the half axes and f*x, f'y, f'z the component forces
of attraction of the cofocal ellipsoid.
Since point P is located on the surface of the cofocal
ellipsoid, it follows that according to formula (0.66):
f'x = L1 x; fly = M1 y and f'z = N1 z.
where LI, M land N1 are constant quantities defined by formulas (0.65),
in which the half axes a, b, c are replaced by half axes al, bl, c1.
Consequently,
fx = abc/a 1 b 1 c 1 L 1 x, fy = abc/a 1 b 1 c 1 M y, fz = abc/abc N Z.
I 1 1 1 1
U e = abc/alblcl(JLIx , J Nly + JZNiz) (0.67)
The half axes al, bi, cl are determined on the assumption that the 2
ellipsoids are cofocal and that the point P is located on the surface
of the ellipsoid having the axes al, bl, cl. The first condition gives: 2 al  b2 a a2  b2 = q2; a2  c2 = a2  c2 = q2 (0.6$)
1 1 1 2
and the second:
x2 2 Z2
2 +' + 2 = 1? (0.69)
a1 b1 cl
These 3 equations enable the desired determination of the half
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Section 15. liagnetizatioa of Rocks in a Unifora Yapetic Field
magnetization of pare and ferromagnetic bodies in a uni
form field is of great importance in the theory of magnetic pros
pecting since magnetic anomalies are caused by rocks occurring at
relatively shallow depth in the form of individual masses of dif
ferent shapes and possessing the property of being magnetized under
the influence of the magnetic field of the earth, which because of
the relatively small volume of these rocks may be regarded as uni
form. T erefore, the magnetization of a rock of susceptibility 11
and lted within the uniform magnetic field of the earth will be
discussed below.
The magnetic potential U at a point p within the rock is
equal to the sum of the potential of the external field Ue, and
the Potential Ui, produced by the rock itself, i.e.,
U=Ui +U3.
Hence the field intensity H at point P is:
H  grad (Ue + Ui) * Hs + Hi.
where Be is the intensity of the external magnetizing field, and Hi
the intensity of the internal field.
On the other band, the magnetization J is expressed by the
equation:
J  xHs .,. xHi .
(0.70)
consequently, magnetization due to the action of the external
uniform field depends not only on the intensity of this field but
also upon the field produced by the nagnitized body itself. The
quantity Hi is the intensity of this field and, as is shown by theory
and experiment involvinguniform magnetization, this quantity always.'
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negative, i.e., Hi has a direction opposite to the external field,
and therefore is designated the intensity of the internal demagni
tizing field.
The internal demagnetizing field is nothing but the field
in which the magnet is located and which is produced by the magnet
itself. Indeed from Figure 9, which shows the lines of force aromd
the magnet, it is clearly apparent that near the surface of the
sample the direction of the lines of force is opposite to their di
rection inside the magnet, and their density is inversely propor
r
tioaal to the length of a magnet of constant diameter. The p cture
is exactly the same as if this flux of the lines of force represented
a magnetic field produced by external sources and the magnet were
places in this field. It is clear that such a field will produce a
demagnetizing action, i.e., will reduce the magnetization.
In the general case, at different' points within the rock Hi
may have different directions in relation to the magnetizing field
met with the result that the magnetization J' itself:;`'iay have dif
ferent directions, which correspond to nonuniform magnetization.
Since Ni is a function of magnetization J, determination of J
requires the expression of Hi in equation (0.70) in terms of J, and
this equation is solved for J.
The expression for Ni is given in Section 11, however it can
not be utilized to determine J it the general case. Therefore,
several particular cases are discussed below.
Assuming' that the rock is magnetized uniformly; than in view
of equation (0.59) we have:
 59 
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J = x[S, + grad (J grad V)J.
The components of jkvAgaetization vector J are determined
from the equation
3
Jk = x [Hek NkiJi],
where the indices k and i denote one of the coordinates x, y, x,
and Nki denotes the partial derivatives of V of second order on the
corresponding coordinates. The values of Nki are the components of
a certain tensor N. which is called the demagnetization tensor. Since
Nxy = Nyx, Nxz= NZx and NNZ= Nzy the tensor N is symmetrical.
In order that J = coast, the components of tensor N must be
constant, and not dependent upon the position of point P, The equa
tions show that at uniform magnetization of rocks of arbitrary shape
the magnetization vector does not coicide with the vector of the mag
netizing field Ne, but forms a certain angle with the latter, which
depends upon the shape of the rock stratum.
in the case of a sphere the magnetization is determined in
accordance with equations (0.61) and (0.70) by the following relation
ship:
J = xRe  4/3rxJ,
which gives
J = x/l + 4/3ux Nev
(0.71)
(0.72)
(0.73)
i.e., in a sphere magnetization coincides with the direction of the
magnetizing field. From the saw equations it follows that in a uni
form field the sphere is magnetized uniformly, siege the assumption
of uniform magnetization of a sphere,, made in the derivation of its
potential is not contrary to equation (0.73), and from which it also
follows that j = coast.
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Is addition to the sphere, ellipsoids having any ratio of
axes, which also possess the property of uniform magnetization, is
readily shown below.
In accordance with the derivations of the preceding sectjon
the magnetic potential inside an ellipsoid is of the form:
UJN xJN y+JN x.
X xx y yY z zz
Denoting by K a constant vector of components
JXNxx*jyNyy,JZNZZ,
the magnetization of an ellipsoid by the action of a uniform field
He is expressed by the equation
J  xHs  x grad (K, r),
or, since K = coast,
xH  xK.
e
Since He and K are constant quantities, the vector J also is
a constant quantity, i.e., the ellipsoid actually is uniformly mag
netized by the action of a uniform field, but the vector J does not
coincide with vector N.. Its components on the coordinate axes are:
ix = x/1 + xNxx Hx, Jy = x/l + XNyy NO Jz  x/1 + xNxz Hz.
The coefficients N'xx, Nyy, Nzz are components of the ellip
soid demagnetization tensor, which are expressed by formula (0.65)
L = Nxx' H = N yy and N  Nzz.
If any 2 axes are equal, i.e., the ellipsoid is an ellipsoid
of revolution, the coefficients of demagnetization may be expressed
in the sisplest functions.
soid in which a = b R + Z S Z/1, 000
Characteristic In this equation
which has been named the numerical
g and Z represent the mean ^on'thly values of the vertical and horizontal
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components, while S11 and %Z are the differences between the maxi
mum and the minimum values of those elements during the given inter
val of time (hour, day, etc), expressed in gammas.
Although they are objective, the numerical characteristics
still cannot provide a correct idea concerning magnetic activity
during any given length of time since 6H and 4 Z are taken as the
difference between maximum and minimum values. Those differences
can be small even though there may be substantial changes during
the interval and conversely, the differences can reach large values
although the changes during the interval may be of monotonous nature.
The intervals of time equal to the periods of random fluctuation of
the magnetic field constitute an exception, and may assume values
of several minutes or less. But even in these instances this evalua
tion method has little practical value.
In 1939 the International Association on Terrestrial Magne
tism and Electricity adopted the k index for evaluation of the
degree of magnetic disturbance and which at the present time is used
by the observatories of the Soviet Union. The k index represents the
numerical characteristic expressed in points of a scale. Each point
of"the scale corresponds to the amplitude of oscillations of the mag
netic elements over a 3hour interval, corrected for the undisturbed
diurnal progression. The k index scale is made different for each
observatory in order to eliminate the effect of latitude, since at
any given activity of the sun the degree of disturbance differs at
different latitudes.
A very conveniem~+ characteristic of the degree of disturbance
is the length of the curve during a given interval' of tine  of the
magnetogram recording of a given element. Such a characteristic is
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utilized by A. P. Nikol'skiy in his study of variations at high?ldti
tudes, in which he measured the length of the curve on recordings of
the declination variometor. Although this length does not express
any physical quantity, it reproduces very closely the nature of the
state of disturbance. Its only flaw is the fact that it does not
permit differentiation of monotonous from rapid periodical changes
of elements, but it does reveal any change of the elements over any
interval of time.
The above measurements of magnetic activity are convenient
for characterization of a short intervalof time (not exceeding 24
hours) but are of little use in characterization of large intervals
such as a month or a year. Therefore in 1932 a specific standard of
activity u [431 was proposed, which constitutes the mean magnitude
during a month or a year of differences in the successive mean daily
absolute values of the horizontal component at or near the magnetic
equator. This standard was selected on the basis of the following
observed fact. During strong magnetic disturbances the horizontal
component undergoes similar changes over the entire terrestrial globe.
During the initial period of the magnetic storm the horizontal compo
nent decreases everywhere, and thereafter it returns slowly to its
nmtl state. This measurement standard is expressed in units of1074 Oe_.
In addition to standard u, standard u1 also is used, and is
correlated with u by means of the following table:
u  0.3 0.5 0.7 0.$ 1.2 1.5 1.8 2.1 2.7 3.6 and higher
ul= 0 20 40 57 79 96? p108 118 132 140
The state of magnetic disturbance is a quantity reflecting
definite phenomena in the course of magnetic variations and must in
elude regularities which follow from the regularities which govern
the phenomena of magnetic variations.
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However, these regularities may be revealed only through
statistical processing, since during every day and hour the state
of disturbance is of random nature and the extent of this distur
bance (magnetic activity) can assume different values during the
day, month or year, which do not conform to any regularities.
With a sufficiently large number of recurring activity values co
inciding with some definite point of time or phenomenon, the
meanof these values shows definite regularities. Thus, the mean
monthly activity values derived by processing of observational
data relating*to a period of several years show a clearly mani
fested yearly progression. The amplitude of this yearly.progres
sion increases with an increase in overall magnetic activity, as
is apparent from Figure 64 which shows the yearly,arogression of
magnetic activity derived by statistical processing of observational
data covering a period of 59 years (from 1872 to 1930). In the pro
cessing all the data were subdivided into 3 groups corresponding to
years of strong disturbance and years of slight disturbance, and
the mean annual and mean monthly values of magnetic activity u were
calculated for each group.
The drawing shows that the activity has 2 welldefined maxima
coinciding with the equinoctial epochs and 2 minima coinciding with
the epochs of solstices. This may be interpreted to moan that in
the northern hemisphere magnetic disturbances are more likely to
occur during spring and autumn, and are least, likely to occur during
summer and winter. Therefore the cause of these regularities must
be connected with phenomena associated with the relative position of
the plane of the? terrestrial equator and the plane of the ecliptic.
Such ph?nomena include the occurrence of sunspots which appear
mostly within zones 100 to 300 of northern and southern heliographic
205
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latitude taken from the plane of the ecliptic, and their maximum
number occur from 100 to 150 latitude. Therefore, during equinoctial
epochs when the planes of the terrestrial and solar equators are in
alignment, the surface of the earth is subjected to the greatest
action of radiation emitted by the spots. Consequently, the
cause of maximal activity during the equinoctial epochs appears
most likely to be due to the maximum radiation emitted by the sun
spots during this period.
The relationship between sunspots and magnetic activity mani
fests itself most characteristically on comparison of the graphs.of
solar activity and mean annual magnetic activity over a prolonged
interval of time. Solar activity usually is taken to mean the sum
of the number of sunspots f and of the 10fold number of groups g
of these spots. This standard of measurement of activity is called
Wolf's number W and is defined by the formula
W = f + 10 g.
Figure 65 shows graphs of solar and magnetic activity for
the interval 1830 to 1930. The curves of this drawing indicate
that during the years of sunspot maxima the magnetic activity also
is at a maximum, and exhibits an 11year period coinciding with the
maximum of sunspots. However the magnetic activity lags somewhat
behind the solar, so that on an average this lag amounts to one
year for an 11year period.
Furthermore there is a,recurrence of activity after 27 days,
corresponding to the period of revolution of the upper layers of
the'sun about the axis. This periodicity also is associated with
the occurrence of spots upon the sun in the following manner. So..
sunspots .persist during several pe`iiods of the sun's rotation and
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when the sun rotates 3600 they are found on the side which faces
the earth. If the spots did not disappear and remain unchanged
there would be a strict periodicity in their appearance every 27
days. But since the spots appear and vanish at irregular inter
vals, there is only a certain tendency toward the 27day periodi
city of their appearance. M. S. Eygenson [44], on investigating
r
the correlation between the duration of groups of sunspots and the
phase of the 11year cycle, found that on the basis of data of the
Greenwich Observatory for the perriod 1870 to l)32, only 20 or 30
out of 100 sunspots persist longer than one revolution oftthe sun,
with most of these persisting for 2 revolutions, although excep
tional spots persist for 7 revolutions of the sun. The spots
which appear during the years of maximum occurrence exhibit unstable
duration. Observations show that magnetic activity similarly ex
hibits only a tendency to recur. This tendency is illustrated by
the graph of Kri (transliterated) (Figure 66), which shows the pro
gression of the activity during 35 days following, and 5 days prior
to a maximum. The graph shows that a second, less pronounced peak
occurs 27 or 28 days after the first peak.
The Kri graph is derived in the following manner. Over the
periods 1906 to 1911 and 1890 to 1900 a selection was made of the
days having the characteristic "2," and the values of activity
during the 35 subsequent and the 5 preceding days were written in
series, after which each column of these series was summated and
the mean values of the activity were determined, which were then
plotted on the graph.
This method later was applied to more extensive dataon the
period from 1906 to 1924. In this case 108 days were taken, right
and left, instead of,the previous practice of taking 27 days from the
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beginning of the disturbance. it was found that the state of
disturbance recurred 4 times at 27day intervals up to the one
hundredth and eighth day. A detailed investigation of this
phenomenon was carried out by N. P. Ben'kova on the basis of data
of the Pavlovak observatory. N. P. Ben'kova [45], made a list of
the sequences of storms observed at Pavlovak, assuming that the'
storms are part of a sequence if the interjal between them is 26
to 28 days. As a result it was found that of the 1,073 storms,
576 werepart of a sequence, and of these 198 storms were part of
2 simple recurrences, 129 of 3., 72 of 4, 65 of 5, 36' of 6, 49 of 7,
8 of 8, and 20 were partof 10 recurrences.
In order to determine whether these recurrences are acci
dental or have a physical significance N. P. Ben'kova calculated
the probability of an nfold accidental recurrence of the storms
and compared it with her empirical probability results.
This comparison is presented in Table 17 and shows that with
the exception of the value z, the empirical probability greatly ex
ceeds the random distribution probability which indicates a definite
regularity in connection with this phenomenon.
TABLE 17
PROBABILITY OF nFOLD RECURRgJICE OF MG1I TIC STORMS
Probability
Number of Recurrences of StqVms
2
3 4 5 6
7
8
9
10
Empirical
0.370
0.130 0.100 0.053 0.026
0.014
0,006,,
0.003
0.001
Random
0.260
0.069 0.018 0.05 0.01
34 105
9 105
2 105
8 ? 106
However, a rigorous functional correlation still was not as
certained between magnetic activity, or disturbance state, and the susts
activity. It is of interest to note that the duration of sunspots over
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the period 1879 to 1932 did not exceed the 7fold period of sun
rotation, while the state of magnetic disturbance can recur at
periods up to 10 solar cycles.
In addition to the above regularities, there is a diurnal
progression within the course of magnetic activity. Figure 67 shows
the progression of this activity over 24 hours, at Kew observatory
near London, during winter, summer and the equinoctial opochs on the
basis of the. international 012 scale, and also the number of hours
having the characteristic "2" during the period 1913 to 1923. It is
seen that the minimum of activity occurs at 1,000 to 1,100 hours and
the maximum at midnight. This indicates that increased activity
during noctwditnal hours cannot be caused by wave radiation from the
sun, and it is more probable to assume an influence of corpuscular
radiation, which under the action of the magnetic field of the earth
can penetrate the atmosphere from the nocturnal side.
~209
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CHAPTER VI
MOXETIC VARIATIONS AND THE AURORA BOREALIS
Section 1. The SolarDiurnal Variations
The 'solardiurnal variations, which we shall denote by the
letter S, consist of periodic variations of the elements of ter
restrial magnetism with a period equal to the length of the solar
day.
A characteristic feature of these variations is their oc
currence according to local time. For this reason, at 2 different
longitudes, the phases of the fluctuations of one element or another
will differ by the difference in longitudes between the 2 points.
Thus, if we represent the deviation from its mean value, I. e., the
variation, of any element at a given point of the earth's surface
in the form of a simple harmonic oscillation:
S = So sin 2T ,
where So is the amplitude, T are the solar days and t the local time,
then the variation at another point whose longitude differs by
is represented by the equation:
S' = So sin ?T' #7~+
Figure 68 represents the mean annual diurnal march of the
declination, the horizontal and vertical components, i. e., the re
lation of the variations of these elements and the local time during
a 24hour period, according to the observations of the Pavlovsk mag
netic observatory.
The mean annual values of the variations of the various
elements respectively are plotted along the axis of ordinates and
the local time along the axis of abscissas.
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On considering the curve for the variations of the declina
tion, it will be seen that the magnetic needle which remains quiet at
night (curve 6 D), is deflected in the morning towards the east, and
by 0800 hours reaches the maximum deflection, after which it begins
to move in the opposite sense, and by 1400 hours it reaches its
maximum deviation towards the west. The remaining curves show that
the horizontal component (the curvo U H) has a minimum at about 1100
hours and a maximum about 2000 hours, while the vertical component,
which remains almost unchanged during the night, begins in the
morning to increase, and after noon reaches its maximum value. As
shown by observations, the diurnal march of the elements of ter
restrial magnetism does not remain constant but varies irregularly
from one day to another; in this case the amplitude of fluctuations
are mainly subject to change, while the phases themselves remain al
most unchanged.
Table 18 gives the values of the differences between the
maximum and minimum values of each element at different seasons
at Pavlovsk.
TABLE 18
Difference between the maximum and minimum
D
$
Z
Winter
4.1'
09
7~e
Spring
7.9
27
12
Sumer
12.0
44
20
Autumn'
9.2
36
12
The Table shows that the variations in the diurnal march
increase from the winter months, when the declination of the sun
is smallest (to the summer months, when the declination of the Sun
is greatest (+ 23.50).
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The next feature of the solardiurnal variations is their
dependence on the value of the magnetic activity on one day or
another. For this reason 2 forms of solar diurnal variations are
distinguished: The variations on quiet days when which are obtained
by working up the observations only for quiet days, and variations
in stormy days, which are called disturbed variations and are ob
tained by working up the observations on stormy days. The former
are denoted by Sd, the latter by S
q
Variations on stormy days differ markedly from the variations
on quiet days. This difference particularly effects the march of
the variations of the vertical component, where not only the ampli
tude but the whole character of the curve changes. In addition, the
amplitudes of the quiet diurnal variations Sq vary during the course
of the year, taking their maximum value during the summer solstice
and their minimum values during the winter solstice. During the epoch
of the equinoxes, the amplitude is the mean between the winter and
summer and is the same in both hemispheres.
Moreover, observations show that the solar diurnal variation
at various points of the earth's surface is of different character.
For points located on one and. the same parallel, however, the diurnal
march is almost the same, but for points located along a meridian,
it varies according to a certain definite law. Figure 69 represents
the curves of the diurnal march on quiet days at different latitudes
,for the 3 elements X, T and Z during the period of the summer solstice
in the northern hemisphere (Figure 69, a),?for the winter solstice
(Figure 69, b), and for the epoch of the equinox (Figure 69, c)5 both
spring and autumn. These curves show that the variations of the
northern component E have approximately the same character in northern
and southern latitudes, since for the Individual elements in the southern
latitudes, they are inverted on passage across the magnetic equator,
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I, e., they are mirror pictures of the variations in the northern
latitudes. On the equator itself, the variations on the eastern
and vertical components are close to'zero. For the northern com
ponent, such a reversibility of the curves takes place at magnetic
latitude of about 300 in both southern and northern hemispheres.
The variation of the amplitudes during the course of the
year will be clearly seen by comparing Figures 69, at 69, b, and
69, c.
Figure 70 gives analogous curves of the diurnal march of
the variations on stormy days Sd for various latitudes. The ordi
nates of, these curves represent the difference between the variations
on stormy days and variations on quiet days (Sd  Sq). These curves
show that in the low latitudes the differences Sd  Sq are small and,
consequently, the Sq variations predominate in them, while in the
high latitudes, on the contrary, the variations SD = Sd  Sq, play
a predominant part.
The most striking idea of the march of the diurnal variations
given by the construction of the socalled vector diagrams which
represent the projection of the vector variations Sq on the horizontal
and vertical planes.
Such a diagram of the projections of Sq of. the horizontal,
plane is shown in Figure 71 in the period of the equinox for a por
tion of.the earth's surface turned towards the sun and bounded by
the geographical latitudes from +60 to 60? and the longitudes from
6 to 18 hours. In this case, the longitudes coincide with the local
time from 6 to 18 hours.
The following may be noted from this diagram: the vector Sq
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in the northern hemisphere during the daylight hours is always
directed towards a certain center located on the forenoon meridian
at the parallel +300, while in the southern hemisphere it is directed
from a center located on the same meridian and the parallel 300.
Figure 72, a and 72, b show the diagrams of projections of
the vector 3q on the vertical planes: one of them on the plane of
the principal meridian, and the other on the plane of a great circle
making contact with the 30th parallel at the meridian point. Both
diagrams show that the ceeers towards which the Sq vectors are
directed lie above the earth's surface roughly above the parallels
+30 and 30 near the principal meridian.
Formally, these centers may be identified with the axis
of the eddy current whose sense is counterclockwise, viewed from
above, in the northern hemisphere, and clockwise in the southern
hemisphere. Thus the diurnal variations any be explained by the
existence in the atmosphere of a system of closed eddy currents
which remain fixed in space, and within which the earth rotates.
Since the maximum value of the vector of the variations come during
the daylight hours, the maximum current strength must be in the
space between the sun and the earth.
The system of electrical currents corresponding to the
fields of diurnal variations. The magnetic field of the solar
diurnal variations, 1. e., the field corresponding to the distribution
of the vector of variations must have its sources which most probably
may consist of a certain system of electrical currents which most
probably may be represented in the form of a certain system of
electrical currents. The general distribution of the vector of
variations on the earth's surface for a given moment of time,
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represented in Figure 72, a and 72, b, indicates that the system
of currents with their center at latitude 300 and on the principal
meridian, remains fixed in space between the sun and the earth,and
an observer on the earth, in rotating with respect to this system,
passes during the course of a 24hour day through all values of the
vector of field strength of these currents, distributed along the
parallel of the observer. Since the electric current and its mag
netic field are connected by the BiotSawra law, the, if we know
the field, the current may also be determined, provided that the
distances between the current and those points at which the field
is known are also known. For this reason to find the system of cur
rents according to an assigned distribution of a magnetic field, we
must start out from the law in question or from its consequences.
Such a consequence, which is the most convenient to calcu
late, is the equivalence of a closed current with a dual magnetic
layer. For this reason, by replacing the system of currents by a
dual magnetic layer of variable density, disposed o> a sphere con
centric with the earth's surface and having a radius R greater than
the earth's R radius, we find the magnetic potential U of such a
layer at the earth's surface. We know from the theory of potential
U =~ak 1)dS,
where)( is the surface density of the magnetic moment of a double
layer corresponding to the element of surface d0; p is the distance
between the point of the earth's surface where the potential U is
sought ; m ,is the direction of the normal to the surface of the sphere
coinciding Frith' ?the radius R. and the integration is taken over the
entire surface of the sphere.
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Consequently, 00
U=~(n+1) Rn
n+3
n=1 r k = 1 jjJakPn(cosY) dS.
It is known from the theory of spherical functions that
j'J)k Pn(cos Y )dS = 4nr2 2n + 1 n for k a n
?
JJi k Pn(cos y )dS = 0, for k= n.
For this reason 6 0
n + 1 R n
U = n = 1 2n + 1 ( r) n Je
On the other hand, the magnetic potential of the field of
variations, as we know, is expressedin the form of a series:
Since p2= Ra + ra 2Rr cos \1 , where y is the angle
between R and r, and r) R,thin
1 +n iP(cos)
n=p r h 0
and, consequently,
I. Rn
U u+I Pn (cos )dS,
a n n=1 4
R
n (n + 1) t P Pn (cosy W.
n i
It is proved in potential theory that every function of
two variables (the latitude 0 and longitude /\), assigned for the
points of a sphere, may be uniquely expanded into a series in
Laplace spherical conditions; and for this reason,/, may be
represented in the form of the sum:
k
~c s P k, w$erep!1/ .n k = 21, (an cos ml + bn sin m~~) P (cos (6.1)
k = 1
U = RZ Un, where Un : 2 (P*n cos m/t + qn sin s, )Pn (cos8 ), (6.3)
where the coefficients e and qs are known from observations and
n n
represent the external part of the potential.of variations.
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On comparing the expressions 6.2 and 6.3, we find that
RUB = 4x i 1 (11 ) n n,
2n + 1 r
R 2n + 1 r n
49  n+1 (R) Un,
and, by substituting n in equation 6.1 and replacing Un by its
value, we have
R 2n r%n
43T n + 1(R
(pm cos m11 + qn sin m/'1 ) en (cos 6) . (6.4)
0
The density of the magnetic moment fy of the double layer is
equivalent to the current strength of a closed circuit, and for this
reason the expression for the strength of the current at any point of
the surface of the sphere with coordinates 8 and ', will have the same
form as equation 6.4.
This equation shows that for the calculation of the current
strength at any point of a sphere it is sufficient if we merely
multiply each term of the expansion of the potential of order n by
the term containing one unknown quantity, the radius
of the external sphere through which the current flows. This radius
must be determined from any other considerations. In the first paper
devoted to the calculations of the currents in the atmosphere, this
radius was fortuitously taken at 100 km more than the earth's radius,
which corresponds very closely to presentday data on the height of
the conducting layer in which the existence of such currents is possible.
At the present time there is ground for holding that the system
of currents causing the solar diurnal variations of the magnetic field
is located in the E layer of the ionosphere, the height of which, ac
cording to observations on deflection of radio waves, varies between 100
and 120 km. For this reason, if we. assume the height of the Z layer as
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equal to 109 ka, as has been done in Ben'kova's paper and t is ex
pressed in thousand amperes, while Pn and a
n qn are expressed in gammas,
then equation 6.4 takes the form:
00 00
I 8,0707 (1,02)n Z a a
n n + 1 a : 0(pn cos a~+ qn sin m'A) Fl
P11(cos d).
It is this formula that will serve for the calculation of the
system of current. On determining from it the value of I for various
values of ~ and , and plotting these.values on a. map, we may draw a
series of isolines (lines of equal current strength), which will
represent current lines, I. e., the lines along which the current
flows. The difference between the values of the current strength on
two adjacent isolines gives us the value of the current flowing
between these lines.
Such a system of currents, correspondent to the solar diurnal
variations in the equinoctial period and for heights of 100 ka, is
shown in Figure 73, while Figure 74 shows the system of currents cor
responding to the summer solstice.
The closed curves showing the direction of the currents are
drawn in such a way that a current of 1,000 amp flows between two
adjacent curves. The currents flow in four main systems of circuits,
two northern and two southern. In this case, two systems of contours
are located on the lighted hemisphere and the two others on the night
hemisphere, the former being more intense.
The total current in the daylight circuit is equal to 62,000
amp during the time of the 'equinox and 89,000 amp during the tine of,
the solstices.
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As stated in Section 3, about a third of the field of varia
tions is due to internal causes, i. e., to currents flowing within
the earth. It may be assumed that these currents are caused by in
duction of the magnetic field of the external currents in some con
ducting layer of the earth or perhaps4a,the entire earth. If the
conductivity of the earth were known, thou the determination of the
system of currents would reduce down to the above mentioned operation,
analogous to the operation of calculating the external currents. But
we know nothing at all of the conductivity of the interior parts of
the'earth. We know only the conductivity of the upper stone envelope,
which is of the order of 106 ohm71 ca1 and the conductivity of the
oceans, of the order of 4 x 10"'2 ohm`l cm" 1.
For this reason there have been attempts to calculate the in
duction current and the magnetic field caused by then under the as
sumption that the conductivity of the earth is everywhere the same.
It was found that to make the calculated values of the varia
tion agree with the observed values, it would be necessary to assume
a conductivity of \'= 3.6 x 104 ohai 1 cm`l for the earth, that is,
somewhat less than sea water and more than the conductivity of the
upper layers. In addition,.it must also be recognized that the upper
layer is probably nonconducting down to a depth of 300 km.
The system of currents shown in Figure 73 and 74 has been
calculated by'equation 6:5 under the assumption that the solar diurnal
.variations are functions of, the geographical latitude at?a local time.
But a comparison. of the observed curves of the diurnal march of the Sq
variations with the curves calculated by equation 6.5 in which the
arguments are the geographical latitude and the local tine; does not
yield good agreement. For this reason the Sq variations were expanded
into spherical harmonics, taking,the geomagnetic latitude and the
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geomagnetic time as the Independent variables (Bibl. 16). The agree
ment between and observed values was now considerably better. Figure
75 shows diurnal march of the northern component at the Nuancayo Ob
servatory (South America, CC _ 12.00, Am 284.70), observed and calcu
lated by equation 6.5 and by formulas in which? and A are geomagnetic.
As will be seen the latter curve is in considerably better agreement
with the observed curve than the former one.
Figure 76 shows a system of currents corresponding to the ex
pansion in geomagnetic coordinates. Its principal 0ifferenoe from
system in Figure 73 is the asymmetry of the currents of the northern
and southern hemisphere; the current in the southern hemisphere is two
and a half times as great as that in the northern hemisphere. In ad
dition, their centers are also asymmetric with respect to the equator.
It is possible to explain this by the fact that the magnetic field is
symmetric with respect to the magnetic axis not the geographic axis,
which should lead, according to the dynamo theory, to a reduction of
current in the northern hemisphere and its weakening in the southern
hemisphere (sic). But there are no quantitative calculations.
For a greater approximation of the observed diurnal march to
the theoretical, N. P. Ben'kova (Bibs. 47) wades. spherical analysis
of the diurnal variations of Sq at 47 observatories, including some
beyond the Arctic Circle, allowing for the fact that the variation
.depends not only on the latitude but also on the longitude of the
place. In this case the longitude, latitude and time taken were all
geomagnetic. No one before Ben'kova has ever performed the expansion
with such a formulation of the question, since all of the investigators
had considered the diurnal variations to be independent of the longitude.
Since the diurnal analysis assumes the expansion of a function
depending only on two coordinates, Ben'kova assumed, in order to calculate
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the third coordinate, longitude, that the Sq variations are the sum
of two functions, one of which, Sql, depends on the latitude and
the local time t, while the second one, Sq2' depends on the longitude
A and the latitude, I. e.,
Sq : Sgl(I , t), + Sq2( ..&)'
(6.6)
and she performed the expansion separately for the functions Sq1 and
Sq2 .
The determination of the coefficients of the expansion was per
formed bythe method set forth in Section 3, by equation 5.4 and 5.8.
To eliminate the influence of the longitude in the expressions for Sql,
the means for the given latitude were taken instead of the coefficients
calculated for the separate observatory as the initial coefficients am
etc in equation 5.8.
The original material for the functions S q2 was the differences
between the mean values of the coefficients am and those calculated by
equation 5.4. As a result of this analysis, performed for a large
number of stations, it was found that the main part of the field of
variations is represented by the function Sal, I. e., it does not de
pend on the longitude.
The longitude function exists, but its influence is shown only
at low and middle latitudes, so that in the polar regions the intro
duction of longitude terms does not improve the agreement between the
calculated and observed values of the variation.
In accordance with the result of this analysis, Den'kova con
structed a system of points for the summer months (flay August) both,
for the functions Sql, and for the function Sq2. They are shown in
Figure 77 and Figure 78. A comparison with the current maps on?Figure
76 shows that they are in better agreement with the maps of Figure 73
221'
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than with the saps of Figure 76. The eddy of current located on
Rs
the daylight aids of the northern hemisphere has about the mane
form as in Figure 73, and is situated at the same latitude, but
Ben'kova found the intensity of this eddy to be greater than that
of the eddy in Figure 73. As for the intensity of the southern
eddy, located on the daylight side, it is, on the contrary, weaker
on Ben'kova's map than on the map In Figure 73. In addition, the
center of this eddy is at 12 hours 30 minutes, while on Figure 73
it is at 11 hours. Moreover, the night region of negative currents,
as will be seen from Figure 77, breaks down into two distinct eddies,
while in on the sap of Figure 73 it reduces to a single eddy. These
differences in Den'kova's opinion should be ascribed to the difference
in the initial data, frith which we must reconcile ourselves, since
the analysis of Figure 73 was based on the data of 21 observatories
located only in middle and southern latitudes, while that of Figure
76 had only 5 observatories, but Ben'kova uses materials of 47 obser
vatories.
The system of currents of the field Sq2 is represented as
shown for four instants of Greenwich time: 0, 6, 12, and 18 hours.
Figure 77 shows that in the equatorial latitudes at local
Noon there is a region of positive current.' Its development reaches
its saxiaua(20,000 amp) when the 300? meridian is close to the noon
meridian. This region almost disappears and is displaced towards
the north, when the local meridian lies on the Greenwich meridian.
There is a region of negative currents on the night side. They are
most distinct at 3 hours on the 3000 meridian; and their maximum value
in this Case is 21,000 amp.
N. P. Ben'kova in her analysis also separated tbAtthe part
of the field corresponding to internal causes and that part of the
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field (eddy) corresponding to vertical currents. But the latter prob
lean are still controversial, since the value of the vetical currents'
strength obtained by Ben'kova from the analysis is many times greater
than the value of the currents observed on the earth's surface.
Attempts have recently been made to directly prove the
existence in the ionosphere of the currents causing the diurnal
variations (Bibl. 48). On 17 March 1949, near the geomagnetic
equator (C 11? S, ,= 89? W) at 1120 hours and at 1720 hours local
time, two rockets with magnetic instruments were sent up. These
instruments automatically transmitted signals by radio, at definite
time intervals, of the value of the magnetic field strength. The
height of the rockets at these moments was determined by a radar
installation on the ground and by the signals emitted from the rockets.
Both rockets reached a height of over 100 km. The first rocket was
sent up at a moment when the current density, according to theory
(Figure 76), was maximum, and the second when the density was minimum.
The results of the worked up observations in Figures 79 a and 79 b.
The former relates to the observations at 1720 hours, the second to
observations at 1120 hours. The solid lines represent the variations
(decrease) of magnetic field strength with height, calculated under
the assumption that the earth is uniformly magnetized and that its
,magnetic moment corresponds to the first term of the Gauss expansion.
The dashed lines are the results of direct observations. The points
corresponding to the ascent of the rocket are marked by circles and
those to the descent by crosses.
The results of these experiments indicate that currents
actually do exist at a height of about 105 km, since figure 79b'has
a sharp variation of the magnetic field of?~se heights, while
Figure 79 as correspondent to the minimum of current such variations
are not observed and the experimental curve coincides over its entire
length with the theoretical.
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For our final conclusions from these experiments, however, it
would be necessary to perform a theoretical calculation of the
variation of the field due to currents, and to compare it to thi ob
served values, as well as to have a repeated observation of the same
nature.
The noncoincidence of the disagreement between the theoretical
and experimental curves near the earth'
surface may be explained by
the existence of an anomaly in the region of the discharge of the
Section 2. LunarDiurnal Variations
In addition to the variations connected with the positions of
the sun with respect to the earth's surface, there also exist varia
tions of periodic character connected with the position of Moon with
respect to the horizon. The period of these variations coincides with
the time interval between 2 successive crossings of the local meridian
by the moon, i. e., with the lunarhalfdays.
The lunar diurnal variations are found on working up the
records of magnetographs with respect to the lunar days. Since the
lunar days differ from the solar days by only 50 min, 28 sec (the
lunar days equal*24 hours, 50 min, 28 sec mean solar time) it follows
that to eliminate the lunar, that is to isolate the'lunar diurnal
variations there is no need to work up the magnetograms according to
lunar days by taking the ordinates for each lunar hours, but that it
is sufficient to use the data obtained in working them up according
to solar days and a rearranging of then according to lunar time.
This rearrangement consists in taking the lunar days as.equal
to 25 hours of solar time, and, for each hour, entering the values
of the ordinates from the tables prepared for the solardiurnal vari
ations. Their vales for the 25th hour are taken as equal.to the value
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of the first hour of the following day. In this way each successive
lunar day begins one hour later than the solar day. The beginning
of the lunar day, which is established by the Astronomical.Annual
Yearbook is taken as the moment of the passage of the moon through
the upper meridian (the instant of upper combination). The upper
lunar combination does not usually correspond to an even solar hour,
but owing to the large number of days being worked up, it is suf
ficient, without leading to large error, to take the next solar hour
as the beginning of the lunar day.
In view of the fact that the lunar day does not contain
exactly 25 hours, but is 9 min, 32 sec shorter, every sixth lunar
day should have 24 instead of 25 hours; in this case the value of
the ordinate for the 25th hour repeats the value for the preceding
hour (cf. Appendix 2).
In addition, in order to eliminate the solar diurnal variations,
up to the time of the rearrangement its mean monthly value is sub
tracted from each ordinate. In this way the process of "taking a
mean" for the lunar day consists in eliminating all irregular vari
ations. Since the amplitude of the lunar variations is very small
by comparison to the nonregular part of the variations, a considerable
time interval is required to eliminate those variations. The first
detailed study of the lunardiurnal variations was made by Chapman
(Bibl. 49) in 1913. That author worked up by the above method the ob
servation from the observatories at Pavlovsk, Pola, TsiLaWei, Manila
and Batavia for 7 years (1897 to.1903 inclusive).
The results of the statistical workup, and of the subsequent
spherical and harmonica) analysis, allowed Chapman to establish a
number of regularities in the march of the lunardiurnal variations,
which differed from the regularities of the solardiurnal.
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The principal regularity is the semidiurnal character in
the changes of these variations. The curves of variation of all
.,'teats during the course of the lunar day have two maxima and
two minima whi14 the time of occurrence of the maximum and minimum
vary daily during the course of the lunar mouihs. The mean monthly
curve, however; has the form of a regular double wave with maxima
at 6 and 18 hours lunar time and minima at 0 and 12 hours for the
northern hemisphere. Figure 80 shows the curves of the lunar
diurnal variations of the declinations at Batavia and Greenwich for
the four phases of the.moon, accompanied by the mean monthly curves.
The displacement of the extreme values during the course of the
month may be clearly seen on the figure. In addition, the curves
for Batavia, which is located in the southern hemisphere, are al
most a mirror image of the curvesfur Greenwich, which is in the
northern hemisphere.
Chapman's expansion of the curves of the diurnal march into
a harmonic series shows that the harmonics with a semidiurnal period
remain constant throughout the course of the entire month, but that
the harmonics of the remaining orders change their phase while the
amplitude remains unchanged, thus also resulting in the. displacement
of the extreme values, Thus the change of phase for the first har
monies during the course of the month was 30?,for the third harmonic,
+300 and for the fourth harmonic, +500.
The lunar diurnal variations have annual march depending on
the position of the sun. During the time of the summer solstice
the amplitude of the lunar days reaches the maximum values in.the
northern hemisphere and the minimum values in the southern, and is
the time of the winter solstice,on the contrary, they reach their
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minimia in the northern hemisphere and their maximum in the
southern. For the vertical and horizontal components, the maxi
mum amplitudes reach only 12 gammas, while for the declination
they reach 40".
The dependence of the lunar diurnal variations on the
latitude and longitude are the same character as those of the
solar diurnal. Thus, with variation of the latitude, the phases
remain constant, but in the northern component, on crossing the
parallels +200, 20?, they change signs to the opposite. With
eastern and vertical components the variation of phase also takes
place on the equator. Tho amplitude of the observations reaches a
maximum in the northern component on the equator and at latitude
450; while the eastern and vertical components' maximum of amplitude
is reached at the parallel of 200
.
The lunar diurnal variations are almost independentsthe
longitude. It is an interesting fact that the amplitude of the
lunar variations is dependent on the distance of the moon from the
earth; more specifically the amplitude of the variations is about
inversely proportional to the cube of this distance. Just as for
the solar variations, spherical harmonic analysis allows the lunar
diurnal. variations to be explained by the existence of horizontal
eddy currents, whose distribution for the new moon is shown in Figure
81: the upper figure relates to equinox, the lower to the suer
solstice. The mtdians correspond to local lunar time and for the
new moon the sun and moon are on the meridian 12, The total current
flowing in the main circuit reaches 5,300 amp for at equinox and
11,000 amp at solstice,
Section 3. Magnetic Disturbances
Variations without definite periods obtained as a result of
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u*traeting the solar diurnal and lunar diurnal variations from the
observed variations, and which at first glance appear entirely arbi
trary, as a result of their random march, have received the name of
magnetic disturbances, and at great intensity, of magnetic storms.
While the amplitude of the periodic variations is expressed by a
few tens of gammas, they may reach a few hundrecb or thousands of
gammas during the time of magnetic storms.
Magnetic disturbances may be calssified according to intensity,
duration, andspatial distribution, into four types.
The first type includes disturbances of very great intensity,
magnetic storms occurring simultaneously over the whole earth. The
amplitude of the fluctuations of the elements of the terrestrial
magnetism of such 'storms may reach a few thousand gammas, and their
fdtion, a few days. The second type includes disturbances of local
character limited to definite regions, mainly the polar region.
Local disturbances may last for one or a few hours and their intensity
will exceed hundreds of gammas. While magnetic disturbances of the
first type commence simultaneously over the whole earth and proceed
in a single phase, disturbances of the second type, even at two nearby
points, may proceed entirely differently.
The third type, the bays, is the name given convexities or
concavities on the magnetograms, recalling the shape of marine bays.
The baylike, disturbances, occurring simultaneously may stretch over
the entire earth or may belimited to a certain region near the auroral
zone.
Finally, the fourth type of periodic magnetic disturbance con
sists of the socalled pulsations, which are sinusoidal fluctuations
in field strength with an amplitude of the order of few gamma and a
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period of & few minutes. The pulsations may take place simultaneously'
over the entire earth but may also be limited to individual regions.
In most cases thedisturbances of the first and second group
usually occur simultaneously, one being superimposed on the other and
causing side effects such as, for example, the induction current in
the. earth which in turn yields an additional component vector of the
disturbance.
Figure 82 shows magnetograms of the horizontal component
during the time of the magnetic storm of 14 ).arch 1922 according
to the records of five observatories located in middle and low lati
,;ruder, while Figure 83 shows magnetograms of the horizontal compo
nont from the records of observatories in high latitudes during the
magnetic storm of 19 February 1933.
The curves show that no complete parallelism is observed in
the march of the elements, but that a few maxima and minima do occur
simultaneously at all stations. The amplitude of the fluctuations
as will be seen from Figure 82, increases with increasing latitude
of the station. Thus this storm represents a storm of worldwide
character with the superimposition of local disturbances.
Statistical processing of magnetic storms has allowed estab
lishing the existence in them of at least 3 components differing in.
character and the laws of. occurrence and the laws of their course.
The first of them, SD, representing the difference Sd  Sq, of
periodic character, with the,,Iperiod of a solar day, has already been
considered in Section 1 of this chapter, The second, the aperiodic
variation Dap is found as a result 'of averaging a large number of world
wide storms located in columns, during the course of'a storm, i._ e.,
when the instant of origin of a storm is taken as the initial moment
of time t : 0.
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A characteristic feature of most magnetic storms is the
suddenness of their appearance. Against the background of a
rather quiet magnetic field, almost at one and the sane instance
of the entire earth all elements of terrestrial magnetism suddenly
vary their values, and their subsequent course undergoes very rapid
and irregular variations. For this reason it is possible to deter
mine the beginning of a magnetic storm of the magnetograms of all
observatories within 1 to 2 minutes.
The third component, which is obtained by subtracting the
aperiodic variation and the disturbances of the diurnal variation
from the observed ones, is really that magnetic disturbance which
we term a magnetic storm and which is manifested in the form of ir
regular rapid variations of all elements of terrestrial magnetism.
This part of the variation is termed, by general agreement, the ir
regular component, and is denoted by Di.
The aperiodic disturbed variation Dap. A characteristic
feature of this variation is that it`is very distinctly manifested
in variations of the horizontal component, to a lesser degree in
the vertical component, and has entirely no effect on the declination.
The general character of the course of Dap is as follows.
The beginning of the disturbance is a shortimpulse which increases
the horizontal component and decreases the vertical one. These
variations amount to +20 gammas for the horizontal component and '3
gammas for the vertical one.
The increases of values of the horizontal component lasts
only a short time, from 1/2 to Z hours. There is then a sharp fall
in thehorizontal component lasting for about 6 hours and going up to
r 8 gammas, after which begins a long process of return to the normal
slate, occupying a period of time up to 2 days. This process has re
ceived the name of after disturbance.
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The vertical component, after a 5 gating drop, which like 
wise lasts from 1/2 to 2 hours, then begins to increase and during
the course of the entire storm remains 5V higher than its normal value.
The declination experiences small deviations (from 1'to 2")
from its normal value towards one and the sane side, and therefor*
no regularities in its variation must be spoken of.
The next characteristic feature of Dap is the dependence of
these disturbances on the geomagnetic latitude of the place, and
their independence of the longitude. The maximum intensity of Dap
is found on the magnetic equator, where the variation in the hori
zontal component reaches 60 gammas and more. To the north and to the
south of the equator, the intensity decreases, and at latitude 600
(Pavlovsk) the variation in H amounts to 40 gammas. Further to the
north, Dap again begins to increase, reaching a maximum in the zone
of maximum aurora.
Figure 84 shows the march of Rp at three latitudes from ob
servations of the observatories at Batavia, Puerto Rico and Honolulu,
located at the geomagnetic latitude 0?, ZiKawei, San Fernando and
Cheltenham, on the latitude 40?, polar, Paviovsk, Greenwich, and Potsdam,
at latitude 60?. The graph of Dap shown on this Figure clearly shows
all the regularities in its course that have been pointed out above.
The variations are. easily' traced up to lati"ode 6070?.
the higher latitudes it is stillnot possible to i4late them so
distinctly, owing to the complexity of phenomena that take place there.
The local variations of high latitudes are so predominant in influence,
that, superimposed on the worldwide magnetic storms, they masked the
variations of D which are of lesser intensity: This will be taken up
aP
in greater detail in the following section.,
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The regularities found in the course of the Dap variation,
indicates that their source must be the ring current flowing in
the plane of the earth's magnetic equator. In this case the di
rection of the current in the first phase of the storm must be
from east to west, and in the second phase, from west to east.
04 A. Bourdeaux was the first to sake a spherical analysis
of tho disturbances of the diurnal variations. Be succeeded in
constructing a system of electrical currents in the upper layers
in the atmosphere, responsible for the variation SD. The form of
this system is shown in Figure 85a and 85b, 851 represents the
system of currents over the entire surface of the earth and Figure
85b a view from the geomagnetic polo. As will be seen, the maximum
density of the current lines is reached in the zone of maximum
auroral frequency, where the current strength reaches 200,000 amp,
while the total current over an area equal to onequarter of the
earth's surface does not exceed 40,000 amp. The strongcerowding
of the current lines is also observed in the polar regions between
the auroral zone and the pole where the total current amounts to
270,000 amp. Figure 85a showed the system of current consists of
eight closed currents of which four are located in the eastbrn hemi
sphere and four in the western. The system of currents constructed
by Bourdeaux is in rather good agreement with the observed diurnal
march of the variations. Thus, for example, it reproduces the in
version of the X component of the latitude 35?, the hours of maxima
and. minima of ' the X and Y co.ponents, and' the general march of the
variations in high latitudes. There is still no theory of the origin
of this system of currents.
The system of currents corresponding to the aperiodic disturbed
variation, viewed from the sun and viewed from the pole, is shown in
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Figure ?6, from which it is clear that the system forms a current,
flowing along the surface of a sphere parallel to the equator, its
current density declining from the equator to the muroral zone,
and increasing on passage through the auroral zone, reaching a
maximum between the pc's and the zone. The theory of the aperiodic
variation has been developed by Chapman and Ferrari; and it will be
set forth in the following Section.
Bair shaped variations. As stated above, the bay shaped
variations which we shall denote by DB, have the shape of the shore
line of marine bay on the magnetogram records, with an amplitude
reaching a few hundred gammas. The bayshaped disturbances appear
most distinctly in the horizontal component. In this case DB may
arise as a solitary disturbance amidst a quiet field, or may be
superimposed one on the other, or may also be present in a general
magnetic disturbance. Figure 87 shows the character of a record of
such disturbances observed at the observatory at Kew, England, in
February 1911. Owing to the existence of individual bayshaped dis
turbances, undistorted by other variations, in the records of the
observatory, their more detailed study was possible. The observations
show that D, occurring simultaneously at all stations of the world,
have a maximum intensity in the auroral_zone'where the amplitude of
the variations of H is tens of times as great as in the low altitudes.
If a graph of. the variations of H and Z during bay disturbances
is plotted against the latitude, laying off the maximum aptitude of D$
along the axis of ordinates and the latitudeof the stations located
close to one and the same meridian.. are laid offlon the axis of abscissas,
it will take the form shown in Figure 88. This graph is analogous in
its shape to the graph of the magnetic field excited by a magnet with
its axis.parallel to ti,,meridian and its center' at latitude 700.
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In view of the fact that such graphs are followed through*
simultaneously on other meridians as well, it is necessary, for the
formation of DB to postulate the existence not of a single magnet,'
but of a series of magnets parallel to each other and, located along
the parallels, so in the alternative, the existence of an electric
current equivalent to it and flowing in a narrow bean which may be
taken as a linear current along the 70th parallel.
Indeed, a linear current flowing at a distance R from the
earth creates on the earth's surface in a direction perpendicular
to the current, a magnetic field determined by the BiotSawara law.
The components in this field along the vertical and horizontal
will obviously be:
S = 21R Z 21x
R2 + x2 ' U
These formulas coincide with equation 8.45 for a singlepole
filament (page 349), the graph at which are given in Figure 167. By
comparing these graphs with the curves without the DB variations in
Figure 88, and obtained from observations, it will be seen that they
are in close agreement with each other.
If it is postulated that B is caused exclusively by current
flowing in the upper layers of the atmosphere, then the height at
which it flows, and the magnitude of the current strength, can both
be easily determined.
The height h of the linear current in found graphically by
constructing the field strength vectors at points located along the
meridian near the 70th parallel, and producing perpendicular tothen..
The point of intersection of the perpendiculars will give us the
linear current on the scale of the height, since for the linear current
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the direction of the magnetic field is perpendicular to the distance
between the linear conductor. and the given point of the field.
The current strength I is determined from the BiotSawara
6H = 0.21
T h
where,8 aT is the field strength of DB at the 70th parallel.
Such determinations gave a value of the order of a million`
amperes for the current strength and a value ranging from 100 Sun to
a few hundred kilometers for the height.
Further refinement showed that the sources of the DB varia
tions do not consist only of the currents in the atmosphere, but
also of induced currents within the earth, which is responsible for
about 40% of the entire field of variations.
Magnetic pulsations. Pulsations represent regular fluctua
tions of the elements of terrestrial magnetism, mainly of the de
clinations and the horizontal component, with a period from 20 sec
to a few minutes and an amplitude of a few gammas. At the usual
rate of rotation of the drum of 20 an per hour, they are found on
magn6tograas in the form of a sawtooth curve with small peaks. A
a higher rate of rotation of the drum, however, (3 mm per minute)
they are recorded in the form of a regular sinusoid. A characteristic
feature of these variations is that they are observed Uthly around
midnight, often from 22 to 2 hours of the following day. The'usual
amplitude of these variations does not exceed a few gammas, although
pulsations with larger amplitudes are sometimes observed. Thus, for
example, on 12'September 1930 at the observatories of Abisko (Lot =
68.4?, Long = 18.8?), and Troemso (Lat = 69.7?, Long = 18.9?) in
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Norway, pulsations with amplitude as high as 30'gammas were registered.
Figure 90 gives the records of these observatories showing the charac
ter of the pulsations themselves. Since pulsations are of a sinusoidal
form, it has been postulated that they are not, the thought has bihnee 't
pressed that 'they might perhaps be natural oscillations of the vario
meter magnets themselves (cf. Chapter XII), due to the seismic vibra
tions of the soil. But pulsations are also observed with magnets with
periods of natural oscillations far from those of seismic vibrations.
An important factor is also that the. pulsations, are sometimes
observed simultaneously at several stations, such as, for example,
the pulsations recorded at Abisko and Tromso and:sho n in Figure 90,
which are also observed at a number of other stations. This has given
ground for postulating the worldwide character of the causes responsible
for pulsations. A more detailed study of the special distribution of
pulsations, however, showed that the region which the action of these
causes, extends is limited to a radius of not more than 1,000 km.
Even such a gigantic pulsation as the one observed on 12 September 1930
at Tromso, which is at a distance of not more than 100 km from j tlkips
declines by & factor of several kinds by comparison with Abisko.
A proof that the pulsations constitute a real phenomenon of
nature'is'also their record by instruments based on the conduction
principal, i. e., instruments which react to a variation of the mag
netic field with time.
In spite of the large number of works devoted to the study of
1
pulsation, their causes have still not been established. and there is
no theoretical explanation for them.
Section 4, Variations at High Latitudes
' s.
The systematic study of the variations in hign latitudes began
somewhat more thin 20 years ago, when in connection with the Second
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International Polar Year (19221933) a number of magnetic observa
tories were opened in these latitudes. The extraordinary complexity
of the variational phenomena in these parts of the earth, however,
demands for its study a large observational material, both in time
and in a number of observatories. For this reason the relatively
short period of observation and a small number of observatories still
fail to make it possible to establish definite regularity in the
course of the magnetic variations at these latitudes, as for the low
and middle latitudes, even the material that is available at the
present time still allows us to point out certain features and to
find regularities of one kind or another in these phenomenon. The
most complete material for the past 25 years has been collected by
the Arctic observatories of the USSR. This material enabled A. P.
Nikol'skiy to find new phenomena in the diurnal march of Magnetic
activity in high latitudes. Very valuable material was secured during
the Second International Polar Year, when a whole system of temporary
stations operated on a single common program. The results of the ob
servations of this sphere were worked up by E. Vestine and published
in 1947 in the form of many different graphs with explanatory text
and a few conclusions.
The results of this workup are the materials which mist serve
as the foundation for future studies and deductions. But certain con
chilons may may already be made, even on the basis of this material.
The solardiurnal variations . Sq. One peculiarity of the diurnal
variations in high latitudes is their dependence not on the geographical
latitude, but on the magnetic. This dependence is manifested with
particular sharpness onpassage through the zone of maximum aurora
frequency. A second peculiarity consists in the considerably higher
value of the amplitude and phase variation in the vertical and northern
component on passage across certain magnetic paaall!ls.
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Figure 91 shows the wean diurnal course march of the varia
tions on quiet days in the winter and summer months, as well as the
mean annual values at the stations of the Second International Polar
Year, locatedtn:>the high latitudes, and at the same time at a few
permanently operated observatories in the middle latitudes. These
curves clearly indicate the latitudinal dependence of the variations,
as well as their dependence on the season. In the winter months the
variations are considerably less than in the summer. In addition,
our attention is struck by absence of that regularity in the march
of the variations which is observed in the middle and low latitudes,
which phenomena is prehably due to the superimposition of irregular
disturbances, since in the high latitudes, disturbances are observed
even on the quietest International days.
As a result of this, the variations SD are of extremely great
interest, i. e., the diurnal variations on disturbed days minus those
on the quiet days.
Figure 92, showing the mean annual.march of the SD variations,
shows that the SD variations are of more regular character than the Sq.
Their peculiarity is the change in phase of the Z component in passing
through the zone of maximum aurora frequency, located approximately
at the magnetic parallel 68?, and also the maximum value of the ampli
tude of the X component in this zone. The change of phase in the X
component takes place at magnetic latitudes 81?, and then again at
latitudes 72?. The eastern component likewise changes its phase twice,
once at latitudes of about 60? and afterwards in the auroral zone.
The amplitude of all the components reaches values in the auroral zone
that many times exceed the values in the middle latitudes.
While the variations Sq have a strong seasonal dependence, the
variations SD remain almost the same throughout the entire year.
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The aperiodically disturbed variation. The least studied
variation in the high latitudes is the aperiodic variation (Dap),
as a result of which there is still no well established regularity
in its course. But the result of the workup of 11 storms registered
at the Arctic stations during the second polar year allow us to give
a few conclusions. In view of the small number of storms, they have
been worked up for a group of stations located close to one and the
same magneticlatitude. The result of this workup is given by
Figure 93 in the forte of graphs, which show that the northern com
ponent at the beginning of the storm has a value somewhat lower than
normal, and which then during the first 1012 hours, falls to a mini
mum, after which it begins gradually to increase to its normal level.
The absolute value of the minimum increases on approaching the auroral
zone, where it reaches SON,,. Inside this zone it declines, and at
latitude 84? it reaches a value of 30 ?, in this case the character
of the curve changes completely and takes on the form of a periodic
curve.
The eastern component varies periodically, now increasing,
now decreasing, within the range of 1020'' , indicating the absence
of any definite regularities, and in all probability, also indicating
the absence of components of;~ Bapin this direction, since the fluctua
tions might be caused by the superimposition of random variations.
The vertical component, up to the auroral zone, has the same
character and the same value of the amplitude as in the middle lati
tudes. On passing through the auroral zone, the vertical component,
10 hours after the onset of a storm, begins to increase sharply,
rettichinr_ a maximum of 60 to 100 gammas within a few hours. The
highist'.vslue of the maximum is assumed at latitude 710.
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Magnetic activity or degree of disturbance. The existence
of a large number of magnetic disturbances in the Arctic latitudes which,
being superimposed on the periodic variations make it?impossible to
isolate thou by an averaging, force us to seek other methods of studying
these disturbances than the study of the diurnal variations. Such a
method, proposed by A. P. Nikol'skiy (Bibl. 50) in 1935, consists in
taking the mean values of the hourly values of some measure of activity,
either the international characteristic k, or the characteristic pro
posed by Nikol'skiy himself, the length of the curve D. The mean value
of this activity over a certain time interval was termed the degree of
disturbance by Nikol'ekiy, who studied it in its relationship to the
time, latitude, and longitude of the place, season, and the 11year
cycle of solar activity. These studies yielded new material for the
elucidation of the causes of magnetic disturbances, and established a
number of new regularities in the course of these variations.
The principal result of Nikol'skiy's work was the establishment
of a definite regularity in the diurnal march of the magnetic of the
degree of magnetic disturbance, which was as follows: the degree of
magnetic disturbance during the course of the 24hour day has two maxima,
one of which comes in the morning hours and occurs according to universal
time, the other. in the evening hours, and occurs according to local time.
In addition, the instant of onset of the morning maximum in the
eastern hemisphere depends linearly on the geomagnetic latitude of the
station while in the western hemisphere this, instant of onset takes
place at ooe And the same hour,(1530163W bours Greenwich mean time).
Figure 94 shows a typical curve of the diurnal march of distur
bance at the Tikhaya?Bay Observatory (Let 71.5?, Long  153.3?) while
FigurS 95 shows the relation of the morning maxitium to universal time
(Greenwich mean time). These curves thus giveus grounds for asserting
the correctness ofthis regularity.
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on Investigating the behavior of these maxima in relation to
various parameters, Mikol'skiy found that each of then behaves dif
ferently, which gave him reason to enunciate the lypothesis that
these maxima were of different nature.
Thus, for example, the value of the morningmaximum is almost
independent of the season, while the evening maximum has its lowest
value in winter and its highest value in summer, which is illustrated
by the curves of the diurnal march of disturbance at Tikhaya Bay,
worked up by seasons  winter, summer, and equinoxes (Figure 96).
The value of both maxima of disturbance depends further on the
general state of the magnetic field.
With the increase of the general degree of disturbance, the
value of the maximum increases but in different ways. Figure 97 shows
the diurnal march at Tikhaya Bay for six groups of days with varying
degrees of disturbance. It will be seen that at the beginning, when
the total disturbance is small, the morning maximum is predominant,
but later, with the increase in the total disturbance, both maxima
increased, but the night one increases considerably faster and on
stormy days is almost twi O as large as the morning maximum.
Finally, we point still another peculiarity, the dependence of
the value of the maxima on the geomagnetic latitude. This' dependence
is shown in Figure 98, which indicates that with. increasing latitude
the maxima also increases,'but that the evening maximum reaches its
highest value in the auroral zone while the aorr ,;maxim= reaches
it at the geomagnetic pole.
All those facts confirm beyond the doubt the thought that these
two maxima are due to different causes. 8ow6ver, attempting to find
these:caus?s and to explain the diurnal march of the degree of the
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magnetic disturbance, Xikol'skiy constructs a very simplified scheme
of currents postulating that the variations are due to a rectiUnear
current of corpuscles of different signs, with the corning maximum
attributed to a stream of corpuscles of one sign, while the evening
maximum is attributed to a stream with the opposite sign, although
he adduces no quantitative calculations to confirm this hypothesis.
Nikol'skiy further attempts to cast doubt on the methods of
segregating the diurnal and aperiodic disturbed variations, and
claims in his work that, at least in the high latitudes, the diurnal
variations and aperiodic disturbed variations do not exist as actual
processes and that they represent fictitious phenomena obtained as a
result of the statistical treatment. It is premature to agree with
such theories and conclusions, since they are constructed with no
quantitative analysis whatever and are unconfirmed even by elementary
mathematical calculations.
Section S. The Aurora
The magnetic disturbances observed on the earth, and especially
in the Arctic regions, are closely connected with the aurora. This
,connection is not merely external, but is also internal and physical
and is due to the common causation of both phenomena. For this,
reason, in considering the causes of magnetic disturbances they cannot
in any case be abstracted from the fact of the aurora, which allows us
to understand more clearly and more profoundly the nature of'the mag
netic disturbances.
1, The forms of theaurora. The. aurora may be,iAi Csified ac
cording to its form 'into two "great groups : auroras of non'r,.dial
structure and auroras, of?radial structure. Each of these groups in
turn : subdivided into a nu iber. of subgroups.
Those of nouradial structure include the following:
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1. uniform quiet arcs the lower edge of which is sharply
pronounced, while the upper edge is blurred. A dark segment is
observed between the lower edge of the arc.and the horizon;
2. uniform bands usually extending in the same direction
as the arcs, but less regular in form. The lower edge is often
sharply defined, but is irregular in form. In most cases the
bands are broken up by dark spaces and for this reason have the
form of feathery clouds;
3. pulsating arcs, which rhythmically appear and disappear
with a period of a few seconds;
4. diffusely luminous surfaces having the form of a veil
covering a large part of the sky;
5. pulsating surfaces consisting of diffusely luminous
parts of the sky, appearing at one and the same place.
The radialstructure auroras include the following:
1. arcs with a radial structure;
2. bands of radial structure, seemingly uniform, but
consisting of a number of streamers;
3. drapery, consisting of a few bands with very long
streamers, similar to a folded curtain. The lower edge of the
drapery is usually more illuminated. Near the magnetic zenith
(the projection of the magnetic pole onto the celestial vault;
it has the fosw?of a fane;
4. isolated streamers which may be narrow or wide, short
or long, isolated or in the form of beaus;
5. a Corona, consisting of streamers, bands, or drapery,
converging to a?single point near'the magnetic zenith.
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2. The direction of the streamers of the aurora. A sub
stantial factor in the observations of the radial structure is
a su1fkc#.ttly close coincidence of the direction of the streamers
with the direction of the lines of force of the earth's magnetic
field. This fact is one of the basic facts for the construction
of the modern theory of the aurora, based on the motion of charged
particles in the earth's magnetic fields.
Indeed, by observing visually the streamers of the aurora
it may be noted that all of them converge on a single point located
near the magnetic zenith. The results of precise determinations of
Fy
the direction of the streamers by photographing the Corona of"the
aurora, are given in Table 19 which gives the observed height h
and x the azimuth of the point of convergence of the streamers and
of the magnetic zenith.
As will be clear, the coincidence between the point of con
vergence of the streamers and the magnetic zenith is observed within
the limits of accuracy of the observations themselves.
Table 19
Station Observer Year
Holde Vegard and Krogness 1914
Oslo Stoermer 19171921
Number of observations Point of convergence magnetic zenith
h a n a
ll 75.40 2.70 76.70 2.50
9 0:0 9.8 TO.8 9.7
Yegard, studying tha structure and distributionof the light
along,a"iwndle of rays, came to the conclusion that the rays always
follow the direction of the lines of force and that the point of their
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intersection corresponds to the magnetic pole, which may be dis
placed owing to the appearance, during the time of strong auroras,
of an additional magnetic field formed by the ring currents around
the earth.
`a~S
3
3. The height of the aurora. The height and position of
the aurora in space may be determined by the simultaneous obser
vation of their coordinates (altitude and azimuths) at two points,
the distance between which is known.
The first determinations for this method were made over 200
years ago and they have since been repeated by many investigators,
who have given the height within the range of 80 to 200 km. But
exact determinations of the height of the aurora became possible
with the introduction of the photographic method.
This method was first used by Stoermer (Bibs. 51) in 1910,
y?
and in 1913 at the Observatory at Vessekop in I,apland,;:and' is as
follows at two stations, working simultaneously and having a tele
phone connection between them, the aurora is photographed on
motionpicture film. Stars are photographed together with the
aurora on 'this film. By determining the position of the. aurora
among the stars on the film, and knowing the distance between the
stations and the azimuth of the aurora, its position in space can be
determined with fair accuracy and its height can be calculated. The
length of the base, or distance between the stations, was only 4.5 ka
in 1910, but in 1913 it was as such as 27.3 km, which made it possible
to make more exact determinations.
Thanks to the short exposure (half a second or less), this
method allowed the observation not. only of quiet auroras but also
of rapidly pulsating ones.
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The result of the first determinations in 19101913 show
that the height of the aurora ranges from 87 to 350 km, with two
maximum frequencies being observed, one at height 101 and the
other at height 106 km.
These results were confirmed in 1920 at the Holder obser
vatory by more systematic determinations of the upper and lower
boundaries of various types of aurora.
The results of these determinations are shown in Figure
99 in the form of curves showing the probability of the appear
ance of the lower boundary of an aurora at one height or another.
These curves show clearly that the most frequent lower boundary
of the aurora is at height 100106 km.
During the period from 1910 down to the present, thousands
of determinations of the height and positions of the aurora in
space have been made. This material has allowed the following
conclusions to be drawn. The quiet forms of the aurora, diffuse
arcs, bands, and pulsating areas have on the average their lower
boundary at a lower height than the pronounced radial structures.
Moreover, the mean height of the lower boundary of the quiet forms
and drapery is almost independent of the latitude of the place,
but for the forms of radial structure the mean height increases
from the auroral zone (cf infra) towards lower latitudes. The re
sults of measurements of the lower boundary of various forms of
the aurora at various observatories is givin by Table 20.
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Holder, (Lat 69?56' Tromso (Lat 69?49' Oslo (Lat 6000'
Lccg 62?55'
Long 16?57' Long 10040'
Type of aurora 
Height, k*
Number of
observations
Height, k*
Number of
observations
Height, km
Number of
observations
:'Stsars
113.2
61
117.0
127
146.9
119
Drapery
109.8
409
112.9
1,039
Draperylike area
106.6
888
106.7
1,175
100.0
150
1
Pulsating areas
160
107.3
66


Diffuse arcs
r
['1 J
409


118.5
201
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The upper boundary of various types of aurora lies within
the limits of 140 to 250 km. The auroras in the form of streamers
are the longest. Thus the mean height of the upper boundary of
some forms are as follows: streamers, 250 km; drapery, 176.3 km;
Thus the length in a vertical direction is 137 km for the
streamers, 68 km for the drapery, and 34 km for the diffuse arcs.
But auroras are also observed in the form of streamers
whose lower boundary goes'up to a few hundred kilometers and its
upper to a thousand.
Thus Stoermer registered on a September 1926 photogram at
Oslo an aurora in which the lower boundary lay from 200 to 400 ka,
and the upper boundary from 1,000 to 1,100 km.
4. Theo4geographic distribution of the aurora. As indicated
above, the aurora is observed not only in the Arctic regions but
also in the middle latitude and even lower. Thus, for exampl*, in
the 1870's anaurora extended to Egypt and even. to India. But the
frequency of appearance of the aurora (the number.of displays per
year) is very small in the middle and low latitudes, while in the
high latitudes the aurora is_observed almost daily.
As early as 1881 (Bibl. 52) a map of the isolines of auroral
frequency (isochasas) was prepared and showed that the aurora was
most frequent in a region 0 out '23? distant fromthe geomagnetic pole.
This zone has received the name of none of maximum aurora.
'For 60 years frog 'the publication of this mapan immense
material was accumulated on observations of the aurora, which made
it posiiblb in 1947 (Biol. 53) to?.construct'a now map of the isochasms
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(Figure 100) which on the whole repeats the 1881 map. A similar
zone of maximum aurora exists also around the south geomagnetic pole.
on the map, the isochasm corresponding to the maximum fre
quency is shown by a thicker line. The values of the isochasms are
shown in arbitrary units, in percent related to the maximum zone.
Figure 101 is a reap of the earth on which the northern and
lower zones of magnetic of maximum auroral frequency and the geo
magnetic equator corresponding to the uniform magnetization of the
earth, are shown.
The following we shall not indicate a few of the most
strongest auroras during the last 100 years. The strongest of them
was on 4 February 1872 when it was visible at Bombay, Lit 19? N, at
a distance of 80? from the magnetic pole. The magnetic zenith of this
aurora was observed at Constantinople and Athens. The aurora
australis was observed simultaneously at Lat 200 S, at a distance of
72? from the earth's magnetic pole. The next intense aurora was ob
served on 1415 May 1921 when the aurora australis reached the Islands
of Samoa (13.80 S). This aurora was accompanied by strong magnetic
storms.
Strong auroras were also observed in January and April, 1936.
5. The diurnal distribution of the aurora. By observing the
aurora daily it may, be noted that during the 24hour day it appeared
not at random, but had a tendency to group itself about a certain
moment about a certain time. Thus, the observations during the First
IntOrnationalPolar Year, 18821883, showed that auroras in the form
of streaimers, draperies and corona have a distinct maximum in the
evening hours, and a winter maximum in the morning hours. In this case,
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the moments of maximum appearance of the aurora, according to local
time differ at the various stations, depending on the geographical
coordinates. Thus at Bossekop station (Lat 69.57? and Long 23.15? E)
this maximum was at 2125 hours local time, while at Fort Rae, Lat
62.39?, Long 115?49' W it was at 2400 hours.
If, however, the local magnetic time is taken instead of the
local solar time, then the times at maximum auroral frequency will
be the same for all stations, and correspond to 23 hours magnetic
time. In this case, "by magnetic time" we mean the angle between a
plane passing through the magnetic axis through the earth and the
sun and the plane passing through the same axis and the given station.
6. The spectrum of the aurora. The spectroscopy of the
aurora, owing to its low intensity, requires special spectrographs
with a large f number and an exposure measured in tens of hours.
For this reason it was long impossible to identify some of the lines
observed with the lines of any specific element. In 1912 Yegard
succeeded in obtaining 33 lines by the aid of such a spectrograph
and an exposure lasting a month.
Subsequent studies showed the presence of a large number of
lines in the auroral spectrum in the visible, infrared and ultra
violet regions. Among these lines, the most intense is the green
line at 5557.3 A. This line does not correspond to any of those
observed under laboratory conditions.
Since this line is observed not only in the spectrum ofthe
aurora but also in the spectrum of.the luminescence of the. night
sky in the absence_of:thp aurora, it gave Vegard cause for enunciating
the hypothesis that solid cryitaline particles of nitrogen were
present in the upper layers in the atmosphere and on bombardment by*
electrons luminesced and radiated this line.
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But this green line at 5777 A (sic) was later obtained under
laboratory conditions from atomic oxygen, and in this way the prob
lem of the origin in this line of the spectra of the aurora and the
night sky was solved.
Of the other brightest lines, we may note the lines at 3914,
4278, 4708 and 5225 A, belonging to the ionized nitrogen molecule
N2+ and the lines 3997 and 4059 A, belonging to the neutral nitrogen
molecules.
The discovery of the green line in the spectrum of atomic
oxygen permitted the prediction of three red oxygen lines at 6300,
7364 and 6392 A, of which two were found in the auroral spectrum.
Thus the aurora on the whole is the luminescence of atomic
oxygen and molecular nitrogen of which the ionosphere is composed.
7. The connection between the aurora and solar activity.
The connection between the magnetic disturbances and the aurora
was first established as far back as the beginning of the 18th
Century, when it was noted that the aurora are accompanied by mag
netic storms.
Further observations completely confirm this discovery,
but it has still not been possible to establish the essential
functional connection between these phenomena. All the studies in
this direction lead merely to a statistical correlation which al
ways was found to be high. Thus the regular observations at the
magnetic observatory at Tikhaya Say made in 19321933 allowed a
Soviet invisttgitor (Bibs. 54) to compare the magnetic characteristics
0 1 2 3 4 with the auroral characteristics 1 2 3 4. In the auroral
characteristics, snit one corresponds to the absence of an aurora on
that day, two to the presence of moderate aurora, but without a radial
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structure, three to the presence of bright auroras of radial struc
tures, and four to very bright auroras. The results of these com
parisons are given in Table 21, which indicates the number of days
with the corresponding auroral and magnetic characteristics.
TABLE 21
COMPARISON OF MAGNETIC ACTIVITY AND AURORAL ACTIVITY FROM OBSERVATIONS
AT TIKHAYA BAY DURING THE PERIOD FROM OCTOBER 1932 AND MARCH 1933
Number of cases when the magnetic
ntensity of aurora
(characteristics)
Total number
of cases
0
Faint (1)
186
10
Moderate (2)
Bright (3)
Very bright (4)
10
0
cnaracti?ra6u+%,;, 3 4 characteristics
1 2
114 46 7 3 1.3
3 2 2 3 2.5
These results, in general, confirm the absence of magnetic
disturbances on those days when aurora are not observed and their
appearance with the appearance of the aurora, although there are
also exceptions.
Like magnetic activity, the aurora have.a tendency to a 27day
cycle, and,,finally, also have an 11=year cycle of maximum frequency.
Figure 102 gives the curves of a number of sunspots (upper curve) and
a number of days in the year with auroral displays (lower curve), ob
served during the period from 1840 to 1896.
Except for the firs cycle, the maxima and minima of the
curves are;almost"in the very sari years, which indicates the close
connection between thssephencuena.
All this forces us to assume that the aurora, the magnetic
disturbances, and the appearance of spots on the sun are interrelated.
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CHAPTER YII
THEORY OF MAGNETIC VARIATIONS AND AURORA
8dction 1. The Ionosphere and Its Properties
The results of mathematical analysis of the curves of magneto
grams give reason for holding that the cause of both solar and lunar
diurnal variations is the existence in the upper layers in the atmo
sphere of a system of electric currents with a distribution that gust
be roughly about what is shown in Figure 73 and 74. For this reason
all theories of diurnal variations, starting out from this proposition,
attempt to give the mechanism of origin of these currents, under the
assumption that the upper layers of the atmosphere possess a correspond
ing conductivity. Until the 1920's, the question of the conductivity
of the upper layers of the atmosphere remained a pure hypothesis, and
consequently the theories of the magnetic variations based on this
hypothesis could not claim to be reliable. But.observations on the
propagation on the radio waves, especially short waves, showed the
existence at a height of 100 to 300 km of conducting layers having
the ability to reflect radio waves, as occurs with metallic conductors.
In this way the hypothesis of the conductivity of the upper layers now
has experimental confirmation. The existence of conductivity is ex
plained by the ionization by the atmosphere under the action of the
ultraviolet and corpuscular radiation of the sun. This region of the
atmosphere has received the name of the ionosphere. Since the iono=
sphere plays an immense rule in the formation of the variationsof the
magnetic field of the earth, it is necessary, even though only briefly
for us to dwell on its properties, and the methods of studying it.
1. The propagation of radio waves in the ionosphere. The de
pendence of the index of refraction on the density of ionization.
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Let us assume that in an ionized medium located in the magnetic field
of the earth, there is propagated a monochromatic plane electromagnetic
wave with angular velocity w in the direction of the z axis, which
makes the angle B with the direction of the magnetic field H. In this
case, under the action of the electric vector E of this wave, the free
electrons are put into motion, and the equation of one of this will
obviously be:
d 2 r _
E _e (drH), (7.1)
2 dt
.if we consider that the electrons do not lose their energy under the
action of the collisions, where r represents the vector of displace
sent under the action of E, and c is the speed of light.
If denotes the dielectric constant of the ionized aediuW,
then
E = E + 41tP,
where p is the vector of electric polarization, which may in turn be
represented in the following form:
P  Nor,
where N is the number of free electrons in 1 cc.
From this, expressing E and r in terms of P. and substituting
in equation 7.1, we obtain:
_ a d2P 4xe 1
Ne t3  & 11 P + Nc ~dt H)
(7.2)
Since,the vector E is a harmonic function then E = Ea sin wt,
equation 7.2 allows us to find the value of c connected with the in
dex of refraction, which is, as follows:
n2 1 _ 2a (1  a) (7.3)
2(1=a)b2H2+ b,H + 4b2H. (1a)2
X x x
4xNe
2
aw
and b _ ,H+
mew
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Let us now consider the case where the magnetic field is
absent, or where it may be disregarded. Then, putting H = 0 in
equation 7.3, we shall have:
2
8=n2=1az14*Ne
nw2
(7.4)
This expression has a physical moaning only for a 1. At
a = 1, the index of refraction becomes zero, which corresponds to
the phenomenon of total internal reflection. Consequently, at
constant ionization density,i. e., when N = const., it is always
possible to select.a frequency of oscillation of the electromagnetic
wave w, at which the index of refraction shall be zero. In this case
the total internal reflection of the incident wave takes place, and
with normal incidence it returns to the place from which it came. It
is from the difference between the time of the emission of the signal
and time of its arrival that the height of the reflecting layer can
be determined. The frequency wc, at which total internal reflection
takes place is termed critical; if we know it we can determine the re
lation of the density of ionization to the mass of an ionized particle
from the equation:
w 2 4nNe2
s7.5)
(7.6)
2 n
N c = fc2
ti 4xe2 = 2 '
where fc is the critical frequency of the oscillations expressed in
cycles per second.
In this way the method of reflection of radio waves of variable
frequency from the ionized layer allows us simultaneously to find the
height of the reflecting layer. and the density of ionization of that
layer, provided we know the species. of ionized particles. Thus if the
particle is an electron whose mass a = 9 x 10`28 g, then
Ni. 1.24 . 108 2 .
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If, however, the ionized particles are oxygen and nitrogen
ions, the mans sass of which is a  3.7 x 1023 grams, then
K = 2.5 . 104f 3
c
If an electromagnetic wave of constant frequency penetrates
into an ionized layer of gas of variable ionization density, then
it may reach a level at which n becomes zero. At this height,
then, there is total reflection from the layer.
For the case when the magnetic field is not equal to zero,
the index of refraction; as shown by equation 7.3, has two values.
For this reason, when an electromagnetic wave passes through an
ionized medium, it must be splitinto two. One of these waves,
corresponding to the plus sign in equation 7.3, is called the ordi
nary wave, while the second, corresponding to the minus sign, is
called the extraordinary wave, by analogy with the doubly refracting
power of crystals.
If expression 7.3 is equated to zero, then we obtain two
equations each of which determines the critical frequency of the
ordinary and extraordinary waves.
One of the roots of this equation a = 1, while the second
a =1 ? bH. The first of then corresponds to the ordinary wave,
the second to the extraordinary. If we replace a and b by their
values and denote the critical frequency of the ordinary wave by WC
and of the extraordinary wave by we , then. we obtain
(!f ,0 4__
a a 9
eH 4xNe2
ae C a c
Y
The quantity e0/ac represents the an r velocity of rota
tion of the charge a about the lines of force of the aagn.ttic field.
If we denote it.by wH, we shall have
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or, passing over to frequencies' expressed in cycles per second:
ft2? Hfcf2=0,
.f ' 11 ?'H
C C J '
As shown by experience, fH is always less than fc, and there
fore we may write, approximately:
the ordinary wave,
but only the plus sign has a physical meaning, i. e., the frequency of
the extraordinary wave is always somewhat higher than the frequency of
of 0.7 megacycle.
frequency of the extraordinary and ordinary wave will be of the order
For middle latitudes, where H = 0.5 ?e' the difference in the
c
1.4 H megacycles.
represented by electrons:
For a layer of the atmosphere the ionization of which is
and consequently also the difference f'  fc, will liketise be a
a few thousand tines as great as the mass of an electron, the?fH,
But if the ionization is represented by ions, whose mass is
few thousand times as great, i. e., of the order of a few hundred cycles.
It followiVioa this that in the presence of double refraction,
when the difference between the frequencies of the ordinary and extra.
ordinary rays is of the order of 11.5 megacycles, wehave an ionised
however, the opposite conclusion cannot be'drawn, since theextraordinary
ray, after being reflected, may also fad as a result,'of absorptionto:
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reach the earths surface.
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Equation 7.7 shows that the ordinary wave coincides with a
wave that is with the wave propagated in the absence of the magnetic
field.
2. Methods of studying the ionosphere. The theory of propa
gation of radio waves in the ionosphere allows methods to be estab
lished by the aid of which it is possible to determine the height of
the ionized layers, the density of their ionization and the character
of the ionized particles. The principal method in the study of the
ionosphere today is the pulse method.
The pulse method is as follows. A radio installation, con
sisting of a transmitter, receiver, antenna, pulse modulator, relaxa
tion oscillator, and cathode oscillograph (Figure 103) periodically
emits short pulses with a duration of the order of 104 to 105 seconds.
Since the frequency of the oscillations of the transmitter is of the
order of a few megae$cles, a few hundred oscillations will go into
each pulse. Thk% time interval between 2 successive pulses, or in
other words the pulse frequency, is so chosen that the radiated im
pulse shall be able to reach the reflecting layer and return from it
before the second impulse begins. With a height of the F2 layer of
the order of 300 km, the travel time t will be 2 x 10'"3 seconds. Usually
the interval between the pulses is taken ten times as great, i. e., 1/50
saconds. The direct and reflected pulses are received by the antenna
and enter the radio receiver, at the output of which an alternating
voltage is produced. This voltage is fed to one of the pairs of plates
of a cathode oscillograph, while the other pair is connected to the re
laxation oscillator of the same frequency as the pulse modulator, 50
cycles.
The relaxation oscillations are synchronized with the pulse
modulator in such a way that at the time of minimum amplitude of the
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relaxation oscillations, I. e., when the voltage across the oscillo
graph plates is at a minimum, the oscillators shall receive a pulse.
In that case the reflectedpulse arriving after a certain time inter
val and entering the oscillograph plates, meets the cathode beam de
flected in a direction perpendicular to the direction of the oscilla
tions of the pulse, and on the oscillograph screen we shall see two
luminous bands a certain distance apart. Since the relaxation oscil
lations are proportional to time over most of its halfperiod, the
deflection of the reflected beam will likewise be proportional to its
time lag. In this way, by measuring the distance between the luminous
bands, if we know the time scale, the height of the reflecting layer
can be determined.
To photograph these bands, the oscillograph screen is covered
by an opaque screen with a narrow slit in the middle, and the photo
graphic film is displaced perpendicularly to this slit. It the height
of the reflecting layer is constant, then a series of straight lines,
correspondent to a different number of reflections, will be obtained
on the film. But if the height varies, we shall obtain a curve showing
the relation of the height of the layer to the time.
The pulse method allows us to determine only the height of
the reflecting layer corresponding to a given frequency of the trans
mitter. To find the critical frequencies by this method, the pulse
observations and measurements are made with a smooth variation ofthe
transmitter and receiver frequencies from low frequencies, up to the .41
frequency which the reflection stops. In modern installations, these
variations of frequency are produced automatically, and such an instal
lation has received the name of ionosphere station.
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3. Measurement of the height of the reflecting layer. If the
velocity of propagation of waves in the ionosphere did not differ from
the velocity of propagation in a neutral medium, i. e., if it were
equal to the ppeed of light c, then the height hg of the reflecting layer
could be calculated by the formula
h =1c
g 2
where ( is the time interval between the moment of emission and the
moment of arrival of the reflected Wave. In view of the existence of
dispersion, the velocity for each frequency in the ionosphere will be
different and will vary with the variation of the ionization density.
For this reason the true height h will differ from the height hg termed
the effective height, and must be calculated by the formula:
h h
h= h + J u dt = h + f dh, (7.8)
0 h 0
0 h
where h0 is the height of the beginning of the layer above the
earth's surface and is the velocity of propagation of the waves in
the ionosphere layer.
By means of appropriate transformations, equation 7.8 is
brought into the following form:
2 nga l
2 2
.0 401
Since WI is an arbitraryvariable, it may be taken as equal
to w sin q , i. e.,
111
and in that case
lv 1 : Wsin 1
xQ2
hgd cT ,
. (7.9)
where h is a function of w sin 4 . Consequently the true height is
Q
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numerically equal to the area between the curve hg = f (t~) and the
axis of abscissas. Since the photograss give us h as a function of
w, it follows that to find hg as a function of (R , must be calculated
by equation 7.9 for the given frequency w and for various values of
w it and the values of hg corresponding to the values of q must be
found from the photogram, and this will in fact give us hg as a func
tion of
If we now construct a graph of hg against T and integrated by
a plaiiism*ter, we shall find h for the given frequency w.
Observations of the reflections of the radio waves by the
critical frequency method, that is, the determination of the frequency
at which the reflected wave vanishes, have allowed the existence of
three ionized layers at mean heights of 100, 200 and 300 km to be es
tablished. These are termed respectively the E, F1 and F2 layers.
The critical frequencies corresponding to these layers depend
spective years.
queneies are at maximu, as well as the number of sunspots in the re
of the mean annual critical frequencies at noon when the critical fre
tivity and as well as the solar altitude. Table 22 gives the values
both on the latitude and longitude of the place'and on the solar ac
The frequencies are expressed in megacycles and are given for
various points of the earth, as well as for various epochs within the
limits of a single cycle of solar activity.,
As indicated by the Table,.the critical frequency, and conse
quently.also the ionization density in all layers increases with the
solar activity and with decreasing latitude of'the places.
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IRAN ANNUAL CRITICAL FREQUENCIES AND NUMBER OF SUNSPOTS
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point
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Latitude Longitude
Number of Critical frequency at noon
A Year Sunspots E F1 F2
1933
6
3.03
3.34

1934
9
3.10
3.28
5.71
1935
36
3.35
3.49
6.43
?
?
1936
80
3.72
4.34
9.34
Washington
38
50' N
77
0' W
1937
114
3.75
4.61
10.0
1938
110
3.74
5.21
10.03
1939
84
3.73
5.26
9.57
1940
69
3.52
4.58
8.43
?
1941
48
3.37
4.43
7.36
Huancayo, Peru
22
3' S
75?20' W
1939
84
4.04
5.46
10.86
Watheroo, Australia
30?19' S
?
115?53' E
?
1939
84
4.44
5.16
10.52
Tomsk
m
56
30' N
?
84
54' E
?
1939
84
3.3
5.2
9.5
Tro
so, Norway
89
40' N
18
95' E
1939
84
3.07
4.87
7.48
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In exactly the same way, the ionization of the Fl and E layers,
as shown above, depends on the zenith distance of the sun, and there
fore has a diurnal and seasonal march. Thus, in the course of a day,
the ionization increases from midnight, reaching a maximum at noon,
and then symmetrically declines to reach a minimum at midnight again.
The F1 and E layers are observed only in the daylight hours
and at latitudes where the zenith distances of the sun are less than
70 . At great zenith distances, i. e., in the evening and Might hours,
and also at high latitudes, the F1 layer disappears,:szbdmerges with
the F2 layer.
She region in which layers are observed is termed the F
region.
By using equation 7.6 and the data of Table 22, it is easy to
obtain the value of the ionization density in the E, F1 and F2 layers.
Thus, if the ionized particle is considered to have the mass of an
electron, then its mean ionization is of theorder N = 1.5 x 105 for
5
the layer; N = 3.0 x 105 for the Fl layer; and N = 10 x 10 for the
F2 layer.
The fact of double refraction in the Fl and F2 layers and its
absence, in most cases, in the Elayer is of importance, for it gives
us grounds to hold that the F1 and F2 layers consist of electrons while
the question of the composition of the ionized particle in the E layer
still remains open, since double refraction in the layer is alet very
seldom,. excepting in the high layers.'
Figure 104 shows a typical photogram of the heightfrequency
characteristic with a reflection from the E layer, and from the F1 and.F2
layers, obtained by means of an automatic record of the reflected waves
of various frequencies. The frequencies areplotted along the abscissa
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axis in megacycles, and the effective heights in kilometers on the
ordinate axis. As will be seen, the low frequencies are reflected
from the lower layer at a height of the order of 100 km (the E layer);
at a frequency of about 400 KZ the reflection is interrupted and be
gins again from a higher layer at a height of about 150160 km (the
F1 layer). With increasing frequency, the height of the layer increases,
and at a frequency of about 5000 KZ a new reflection corresponding to
the F2 layer at a height of 280 ka, begins. At a frequency of over
6000 kz, the height of the layer increases sharply, and the curve, by
bifurcating, moves upward into branches] and the reflection stops.
The bifurcation of the curve corresponds to the reflection of
the ordinary and extraordinary wave. The upper, fainter curve is the
result of being twice reflected, as a result of which its ordinate,
I. e., its height, is double that of the lower curve.
FigurO 105 gives a graph of the ionization density against
height obtained by working up a photogram, and indicates the paths
of rays of various frequencies emitted from one point of the earth's
surface and'received at another point.
4. Composition of the ionosphere and formation of ionized
layers., On the basis of data on the spectra of the aurora and
luminescence of the night sky, it will be assumed that at heights
100120 km (the E layer) that the ionosphere consists of molecular
nitrogen N2 and molecular oxygezt, 02. Above 120 km, the dissociation
Of the oxygen begins, and in atomic oxygen, 01, begins to predominate
in the composition of the ionosphere. For. this reason it is usually
considered. that the E layer consists of ionized nitrogen,.N2, when
it bifurcates into two layers, layer F1 consists of N2 while layer F2
consists of O1. In higher regions dissociation of nitrogen begins
and atomic nitrogen, N1 begins to play a role.
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An entirely definite energy (the .ionization potential) is re
quired for the ionization of oxygen and nitrogen molecules or atoms.
This energy has the following values expressed in electron volts
(1eV = 1.59 x 1012 erg):
energy 12,5 13.5 15.8 14.5
This energy may be obtained from solar radiations which con
sist of two forms: corpuscular and waves. The role of the corpus
cular radiation, however is very insignificant and is manifestedonly
at moments of enhanced solar activity. on normal days, however, as
shown by research, the principal ionizing factor is the wave radiation
in its shortwave portion. Indeed, from the condition that the energy
of a light quantum P must be not less than the value of the ionization
potential above mentioned, it follows that ionizing radiation must have
a wavelength between 850 and 1,000 A.
Alongside of ionization the reverse phenomenon also takes
place, namely the recombination of ions, which proceeds faster the
greater the density of the atmosphere. At a height of the order of
100 km, the equilibrium state between the processes of ionizationand
recombination is rather rapidly established, and therefore the density
of ionization (the number of free electrons per cc) at these heights
must be a function of the solar altitude. At great heights, *win
the low density of the atmosphere, there is alag in the processes
phase with respect to the solar altitude.
i L
Since the density of the atmosphere decreases with increai ng
height, while the ionizing radiation cannot penetrate into the low
layers, the ionization density must reach a miximum;at a certain height\,
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leading to the layered structure of the ionosphere. The existence
not of a single layer but of several is explained by the complex
composition of the atmosphere and by the different height distribu
tion of differenttelements.
The principal ionizationrecombination equation determining
the state of the ionosphere has, as is well known, the following
form:
ation, and q' the number of electrons recombining per second.
newly formed per second under the action of the ultraviolet radi
where ne is the density of ionization, q the number of electrons
Obviously the value of q is proportional to the intensity
of the incident monochromatic radion W, that is, to the quantity
of energy passing per second through lcm2 of surface and normal to
the radiation, and to the number of neutral particles n  ne, and
is inversely proportional to the quantity of energy w0 which is
necessary for a single event of ionization, i. e.
q s P(n  ne) r'
TO (?.11)
?
where T is the, coefficient of photoabsorption, n the number of
neutral particles per cc in the unionized atmosphere, and ne the
number of electrons per cc.
The number of q is proportional both to the number of
q' : xnen+,
where xis the coefficient of electronic recombination.
Since ne xn.~, then
266
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equation 7.10 may thus be written in the form
dne = q  xn2.
d e
(7.12)
In additionto the direct processes of photoionization of
electrow and of their recombination, a considerable number of other
processes must also exist, as a result of which electrons appear
and disappear; these processes include: the adhesion of electrons to
a neutral particle, leading to the formation of negative ions; the
separation of electrons from negative ions; and the recombination
of ions. In reality, therefore, equation 7.12 is more complex in
form, but, as shown by calculations, it may be brought into the same
form 7.12 if alpha is taken to mean a certain effective coefficient
of recombination depending on the coefficient of adhesion, separation,
and recombination of ions.
Ya. L. Al'pert gives the following values of q, ne and alpha
for various layers of the ionosphere:
Layer
ne (electrons) per cc
a (cc/sec)
q
E in summer
F1 in summer
F2 in winter
1.5
x 105
3 x 105
2.5
x 106
108
3 x 109
.1.5
x 1010
200
300
800
These vales relate to a time close to noon, to maximum
solar activity,,and measurements in the middle latitude of the
northern hemisphere.'
To find the height of the layer where ionization is maximum,
let us now return. to equation 7.11 and substitutein it, for the
quantity W, the int nsity'of radiation beyond the boundaries of the
ionosphere. For this purpose, let us consider the passage of radi
ation at the angle z to the vertical through a plane layer of an
atmosphere.of thickness dh. The losses of energy in this case will be
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dW =6 (n  ne) W sec^;dh.
On integrating this equation between the limits from height
h to infinity, we obtain: C:f M
x
We h h
If we substituted this value in the equation 7.11 we have
W  S sec z (in dh  cnedhl.
q= _2. (n  ne)e h
wO
(7.13)
The integrals in the exponent of the power of e represent
the total number of neutral particles and electrons in a column
1 sq cia2 cross section extending in height from level h to infinity.
The maximum number qm and the height hm at which this forma
tion takes place, the finding of the maximum number qm and the height
hm at which such formation takes place is very much simplified if it
is assumed that the layer is almost unionized, i. e., if we neglect
the value of na.
In that case: m
q "t Ae19 sec z ~ n dh
wO h
(7.14)
The number of neutral particles, n, in this equation, is
determined by the barometric formula
ah
n = nOe
in which the constant a has the value:
A : , IYA
where his the molecular weight of the gas, R the, gas constant, g
the acceleration of gravity, and T the absolute teviperature.
The height hm atwhich q reaches its maximum value, is found
from the equation
d q
dh
 t sec
[in dh  fuie('hi.
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This equation gives
1 nesecz
I'm  a log 6
a
(7.16)
It follows from equation 7.15 that at this height n takes
nm a..'
sec z
on substituting the expressions for ha and nm in equation 7.14, we
obtain the value of the maximum of q:
w0 a see z
Equation 7.16 shows that the maximum formation of photoelectrons
at a given zenith distance z is determined by the value of T and the
molecular weight f of the ionized gas, since the coefficient a depends
at constant temperature only on .
To find the height hm of the ionized layer at which the maximum
formation of electron qm takes place, the term containing ne in equation
7.13 must also be taken into account. Ili that case the condition of
the maximum a h = 0 yield:
'n ~ ne)2 sec z ,
h d h
whence as it is from this that We find the value of h
But the maximum number of electrons formed is still insufficient
for the appearance of a layer with maximum electron density me. For
this it is necessary that the disappearance, I. e., the process of',re.
combination, shall not coepensate the maximum ne. The existence of
layers with a maximum of density shows that such compensation in
reality.does not exist.
To find the heightof the layer the maximum density of
0 with respect
ionization it is necessary to solve the equation
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to h, but for this we must know the function of no, and this is
found from the ionizationrecombination formula of equation 7.12.
It must be borne in mind that the maximum n0 must not by
any means coincide with the maximum of q.
But the solution of equation 7.12 meets with difficulties,
since up to now the numerical value of the coefficients characterizing
one process or another of ionization and recombination has not yet
been established. As exactly the same value of the coefficient ?t
has been established on the basis of quantum calculations, only for
atomic oxygen 01.
5. The conductivity of the ionosphere.
Thanks to the presence of free N ions in the atmosphere, it
becomes electrically conductive, and by the electronic theory, its
conductivity,is expressed by the relation:
I
(7.18)
wherer is the radius of eddying vertical motion of the electrons
about the""Lines of force of the magnetic fields'N, determine the equation
Y We 2 (JI17)
,
2mv
where t is the mean free path of a particle, v the mean velocity of
the thermal motion of the particle, and m its mass. But if the
ionized layer is in a magnetic field, then the conductivity in the
direction of the field R does. not remain the same as in the absence
of the field, and in the direction perpendicular to the field, the
conductivity ' now becomes equal to
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The mean free path of a molecule is determined, according to the
kinetic theory of gases, by the relation
\2rcnd2 x 1, (7.20)
where n is the number of molecules for 1 cc cm3 and d the diameter
of those molecules.
Table 23 gives the values of n and 1 for various heights,
calculated from equation 7.20 and the barometric formula 7.15.
TABLE 23
NMER OF MOLECULES AND HEAN FREE PATH AT VARIOUS HEIGHTS
140 200 300
0 100
6
h, km 15
1.6. x 104 4.4 x 1012 2.5 x 108 7.7 x 106
U 9. 9 x 10
7
1
10
57 10
1, cm 0.03 1.5
The radius of vertical motion of the electron depends on the
magnitude on the magnetic field strength, and therefore, for particles
of the moving at the same velocity, they will differ at different lati
tudes. It takes the greatest value at the magnetic equator by H = 0.3
0e. Therefore, for electrons on the equator, r = 1.3 cm, while for ions
r = 20 cm.
Consequently the ratio 1/r for electrons takes greater values
for height over 100 ka, and for ions, for heights over 150 km.
Table 24 gives the values of the specific conductivities of
various layers, calculated by formulas 7.17 and 7.18 in the CGSV6ystem?
It was assumed for this caatlation that the velocity of thermal
motion of particles v corresponds to the absolute temperature T = 360?,
which, as' shown by radio measurements, the E layer does have.
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VALUES OF THE SPECIFIC CONDUCTIVITY IN VARIOUS LAYERS
Ye11
Yu11
Yu 1
0.87.
1032
3.66
?1014
1.25
1011
0.83 .
1011
0.62
'10 11
F1
1.6
1032
3.3
1014
2.48 ?
10 11
...~
1022
P'
1022
F2
6.4
1032
12.2
1014
9.1
10 11
1023
N
1023
At greater heights the temperature in all probability is higher,
but the experimental data on the temperature of the upper layers is still
inadequate, and therefore it has been taken at one and the same for all
layers. This temperature corresponds to a velocity v = 6.8 x 106 cm/sec.
6. The sporadic layer.
Besides the E, F1 and F2 layers a layer is often observed at the
height of the E layer with a critical frequency exceeding the critical
frequency of the E layer, and sometimes even the critical frequency of
the F2 layer which points to the existence of reasons with an enhanced
degree of ionization in the E layer. Since this layer is not found
everywhere nor is it found at different times of the day and year, it
has been termed the sporadic layer or E. layer. The sporadic layer has
been found to appear most often in the middle and high latitudes and to
have a local character. In the middle latitudes the E. layer appears
more often in summer, from May to September.
Observations show that the waves reflected from the sporadic
layer contain both an ordinary and an extraordinary wave. This gives
grounds for assuming that the sporadic layer has an electronic structure
but the nat~ln of its origin has notyet been elucidated.
7. Tidal phenomena in the ionosphere.
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Observations on the reflection of radio signals from the E
layer have allowed the establishment in this layer of the existence
of semidiurnal fluctuations at height corresponding to the tidal
phenomena caused by the moon. The am 1itude of such fluctuations,
as shown by the observations, reach 1 km, and the maximum occurs
3/4 hour before the superior combination of the moon, as will be
seen from Figure 106 which represents the results of direct obser
vations of the fluctuations of the E layer.
On the earth's surface tidal phenomena of the atmosphere
caused by the moon are also observed, but these phenomena are mani
fested in the form of semidiurnal fluctuations of the atmospheric
pressure p with amplitude of about 0.0000115 p.
If the fluctuations of height in the E layer with amplitude
4 h = 1 km are converted into fluctuations of pressure, then, ac
cording to the barometric formula of equation 7.15, we obtain
? p = paL h = 0.08 p,
i. e., 7,000 times greater than the relative fluctuations at the
earth's surface.
Recent investigations show the existence of such fluctuations
not only of height but also of the critical frequency.f, i. e., the
density of ionization. Figure 107 shows the semidiurnal tidal
fluctuations of the equivalent height and critical frequency in the
F2, layer. It will be seen that the amplitude of the fluctuations
of height in the ,F2, layer are twice those in the ,E layer, and that
in general, as shown by observations, the amplitude of both A h, and
f increase with the height of the layer.
The lunar tidal motions of the atmosphere may be regarded as
the propagations of plane waves in a medium with a certain coefficient
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of refraction. If such waves at a certain height meet a region of
the atmosphere where the index of refraction is equal to zero or has
become negative then these waves will be reflected and an oscillatory
motion will be set up between the. earth's surface and this region.
The region of the temperature maximum at a height of the order 1530 km
is believed to be such a reflective region for oscillations at the
earth's surface.
To explain the lunar fluctuations in the E layer it is assumed
that in addition to the second region of temperature minimum at height
80 ka, there is also a third region at height 140160 km and the fluc
tuations here take place between the second and third regions.
The existence of fluctuating motions in the ionosphere is very
important for the construction of the theory of the diurnal variations
of the magnetic field of the earth since one of the possibilities of
explaining such variations is the motion of ionized layers in the
earth's magnetic field.
The existence of the conducting layers of the atmosphere at
heights of 100 to 300 km gives ground for asserting confidently that
cause of the diurnal variations and in general of all the magnetic
variations are the electric currents in these layers. But to explain
the causes of these currents we still have no well established fact
which would allow us to take one explanation or another and consider
it correct. All the existing theories of the diurnal variations thus
reduce to a choice of the mechanism of,the origin of the current each
of which mayclaim's more or less degree of authenticity.
Up to now there have been 3 principal theories of the diurnal
variations: the theory of the atmospheric dynamo proposed in 1872
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and worked out in detail somewhat later (1889), the diamagnetic theory
(1928), and the theory of drift currents (1929).
Recently in connection with the work of I. Ye. Tamm (cf. infra)
the diamagnetic theory has already lost its importance, while the
theory of the atmospheric dynamo as a result of the periodic notions
movement discovered in the ionosphere has acquired a higher degree of
credibility.
The theory of drift currents thanks to the same work of Tame,
has found justification and may on the same level as the theory of
the atmospheric dynamo claim a certain standing of credibility. It
is therefore necessary to dwell on each of these theories and to point
out their shortcomings.
1. The theory of the ata ,s1heric dynamo. This theory proposed
by Stewart Shuster (Bibs 55) is based on the principle of the induction
of electromotive force on the motion of a conductive in the earthIs
magnetic field in a manner similar to what takes place in a dynamo
and for this reason the theory is called that of the atmospheric dynamo.
Since the high layers of the atmosphere possess conductivity, it
follows that when they move in the magnetic field of the earth H, an
electric field perpendicular to H and to its direction of motion is
produced and its strength is determined by the Faraday law:
E = [uR],
where us is the velocity of motion of the ionized layer. Under the
influence of the field E a current whose direction is perpendicular
to u and H,horizon in the atmosphere while the density j is determined
by the equation
j E,
where y' is the conductivity determined by equation 7:,18 since j is
directed perpendicularly to H.
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By substituting for y ' its value taken from equation 7.18
surface of the earth.
of the diurnal fluctuations of atmospheric pressure observed on the
displacement of the upper layers of the atmosphere under the influence
j Ne2
2mv 1 + (r)2
Shuster assumed that the velocity u was due to the horizontal
Barometric observations show that the pressure has 2 fluctua
tions during the course of the day: a regular one with a semidiurnal
period and an amps tude of ^ar1egagregular one with a
diurnal period and an amplitude of 0.3 mm. The causes of these fluc
tuations may be either the tidal or the thermal effect. The air
currents leaving the high pressure regions located in the meridian
at the equator form electric currents of the same character as shown
in Figure 73. During the daytime the currents moving northward in
the vertical magnetic field Z, create currents directed westward
which as a result of the reduction in conductivity in the atmosphere
with increasing latitude lead to the formation of closed currents
causing the diurnal variations of the magnetic field.
while according to this theory they should occur between 14 and 16 hours.
of the; northern component are observed Eros 10 to 11 hours (Figure 69)
still markedly different from them in phase. Thus the extreme values
incide with those calculated on the basis'of magnetic observations are
ation of barometric pressure although they do in their character co
as a result of the notion of the atmosphere owing to the diurnal vari
Investigations have shown that the electric currents obtained
Such a disagreement may be explained by the horizontal displace
cents in the upper layers due to the fluctuations of the barometric
pressure are assumed to be the same as the displacements observed on.
the surface of the earth, which might not in. reality be the case at all.
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The contradiction with theory was found in the value of
the conductivity. if the velocity u is considered a consequence
of the diurnal barometric fluctuations, the amplitude of which
amounts to 1 mm, then for the equator it should have a value of
the order of 30 cm/sec.
On the basis of the current distribution map (Figure 73)
the strength of the current f lowing through a cross section of
the layer 1 cm wide and with the width equal to the thickness of
the layer is equal to 3 x 10 _g MY, while the vertical component
of the magnetic field Z at latitude 300 equal to 0.3 Oe, and thus
the conductivity of a vertical column of the ionosphere 1 cm2 in
cross section and with the height equal to the thickness of the
layer is determined by the equation:
5
3 Y r" 30?10.3 3 ' 103 CGS .
The thickness of the E layer is of the order of 50 km and
therefore the specific conductivity of the E layer should be
".( 1011 CGS +?t .
On comparing this value with the values for the conductivity
in Table 24 we see that they agree with the value of the ionic con
ductivity but area thousand times smaller than the electronic con
ductivity. For this reason in order to reconcile the theory with
the observations it would bie necessary to assume either the existence
of a predominance of ions in the E layer, or the existence of
velocities of displacement of the ionized masses of the atmosphere
that are many=times as.great as the velocities observed at the sur
face of the earth (30 cni/sec).
At the present time neither of these possibilities are ex
cluded, since occasionally we do succeed in observing the splitting
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of a reflected ray in the E layer into 2, an ordinary and an extra
ordinary with a frequency difference corresponding to the ionic
nature of the layer. But the existence of fluctuations of pressure
in the E layer 7,000 times as great and the fluctuations at y the
surface of the earth give grounds for considering that the velocity
of displacement must also be many times greater than 30 ca/sec.
2. The theory of the drift current;(Bibl 56). According to
this theory the charged particles in the region of long free paths,
moving in the magnetic and gravitational fields of the earth must
experience a translatory motion (drift) in a direction perpendicular
to the force of gravity mg and to the magnetic field H. The velocity
of this drift would be :
(7.21)
where ~ is the angle between H and the force of the gravity or the
vertical. Here H and a must be expressed in absolute electromagnetic
units.
U sinC ,
strength H may be represented in the form:
H H0 \11 + 3 sin 2
since the magnetic field of the earth may be considered in
first approximation as the field of an elementary magnet its field
(7.22)
where H0 is the field strength at the equator and, is the latitude.
Moreover for an elementary magnet the following relation
holds true:
cote = 2 tan4 .
For, this reason expressing sin ~ in terms of tan in equation
7.21 and substituting tan according to the following formula, we
eH rl sin
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On substituting for N its expression in equation 7.22, we obtain:
. cos p
ONO
1 + 3 sin2i~
Thence the velocity of drift of the electrons at the pole,
u9o = 0, and at the equator uo = 1.2104 ca/sec, while the drift
of the ions of the atmosphere, the mass of which a = 5 x 10'23 g,
is zero at the pole and 10.2 ca/sec at the equator.
The current density is obviously expressed as:
j = eNu MILN cos L~ IF CGS J.
NO 1 + 3 sin
Since both electrons and ions participate in the formation
of the drift currents, it follows that if the number of ions is
considered equal to the number of electrons, and the density of
ionization in the upper layers N = 106, we have for the density of
the dint current on the equator the value j = 1.7 x 1013 CGS~.
Such a drift, or, to put it differently, displacement of
charged particles, produces a current in the direction from west
to east, since the magnetic field is directed from south to north.
The drift of positive ions to the east causes the accumulation of
positive charges on the morning side, whilethe drift of negative
ions causes their accumulation on the evening side. As a result,
in the region of long free paths an electrical field arises which
is directed toward the west and which, however, may be electrically
connected with the lower region of short free paths, since the ions
may move freely under the influence. of the force of gravity along
the lines of force,of the magnetic fields, especially around the
pole. For this reason, in the lower, part of parts currents arise
which with acorresponding distribution?of activity say fora a
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ably lower than that required by the diamagnetic theory.
Figure'73 indicates that the conductivity may be considered consider
currents corresponding to the calculated system of currents shown in
culation of the conductivity necessary for the formation of drift
system of currents similar to the one shown in Figure 73. The cal
The theory of drift' currents, however, encounters obstacles
in explaining the forenoon extreme extremum of the northern and
eastern components, andalso in explaining the intensity of the ob
served variations. In addition some authors have expressed doubts
that drift currents could lead to the formation of so complex a
system of currents as the theory requires.
pressure of the ion gas p',which is defined by the equation
the atmosphere, the own weight of the ions is balanced by the partial
force of gravity. In reality, however, in the equilibrium state of
absence of a magnetic field, fall downwards under the action of the
which are free and independent in their motion and which may, in the
an equilibrium system, ionized gas consists of an aggregate of ions
that were established by Chapman. According to Chapman's belief, in
unsoundness of this theory if we start out from the sane propositions
drift current theory and for the first time pointed out the complete
I. Ye. Tamm (Bibl. 57) who has given a criticism of the
p' = kTN
where T is the absolute temperature, If the density of ions, i. e.,
their number in unit'volume, and K is the Boltzman constant. For
this reason, in the equilibrium state of the atmosphere, when the
density of the ion is determined by the barometric formula, there
can be no drift, and, consequently, the Chapman theory,just like
the diamagnetic. theory, is a simple misunderstanding.
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But in this saws work, Tama also pointed out the possibility
of retaining the drift current theory explains the diurnal variation
if we start out not from the equilibrium,cofdition of the ionosphere,
but from the existence in it of a "nonuniform" distribution of
densities that is, of densities varying otherwise than by the baro
metric formula. Such a distribution, in Tam's opinion, corresponds
moreto the real conditions of the state of the ionosphere, which
also makes possible the existence of drift currents analogous to
those of Chapman. Tam gives a derivation of the formula for the
density of the drift current on thebbnsis of a statistical model of
the ionosphere obtained by solving the Boltzman kinetic equation.
Since in its general form the Boltzman equation is very complex,
Tama introduced 2 assumptions to simplify it: first that the distri
butions of ions and electrons by velocities in each element of volume
of the ionosphere differs little from the Maxwell distribution cor
responding to the temperature T, and secondly that the ordered motion
of ions of the atmosphere is quasi stationary, that is, at each
given moment the motion is determined by the instantaneous distribu
tion of the ion density and the temperature, as a result of which
the derivatives of density and temperature with respect to time may
be neglected and the density and temperature considered assigned
functions of the coordinates.
v grad f + (g + ! B + ~~ [fl]) grad f =4 , (7.23)
motion of each species of ion will have the form:
Under such assumptions, the kinetic equation of the stationary
gravity, B the electric field strength, H the magnetic field strength,
where v is the velocity of motion of the ions, g the acceleration of
0
differs from the Maxwell function by the small quantity l' e.,
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e/m thslchargeratio, f a function of the distribution of ions; which
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f = f0 (1 +tp ,
while 3 3' 2 2 a
f 0 = N it 2e  v , and " YET .
The term Pf, taking account of the collisions between the
ions themselves and with neutral and between ions and neutral mole
cules, is assumed to be proportional to the deviation of the distri
bution function f from its equilibrium value fo, 1. 0.,
Pf w =& (f  f0)=  ~f0 4).
where \ is the length of the free path.
After substituting the values f, f0 and Pf, equation 7.23
(e [vi] grads) + =' [vq]
 ,2v2 b);
adT
a = g ?. ! B  21 grad In (~ 3/I , b gr
p T
The solution of this differential equation is in the follow
ing form:
2
.:q(2 a
S24g2v2
[~v(vq + tv [qH] + v (vqt
where q_L is the vector perpendicular to the magnetic field H, q,` is
the vectorparallel to that field, and.H is the ratio between the free
path and. the diaiseter of the circular orbit of a particle. in the mag
netic field, I. a.
aIN
2cvt2mkT
.The density of the current forced by the motion of charges of one
sign is determined by the' laws of statistics as:
+ 4 )dvx dvy dvx .
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For the case X p,
.
s2
2
c s
R
2
34)
(7
3
g= ( ..
)
.
Consequently the integration of the system of equation 7.25
reduces down to the integration of equation 7.33 and 7.34.
Equation 7.33 shows that they determine the trajectory of the
notion of a charge in the plane passing through the axis z and the
radius vector R. The equation 7.34 however determines the rotation
of this plane as a whole about the z axis.
Thus the problem of determining the notion of a charge in
space reduces to 2 problems: determination of its motion in the
plane xR and of the rotation of this plane about the x axis.
However, without solving the problem, but merely using the
properties of the function Q, it is easy to find those regions of
space in which the charge my nova.
Since, according to equation 7.34, Q cannot assmse negative
values, the coordinates of thecharge aunt always satisfy the in6
equality:
iO + R r2
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(7.32)
It is not hard to show that the righthand sides of equation
7..31 represent respectively the partial derivatives with respect to
R and z of the function Q, separated into 2, and thus the system of
equations in equation 7.27 may be replaced by the system:
or, substituting ~ (f/ a s from equation 7.30:
( )2, (~Z )2 1  ( + R3)2
~s bs R r
b2R 7Q; 2 : l
2
s'3 2 a R b s2
(7.33)
6z
BR 2 ( bz )2
s bs
(7.34)
Consequently the integration of the system of equation 7.25
reduces down to the integration of equation 7.33 and 7.34.
Equation 7.33 shows that they determine the trajectory of the
notion of a charge in the plane passing through the axis z and the
radius vector R. The equation 7.34 however determines the rotation
of this plane as a whole about the z axis.
Thus the problem of determining the notion of a charge in
space reduces to 2 problems: determination of its motion in the
plane zR and of the rotation of this plane about the z axis.
However, without solving the problem, but merely using the
properties of the function Q, it is easy to find those regions of
spaci',in which?the charge may move.
Since, according to equation 7.34, Q cannot assume negative
values, the coordinates of the charge must always satisfy the in
equality:
R R
+ 2 f a. magnetite crystal at room temperature. It may be seen that
saturation along the axis of easiest magnetization is attained in a
= 318 
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field of approximately 300 Oe intensity and is equal to 430 CGSJ
i.e., almost 4 times less than the saturation of iron. Along the
axis of difficult magnetization saturation requires a much greater
field. The initial magnetic susceptibility of pure magnetite is
of the order of 8 CGS). units, while the residual magnetization is:
for the triad axis Jr = 80, for the twinning axis Jr = 70 and for
the tetrad axis Jr = 90 CGS.
Magnetite ores consisting mostly of magnetite have a poly
crystalline structure and therefore their magnetic properties do
not depend on the direction of magnetization. Due to the different
admixtures of such ores obtained from different deposits their mag
netic properties also have different characteristics.
Figures 119 a, b, c and d show the magnetization curves of
magnetite obtained from 4 different deposits: Figure 119a relates
to a sample obtained from the Kola peninsula, Figure 119b to a sample
from the Vysokaya Mountain (Urals), Figure 119c to a sample from the
TemirTau Mountain (Urals) and Figure 119d to another sample from the
";r.als the exact origin of which is not known. The first sample con
tains 43%, the second 40%, the third approximately 60% and the fourth
50% magnetite. The remainder consists of crystalline inclusions of
nonmagnetic nature.
Comparison of these curves with the curve of pure magnetite
shows the extent to which":the magnetic properties of ores vary, de
pending upon their mineralogical composition. The basic magnetic
chairaicteristics listed in Table 25 for the 4 samples of magnetite
correspond to the .~.,CMr'ves .
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Samples
Magnetic Characteristics
No 1
No 2
No 3
No 4
Initial magnetic susceptibility
0.2
0.9
1.9
0.25
Residual magnetization J (CGS P)
12
20
4.1
7.0
Coercive force in Oesteds
30
20
7.2
40
Saturation (CGS P)
100
150


As shown by S. V. Lipin [61] pyrrhotite, the general formula
of which is Fen Snki constitutes a solid solution of ferrous sulfide FeS
with ferric sulfide Fe2S3 or ferosoferric sulfide Fe3S4.
Analyses show that the atomic ratio S:Fe varies from 1.20 to
1.00 assuming all the intermediate values, from approximately Fe5S6 to
Fe21S22?
Crystals of pure pyrrhotite belong to the rhombic system and
have the appearance of rectangular prisms, but regardless of the di
rection of the magnetizing field are magnetized only in a single plane
perpendicular to the axis of the prism, called the magnetic plane.
Within this plane there is a direction in'which magnetization occurs
readily (OX). Perpendicularly to the latter (direction OY) magneti
zation takes place with difficulty. These directions are called the
main axes of the crystal. Magnetization perpendicular to the magnetic
plane is detected only in very strong fields.
Magnetization curves corresponding to these directions for a
sample of pyrrhotine are shown in Figure 120. While, saturation along
the OX axis takes place at H = 1,000 Oe, saturation along the OY axis
is attained in a field H = 7,300'0s, and for a saturation along the OZ
axis a field of approximately .176,000 Oe is required. The magnitude
of saturation may vary, depending on the sample, from 17 to 70 CGS ?,
and the coercive force may vary from 15 to 30 0e. Crystals of pyrrhotite
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possess very sharp anistropic characteristics and are of greatest
interest from the standpoint of the theory of ferromagnetism.
Because of their polycrystalline structure natural pyrrhotites
like the magnetites, lost their anisotropic properties and are mag
netically isotropic. Their magnetic properties depend upon the com
ponents characteristic of any given deposit. Figures 121a and 121b
show the magnetization curves of 2 pyrrhotites, one of which (Figure
121a) was determined by N. I. Spiridovich [62) using a sample contain
ing 7.5% Fe203 and the other curve was determined by T. N. Roze [63)
using a sample containing approximately 20% magnetite and 10% chalco
IT,
pyrite. In the first sample the residual magnetization was Jr = 1.12
CGSJM and the coercive force He = 110 0e; in the second J2 = 8 COSj1
and H = 46 Oe. The experiments of T. N. Raze have shown that the mag
c
netic properties of pyrrhotites are greatly altered upon heating to
temperatures above 270 0 C. Thus Figure 122 shows the hysteresis curves
of pyrrhotite from a deposit located on the Kola peninsula, containing
75% pyrrhotite and 20% magnetite before heating, and after heating
above 270?.
The experiments have demonstrated that at 270? pyrrho
r
tite is irreversibly converted into a different phase which is retained
upon reheating.
The curves show that heating causes a sharp increase in
magnetic susceptibility and residual magnetization, while the coercive,
force has remained almost unchanged. On comparing magnetization curves
of natural rocks with those of crystalline pyrrhotite it becomes ap
parent that the magnetic properties of natural pyrrhotites are inter
mediate to the properties of the crystalline material along the 3
mutually perpendicular directions corresponding to their polycrystal
line structures in which the axes of easy acid difficult magnetization
are disposed in every possible direction.
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It was shown by studies of its crystal lattice that hema
tite (Fe203) occurs in 2 varieties: hematite, which has a rhombo
hedral system and a ntiferromagnetic properties, and \ hematite,
which has a cubic system and ferromagnetic properties.
An tiferromagneties belong to the group of paramagnetic
bodies but differ from the latter in specific magnetic properties,
namely a specific Curie point below which the magnetic suscepti
bility depends on the magnetizing field. For ckFe 203 this point.
is in the high temperature region, equal to 675? C [64].
Natural hematites always contain admixtures of other minerals
and therefore their magnetic property can vary depending on the
content of these admixtures, and depending also upon the ratio of
magnetic and paramagnetic varieties.
According to the investigations of N. I. Spiridovich and T. N.
Roze hematite ores, natural hematite isolated from the ore, and syn
thetically produced iron oxide have the magnetic characteristics
shown in Table 26. It is seen that:anspecific feature of hematites
is lp!sresidual magnetization and a high value of coercive force,
which becomes greater with increase of the magnetizing field.
Figure 123 shows the hysteresis curve of hematite ore from
Tuloaozero;,determined with a maximum magnetizing field value of
400 Oe. The curve reveals a sharp increase in,iduai'magnetiza
tion and coercive force with increase of the. magnetizing field.
Ilmenite (FeO ? TiO )'is a mineral componentof titanium
magnetite ores, i.e., a combination of ilmenite and magnetite at
various ratios, and is a paramagnetic mineral with a magnetic sus
ceptibility of approximately 60 ? 105CGS ? and a magnetic field in
tensity of 2,000 oe.
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TABLE 26
MAGNETIC CHARACTERISTICS OF HEU TITBS
Chemical Composition
Hm
Rocks and Deposits
Formula
%
Oersted
Hematite from Sareliya (Tuloaozero)
Fe203
69.8
1,880
FeO
0.8
900
TiO2
0.5
Iron ore froo Mount Magnitnaya
Fe203
92.76
3,500
FeO
0.98
280
TiO2
0.16
Iron quartzites froman unknown deposit
Fe203
40.5
3,500
FeO
0.46
280 <
TiO2
0.02
Magnetite powder isolated from ore
400
Synthetic Iron oxide
Fe?03
99.98
400
r
He
CGS;JJ
Oersted
Investigator
0.3
480
N. I. Spiridovid
0.037
188
0.67
185
T. N. Roze
30
0.24
'> 185
T. N. Roze
0.03
40
0.09
130
N. I. Spiridovich
0.03
80.0
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C
However in combination with magnetite it acquires strongly
manifested ferromagnetic properties which are distinct from those
of pure magnetite. A differentiating characteristic of titanium
magnetites is a greater coercive force than that of magnetite.
This is confirmed by the experiments of N I. Spiridovich and the
data of T. N. Roze.
Figure 124 shows the magnetization curves of titanium mag
netite samples containing 4% Ti02 and 50% and 70% Fe203 according
to the data of N. I. Spiridovich and T. N. Roze. The coercive
force of the first sample is 31 Oe, and of the second sample 20 Oe,
whereas in magnetites it is approximately 10 to 15 Oe. The basic
magnetization curve is somewhat lower than that of magnetites and
therefore the magnetic susceptibility of titaniummagnetites is
somewhat less approximately 0.20.4 CGS Jl. On this basis it may be
asserted that titanium oxide TiO2 although not magnetic, exercises
an influence on the magnetic properties of ferromagnetic minerals.
Limonite (Fe203? nH20) is a hydroxide of iron which forms
deposits of brown iron ore widely encountered in nature. Limonite
is only slightly magnetic, having a maximum susceptibility of ap
.proximately 100500 ? 10 6 CGS,/ but husclearly manifested ferro
magnetic properties. According to the investigations of N. I.
Spiridovich a sample of limonitecontaining 6% admixtures showed a
coercive force of'10 Oe and a residual magnetization of J = 0.011.
Igneous rocks are subdivided into acidic igneous rocks rich
in solicit'acid, including granites, granodiorites, diorites; basic
and ultrabasic rcks containing little solicic acid, including
gabbro, diabases,?basalts serpentines, peridotites, dunites,;vonp&y
rites and others, and rocks rich in alkalies such as 'syenites. The
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magnetic properties of each of these rocks vary within such wide
limits that no definite magnetic susceptibility or definite mag
notization curve can be given for them. Their magnetic properties
vary sharply, depending on the content of any given mineral compo
nents. However, numerous observations by various investigators
permit certain definite conclusions. Of the entire series of ig
neous rocks the most magnetic are the basic and ultrabasic rocks,
including gabbro, diabases, porphyrites, serpentines, basalts and
periodites. Most granites and granodiorites are practically non
magnetic. As is shown by observations the magnetic properties of
igneous rocks are due to the presence of ferromagnetic minerals,
but these properties are not additive since it is most likely that
the magnetic properties are affected not only by the amounts of
ferromagnetic minerals, but also by the form in which they are in
cluded in the rocks, whether they constitute individual inclusions
or form an intimate compound.
The experiments of N I. Spiridovich confirm the relation
ship between the magnetic properties of igneous rocks and the pre
sence in the latter of ferromagnetic minerals. The results of these
experiments are listed in Table 27.
This table shows there,is no definite regularity between
magnetic properties and the content of iron pr ferromagnetic material,
but there exists a general tendency of increased susceptibility with
increasing content of ferromagnetic components. In the absence of
ferromagnetics the rock becomes paraagnetic, which is confirmed by
one of the samples of metamorphic gabbro.
In view of this fact every known rock, may be characterized
only by the limits within which its properties may vary, these properties
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being manifested by the magnitude of maximum magnetic susceptibility
or merely by a qualitative evaluation as nonmagnetic, slightly mag
netic, magnetic, or strongly magnetic.
Table 28 shows the limits of magnetic susceptibility accord
ing to the data of various investigators.
Although the magnetic properties of igneous rock are charac
terized generally by the magnitude of magnetic susceptibility it
must be borne in mind that all igneous rocks are ferromagnetic and
therefore the susceptibility may vary rather widely for each igneous
rock type, depending upon its magnetization. Moreover every type of
ferromagnetic rock has hysteresis, as a result of which it may have
a residual magnetization. Figures 125, 126 and 127 show magnetization
curves and the correlation between magnatIz susceptibility and magne
tizing field for samples of diabase, gabLronite and gabbrodiabase,
according to the data of N. I. Spiridovich. The drawings show that
all 3 samples are strongly ferromagnetic
Sedimentary Rocks. Sedimentary rocks include various clays
and sandstones, limestones, gipsum, chalk, marls, dolomites, and
rock salt. With the exception of some varieties of clay and sand
stone, they are all practically nonmagnetic since their susceptibility
does not exceed 10"5 COSI~J .
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Rocks
Diabase magnetite 10.2 12.1
Diabase magnetite 27.5 30.6 88,006 7.34 75.0
Gabbro diabase
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magnetic mineral in % Total iron content :tn % X ? 106 J H
Jr c
magnetite 6.37
Gabbro diabase magnetite  4.0 12.1
Diabase
85,000 0.91 89.0
5,000 0.23 32.6
5,100 0.43 57.0
3,280 0.02 5.3
Metamorphic diabase limonite 1.8 240
Gabbronorite pyrrhotine 3.0 2.5 1,700 0.22 78.0
Metamorphic gabbro magnetite 12.1 15.0 12,400 0.3 21.3
Metamorphic gabbro ilmenite 2.12  510 0.04 58.0
Metamorphic gabbro none C 92
Pyroxenite pyrrhotine 2.5 7.7 820 0.05 46.0
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VIRG
Putsikha
Koenigsberger
Rocks
Number of Samples
x  106
Number of Samples
x ? 106
Number of Samples
x X106
Granite
171
04555
3
82720

4200
?Granodiorite
6
2002000
Diorite


1
47
Diabase
97
013820
2
64106
2
700
I
2700
,w
00
Gabbro
33
10007470
2
692370
4
3104100
Basalt
43
12515500
Peridotite
36
40072800
Porrphyrite
136
022700
3
45120
Syenites
38
06590
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Among sedimentary rocks of significance in magnetic pros
pecting are certain varieties of clay containing iron oxides, and
sandstones containing ore in the form of magnetite granules, or
containing other minerals, the magnetic susceptibility of which
may reach 103 CGS J, and therefore both can cause magnetic anomalies.
Table 29 lists the magnetic susceptibility of some sedimentary
rocks recovered from breholes in the Western Urals based on the
data of B. M. Yanovskiy and Ye. T. Chernyshev [651.
TABLE 29
ne
Susbagagnetic
MAGNETIC SUSCEPTIBILITY OF SEDIMENTARY ROCKS
biity
Borehold # Depth of occurrence
x ? 106
92.6  96.6
Marl, brown, with sandstone interlayers
85
140.5 146
Clay, brown, browngrey
239.6 241.5
Clay, browngrey, lime containing
157
318.1 320.2
Clay, brown
491.2 492
Clay, dark brown
5
1 558
Gypsum, white
.
588.
182.8 184.2
Clay, brown, soft and sandy
309.8 310.4
marl, PunpitzgreY, with calcite inclusions
0
351.8353.9
Sandstone, brown, many
365.4366.1
Marl, redbrown, dense
513.2516.9
Marl, pink, dense
553.8555.
Marl,, dolomitized
581.6583.6
Sandstone, reddish browngrey
Section 3. Residual Magnetization of Rock Formations and the Causes of
i18 Occurrence
At the beginning of the preceding section it,was stated that
most rock formations show magnetization which cannot be explained solely
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by the action of the magnetic field of the earth, since in such a
caso the magnetization of the rock would change if the rock were ro
tated 180? in relation to the terrestrial field. Moreover, observa
tions show that the magnitude of residual magnetization sometimes is
several score times larger than the magnitude of inductive magnetiza
tion j = xH. This is apparent from Table 30, which shows the values
of residual magnetization J , and its ratio to the inductive magneti
r
zation for some rocks.
Thus the question naturally arises as to how the rocks could
become magnetized if one does not assume the influence of extraneous
magnetic fields caused by magnetic fluxes, considering that nothing
isknown about the latter at the present time.
TABLE 30
j ? 103
Jr/JB
3.2
2.4
Diabase
2.9
1.4
Gabbro, postsilurian
42.0
28.0
Gabbro, tertiary
12
25.5
Gabbro, tertiary
14.0
9.3
Basalt metamorphic
1.6
6.4
Graniteporphyry
4.4
3:2
Basalt, tertiary
This problem was first encountered inthe interpretation of
the Kursk magnetic anomalies. Samples of magnetite from boreholes
in'the anomalous area showed such a magnitude of residual magnetiza
tion which could not be attributed to the action of the terrestrial
field. The first attempt to explain this phenomenon was made by P.
V. Lazarev, who assumed the existence of a more intensive magnetic
field of the earth during the epoch of the formation of these ore deposits.
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However at the present time such a hypothesis is disproved by a
large number of facts derived by measurement of rpsidualaagmeti
zation of sedimentary and igneous rocks of known ago. In &.1.1
probability the magnetic moment of the earth has undergone hardly
any change since the time the Kursk magnetites were formed.
More probable is the hypothesis of the influence of some
kind of physical or chemical factors on the process of magnetiza
tion of rock formations at the time of their origination. This
idea was first expressed by A. F. Ioffe, and the ti wst experiments
along this line were conducted in 1926 by A. N. Zaborovskiy, who
observed the magnetization of magnetite while it cooled from the
Curie point in the earth's magnetic field. T. N. Roze [63.] conducted
detailed investigations on the influence of temperature in the magni
tude of residual magnetization in a large number of rocks. These in
vestigations have shown that any ferromagnetic rock heated to the
Curie point and placed in a weak magnetic field acquires a residual
magnetization on cooling to room temperature within this field, the
magnitude of which is severalfold greater than that corresponding
to the given field.
Thus figure 128 shows the correlation between the magnitude
of residual magnetization Jr and the magnetizing field when magnetite
was listed to the Curie point and then allowed to cool (curve Jrt )?
The same ', correlation also is given for magnetization at room tempera
ture (curve''Jr). Comparison of these curves reveals that the influence
of temperature is greatest in weak fields in which Jrt is several tens
of tiffs greater. than Jr.
It is of interest to note that the residual magnetization
produced by cooling of the sample in a magnetic field almost coincides
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with the ideal magnetization obtained by the superimposition of a
variable with an amplitude decreasing to zero upon a permanent field.
Consequently, the action of temperature decreasing from the Curie
point is somewhat analogous to the action of a variable field of de
creasing'amplitude. The same properties also are possessed by some
of the presentday alloys used for permanent magnets, such as magnico.
The theory of this phenomenon has been developed by D. A. Shturkin
and Ya. S. Shur [66),
Figure 129 shows the curves of residual magnetization of a
specimen of pyrrhotite from the TemirTan deposit, which shows this
effect very clearly. From the drawing it is apparent that the ratio
Jrt/Jr in pyrrhotite has a value of hundreds of units in small fields,
i.e., magnetization accompanied by the action of temperature is hundreds
of times greater than under the action of the magnetic field alone.
Analogous curves also are obtained for other rocks such as
hematite, titanium magnetite, diabase, basalt, gabbro, serpentine,
andesite, pyroxene, and druserite, which have been investigated by
T. N. Itoze.
Another important factor affecting the magnetization of rocks
can. be mechanical action to which the rock is subjected, such as con
pression, elongation, vibration of seismic oscillations, which in
duce internal stresses in the rocks.
The influence of elastic stresses on the magnetic properties
of ferromagnetics has long been knownand is the subject of numerous
investigations, the theoretical foundations of which are found in the
work of N. S. _Akulov [67]. However, no investigation of rock forma
tions from this point of view has yet been carried out, and therefore
no quantitative data are available concerning the effects of mechanical
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action on any given type of rock. H. A. Grabovskiy (68] recently
conducted the first investigations of magnetite specimens from the
Blagodat' and Vysokaya mountains, including study of the effects of
unilateral compression on magnetization curvesand on the magnitude
of residual magnetization.
The results of his research are contained in Figure 130 which
shows the relationship between residual induction and magnetizing
field in the case of a specimen not subjected to deformation and of
a specimen subjected to the greatest unilateral ecnpression permit
ted by its durability. It is seen that the residual induction of
magnetite decreases on compression, i.e., the magnetite has positive
magneto striction, and therefore on elongation the residual induction
should increase. However, because of the brittleness of the specimen
it was not possible to confirm this conclusion experimentally. Due
to the lack of experimental material it is not known whether there are
rocks having negative magnetostriction, the residual magnetization of
which would increase on compression, but such a possibility is not
excluded.
Experimental data on the magnetization of rooks cooling down
from the Curie point, and the initial experiments on their magnetiza
tion during compression give reason for assuming that the observed
high magnetization of rocks is the result of their cooling, following
intrusion, in?the magnetic field of the earth, since there is every
reason for believing that`,crystallization'of the rocks took place on
their transition from the liquid to the solid phase. In addition,
the magnitude.of residual magnetization was undoubtedly affected by
mechanical deformations of the rocks induced by uneven compression
and bending. Since the earth's crust still is in a state of con
tinuous movement, which is manifested by the slow rising and sinking
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of dry land and by rapid earthquakes or volcanic eruptions, it fol
lows that the mechanical stresses which act upon magnetized rocks
are undergoing continuous changes, and consequently the magnetiza
tion of the rocks also must change. Therefore one of the causes
of secular magnetic variations must be the change in magnetization
of rocks due to changing mechanical stresses such as comparison,
elongation, etc. These phenomena must be manifested most strongly
within seismic areas in which the internal strains in the rocks
are suddenly changed during earthquakes. However, because at
present there is very little experimental material available on this
problem nothing is known of the regularities which are inherent in
these phenomena.
Section 4. Effect of Variations on the State of Magnetization of
Rocks
Changes in the magnetic field of the earth with lapse of time,
i.e., magnetic variations, must have an effect on the state of mag
netization of rocks if this magnetization is caused by the action of
the terrestrial field. This problem was investigated theoretically
and experimentally by the present author [69], in respect to the
Kursk magnetic anomalies, which are due to the occurrence of a sharply
dipping stratum of quartzites withvagnetite interlayers at a depth of
several hundred meters.
This stratum may be roughly compared to a rectangular prism
(Figure 169), the horizontal side of which (length bb') has infinite
dimensions and the vertical side has a(hight ac).
The prism is uniformly magnetized in the vertical direction
and its width ab is small'in comparison to the height, and thus the
influence of the bottom side cd may be disregarded.
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According to the deductions of Section 8 of this chapter the
vertical and horizontal components of the intensity of the magnetic
field of such a prism will have the form:
0110 7 l
wherein R is the depth of occurrence, i.e., the distance from the
earth's surface to the upper side of the prism, a the width of the
prism, x the distance from the axis of anomaly and j the magnetiza
tion of the prism.
With a change in the magnetic field of the earth the magneti
zation of the rock J, and consequently the magnitude of the compo
nents Za and Ha also will change. Therefore, on observing the changes
in anomalous field it is possible to ascertain the extent to which
variations of the terrestrial field alter the magnitude of magnetiza
tion of the rocks, and thus the question whether the magnetization is
of residual or inductive origin may be resolved.
Since the anomalous field reaches its highest value at the
center of the'anomaly the effect of variations must be strongest
over its center. Therefore let us consider the variationsof the
magnetic field over the center of the anomaly. At x  0:
and consequently, the variations of the anomalous field will be:
S o? az~, 2 r
where Jr is residual magnetization, x is magnetic susceptibility'of
the rocks and Z is the vertical component of the normal field; hence
n
10,4 )7,
The observed variation will be the sum of the normal variation
and the anomalous variation ( Z., i.e.,
r Zn
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X ,}7
Thus the variations of the vertical component over the center
of the anomaly must be proportional to the normal variation and as
is shown by equation (8.1), the proportionality coefficient always is
greater than one, therefore
If the residual magnetization is small and it may be disregarded,
Z G.
At the Kursk anomalies Za/Zn 4, therefore the variation over
the center of the anomalies in this case should be 5 times greater
than the normal.
On the other hand, if we disregard the inductive magnetization
xZ , it follows that
6 Z : (/ I r
In the case of the Kursk anomalies Za = 1.5, and according to
the findings of N. K. Shchodro the magnetic susceptibility and residual
magnetization respectively are x = 0.1, and Jr = 2.5. Hence
i.e., in this instance variations of the vertical component should be
only 5% greater than the variations in the normal field.
Finally, if the susceptibility of the rocks is very small it
follows that, with any value of residual magnetization Jr:
i.e., the permanent magnetization of the rocks does not alter the
varia*ions of the normal field.
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Observations conducted in 1936 by the present author within
one area of the Kursk anomalies have shown that variations within
the anomalous area are of the same order as those in the normal field.
The observations were carried out simultaneously at 2 stations, one
of which was located at the maximum of Z, and the other at the maximum
of H. These observations were compared with the observations at the
Nizhnedevitsa observatory, located within the normal field approxi
mately 60 km from these stations.
The results of the observations are shown in Figure 131 in the
form of graphs of the mean monthly diurnal variations of vertical,
north and east components. These curves reveal that the variations
of all the components are of the same order both at the stations and
at the observatory. There are certain deviations in the amplitude
of north and vertical components, but the phases are the same for all
the components. However, these deviations must be attributed to
errors in measurements rather than to the effect of the variations
on the change ;;n magnetization of the rocks. Thus these observations
fully confirmed the theoretical deductions that variations of the
magnetic field have a~very slight effect on the magnitude of the
anomalous field, if the latter is causedby residual, and not by
inductive magnetization of the rocks. As was shown directly by the
experiments the residual magnetization of the rocks of the Kursk
anomalies is several score times greater than the inductive magnetization.
Section 5. Methods of Investigation of the Magnetic Properties of Rocks
In view :of the great variety of magnetic properties of different
types of rocks, pre edures for the investigation of magnetic properties:
also are diversified. Certain methods are applicable in the case of
strongly magnetic rocks, while slightly magnetic rocks having para
magnetic properties necessitate the use of other, iaore sensitive methods.
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In addition, the requirements of practical operations dictate specific
prerequisites which necessitate modification of the accepted methods
or the development of new methods. Thus there is no single invostiga
tive procedurett, There are a number of methods and a number of instru
ments of specific design, the use of which depends on the properties
of the rocks as well as on the conditions under which the investigations
are carried out.
Investigation of magnetic properties consists of the measure
ment of magnetic permeability, if it does not depend upon the magneti
zation J, determination of the basic magnetization and hysterosis
curves and determination of the residual magnetization of the rock in
its natural .state.
The ballistic method is entirely suitable for determination
of the hysteresis curve of strongly magnetic rocks such as magnetite,
while the magnetometric method is adequate in the case of less mag
netic rocks. The magnetometric method is more convenient for deter
mination of residual magnetization.
The method of attraction and repulsion within a nonuniform
magnetic field is most convenient for the determination of magnetic
susceptibility of para and diamagnetic rocks which are uniform in
composition and in which the magnitude of susceptibility is of the
order of 105 to 106 CGSJ.
However, rocks which4re not uniform in composition are more
conveniently tested by means of a magnetometer, since in this case
the more characteristic quantity is the mean susceptibility of a
specimen of large volume, the measurement of which is impossible in
principle by the first method.
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The selection of the procedure to be used also depends to a
large extent upon the shape of the specimen. In Section 9 of the
introduction, it was shown that the magnetization curves of ferro
magnetic specimens consisting of the same material, but having dif
ferent shape differ considerably from one another, and therefore
the characteristic of magnetic properties is considered to be the
susceptibility, or the magnetization curves of specimens having a
closed magnetic circuit. For transition from an open to a closed
magnetic circuit the demagnetization coefficient N of the specimen
being tested oust be known.
In Section 9 it also was stated that the demagnetization co
efficients are known only for specimens having the shape of an ellip
soid or cylinder. Therefore the ballistic and magnetometric methods
may be utilized only when the magnetic susceptibility of the rocks
does not depend upon the shape or where the rocks can be shaped into
a cylinder.
The difficulties encountered in working specimens of hard
rocks into a regular geometric shape have necessitated the develop
ment of specific methods.of measurement which do not depend upon the
shape of the specimen. However, these attempts have not provided a
11,,
complete substitute for any of the above measux,#ment methods, in view
of the greater accuracy of the latter.
Thus in addition to the approximation methods used mostly
under fieldl`condltions?, the ballistic and magnetometric methods
remain the fundamental methods for measuring the magnetic properties
of rocks. The method of attraction within a nonuniform magnetic
field. is used exclusively in the case of rocks having pars and dia
magnetic properties.
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1. Ballistic Method. The ballistic method of measurement
of the magnetic quantities of a specimen is based on the generator
of a current within a conductor surrounding the specimen called
the measuring coil, upon variation of the magnetic flux within the
specimen from + # to  # or from + # to 0. The induction current
is measured by means of a ballistic galvanometer, and for this
reason this method is called the ballistic method.
The current intensity I arising within the closed circuit of
the measuring coil at some point of time
I = E/R = W/R ? d #/dt 108 a.
where R is the resistance of conductor and galvanometer and W the
number of turns of the measuring coil.
Hence the amount of electricity Q flowing through the cross
section of the conductor when the flux is changed from + # to  #,
is expressed as the integral
t
Q=J I dt=2. 108W#K.
If the measuring coil is connected in series to a ballistic
galvanometer, upon the passage of Q33i*uits of electricity through this
coil the pointer or mirror of the galvanometer will be deflected over
an angle 0 and thus
Q C 6
where C is a constant which depends upon the design of the galvanometer.
Denoting'the crosssectional area of the specimen by S and the
magnetic induction in the latter by B, we have
B 108 CR/2wS 9. (8.2)
The constant quantity CR may.be determined by measuring the
galvanometerdeflection 9 caused by the change of magnetic flux in a coil
with a dual winding, the coefficient of mutual induction of which is known.
 340 
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intensity in the other (primary) coil is varied from + I to  I.
108 MI = 108
91 = 108 MI Maxwell
Therefore, on varying the current intensity from I to  I
we obtain through formula (8.2):
For this purpose one of the windings (secondary) of the coil
is coniected in series to the ballistic galvanometer and the current
If the coefficient of mutual induction of the coil M is ex
pressed in Henry's the magnetic flux 11 expressed in amperes, con
tacting the secondary winding of the coil when current I flows
through the primary will be
CR1/2 A,
where R1 is the resistance of the secondary winding of the coil and
the galvanometer. Usually the secondary winding of the coil is con
nected in series to the measuring coil which surrounds the specimen.
Therefore Rl = R, and hence
CR = 2Ml/91
The following measurements are possible by this method: (1)
determination of the basic magnetization curve, (2) determination of
the hysteresis curve, (3) determination of coercive force and residual
induction, (4),determination of reversible permeability, and (5) de
The work diagram of these measurements is shown in Figure 132
in which R1 and R2 denote low and high resistance rheostats, respec
tively, for the regulation of the current in the circuit. Al and A2
are ammeters for measuring the current intensity. A 1 measuring high
intensities and A2 measuring any current intensity, G is a ballistic
galvanometer, K is a magnetizing coil in which the specimen is placed,
84 is a standard mutual induction coil for calibration of the galvan
ometer, KB 1 is a variable mutual induction coil for compensation of
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w }.
the flux in the air gap between the specimen and the measuring coil,
P is the main knife switch, 11 1 is a circuit current direction
changeover switch ,FT 3 is a changeover switch for the galvanometer
circuit,( 2 is a switch for connecting either the magnetizing coil
or the specimen coil, and B1 is a single pole knife switch for vary
ing current intensity.
In Section 17 of the introduction it was shown that the
hysteresis curve and the basic magnetization curve are the charac
teristics of a magnetic rock'when magnetization or magnetic induc
tion relate to a closed magnetic circuit. Induction in a closed
circuit can be produced in either of 2 ways: direct observation in
the closed circuit, or observation of induction of the specimen in
an open solenoid and transference to the closed circuit by means of
the demagnetization coefficient, which must be known for the given
shape of the specimen.
The first procedure is not applicable in the case of rocks
due to the difficulty of obtaining a specimen of necessary shape.
Therefore tests of rocks by the ballistic method always are carried
out in an open circuit by placing the specimen in a solenoid.
However, in such a case the specimen must be of a definite
shape, either a cylinder or a rectangular prism with a lengthto
diameter ratio of not less than 15, sothat its coefficient of demag
netization may be determined.
2. The Magnetometric Method of measuring magnetic values is
based on the interactionof 2 magnets. The specimen XS is suspended
from a filament and placed within the horizontal plane inside the mag
netizing coil, perpendicularly to the magnetic meridian in the first or
second Gaussian position (see below Section 7 of Chapter X) in relation
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From equation (8.3) it follows`that
to one of the magnets ns (lower) of the astatic system (Figure 133).
By an astatic system is meant 2 parallel magnets rigidly fastened at
a certain distance )u from each other. The magnetic moments of these
magnets must be equal and directed in opposite directions. Actually,
their magnetic moments differ slightly, both in magnitude and direction.
The astatic system is placed within the magnetic meridian,
and under the influence of the specimen NS it is deflected by an angle
9, which may be measured, and this the magnetic moment M of the speci
men may be determined.
Assuming that the upper magnet is at so great a distance from
the astatic magnetic system that the influence of the specimen can be
disregarded then, denoting the magnetic moment of one of the magnets
of the system by M', the torsion coefficient of the filament by C and
the distance between the centers of the specimen and of the magnet of
the astatic system by R, we have in accordance with formula (10.35)
the following condition of equilibrium for the first and second
Gaussian positions
n NM'/R3 k cos 0 = CO, (8.3)
where n = 2 for the first, and n = 1 for the second Gaussian positions,
K is a coefficient greater than unity depending on the coefficients of
distribution and different for different positions.
N CR3/nM'k 8 sec 6,
at small angles of deflection whichusuelly are encountered in practice
sec 0ma y be replaced by the 2 first terms of the expansion, so that
N m E.9 (1  1/2 82). (8.4)
where the factor (' , equal to
t = CR3/nM'k, (8.5)
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represents the value of one division of the magnetometer scale and is
constant for any given design.
sensitivity above which the system becomes unstable and its use impossible.
It is seen that at the given values of C and M' the magnetometer
is proportional to the cube of the distance and, therefore, the value of
one division may be changed by moving the specimen and thus the sensi
tivity of the instrument may be varied within wide limits. At a given
distance the value of one division is directly proportional to coeffi
cient of torsion of the filament and inversely proportional to the mag
netic moment of the magnet of the astatic system. Therefore, it would
appear that the sensitivity of the instrument could be increased to
any level, but experience shows that every instrument has a limit of
This is due to the fact that equation (8.3) has meaning under
completely astatic conditions of the system and when the magnet of the
magnetometer has no influence on the specimen under study. The first
condition is never fulfilled in practice and the second is possible only
if the specimen is placed at a distance not less than that at which the
magnet begins to exercise an influence on the specimen.
The value of a division , at both positions is determined by
means of a permanent magnet, the magnetic moment No of which is known
and which is approximately the same size as the specimen under study;
Measuring the deflection 6o of the magnetometer due to the influence
of this magnet, we have
MO/00(l  1/2 04)?
To determine the magnetization curve, i.e., the correlation,,
between magnetization J and the magnetizing field H, the specimen under
study is given the shape of an elongated ellipsoid of cylindrical rod,
the demagnetization coefficients of which are found in tables.
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The specimen is magnetized in a coil at least 3 times larger
than the length of the specimen. This is necessary to ensure a uni
form field in the middle of the coil, where the specimen is placed.
A second, identical coil is placed on the other side of the astatic
system so that it is located symmetrically with the first coil and
is connected in series with it. A diagram of the connection of these
coils is shown in Figure 133.
lely to the action of the magnetized specimen.
The second coil serves to compensate the magnetic field pro'
duced by the magnetizing coil. This is done by'moving the second coil
while current is flowing through it, until the astatic system re
sumes its initial position. Thishaving been done, when the specimen.
is placed inside the magnetizing coil, deflection of the system is due
Magnetization of the specimen J, corresponding to the magnetiz
ing field H is determined by formula (8.4), in which the product of
magnetization J by the volume of the specimen V, is substituted for the
magnetic moment, i.e.,
J = ~/V 8 (1  1/2 82)
The intensity of the magnetizing field is determined by the
formula
H=He  NJ
where He is the field intensity inside the coil and N is the magneto
metric coefficient of the demagnetization of the specimen.
Furthermore, at small distance's between the magnetizing. and
compensation coils the intensityof the field producedby the compen
sation coil at the place where the specimen is located must be sub
tracted from He.
.The magnetometric method in this form enables tests of
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ferromagnetic specimens, the magnetic susceptibility of which is not
less than 104 CGS /4 . Atlokwevsusceptibility values this method be
comes unreliable due to inad^quate sensitivity.
As in the ballistic method, the deflection angle 8 is observed
located at distance L from the mirror. Therefore, the angle 6 is re
by means of a mirror connected to the magnet ns, and a graduated scale
s:
placed by the scale reading n, in accokdance with the formula
8 = n/2L (1.  n2/3L2).
idly astatic and that the second magnet of the system was remote from
the specimen.
design of the magnetometer on itsreadings. This is necessary because
in the derivation of equation (8.6) it was assumed that the system was
(8.6)
Let us consider next the influence of defects in the setup and
Influence of the Second (Upper) Magnet. The influence of the
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specimen.
angle. However, this influence varies with different positions of the
instance remains a constant quantity independent of the deflection
upper magnet affects only the value of one division, which in this
In the first Gaussian position the moment of rotation Q acting
on the upper magnet n's' may be readily computed, assuming the speci
men to be an elemental magnet (dipole). Indeed, denoting the compo
the component perpendicular to the latter by H , we have:
cos + H"H~ sin, 3 cos
where H" is the magnetic moment of the upper magnet and Q the angle
between the axis 001 and the direction r (Figure 134a).
nent of the field of this magnet, ? iti the direction r by Hr, and denoting
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Replacing Hr and H %, according to the formulas for dipoles,
and replacing sin 0 and cos 9 by the ratio of the sides we obtain:
It is evident that this moment is equal to zero when
it is opposite to the mo
1 ^1 at R`> . /4 the moment Q is negative, i.e.,
ment acting on the lower magnet, and at R ~/C the moment has a posi
tive value, i.e., the same direction as the moment acting on thelower
magnet.
If the magnetic moments M' and M" are equal in magnitude and
opposite in direction the equation of equilibrium of the system will be:
y 3 9)]c
5
Solving this equation for M, we obtain
(8.7)
The factor appearing before S sec 9 is constant for a given distance
R. Therefore, denoting the former by E 1, and in view of the smallness
of angle 0, denoting sec 0 by 1 ?~ 1/2 82 we obtain
Substituting angle '9 in accordance with formula (8.6) and assuming
we obtain: '7
The quantity i is the value of one division of the magnetometer,
expressed in.units of magnetic moment.
As is apparent from equation (8.7) the value ofone division
depends on the torque of the filament and the distance between the'
specimen and the magnetic system.
As is apparent from Figure 134, if the specimen is considered
a dipole the moment of rotation Q in the second Gaussian position is
given by the equation
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M~'.
3
Consequently, the equation of equilibrium will be of the form
1,1141
Thus
or, analogously to the foregoing,
In this case the value of one division increases with an
increase in R over the entire range from R = 0 to k = C, and the
increase is more rapid than indicated by formula (8.5).
Influence of NonAstatic Nature of the System.
We denote the magnetic moments of the 2 magnets by Ml and M2
(Figure 135), their geometric sum by M, the horizontal component of
the terrestrial field by H, the intensity of the specimen's magnetic
field at the site of the lower magnet byrHl, and the intensity of the
specimenis field at the site of the upper magnet by H2, with H1 and H2
perpendicular to H, the angle of twist of the filament is indicated
by c i and C is the constant.of  torsion. 
We denote, further, the angle between M. and H by 01, the
angle between M2 and H by A2 and the angle between H and H by 8 = 81
in which case the condition of equilibrium of the magnetic system will
/~ # ' /0'
A change in magnetization of the specimen produces a change in
the intensity of fields H1 and H2, as a result of,which the condition
of equilibrium is disrupted and the system is deflected over a certain
angle d6. Differentiating equation (8.8), and taking into accountthat
H2 is proportional to H1, i.e., that H2 = kHl, we have
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) , '41 Hc,' ~' C
Considering kM2 as a certain magnetic moment MM, parallel to moment
U2 and considering the geometric sum 91 + M'2 as a certain moment M',
the foregoing expression may be written as:
where A' is the angle between H and M'.
Disregarding the influence of the specimen on the upper magnet,
i.e., assuming k = 0, it follows that M' = Ml, and 9' = 91, and there
The angle of deflection 91 of a magnetometer never exceeds
several degrees, and thus cos 91 may be replaced by unity, and sin 91
may be replaced by the angle 91, in which case
If dH, changes at values other than H1 = 0, it follows that at
small angles of deflection it may be assumed that dHl = H1 and d6=91,
in which case
Since the second and third terms appearing between the brackets
are small in comparison with unity, it follows that H1 in the second
term may be replaced by the quantity H  C/Y' 9 And therefore
at angles of deflection 8 not exceeding 59, which usually is the case
in practice, the second term 91 is not greater than 0.01, and may be
disregarded. In the third term the angle 9 may be replaced by the sum
91 wherein is the angle between M and Ml. In this case equa
tion (8.9) bacomes : C 61
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Denoting C/U1 where is the value of one division of the
astaMc#zed _$ystera;. it follows that
H1 (:  H/i, cos (,.. + e1) le1?
Consequently, the value of one division of the astatic magnetometer
depends on the angle 9, and the smaller the value of one division, the
greater is this dependence. Since o4% is always close to 900 it follows
that at (/t.+ 91 ) = eI the condition under which this dependence can be
disregarded
U/M1 ? 11/s 91 < 1,
and the value of one division E must be
r > M/Ml H el.
Influence of NonParallelism of Magnetizing and Compensation Coils.
If the coils K1 and K2 are not parallel to each other, their
summative field never is equal to zero, but this field may have no
effect on the equilibrium position of the magnetic system and when
the current in the coils is switched off the magnetic system may re
main at rest. For this it is necessary that the direction of the
summative field zcoincide with the magnetic axis of the lower magnet.
With such a system of compensation the value of one division becomes
greatly dependent upon the intensity and direction of the current in
the coils.
If the intensities of fields H1 and H2 produced by the coils
at the site . of, the lower magnet form an angle, between them which is
different from 1800 (Figure''136), denoting the angle formed by the
axis of magnet M1 with thedirection H1'by 4 disregarding the influence
on the upper magnet and assuming that the system is fully astaticized,
the equilibrium, equation of the system will assume the following form
wig cos 0 + M1H1sin t~ + Y3H2si l (d  )  C
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where 8 is the angle of deflection of the system from the position
of equilibrium due to the action of the summative field of intensity
H, and Y is the angle of twist of the filament.
Differentiating this expression and solving for dH, we obtain
dH = C/Ml d9  [H2 cos (('  S) ! H1 cos T I d e.
Next, taking Hl = H2 and transforming the sum of cosines, we have
dH = C/Ml d8  2Hi cos 6,/2 cos (rj`/2  ~)d9.
The angle 6,/2 ~ is approximately zero, and therefore its co
sine may be replaced by unity, hence
dH = [C/K1 + 2H1 cos /21 d 8.
Since 2H1 cos d,/2 represents the magnitude of the summative
vector (H1 + H2), on denoting it by h, we see that the value of one
division of the magnetometer E, in this instance is expressed by the
El= C/M + h.
h changes its sign upon a change in direction of current in the coil,
and therefore when the magnetometer is deflected in one direction the
value of one division will be 2h times greater or smaller than when
deflection is in the opposite direction.
In order to avoid this the coil axes must be parallel. This
is done as follows: after the coils have been compensated the speci
men is placed into the coil. The deflections of the magnetometer are
observed when the current flows in either direction, and then one of
the coils is rotated around its vertical axis until the deflections
are equal in both directions.
Measuring Procedure. To, determine the basis curve of magneti
zation the cylindrical or prismatic specimen is placed inside one of
the coils of the magnetometer, and is demagnetized by switching an
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alternating current through the coil, which is gradually decreased
to zero. Next, a direct current coinciding with the initial value
of the intensity of the magnetic field is passed through the coil.
In this operation a completely open rheostat is connected to the
coil circuit, and the current is adjusted to the necessary level
solely by a gradual decrease of the resistance. After the current
has been adjusted the magnetic preparation or repeated reversal of
the, current is performed, and the last time the current is switched
on a reading is taken of the magnetometer.
For determination of subsequent points the current is gradually
increased and magnetic preparation is carried out prior to each obser
vation.
There is preliminary demagnetization since in determination of
the hysteresis curve, determination of each point begins with the
switching on of a maximum current and its gradual decrease to the
necessary magnitude corresponding to the intensity of the demagnetizing
field.
3. The Magnetometer of B. M. Yanovskiy and Ye. T. Chernyshev. B. M.
Yanovskiy and Ye. T. Chernyshev have proposed [65] a more sensitive
method which is a variant of the above. The cylindrical specimen A
(Figure 137) with diameter not less than 3 cm., is placed several cm
(13) from the lower magnet of the astat ;c system M. Large diameter
Helmholtz rings K1. are used for magnetization of the specimen, and on
either side of the system so that the specimen and the lower magnet
of the system are within the same field. For compensation of this
field the upper magnet is placed into small diameter rings K2, which
are connected in series to the large rings. Since the small rings re
main stationary final compensation is effected by changing the current
in the secondary winding of these rings which is connected in parallel
to the large rings.
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Taking the intensity of the field produced by the cylindrical
specimen inside the rings as Ho, in such a case the condition of
equilibrium of the astatic system will be:
H0M cos 9 = CO (8.10)
because the influence of the specimen on the upper magnet may be dis
regarded.
The magnitude of Ho is obtained by differentiating equation
(0.63) with respect to R:
Ho=2rtJ[R+Ma2+(R+ )2R/12 +a2 ]
Substituting this value of Ho in equation (8.10), and replacing
angle 9 by the reading on the scale n, we obtain:
r
J n/R+?J/a2+(R +;x )2R/'n
where I = C/2LM is the value of one division of the instrument's scale.
The intensity of the field corresponding to magnetization J in the
case of the Helmholtz rings is determined in accordance with the usual
formula (0.38).
At the experimental shops of the Institute of Physics of the
Leningrad State University imeni A. A. Zhdanov a magnetometer has been
built which enables measurements to be made by the conventional mag
netometric method described on pag 302, and by the method of B. Y1
Yanovskiy and Ye. T. Chernyshev.
It consists (Figure 138) of a base A, on which a suspension
tube B, containing the astatic system is Mounted. Rigidly attached
to the base is the bar C, on which 2 coils Kl and K2, the magnetizing
and the compensation coil are mounted, each 300 an long and having an
internal diameter of 20 mm. The coils are set in the second Gaussian
position. Another bar D is located perpendicular to this bar, on which
2 pairs of square coils K3 and K4 with dimensions 250 x 250 ML. are mounted.
353 
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At the upper portion of the suspension tube is a pair of Helmholtz
rings which compensate the field of the square coils and determine
the value of one division of the scale.
The entire instrument rotates through an angle of 300 about
the vertical axis for adjustment with respect to the magnetic meri
dian. Figure 139 shows the circuit diagram of the coils.
The specimen, in the shape of a cylinder up to 20 mm in di
ameter, or a prism with one side of the base up to 15 mm long, is
placed inside the solenoid coil and the basic curve of magnetization
and the hysteresis curve are determined by the method described on
page 309.
iv _O lc  ltd!.
However, in this case magnetization is effected by the 4 sq
coils instead of the Helmholtz rings, and therefore the intensity
of the magnetizing field at the axis of the coil must be determined
for each coil in accordance with the formula:
7(x0 ~..~
(8.11)
where 2a is the side of the square, w the number of turns, x the dis
tance from the center of the coil and f(x) is a factor of wI depending
on X. The distance between each pair of coils is the same and the
magnetic field along the axis is made as uniform by proper selection of
the number of turns of the external and internal coils. This ratio
must ensure equal intensity at the center of each coil;
The field intensity at the center of the external coil at I = 1,
according to formula (5;11) is given by the equation:
0 _~) 0 1 1 /
__~ ) ( '
and at the center of the internal coil:
Under the condition that. the lefthand portions of these equations are
equal, we have
Thus, at x = a,
071 34Z__f ~" W_
I (V/`"r (`;gJ
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4. The Method of Attraction and Repulsion in a NonUniform Magnetic
Field.
This method was proposed by Faraday and is based on measurement
of the force F acting on a body of small volume, placed within a non
uniform magnetic field.
The th, o shows that the force F is proportional to the mag
netic susceptibility x and the volume v of the body, that is
F = Axv (8.12)
If the coefficient of proportionality A is known and the force F
has been measured it is possible to determine the magnitude of magnetic
susceptibility. Coefficient A may be determined experimentally by
direct measurement of the quantities which constitute this factor, or
by measurement of the force acting on a body of known susceptibility.
in the former instance the method of measuring the susceptibility is
an absolute method, and in the latter it is relative.
In the derivation of formula (8.12) the expression for the po
tential energy U of a body placed within a magnetic field is used,
namely
U = , t xH /2 dv, (8.13)
where H is the intensity of the magnetic field, dv is the element of
volume, and the integration applies to the entire volume of the body.
According to the laws of mechanics the force F acting. on the body is
expressed by the gradient of the quantity U preceded by the opposite
sign, i.e.,
F = grad I xH2/2 dv
Since for paramagnetic bodies x is a constant quantity, it
F  x/2j grad H2dv.
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If the body is of small dimensions it may be assumed that
grad H2 does not change its magnitude within the boundaries of
this body and therefore integration over its volume yields:
F = 1/2 xv grad H2 = xvH dH/dr,
(8.14)
where r'is the direction of greatest change in H, and v is the volume
occupied by the body. Thus the constant coefficient A of formula (8.12)
is the product of the intensity of field H and its gradient.
From equation (8.14) it follows that an equilibrating force
must be applied in the direction r to measure force F.
From equation (8.141:..We .obtain!: .kat a'
x = F/H dH/dr v (8.15)
The magnetic susceptibility'x, which may be designated as
magnetic susceptibility by volume, may be replaced by the specific
susceptibility x, i.e., the susceptibility per unit density. There
fore, dividing both sides of equation (8.15) by the density D of the
body, we have
= x/D = F/H dH/dr in, (8.16)
where m is the mass of the body.
For the case in which the body under study is located in a
medium having magnetic susceptibility x 0 (viz air, a liquid, etc)
equation (8.13) must be corrected as follows:
U =  S xxo/2 H2 dv.
Thus in the general case formula of susceptibility (8.15) is:
x  sto + F/H dH/dr v,
or, substituting volume for susceptibility thefoliowing is obtained
for the specific susceptibility:
7( _ )~ QDo/D + F/aR dH/dr, (8.17)
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where Do is the density of the medium (air), and D the density of the
substance being measured.
This method has very high sensitivity, enabling measurement of
sensitivity up to 1010 CGS f. Indeed, from equation (8.16) it follows
that at H = 2 ' 104 Oe and dg/dr = 104, which is readily attained by
means of electromagnets, and at an attraction force F = 0.01 dyne,
which is registered by a beam balance, x is approximately 10 'OCGS/A.
However, despite its high sensitivity as an absolute method
this method has a substantial flaw in its low degree of accuracy, and
for this reason it has been used mainly as a relative method. Formula
(8.17) shows that for determination of A4 the field intensity H and its
gradient in the direction r must be measured in addition to direct
measurement of the force F and of the mass m of the body being tested.
Although the measurement of H involves no great difficulties, the
measurement of dH/dr is very difficult, especially in the case of a
large gradient. Both these quantities are eliminated with the use
of a relative method and the problem is limited to direct measurement
of F and a. Indeed, taking equation (8.17) for 2 bodies having specific
susceptibilities xl and x2, densities Dl and D2, masses ml and m2 and
placed within similar fields having the identical gradients, we have
y y ~y
0
7W 0&1
Transposing the first terms of the right half of the equation
to the left half and dividing oie equation by the other, we obtain
P a  ~  0 ) F
 4 (~ ;~ 4  J
(8.18)
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From this formula it is apparent that in the case of a
relative method for determining x, the attraction forces F and F'
acting on both the test specimen and on the standard specimen,
and the masses th 1 and m2 of these specimens are measured directly.
The intensity of the field H and its gradient dH/dr can remain
entirely unknown.
An electromagnet with tapered pole shoes,NS, as shown in
Figure 140 usually is used to produce a nonuniform field. If the
axis of the pole shoes AB is directed horizontally, it follows from
the conditions of symmetry that the greatest variations of the
quantity H in the vertical direction will occur along the CD axis
which is perpendicular to AB and extends through the center of the
pole shoes. Therefore if the equilibrating force is applied in the
vertical direction, the point of application of this force must be
on the axis CD, otherwise the specimen will be subjected to the in
fluence of an additional force perpendicular to the axis.
The force of attraction which acts on the specimen may be
counterbalanced by the force of gravitation, by a torsion force, or
by an'electromagnetic force. In the first case the body a under
study is placed on one arm of the balance between the poles of the
electromagnet (somewhat higher) and the force of its attraction is
counterbalanced by counterweights placed on.the other arm (Figure 141).
in such a case F = M. and F' _ .Hlg, where N and Hl. are the masses of
,li p /~I Al Nr
counter weights used to counterbalance the force of attraction of
test body andof the standard body, so that formula (8.18) becomes
(8.19)
In the"second instance the specimen under study is suspended
from the arm of a torsion balance and is placed'laterally of the gap
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between the poles. The attractive force Fx is counterbalanced by
the torsion of the filament, which is twisted a certain angle 9.
Since according to the condition of equilibrium
FL = C 8
where L is the arm of the balance beam and C the coefficient of tor
sion, it follows that:
X = X D /D + (X  X D /D ) 8/0 . m 2 /m 1 . (8:20)
1 0 0 1 2 0 0 2 1
Among the defects of Fa)mday's method as a relative method, is
the variation of the quantity H d?i/dr in the direction r. The formu
las of magnetic susceptibility (8.19) and (8,20) are derived on the
assumption that the product H dH/dr remains constant on equilibration
of force F, as well as of force F', since the specimen under study and
the standard specimen are placed at the same distance from the pivot.
Actually, complete coincidence cannot be attained in practice, and the
slightest displacement of the body under study causes a sharp change
in the force of attraction, since the product H dH/dr changes very
rapidly with a change in the distance from the poles of the electro
magnet. At point 0, on the axis of the pole shoes (Figure 140) the
gradient is dH/dr = 0, and consequently the productis H dH/dr = 0.
Upon vertical displacement along the axis r the product H dH/dr
increases, reaches a maximum, and then gradually approaches zero. The
curves representing H dH/dr as a function of r, in the case of sym
metrically located pole shoes are shown in Figure '142. It is seen
that the curve increases very rapidly from the point 0, reaches a
sharp maximum, and thereafter, tends asymptotically to the ordinate
axis. It is apparent from the curve that the most advantageous con
dition for measurement of x is.obtained by placing the testbody at a
point where the quantity H dH/dr is at a maximum,'since on either side
of the maximum the function H dH/dr changes very slowly.
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Figure 143 shows a diagram of a typical arrangement of a
torsion balance. The balance beam AA is suspended from a metallic
filament F which can be twisted by means of knob G. At one end of
the beam is suspended the glass flask B containing the specimen
being tested, and at the other end are the counterweight E and
mirror Z, which is used to observe deflection. The system is
damped by means of an oil damper H. To prevent the effects of air
oscillations the entire apparatus is enclosed by a sealed housing.
5. Cylinder Method.
Another variant of the method of attraction in a nonuniform
field is based on measurement of the force acting on prism AB or on
a cylinder, one end of which is placed within a uniform field H, and
the other end within a uniform field Ho. This is called the cylinder
method (Figure 144).
The attraction of such a cylinder may be regarded as the re
sultant of the attraction of each elemental volume dv = Sdz, where S
is the crosssectional area of the cylinder and dz its height. Accord
ing to the preceding paragraph the force acting on such an elemental
volume is
dF = 1/2 xS grad H2 dz.
The vertical component is
Ate xSHZ)H/'6 z dz,
dFz = xSHdH.
Hence the total force in the direction z is:
FZ = xS fit dH = xS/2(H2  H2), (8.21)
no
where H is the intensity of the field between the poles and no the in
tensity of the field at the lower end of the cylinder.
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If the i4xis: of the cylinder is half way between the poles,
i.e., coincides with the vertical symmetry axis of the poles, it
follows that ) H/ x = 0 and therefore the horizontal force is ab
sent. However, upon the slightest deflection of the cylinder axis
from the symmetry axis a horizontal attraction force arises. In
the following it is assumed that the cylinder axis coincides with
the symmetry axis, and consequently there are no'forces other than
the vertical forces F. Thus from equation (8.21) we have:
x = 2Fz/S(H2  Ho) (8.22)
The force F. may be counterbalanced as in the Faraday method,
either by means of a beam balance, placing the test cylinder at one
end, and the pan holding the counterweights at the other end of the
beam, or some other method may be used.
In the former which is the most widely used method, the force
Fz will be equal to the product of mass m of the counterweights and
the acceleration due to gravity, i.e.:
Fz = mg
hence,
2mg/SD(H2  Ho)
where D is the density of the substance being tested.
Since the test usually is performed in the air, it follows
from deduction's analogous to those pertaining to the Faraday method,
that corrections must be made for the susceptibility of the air, and
the formula will be:
'( o Do/D + 2mg/SD (H2  Ho) . (8.23)
Thus we see that the cylinder method necessitates that the 'body
under study be of a specific shape, either cylindrical or prismatic,
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which leads to great difficulties in measuring the susceptibility
of solids. Therefore this method is limited mostly to liquids,.so
lutions of solids in liquids, and powders.
The liquid or solution is poured into a length of cylindri
cal glass tubing which is twice as long as the liquid column, to
eliminate the influence of the magnetic field on the glass tubing.
By placing the tubing between the poles of the electromagnet so
that its center is at the center of the pole shoes the top and bot
tom ends of the tubing will be equidistant from the center of the
electromagnet, and the action of the magnetic field on each half of
the tubing will be equal but opposite. A diagram of the cylinder
method, using a beam balance, is shown in Figure 145.
6. Astatic Magnetometer of S. Sh. Dolginov [1.0)
The principle upon which S. Sh. Dolginov's method is based
was proposed by N. I. Spiridovich [71].
Dolginov's method differs from that of N. I. Spiridovich in
that the former incorporates an astatic system of small dimensions,
with the result that it can be utilized under field conditions. Also,
the instrument proposed by S. Sh. Dolginov is absolute, whereas that
of N. I. Spiridovich is relative.
Let us assume that,a specimen of random shape and of small
'volume compared to is placed in the sane plane, and at a certain
distance from the dimensions of the magnet the center of one of the
magnets of the astatic system.
If the radiusvector r (Figure 146) forms an angle other than
900 with the axis of the magnet a torque caused by the interaction
between specimen and magnet will, act on the magnet. This torque is
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proportional to the product of magnetic moments of specimen Y' and
magnet M and inversely proportional to the cube of the distance,
and is counterbalanced by the twisting moment of the filament when
the filament is twisted through the angle , i.e.,
C~  k MMI/r3, (8.24)
where k is the proportionality coefficient and C the coefficient of
torsion of the filament. The coefficient k is smaller than unity
and depends on the mutual positions and on the dimensions of the
specimen and the magnet.
The magnetic moment M' is due to the action of the magnet and
is roughly approximated by the expression:
K' = xHv, (8.25)
where x is the apparent magnetic susceptibility of the specimen, H is
the intensity of magnetic field produced by the magnet, and v is the
volume of the specimen. The quantity H may be written in the form:
H = k' M/r3, (8.26)
where k' is a coefficient depending on the angle between r and M.
Therefore, replacing M' in equation (8.24), from the values derived
from equations (8.25) and (8.26), we obtain
kk' M2v/r6
Taking the same equation for another specimen of volume V.
and magnetic susceptibility xo, and dividing one equation by'~ the
other, we have:
X = xo vo/v ? '~/o
Therefore if the volume and magnetic susceptibility of a
specimen are known, the magnetic susceptibility may be,determined
by observing the angles q and q. through which the magnetic needle
is deflected by the action of the test specimen and that of a speci
men of known susceptibility.
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If this instrument is used for absolute measurements the
specimens must be cylindrically shaped to enable calculation of
the intensity of the magnetic field produced by the magnetized
specimen. Moreover, the instrument must be calibrated by means
of a cylindrical coil, the diameter and length of which are about
equal to those of the specimen under study.
According to equation (0.62) the intensity of the magnetic
field produced by a specimen of cylindrical shape at a distance R?
from its base will be: r
(8.27)
where , is the length of the specimen, and a is its diameter. The
magnetization of the specimen is J = xHm, where H. is the intensity
of the magnetic field of the astatic system. If 2 identical cylin
ders are placed on either side of one of the magnets of the astatic
system, as shown in Figure 147, the torque to which the magnet of this
system is subjected is equal to Dpi, and the equilibrium equation is:
Iii=Cp,
or
In place of the magnets, 2 coils of length and diameter a
having w turns are installed, and at a current intensity I the field
intensity at distance R will be expressed analogously to formula (8.27):
and the equation of, equilibrium of the magne wiiibe
Therefore
U H ~,~ P LL
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In order to determine the magnetic susceptibility of the
specimen, in addition to recording the angles k and coil, the
intensity of the magnetic field HU produced by the magnet of the
astatic system also must be known. The intensity Hid is deter
mined by rotating a small coil of known area and number of turns
at a definite rate of speed and measuring the omf.
Because this instrument permits measurement of only the
apparent magnetic susceptibility, the error in measurements due to
the shape of the specimen increases in the more magnetic rocks.
Thus in the case of specimens having a susceptibility of about 0.1
CGS / the error may reach 20% or more. An appreciable error is
introduced by the great dependence of the instrument readings on
the position of the specimen under study since the angle "''r' is in
versely proportional to r6 and the coefficients k and k' are not
constant, but depend upon the mutual positions of specimen and magnet.
However, with careful positioning of the test specimen and the coil,
the author estimates the error in measurements to be 6 or 7%.
The advantages of this instrument include its small size and
the convenience and simplicity of the measurements, which permit its
use under field conditions.
7. The Iftthod of T. N. Roze [72)
Among, the methods which permit a large number of determinations
of magnetic susceptibility to be made without requiring knowledge of
the demagnetization coefficient of the specimen is the method proposed
by T. X. Roze. This method is based on the change in residual induc
tion of a permanent magnet when a specimen having a magnetic suscepti
bility other than unity is placed into its interpolar gap.
This method is embodied in an instrument consisting of a permanent
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M
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horseshoe magnet iK (Figure ii8), with a winding K consisting of
several thousand turns of fine, insulated wire at its bend, and
a ballistic galvanometer 0.
The winding is connected to the galvanometer, and the ob
long specimen A is placed between the poles of the magnet.
Under the influence of the field of the permaent magnet the
specimen becomes magnetized in the transverse direction, increas
ing the magnetic flux cutting through the winding k. When the
specimen is removed from the interpolar space the flux decreases
and the galvanometer will show a deflection proportional to the
change in flux Q a.
Denoting the cross sectional area of the specimen by S, the
intensity of magnetic field in the interpolar gap by H, and the
number of turns on the magnet by w, we have
Q O = 4n x' HSw
where x' is the apparent magnetic susceptibility of the specimen.
The change in flux 01 corresponds to a galvanometer deflection v(.
which is proportional to A 0, and thus
4,t x' HSw = Cy. ,
where C is the galvanometer constant.
In the case of a specimen of known magnetic susceptibility x0
and cross sectional area So, we have:
41% XOHSDw  C a o
Dividing one equation by the other, we have
X' =x0?soc./s(3o
Consequently, in order to determine the susceptibility of
some rock a specimen with known susceptibility xo and cross sectional
area So is needed and 2 observations must be madeby removing the
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specimen under study and then the specimen of known susceptibility
from the interpolar gap. Although the shape of the specimen is un
important in these measurements its middle portion must be prismatic
or cylindricalin rder that the cross sectional area of the speci
men may be measured.
As in the preceding case this method yields not the actual
but the apparent susceptibility, and thus specimens with low mag
netic susceptibility must be used, in which the apparent approximates
the actual magnetic susceptibility. With proper selection of the
number of turns, galvanometer constant, and dimensions of the perma
nent magnet this method may be used for the determination suscepti
bility of the order of 10 CGS J.
For susceptibility values of several tenths and higher in
addition to the influence of the demagnetizing field, the magnitude
of magnetization of the magnet itself is affected by the magnetiza
tion of the specimen, which causes an increase in A 6 and also in x.
8. Contact Magnetometer for Measuring the Magnetic Properties of
Rocks and Materials of Small Volume
All the above methods for measuring the magnetic properties
of rocks involve determina~4on of the mean value of some magnetic
quantity of the tested specimen.
mean values are quite adequate for interpretation of magnetic
anomalies, and therefore measurement procedures were developed for
the purpose of obtaining these values. Yet, the mean values of mag
netic,properties do not permit correlation of these properties with
the mineralogical composition of rocks necessary for thorough under
standing of the geological nature of anomalies. It is also difficult
to determine which of the materials included in the composition of the
rock cause its magnetic properties.
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In recent years A. S. Semenov and A. P. Ochkur have designed
an instrument called a contact magnetometer, which makes possible
determination of the magnetic susceptibility of individual particles
ingrained in rock, thus enabling determination of its mineralogical
composition.
The instrument consists of a magnetic needle M (Figure 149)
6 or 7 cm long and one mm in diameter suspended on a horizontal
filament C. The ends of the filament are held in clamps P and Q,
with which the filament may be twisted to adjust the position of the
needle, With the needle in an oblique position, if a magnetic speci
men is brought near acne end of the needle the latter will be drawn
toward the specimen with a force of attraction proportional to the
magnetic susceptibility of the specimen. Therefore, by measuring
this force in arbitrary units the magnitude of magnetic susceptibility
also may be determined by twisting the filament C through an angle
equal to that which separates the needle from the specimen. The
magnitude of this angle provides a measure of the magnetic suscepti
bility.
Tomeasure the magnetic susceptibility of individual particles
it is necessary.. only to bring visually identified particles on a
plane polished surface of the specimen close to the tip of the magnet.
It has been ascertained by tests that the average diameter of these
particles must be not less than 5 mm in the case of slightly magnetic,
and not less than 0.5 mm for. strongly magnetic minerals.. This instru
lment permits measurement of magnetic susceptibilities from x  102 to
x =e1 CGS The measurement error is in the range of several percent.
Section 6. lfagnetibmetric procedures
In conducting magnetometric operations the primary prerequisites
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are congruence between the number of points and nature of the anomaly,
and correct choice of a sufficiently accurate observation procedure
for determination of the valuesof anomalous components of the in
tensity of the magnetic field. Since the choice of procedure also
depends upon the nature of the anomaly the maximum values of the
anomalous field components and their gradients must be determined, or
approximated first. Also, for correct distribution of the observation
point network it is necessary to determine the trend of the anomaly,
that is, the direction of the axis of the anomaly and the approximate
area it covers.
Such determinations are made by preliminary reconnaissance of
the anomaly and recording the points of maximum value of the vertical
component on a map or marking these points on the terrain.
If the anomaly has a longitudinal trend, i.e., the maximum
vertical component values are distributed along a straight line, such
as at the Kursk magnetic anomalies, the points of observation are dis
tributed over lines or profiles perpendicular to the line of maxima,
or, as it is called, crosswise to the trend of the anomaly. This
distance between the profiles and the distances between observation
within this interval is not constant the measurements at terminal
points xl'and x2 do not permit determination of the values of z within
magnitude of Z at any point situated between xl'and x2. If the gradient
to.carryout observations at points xl and x2 in order to ascertain the
points along the profiles are selected according to the magnitude of
variation of the gradient of the vertical component. If the gradient,
i.e., dZ/dx, over the distance x2xl remains constant it is sufficient
this interval. Thus for a true picture of the distribution of the mag
netic field of the anomaly?the distance between points must be chosen in
a manner which assures a linear change in the vertical component over
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For this, the nature of the magnetic field within the area
under investigation must be roughly determined during the prelim
inary reconnaissance.
The location of the observation points is determined by topo
graphical methods, by tying the points in with topographical, control
points selected beforehand at the center of the anomaly, Usually
these points, indicated by tall stakes or pyramids, are distributed
along the axis of the anomaly. The geographical coordinates of these
markers are determined either by tying them in with the nearest topo
graphical points or by means of a largescale map.
For ready tying in of the observation points with the control
points, the former are distributed along straight profiles perpendi
cular to the main line, i.e., the line connecting 2 control points,
if the main line follows the axis of the anomaly. If the main line
departs from the axis of the anomaly it is more convenient to plot
the profiles at a certain angle to the main line, so that they are
perpendicular to the trend of the anomaly. Distances between points
of observation are measured with a tape and the angles between the
main line and the profile are measured with an anglemeasuring instru
ment (theodolite). If the nature of the terrain makes the plotting of
straight profiles impossible they can be laid out as broken lines,
but in such a case the angle of deflection must be determined at
each bending point. Such a distribution of points and the method of
tying them in with the control points is illustrated in Figure 150.
In the investigation of magnetic anomalies the accuracy of
magnetic observations must correspond with'the accuracy of determin
ation of the coordinates of the observation points. Regardless of
how accurately the elements of terrestrial magnetism have been measured,
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if the position where these observations were made is unknown, or is
known only very approximately, these measurements are without value
since they cannot be utilized for interpretation purposes.
The same is true of the converse situation, in which the posi
tion?of a point has been determined exactly but measurement of the
elements of terrestrial magnetism are approximate.
The proper correspondenceof accuracy is determined by an
equation correlating a change in some element of terrestrial mag
netism and a change in location on passing from one point to another.
In the case of the vertical component this equation is
Z Z/) x L5 X.
where J Z/ ) x is the gradient of Z in the direction x.
if A x represents the error involved in determining the
distance between points,.6 z will be the error due to inadequate
knowledge of the precise position of the point, and consequently
it will be the minimum error beyond which any measurement is mean,'ngless.
For example, in the region of the Kursk anomalies the gradients
of Z attain 200 4 /m, while the error in determination of distancas was within
one meter,. consequently the error in, measurement of the vertical component
had to be not more than ? 200 \r.
This actually was the case, since the
measurements were made by means of a deflector magnetometer, in which
the error is approximately of this order of magnitude.
section 7. Aerial Nagnetic,Surveying
During the past 10 years the aeromagnetic method of investigating
the magnetic field of the earth has been used extensively in geological
prospecting. This method, which wasdeveloped and put into practical
use byA. A. Logachev [731, has been termed aeromagneti surveying.
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Aeromagnetic surveying consists of continuous measurement of the
vertical component by means of a semiautomatic instrument, called
an aeromagnetometer, installed in an aircraft. Because of the high
speed of aircraft this method enables a several hundredfold increase
in the productivity of magnetic survey operations. While the use of
a magnetic balance in magnetic survey ground operations enables a
single observer to complete a detailed survey of area during a
summer season (100 1tm2), the aeromagnetic survey method covers 30 to
40,000 km2 in the same period with the same intervals between itiner
aries. The aeromagnetic survey has the additional advantage of cover
ing areas inaccessible to conventional ground survey operations
(mountains, tayga, deserts, etc).
Aeromagnetic survey has been applied in locating iron ore de
posits, and in geological mapping. During recent years a number of
large deposits of iron ore which are of great industrial significance
have been discovered by aeromagnetic survey.
At the present time this method still does not equal the accuracy
of ground instrument surveying which is necessary for exact determina
tion of the boundaries and depth of occurrence of rocks. Therefore,
when an anomalous area has been discovered by aeromagnetic survey,
instrument surveying is necessary for its interpretation. However there
are no fundamental hindrances to increasing the accuracy of magnetic
measurements and tying them in with landmarks and it may be expected
that in the near future new aeromagnetometers will be developed which
willcompletely supplant ground surveying operations.
Itshould be noted that the idea of a survey conducted by an
obser~4 in continuous motion and the first practical application of
'`
this method must be credited to E. I. Val'skiy [74]', an associate of
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the Department of the Physics of the Earth's Crust, of the Leningrad
State University. In 1930 Val'skiy made the first observations from
a railroad train in the area of the Kursk magnetic anomalies, using
an instrument of his own design, the magnetr9i.
Because of the specificity of the conditions under which
aeromagnetic survey is conducted, namely the high speed and consider
able altitude of the observer, this method may be utilized to investi
gate only anomalies which exceed a certain minimum size and intensity.
Thus, for example, operational practice shows that the time required
for visual measurements cannot be less than 45 seconds, which at an
air speed of 100120 km per hour corresponds to an anomaly with dia
meter not less than 150 a. It is known also from experience that the
minimum vertical component which can be detected in aeromagneti survey
is 500 ' ,
1P4
Aeromagnetic survey is conducted over straight itineraries
perpendicular to the course of the principal local geological struc
tures. However, because of the availability of landmarks for tying
in itineraries, the course of the flight path may be altered with re
spect to the direction dictated by geological factors. The lateral
interval between itineraries is determined by the altitude of flight,
which in turn is chosen in accordance with the depth of occurrence of
the rock formations which cause the anomalies. To discover an anomaly
it is sufficient to trace the change of the vertical component within
the limits of. onehalf ofits maximum value, i.e., from Zia/2 to + Zm/2.
In the case of monopolar occurrences such as a vertical cylinder
or a vertical stratum of very great length, the distance between
 Zm/2 and + Zm/2 is equal to twice the dV1h=of occurrence of the upper
pole.
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In the case of a spherical deposit or a cylindrical deposit
of infinite length, the axis of which is parallel to the earth's
surface, this distance is equal to the depth of occurrence of the
sphere center or cylinder axis.
In the case of rocks having a finite course in the vertical
direction, i.e., having poles at a finite distance, the distance
between  Zm/2 and . Zm/2 is greater than the depth of occurrence
and less than twice its value.
Therefore, if the flight altitude is great in comparison
with the depth of occurrence of the pole, and the altitude may be
considered equal to the distance between the pole and the aircraft,
in order to discover the anomaly it is necessary that the distance
between itineraries be not greater than twice the flight altitude.
However, the depth of occurrence of any given type of rock
within an uninvestigated area is not known and may be only estimated
on the basis of geological factors. Thus the altitude must be
selected in accordance with geological data relating to the area
under study and the problems involved in?carrying out the aeromag
netic survey.
The results of an aeromagnetic survey are plotted on the map
in the form of the graphs of Z as a function of itinerary distance
for each of the itineraries. The plotting of the itineraries is based
on local landmarks over which the aircraft will travel. Such land
marks may be road intersections, river bends, inhabited localities,
summits of hills or mountains, etc.
As an example Figure 151 shows a map of the magnetic field
in the area of the Krasnokamen iron ore deposit (Eastern Sayans),
plotted by A. A. Logachev on the basis of the results of an aeromagnetic
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survey. The shaded areas between the curves and the directions of
the itineraries represent the positive values of anomalous vertical
component and the solid black areas represent negative values.
While it has no particular advantages over the isoline method
with respect to clarity of representation of the magneti field, this
graphic procedure does have the advantage that it requires less time
and effort in reproduction, and for this reason it has been generally
adopted for both aeromagnetic and ground surveys.
Figure 152 shows a geological map of the same area of the
Krasnokamen deposit to illustrate the practical importance of the re
sults of aeromaguetic survey. Comparison of the magnetic and geo
logical maps reveals that the regions of increased vertical component
are in good agreement with the areas of occurrence of igneous rocks:
gabbrodiabases, prophyrites, syenites, etc, which exhibit increased
magnetic susceptibility. It is this fact which permits utilization
of the results of aeromagnetic surveys in geological mapping.
At the present time large areas of many regions of the USSR
have been covered by aeromagnetic survey and numerous magnetic
anomalies have been discovered, which in addition to enabling identi
fication of the geological structure of these areas has also resulted
in the discovery of a number of iron ore deposits.
Section 8. The Pain Problem of Hagnetbmetry. Magnetic Fields of
Re ularly Shaped Bodies.
The main problem of magnetometry is the determination of the
magnetic field produced by magnetized rocks of different shapes. Al
though the infinite variety of shapes of rock occurrences does not
permit their expression by any one equation, still in many instances
each of then can be generally likened to some geometrically regular
shape (sphere, prism, etc).
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This is based on the fact that at distances much greater than
the dimensions of the rock formation involved the magnetic field is
equivalent to the field of a uniformly magnetized sphere.
Indeed, from the expansion of the magnetic potential to a
series by spherical functions (equation (2.20)] it follows that on
recession of the point P at which the potential is being considered
the terms of higher order become small in relation to the initial
terms and may be disregarded. The first term of the expansion, on
the other hand, represents the potential of a uniformly magnetized
sphere.
The regular geometric shapes of importance in magnetic
prospecting are: the sphere, circular cylinder, rectangular prism,
ellipsoid of rotation, ellipsoidal cylinder and plane plate, since
most deep rocks which cause anomalies can be ].ikendd to one of these
shapes.
Therefore let us calculate the components of the magnetic
field of the above bodies, assuming that they are magnetized uni
formly over one of their geometric axes.
The solution of the main problem is based on the formulas
and laws which were set. forth in the introduction. in many instances
use will be made of the law of Coulomb, assuming that the magnetized
rocks contain magnetic charges.
,r;
The assumption that the rocks fare uniformly magniti zed is
quite tenable since their magnetic susceptibility in most cases is
of the order of 102 to 103 and changes slightly, depending on sag
netization. Indeed, according to the deductions of Section 15 of the
introduction, magnetization at any point of the rock is defined by
the equations
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Since x is small, and the value N of most rocks is smaller
than unity, the product xN may be disregarded, in which case
If x is a constant quantity, J also will be constant for all
points of a body.
1. Sphere. In accordance with the decutions of section 1
of chapter II the potential of a uniformly magnetized sphere at
any point of its surrounding space having the spheric coordinates
r, 0 and )., taking the origin of coordinates at the center of the
sphere, is expressed by the equation U = M/r2[cos 0 cos 8o + sin 8
sin 9O cos ( /1  ) Q) j where 8o and f\ o are the coordinates of the mag
netic axis of the sphere and M its magnetic moment.
Since we are concerned with the distribution of the magnetic
field over a plane, the spheric coordinates must be replaced by
rectangular coordinates. Taking the z axis perpendicular to the
plane involved (Figure 153) and the xaxis the line 0'P extending
through point P, denoting the coordinates of point P by x, y, z,
the coordinates of the point of intersection of the magnetic axis with
the plane, by xof Y09 zo, and the distance from the center of the
sphere to the origin of coordinates by R, it follows that,
0
and consequently
where x'
O+Q, r = OQ and,
as is readily apparent from the drawing
If2
For practical purposes it is sufficient to con4ider only the
magnetic field along the line O`_Q, which constitutes the intersection
of the vertical plane extending through the magnetic axis of the sphere
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with the horizontal plane. In such a case, the angle )  h o = 0,
and the coordinate y = 0, therefore
o
The components of the intensity of the magnetic field on the
x axis and the z axis, i.e., the horizontal and the vertical compo
nents, denoted by H and Z, respectively, are as follows:
d U (q P 2X' t d 'c %0
I' Q ^,C /L /rsb
U r~
(8.28)
Since the magnetic moment of a uniformly magnetized sphere is
the product of the sphere volume V by its magnetization J, the right
hand portions of equations (8.28) are functions of zero order and
therefore do not depend on the scale used to measure the quantities
R, x, xo or the radius of the sphere. Therefore, taking the distance
R as the unit of length and expressing H in terms of the sphere radius
a and the magnetization J, we hav e : ? 02 ;_ %%V V ~U
jet, 3
(8.29)
In this form the equations in which a and x are expressed in R
units, with a given angle of inclination 9o, are independent of both
the depth of occurrence of the sphere and its dimensions, and the
graphs plotted in accordance with these formulas will hold for any
dimensions of the sphere and any depth of occurrence.
Thus, for example, when the depth of occurrence increases n
fold, and'on''retention of.the same graph, i.e., retention of the same
values.of x,? z'and H, it is necessary only to increase the radius by
n. When the radius of the sphere is changed nfold and the same graph
is retained J must be changed in reverse proportion by the factor n3.
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Graphs corresponding to formulas (8.29) for an angle
8 = 300 are shown in Figure 154, where the x distances are plotted
0
on the abscissa axis and the corresponding values of H and Z are
plotted on the ordinates axis.
From the graphs it is apparent that the horizontal component
has one maximum and one minimum, and the vertical component has one
maximum and 2 minima, with one of the minima having greater absolute
magnitude than the other and is disposed opposite to the inclination
of the magnetic axis in relation to the horizon. On the other hand
the maximum of the vertical component is shifted from the origin of
the coordinates in the direction of the dip of the magnetic axis.
Simpler expressions for H and Z are obtained when the magnetic
axis of the sphere is perpendicular to the earth's surface, i.e.,
when magnetization of the sphere is directed along the vertical. In
such a case X0 = 0 and r0 = R, and consequently:
//1 P 111:1z_. I
 (8.30)
fL 5
The H and Z graphs (Figure 155) in this case will be analogous
to those of Figure 154, but will be symmetrical in relation to the
ordinate axis, i.e., the maximum of Z will be at x = 0, and the minima
will be at equal distances X. from the origin of coordinates and equal
to each other in absolute magnitude.
It is readily apparent that
at x 2R I Ist~ei t S 7
`
at x = R/2
It is Also of interest to note that in the case of a sphere the
magnetization of which is para.el to the horizontal plane, i.e.,
CO A
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the components are
i.e., the nature of their change with distance x will be the same
as the nature of the change of Z and ii respectively, with vertical
magnetization.
2. Cylinder Having a Small DiametertoLength Ratio, Magnetized in
the Direction of Its Geometric Axis. The natural occurrence
of intrusion stock with small cross sectional area may be compared
to this geometric type.
Because of the small transverse dimensions of the cylinder in
comparison with the depth of occurrence the magnetic charges distri
buted over the end surfaces of the cylinder may be considered as con
centrated at 2 points, at the ends of the cylinder. The magnitude of
these charges evidently is equal to the product of magnetization by
the area of the cylinder base, i.e.,
m=JS
Magnets in which the charges may be considered concentrated at
2 points (poles) are called schematic magnets. The magnetic potential
of a schematic magnet is determined by the law of Coulomb:
wherein r1 and r2 are the distances of the poles from point P (Figure
156).
As with a sphere,..in practice it is sufficient to consider
the distribution of the field along a horizontal line extending through
the projections of the poles + a and  a. Taking this line as'the x
axis,. denoting the coordinates of the poles + a and  a by xl, Ri and
'R2, respect vely, &j&) the coordinates of pydiatxP x, R, we have :
A,
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Placing the origin of coordinates at one of the poles of the
schematic magnet, i.e., assuming xl = 0, Rl  0 and x2 = xo, R2
Ro, we obtain:
gf= )7
_44rG1
3 3 (8.31)
1
(  q ) z
X , 0)
I
Analogously to the preceding example, the righthand portions
of these equations are functions of zero order, since the magnetic
charge may be represented as the product of magnetization by the cross
sectional area of the center.
Therefore, with given values of cylinder length and angle of
inclination of the axis in relation to the horizon, a graph in which
the abscissas are plotted at a scale of depth of occurrence of the
upper pole will relate to any depth of occurrence of the cylinder and
any dimensions of the cross sectional area, provided that the dimen
sions of this area are small in comparison with the length of the cy
linder and the depth of occurrence.
The graphs of these functions are shown in Figure 157. In
essence they are analogous to the graphs of a sphere with magnetic
axis inclined in relation to the horizon.
The difference resides in, the relationships between the maxima of
H and Z, the maximum and minimum of Z and in the distance of the extreme
values of R and Z from the, origin of coordinates.
In the particular instance when the axis of the schematic magnet
is vertical, i.e., when xl = x2 = 0 and R2 = ) , the length of the
schematic magnet, we have:
'4' J lea 0ON
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These functions are symmetrical in relation to the z axis
(Figure 158).
In the special instance in which the lower end of a magnet
or stock is located at a very great depth in comparison with its upper
end, 'so that the influence of the lower. pole may be disregarded, we
have : Y/i,C y 41
The graph of these functions is shown in Figure 159.
(8.33)
From equation (8.33) it is apparent that at x = 0 the vertical
and horizontal components assume the value
and at , ~ they assume the values
J
Let us determine the conditions under which the 2pole magnet
can be replaced by a singlepole magnet, i.e., the case in which the
field of a 2pole magnet coincides with that of a singlepole magnet
within the limits of observational error. If, in the equations (8.32)
and (8.33), the vertical component of a 2pole magnet is denoted by(Q~Zl and that of a singlepole magnet denoted by Z2 expressing x and N
in R units and Z1 and Z2 in units of their maximum values, i.e., taking
Z 1 and Z ft 1, it follows that:
7 ~ ,7t
x7
where Zo is the value of Z2 at x = 0, i.e.,
Thus the relative difference of the vertical components of 2pole
and singlepole magnets depends not only on the distance between the pole
but also on the coordinate x of the point where the observations are
carried out. The further the point of observation is from the maximum Zm,
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the greater is this distance. The values of this difference, ex
pressed in percent, for the different values of i and x are shown
below.
Y,
4
6
8
10
0
0.0
0.0
0.0
0.0
1
6.8
3.4
1.9
1.3
2
37
201:5
14.0
8.9
The table shows that only at a value of ,,more than 10 times
greater than the depth of occurrence of the upper pole the difference
over the length 2R will not exceed several percent, and therefore the
formulas for a singlepole magnet may be utilized only at Q > 10 R, in
which an error of several percent is committed in the determination of
Z. However, for determination of the depth of occurrence it is suffi
cient to have a portion of the curve, of length equal to the depth of
occurrence, and in such a case the replacement is possible with t > 6R.
Another problem of practical importance is the ratio of radius
of vertical cylinder to the depth of occurrence of its upper surface
at which the cylinder may be considered as a 2pole or as a singlepole
magnet. Determination' of the magnetic field of a cylinder with a
finite base radius is a complex mathematical problem, and therefore the
solution is obtained by the approximation methods described on page 361.
This method shows that the relative difference in percent, between the
vertical component Z2 of a cylinder, the depthof occurrence of the
upper base, of which is 4 times greater than the radius of the cylinder,
and the vertical component Z1 of a punctal pole at the center of the
anomaly, does not exceed 7% and remains almost the same over an area
equal to the depth of occurrence. Therefore, if the upper base of the
cylinder is located. at a depth no greater than 4 times the radius of the
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cylinder the cylinder may be considered a punctal magnet. In such a
case the vertical component has no negative values.
3. Ellipsoid. Among the ellipsoids with different axial
ratios the 2axis compressed ellipsoid i?s'of significance in mag
netometry. Rocks occurring in the form of lenses parallel to the
earth's surface are of this type. An elongated ellipsoid (according
to the theory) of great focal length is equivalent, with a high degree
of approximation, to a schematic magnet the poles of which coincide
with the foci of the ellipsoid. Therefore, complex calculation of
the potential, such as by formulas of the potential of the ellipsoid
and its first derivatives is unnecessary since tfte potential may be
calculated in accordance with the simpler formulas relating to a
schematic magnet with accuracy adequate for practical purposes.
Ellipsoids of small focal length obviously are equivalent
to spheres. The magnetic field of a compressed ellipsoid having a
large axis ratio has no simpler analogy, and therefore its potential
must be calculated and practically utilized. The magnetic potential
of a 3axis ellipsoid is expressed by equation (0.67). If we take
a = b this equation becomes
t _ F: z 1 0 .
V r 't
and assuming that a~ c and that the ellipsoid is magnetized along the
z axis, then
Jz
and the constant coefficient Ni is of the form:
or, after integration, 1)
i?
(8.35)
/'/  f~7,`~u! C'l "C/ 3 ` (. (8.36)
where q ai  cl Noting that 4/3 ,talc J represents the magnetic
moment of ellipsoid K, and substituting N1 from equation (8.36) in equa
tion (8.35) we obtain:
D 1 r (8.37)
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Assuming the ellipsoids are cofocal and that the point P,
with coordinates x, 6 and z, ,is located on the surface of the el
lipsoid with axes al and ci we have:
? I
4
and solving these equations we obtain:
2 (X
Denoting the coordinate z by R?the components of the inten
sity of the magnetic field are determined from equation (8.37):
Z_: M
1 '
Taking the unit of length equal to onehalf the focal distance,
(8.38)
(a
~ G J
 
i.e., if we set q
= 1,
the ratios R/M and Z/M will~?`be independent of
the dimensions of the ellipsoids and the ratios of their axes, since
al and cl depend only on the focal distance 2q and the coordinates of
point P.
Therefore, to keep H and Z constant.at any dimensions of the
ellipsoid the magnetization must be changed by the corresponding
factor. Figure 160 shows the graphs of functions (8.38) for q = 1
(1) and q = 2 (2) with R = 1. These graphs show the degree of change
in the Z and H curves corresponding to a change in the horizontal
axis of the ellipsoids (q in 1 and q = 2). From the curves it is ap
parent that the ratio of maximum H. toaaxiaum Za,, when q = 1, is
0.46, and when q = 2 it is 0.56. The.ainiaua of the vertical compo
nent is only slightly more than 0.02 of the maximum when q 1 and
when q = 2. These, as well as the precedinggraphs are characterized
by the slow change of horizontal and vertical components.' The curves
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become wider with increase of the horizontal axis or, what amounts
to the same thing, with increasing eccentricity of the ellipsoid.
The graphs also show that the maximum and minimum of the
horizontal component occur almost over the foci of the ellipsoid.
Infinitely Long Circular Cylinder Magnetized Perpendicular)
to the Axis
Many of observed anomalies are narrow regions elongated in
one direction, exemplified by the Kursk anomalies. Such anomalies
are produced by extremely long bodies of rock with transverse dimen
sions very small in comparison with their length. Their geometrical
images are infinitely long bodies parallel to the earth's surface
and in cross section resembling a circle, ellipse, rectangle or
parallelogram. In the first instance the rock body may be considered
an infinitely long cylinder, in the second an elliptic cylinder and
in the third and fourth instances infinitely long prisms.
The magnetic field of any such body is uniform along a line
parallel to its axis, i.e., does not depend on one of the 3 coordi
nates. Therefore the problem of determining the magnetic field of
such bodies is called a plane or 2dimensional problem, since its so
lution consists of finding the potential within a plane perpendicular
to the axis of these bodies.,
We will seek to determine the magnetic potential of a uniformly
magnetized cylinder by means of the theorem of Poisson
U = (J grad V), (8..39)
in this case the vector J must be perpendicular to the axis of the
cylinder.
The quantity V, which is proportional to the gravitation potential,
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is determined from the equation of Laplace, which in the case of
a 2 dimensional problem with cylindrical coordinates is of the form
1 y~ 4 X j~ ~J ~y i
where r and & are the cylindrical coordinates of point P (Figure 161).
Since the cylinder has a constant density the gravitation po
tential must be symmetrical in relation to the axis of the cylinder,
and therefore independent of the coordinate as a result the above
equation becomes: 7 `1 _ du c' V
a,[,
Integrating this equation twice, we obtain
'
where C1 and C2 are arbitrary integration constants.
Substituting this value of V into equation (8.39) we have:
C, Ci r (8.40)
An expression analogous in form is obtained for the magnetic
potential of 2 infinitely long filaments having opposite charges and
separated by an infinitely small distance.
Indeed, considering such charged filaments as a combination
of an infinitely large number of dipoles parallel to one another, the
axes of which are perpendicular to the filaments, their magnetic po
tential may be represented as an infinite sum of the potentials of each
of the dipoles, or at the limit, as the integral of the dipole potential,
However, the magnetic moment of the dipole may be expressed as
the product of magnetic moment J~ of the unit of length of such a fi4a
went, by the element of its length dl, i.e., d9 _ j) dl, while the
distance r' and angle ~ may be a pressed as the distance r (Figure 162),
and angles 9 and0 may be expressed by the formulas
387
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C"W (e
Substituting these values, we have:
~~ ~ 1% ta
tJ J
which, when integrated gives
(8.41)
(8.42)
Since expression (8.t0) for the potential does not depend upon
the magnitude of the cylinder radius, it follows that by reducing the
radius to infinitely small dimensions, we arrive at a 2pole filament.
Comparing expressions (8.40) and (8.41) we find that
hence
where S is the cross sectional area of the cylinder, i.e., the area of
a circle.
Thus a cylinder uniformly magnetized in the direction of its
diameter is equivalent to a 2pole filament, the magnetic moment per
unit length of which is equal to the magnetic moment per unit length
of the cylinder.
Assuming that themagnetization of the cylinder forms an
angle ~ U with the vertical axis (Figure 161), and the radiusvector
r forms an angler with the same axis, then 8 and therefore
/f
Transposing from polar to rectangular coordinates, we obtain
kt ( ~~ , ~'_2    
U J. 'Y
whence we have
l7 _ ?_2
~ ~ fxY L'
The graphs of these functions are shown in Figure 163.
At 0, i.e., with vertical magnetization we have
u
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rte ' (.t,/&
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The corresponding graphs are shown in Figure 164. These graphs
are analogous in form to the graph of a sphere or dipole, but the cor
relation of extremet.1values of H and Z is somewhat different, thus;
II J rf ~ w'
atx=0 L 1 {1_1'_/ I
at x=113R
i/ ir3 R
f
5. Elliptical Cylinder of Infinite Length, Magnetized along the,?llf ybr
Axis of the Ellipse.
An infinitely long elliptical cylinder constitutes an ellipsoid,
one of the axes of which is infinitely long. Considering the case in
which the major axis of the ellipse of section is vertical and the mag
netization coincides with this axis, then according to formula (0.57)
the magnetic potential of an ellipsoid, for one axis of which a = co,
Since a = a1, N1 will be
i
_..ri r yam
.,l
Integration is effected by the substitution
which gives:
and as before, b, and c, are
determined from the equations
substituting the value of N1
f)
where is the magnetic moment per unit length of the elliptic cylinder.
By simple differentiation of potential U the following values
are obtained for the components of the intensity of the magneticfield:
~
Z ~
I
Z
?,
,fir .. G ~
3~ l
in the equation of Ul? we have;
(8.43)
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Figure 165 contains the graphs for q = z = 1, showing H and Z
as functions of y, and the graphs of an infinitely long vertical plate
of width 2q. From comparison of these graphs it is apparent that the
magnetic field of an infinitely long cylinder differs very little from
the field of thistype of plate, and therefore in practice it is en
tirely permissible to utilize formulas relating to an infinitely thin
plate. This enables determination of the position of the upper and
lower ends of the plate, which are equivalent to the foci of an ellip
tical cylinder. This is substantiated by the fact that the dimensions
of an ellipsoid with semiaxes a and b and given q can be of any magni
tude, as is apparent from equation (8.43).
6. A Thin and Infinitely Long;Flaty,elate Magnetized in Width.
Suppose that a rock layer having the shape of a thin plate is
located parallel to the earth's surface, forming an angled, with the
vertical 001 (Figure 166), and the magnetization J coincides with the
side ab = 1 of the rectangle abed of the transverse section of the
plate.
With uniform magnetization of the upper and lower surfaces of
the plate magnetic charges of surface density J will be formed, and
for each unit length the magnitude of this charge m will be m
Disregarding the width be, the magnetic mass concentrated at
some point Q, may be represented as a dy,, where dy is the element of
length of the plate. At any'point of space P this mass produces a mag
netic field with intensity
d$T = m dy/p2 ,
where p is thedistance between the elemental mass and the point P.
The components dX, dY, dZ on the coordinate axis are obtained
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by multiplying dHT by the corresponding cosines of the angles between
PQ and the coordinate axes.
The x axis is extended through point P perpendicularly to the
length of the plate, the z axis is placed in a vertical position, and
the origin of coordinates is placed at some random point of the upper
surface of the plate.
In this case the coordinates of point P will be x, o and R,
the coordinates of point Q>>rill be o, y, o, and therefore
The components of vector dHT are expressed by the equations:
I
.' ~r F ,;2_ Y44'1 (8.44)
J
Integrating these expressions within the limits  &: to + cv we
on Ile,
14 2 '4
(8.45)
These equations show that the magnetic field of an infinitely
long charged, or single pole line does not depend upon the coordinate
y which is parallel to this line.
Similar expressions for H and Z are obtained in relation to the
lower surface of the plate, the coordinates of which are
Therefore, the magnetic field of the entire plate is defined by
components H and Z, which are of the following form:
With a vertical position of the plate, at 0:
(~tj ( 1
7___,_
(8.46)
(8.47)
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Graphs, corresponding to equations (8.46) and (8.47) are
shown in Figure 167, and the graphs of equations (8.45) are shown
in Figure 165.
In view of (8.45) the maximum values m and Zm in the case of
a singlepole line will be:
Hm = m/R, Z = 2m/R
i.e., the maximum Zm is twice Hm.
The geometric image intermediate between a singlepole and a
dipole formation is that of a 2pole magnet (schematic) and thus a
thin plate is the intermediate geometrical image between a linear pole
and a dipole filament equivalent to a cylinder magnetized along the di
ameter. Indeed, with L approaching infinity we have the field of a
singlepole filament or line, and with ,,t.. approaching zero formulas
(8.46) and (8.47) describe a cylinder. To demonstrate this, it is
sufficient to reduce equations (8.46) and (8.47) to a common denomin
ator and in the resulting expressions equate 'L= 0 and mdl =' .
The conditions under which a 2pole plate may be replaced by a
single hole plate are analogous to those of a 2pole magnet, and are
derived in the same manner as the conditions expressed by the ratio (8.34).
In such a case the relative difference between the vertical com
ponents, which,must be loss than a certain quantity, will be defined by
the following expression:
Zx  Z2/Zl = (ZO  1) f (l +2)2 + x2 ) + (1 +1) +x 2 )/
ZO El + ~Q ) + x j ,
Z0 s ) /l +,b.
However the depth at which the second ,pole of the plate must be
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located will be considerably greater than the depth of the second
pole of the magnet. The following table shows the magnitude of the
relative.difference of the vertical components in percent, for var
ious distances between the poles, and for various values of the
coordinate x, assuming'the depth value of the upper pole is equal
to unity.
x
5
10
20
0 .
0
0
0
0.5
3.9
2.3
1.2
1
18
8.8
4.8
The table shows that the depth of occurrence of the second pole
must be not less than 10 times the depth of the upper pole, for the
plate to be considered monopolar, assuming that the relative error in
the determination of Z is of the order of 6 to 8%.
7. Infinitely Long Rectangular Prism, Magnetized Vertically. Verti
cal Stratum. A rectangular prism of infinite length with its
upper side parallel to the earth's surface may be consideredas being
composed of infinitely great number of infinitely thin plates, and its
magnetic potential is calculated as the sum of the potentials of such
plates. In this calculation linear density of magnetic charge of the
thin plate must be formulated as the product of the surface density of
magnetism E "concentrated at the upper side, and the width of the in
finitely small plate dx0. Using the same system of coordinates
described above the horizontal and vertical components of the magnetic
field produced by the charges of the upper side of the prismare ex
pressed by the following integrals:
rte, / (YO x) 4vf?~~DJ (8.49)
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where x is the abscissa of point P at which the field is being con
sidered, x0 is the abscissa of the element of length dx0, and 2a is
the width of the prism.
Integration gives:
V
(8.49)
rti f 4 (8.50)
form with the zaxis by ol.,1 and ,.2, it is apparent from the drawing that
2t(z
If the radiusvectors r1 and r2 are plotted from point P to the
edges of the prism (Figure 168), and denoting the angles which they
i `Y f x
G' f J
r;
(8.51)
where c, is the angle at which the upper side of the prism is seen from
point P. It is evident that the charges of the lower side of the prism
will produce an analogous field, and its components will be expressed by
analogous formulas in which the distance Rl to the lower side appears in
lieu of the distance R. Hence the expressions for II and Z for entire
prism will be:
P
where the values r3, r4 and
01 6 NI l._  :,2
6 t ~c J,
/y ~' 1113 '
are apparent from the drawing.
If ' the ko ktudina'l dimensions of the prism are large in com
parison to the depth of occurrence of the ppper side, the influence of
the lower side may be disregarded. In such a case the magnetic field
will be defined by formulas (8.49) and (8.50), the graphs of which are
as
shown in Figure 169. The. graphs relate to 2 instances: when the width
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'r lf_.~._)  ,
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of thi prism is equal to its depth of occurrence, and when its width
is twice the depth. The graphs show that in the case of a prism with
addition, the maximum H is shifted to the right of the z axis.
equal to 0.5, increasing somewhat with an increase in width 2a. In
occurrence. The ratio of the maxime Hmax to Zmax is approximately
only in the case of a prismthe, width of which is twice its depth of
with the curves relating to a thin plate, and the deviation is apparent
width equal to depth ofcdcurrencethe H and Z curves almost coincide
The curves H and Z of the prism represent a sum of an infinitely
large number of H and Z curves of thin plates which compose the prism.
However, with an error that is admissible in practice they can be rep
resented as the sum of a finite number of curves of thin plates disposed
at a finite distance from each other. Thus the curves of a prism with
width 2a = 2R differ only slightly from the curves representing the mag
netic field of 2 plates disposed symmetrically in relation to the z axis
at a distance R from each other. The curves corresponding to the latter
case are shown in dotted lines in Figure 169. Therefore, given the H
and Z curves derived from observation it cannot be deduced whether they
represent the magnetic field of a prism or the sum of the fields of 2
or several plates, if no additional geological or geophysical informa
tion concerning the nature of the occurrence of the rocks is available.
a,.large;number of linear dipoles, the Magnetic field of which was dis
cussed'.in' section 2 of.' the present paragraph. The potential of such a
plate is represented by the integral
8. A Thin, Horizontal, Infinitely Long Plate. Thin Horizontal Stratum.
A= thin;, verticallyghagnetized plate or a stratum infinitely thin
in ;comparison;;:with.`the depth of:. occurrence, constitutes a combination of
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where the expression within the integration sign is the potential of
a linear dipole or infinite cylinder with linear magnetic moment )dxo.
Further, y is the abscissa of point P at which the potential is being
considered, and 2a is the width of the plate.
After integration, we have:
( 6.11
which is differentiated with respect to the corresponding coordinates,
yielding the following values for horizontal and vertical components:
V /11
V"a C =t (x 6) ZJL ' z+ (x ~,~''
t 62
PC / l1~ l
4~ Jr ~ f ~ .x  JL f; t C4!
(8.52)
L ? ~~ rd C~  ~. z I
/L P
where r1 and r2 have the same values as in the preceding instance of a
prism (Figure 168).
The same correlations are obtained for the components of the,
magnetic field of 2 infinitely long rectilinear currents distributed
along the edges of the plate, since the magnetized plate constituting
a double magnetic layer is equivalent in its properties to such currents.
From this it follows that if the width of the plate is large in compari
son with the depth of its occurrence, or if the distance between the
currents is large, the magnetic field over the edge of the plate will
be greater than over its center.
Indeed, upon analyzing equation (8.'52) it canbe readily shown
that if a ?'3 the vertical component Z has one maximum at
a Z maximum values are
and 3 minimum values are obtained if ', at
Figure 170 shows the Z and B graphs for a = 1/2 R and a  3R. In
the fir*t instance, a "3R, we actually obtain one maximum and 2 minima of
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Z, and in the second instance, when a ) '4 3R the Z curve shows 2 saxi
In addition, it is characteristic of bodies of this kind that
with an increase in the width of the plate 2a the maximum of the hori
zontal component increases, while at a width exceeding a certain value
the maximum of H becomes greater than the maximum of Z and is located
over the edge of the plate.
only slightly, i.e., the magnetic field of a thin plate, the width of
which is less than the depth of occurrence, practically does not differ
from a linear dipole.
of a linear dipole or cylinder (Figure 165), it is seen that they differ
Further, on comparing the curves for a = 1/2 R with the curves
9. SemiInfinite Horizontal Stratum.
Let us suppose that the rock formation is horizontal stratum of
constant thickness and infinite area in a horizontal plane, but bound
on one side by a vertical plane. Such a stratum will be called semi
infinite. The limiting plane is called the contact plane, since at
this plane the stratum contacts other rocks. If such a stratum'is
magnetized vertically it is evident that at a distance from the con
tact the magnetic field underneath it will be constant and close to
zero, and appreciable gradients and an appreciable s#gnitude are found
only above the contact.
The components of the magnetic field intensity of such a stratum
are determined,by expressions (8.48) for H and Z, corresponding to the
upper and: lower side of an infinitely long prism, and only the limits
U, 77 I'll " (If4tiwi"
/z 
jof integration are changed, namely:  eo
?t~a
6Z x) 0' '14
16
Al 0
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After integration, we have 2~ (4c
T ; f Z (8.53)
(8.54)
It is readily apparent that above the contact, where x  0 and
Z = 0, H has a maximum value, and at x >> L both components H and Z
approach zero.
Maximum and minimum values of the component Z occur at x =
? JR(1t + R), and their magnitude is
17
The curves for R = 1 and J = 4, based on formulas (8.53) and (8.54),
are shown in Figure 171.
10. Vertically Magnetized Sloping Stratum of Infinite Length.
Let us assume that a sloping stratum of rock has a vertical
plane section shaped like a parallelogram (Figure 172) one of the
sides of which is parallel to the earth's surface and the opposite
side is located at a great depth, which we will assume to be equal
to infinity. With a uniform vertical magnetization magnetic charges
will be found at the upper side ab, having surface density , while
at the lateral sidebe the density is.S cos , where ~ is the angle
of inclination of the stratum in relation the the horizon. At the op
posite side ad the charge.density is  cos
In accordance withi'the deductions of section 7 the components
H1 and Zi of the magnetic field at point P on the upper side will be:
/~~ (8.55)
where c, rl and r2 have the values shown in Figure 172.
If the origin of coordinates is placed at point P and a new x'
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axis is drawn parallel to the side be, in accordance with equation
(8.48), the components H' and Z' of the sides be and ad, in relation
to the new system of coordinates, will be:
26 7
determined by adding expressions (8.55) and (8.57), i.e.,
In the former system of coordinates the components H2 and Z2
are determined in accordance with the transition formulas
?l f (8.57)
The expression for the components of the entire stratum are
 ?..LGrte ` "
or, in accordance with formulas (8.49) and (8.50), replacing (r
and r2 by their values, we have:
From Figure 173, which shows the Z and H curves in the caseof
a stratum with an inclination of 600, in relation to the horizon, it
is apparent that the inclination causes asymmetry of voth curves al
though the lower boundary is at infinity. In comparison with a ver
tical position of the stratum (Figure 169), the value Z of which is
nowhere negative, the slanting stratum has a vertical component which
passes through zero at values of X approximately twice as large as
the depth of occurrence of the upper side, and at x> 2R it assumes
negative values. Further, the maximum value,of_ the horizontal compo
nent increases sharply with an increasing angle of inclination, which
also is a characteristic feature of rock occurrences of this kind.
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(8.58)
rl
.~ r  i? !lam ( ~(/ ^~l` r 7
t.
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It must be noted that in the case of a sloping stratum which
extends deep into the earth but is magnetized along its axis, the
magnetic field will be identical with the field of a vertical stratum,
i.e., in the case of axial magnetization the inclination of the
stratum is of no significance.
11. The Method of Cards for Solving the Main Problem of Magnetometry
In cases in which exact determination of the potential is im
practicable, the conventional methods of approximate calculations may
be used if they enable the, preparation of suitable cards for determin
ing the magnetic field of sots group of bodies. In geophysics the term
cards is used to designate transparent sheets on which systems of curves
or straight lines derived by theoretical calculations are plotted. In
many instances the superposition of such a card over the experimental
curve enables the necessary result to be obtained without resorting to
additional calculations. Thus D. S. Mikov has proposed a card for
determining the magnetic field of infinitely long horizontal cylindri
cal bodies, the cross section of which may be a plane figure of any
shape, and which are uniformly magnetized perpendicularly to their axis
(75). Due to the uniform magnetization the magnetic field of such a
cylinder may'be regarded as the field of an infinitely large number of
elemental cylinders the magnetic potential of which may be expressed
according to equation (8.41) as follows:
dU = 2dM/r cos 8.
The field intensity components inthe direction of the radiusvector r
and perpendicular to the latter will'be:
dlr .= 2d /r2 coo 8 and dHe 2dWr2 sin e.
Projecting these components on the magnetic axis of the cylinder
and on the axis perpendicular to the latter, we obtain the components
in these directions, which we shall denote by dHn and dHl . It is
readily apparent that:
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Since in these expressions the integrals do not depend upon the
magnetization of the rock and their values are determined solely by the
position of point P and the shape of the cylinder cross section, the
integrals may be calculated once ;lor??*l2:yd finite shapes of cross sec
tional areas found at different distance a r from the point P. It is
most convenient to determine the dimensions of areas of segmentary
shapes, which produce at point P a field intensity component equal to
unity.
A v
The elemental magnetic moment dM may be replaced by dM=J dS,
and the elemental surface may be expressed by the polar coordinates
dS = r dr d9.
Then, replacing dM in equation (8.58) and integrating over the
entire area of the cylinder cross section we obtain:
For segments limited by radii rn and rn 4 1 and the angles 8m
and 9m+1, the equations (8.59) assume the form:
(8.58)
(8.60)
y;
Taking P=100 CGSJ` and equating each of the equations equal to
5
one gamma (1.10 CGSJ ), we obtain 2 equations and 3 unknown differ
ences. Therefore one of them must be arbitrarily taken equal to a
.:~* o,03V7
certain constant quantity. D. S. Mikov assumed
in which case the natural logarithm of,the ratio was found to be
and the common logarithm t7
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Hence the values of angle 0 and radius r which limit the
segment areas are:
8 = 0?; 3.6?; 7.20; 11.00; 15.00; 19.30; 24.3?; 30.50; 45?;
r = 1.00 ; 1.08; 1.17; 1.27; 1.38; 1.49, etc.
The angles of the next octant will be symmetrical with the angles of
If radiusvectors are drawn from some point P at angle 0
and limited by the radii r we obtain a grid consisting of'a number
of segments, each of which produces at point P a component H~,equal
to 1 . 105 CGS P. It is evident that the direction of H,t coincides
with the radius vector corresponding to 0 = 0. Having drawn such a
card, if we wish to calculate the magnetic field of a cylinder having
the cross section shown in Figure 174 the center of the card is placed
over the point P at which the component Hj~ is to be determined and the
card is adjusted so that the zero radiusvector extends in the direction
of magnetization of the cylinder. The number of areas thus brought
within the contour will correspond to the magnitude of Hl[. From the
equations of (8.60) it is apparent that to determine H1 it is sufficient
to turn the card through an angle of 45?. When Ht} and?Hi are known
it is not difficult to switch to the components Z and R. which are,
determined by means of the equation
Z = H t` cos i  HI sin i ; H : H} cos i + H \` sin i
what* i is the angle between the vertical and the vector J.
Instead of counting the areas it is more convenient to count the
points found, at the center of each area. An example of such a card is
shown in Figure 175.
T. X. Rose [76) used the same card method to calculate the vertical
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component of semiinfinite vertical cylindrical bodies, the upper
base of which is limited by any contour. In this process it is as
sumed that the cylinder 'is uniformly magnetized in the vertical di
rection.
In the case of such a cylinder, the upper baseof which is
located at a depth h, the vertical component of magneti "field in
tensity at point P (Figure 176) is expressed by tht.equation:
Z = hJ"J dS/(h2 + r2)3/2,
where J is magnetization of the cylinder, r the distance from the
element of surface dS to the vertical axis extending through point
P. and the integration is carried out over the entire surface of the
cylinder base. If dS is expressed in polar coordinates, it follows
Z  hi ( r dr d6/(h2 + r2)3/2.
As in the previous case the base of the cylinder is divided
into segments of such area that each of them produces a vertical
component of one v, at point P. The dimensions of these segments
must satisfy the equation:
J(om+1gm)(h/ h + r2  h/ 1 h2 + rZ ) = 1  10"'5
n n+l
Having calculated and plotted 0m and rm in accordance with
this formula the contour is divided into a number of areas limited
by radiusvectors and circumference arcs. The number of these areas
will be equal to;,the magnitude of the vertical component expressed
in gammas. Since the dimensions of, these areas are independent of
the magnitude of .J, they may be plotted beforehand for all the values
from r 0_ to r =W andfrom 9 0 to 9 = 2s and may be used as a card
for the calculation of the Z of any plane figure.
To prepare such a card it is necessary to assume values for
the interval Om+liim, and for the quantity J..
403 
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Assuming J = 103, Am+1" em " 10? and replacing the terns
appearing between the second parentheses by the cosines of angles
(Figure, 176), we have:
(cos n  cos (C n+i )  0.0573
assuming cos k n + 1= 0, which corresponds to r = Oo, we obtain
cost = 0.0573 or ~ n = 86.7?. Then, adding 0.0573 to each value of
the cosine in succession, we obtain all the following values. The
last value of the cosine will be 0.9738.
When l nis known, the radius r may be determined according to
the formula rn = h tg T n
The last value to = 0.9738 corresponds to an area with radius
r0 : 0.23 h, for which the quantity Z is determined according to the
formula
Z=29J(lh/)).
0
A card plotted in this manner will have the appearance of rays
drawn from a single point at intervals of 100, and of a number of con
centric circum=erences having the radii r0 = 0.23 h, ri = 0.44 h, jig 
0.60 h, etc.
To determine Z at some point p the center of the card is
placed over the point P and the number of areas included within the
contour is counted.
This.kind of card also may be used for determination of the
vertical componentproduced by a body of any shape, magnetized in the
vertical direction. To do this the body is divided into layers by
horizontal planes; at equal intervalsand the card is used to determine
the Z , for, a, number ~ of plane surfaces : S1, S2  S1; S3  S2, 'etc where
S1 is the surface produced by intersection withthe first plane, S2 that
of the second section, etc (rigors 177).
 404 
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Since the depth of occurrence of each sectional surface is
different, the center of the card must be moved every time so that
the ratio of the distance of point P from the epicenter 0, to the
depth of the corresponding layer is always the same.
The card for determination of the horizontal component is
somewhat different in appearance. Within each sector having an
angle at the base equal to 100, the radii of circumferences rn have
different values for the same values of h, since in this case rn
depends upon the angle 0n.
12. Experimental Method ofSolving the Main Problem
Both theoretical calculation of magnetic fields and the card
method include many assumptions, such as the uniform magnetization
of the rock strata and the geometric regularity of their shape, which
actually are never encountered. Furthermore, in many instances both
methods lead to fairly complex and laborious calculations as a result
of which they are poorly adapted for practical use.
Therefore, in such cases it is more appropriate to determine
the magnetic field by means of direct measurements of the field over
rock formations of different shape, produced artificially in the form
of models. From the very formulas for the components of intensity of
a magnetic field produced by bodies of regular shape, it follows that
with a proportional change of their dimensions; i.e., retention of the
similarity, the intensity' of the magnetic field will remain the same
if the distance to the center of the body is changed in the same propor
tion. However this characteristic of similarity in the topography of
the field of such bodies also is retained in the case of bodies of
any shape, and therefore permits utilization of the method of models
for determination of the magnetic field of rock formations. of any shape.
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The experiment,conjiists of the preparation of a correspondM
1
ing model and measurement of Z and H at different points of space
around this model. Because of the small dimensions of the model in
comparison with the naturally occurring rocks, the gradients of the
magnetic field become very large, with the result that the volume
of space within which the field may be considered uniform is very
small. Hence the measuring instrument must be of very small dimen
sions. This requirement is met by the magnetometer of A. G. Kala
~Shnikov, which is called a field meter. The field meter consists of
a direct current generator, the principal part of which is a coil
provided with a 5 malong and 1.5 mm diameter permalay core wound with
thin enamelcoated wire 0.03 mm in diameter. The coil is connected to
a synchronous motor by a long rod which rotates the coil in the magnetic
field to be measured at a speed of 25 revolutions per second, and the
induction current produced as a result of this rotation is measured by
means of a galvanometer. With the above dimensions the coil measures
the mean value of the field within a spherical space 3 ma in diameter.
This is quite sufficient if the dimensions of themodel are chosen
accordingly.
The coil and the motor can be moved a certain distance in
vertical and horizontal directions.
Using,this type of a field meter A. G. Nalashnikov and S. S.
ponton (77] carried out measurements of the fields over models of
faults eonsistingg of 2 rectangular plates 120to 500 ma long and 20
to 120 mm thick, which were given various vertical displacements.
The models were made of magnetite powder mixed with an equal
weight of pl'astelin;
I I
mately 1002cGSJ%
giving the'product a susceptibility of approxi.
406
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Figures 178 and 179 show the characteristic graphs of Z and
H over faults of different types, Figure 178 relating to a fault in
which both branches extend over a large distance while Figure 179
relates to a fault in which length is commensurate with thickness.
A characteristic feature of these graphs is the inflection of the Z
curve, and also the occurrence of an H maximum over the boundary of
the fault.
The results of these measurements have demonstrated the possi
bility of wide utilization of the experimental method in the solution
of the main problem of magnetometry, which in the case of structures
such as faults is extremely complex in its theoretical aspects.
Section 9. The Possibility of Utilizing the Plane Problem
A boundless expanse of rocks does not occur in nature. A rock
occurrence always is limited in the horizontal as well as in the ver
tical direction. Hence the question arises as to the practical in
stances in which it is possible to apply the formulas derived in the
preceding section for bodies or rocks having boundless dimensions in
the horizontal direction, i.e., when is it possible to apply the plane
problem. To answer this question the magnetic field of the rock or
body having an unlimited expanse must be compared with d body of
finite dimensions. Therefore, as an example we will consider the
magnetic field of a singlepole line and of a linear dipole of finite
length.
Since for mathematical interpretation ofanomalies of limited
area it is: sufficient to know the magnetic field along the profile ex
tending through the middle and throughthe cross section of its area.
it follows that without departing from the general nature of the problem
we can, seek to determine the magnetic field'through the middle, and *er
pendicular to the singlepole line and to the linear dipole.
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In accordance with equation (8.44) the components of field
intensity of a single pole line o ,Ec finite length may be represented
by the following integral expressions%
V
,U
where the limits of integration are the positive and negative values
of the halflength of the pole line. The values of the other symbols
are the same as on page 349. The symbol 11 appearing next to X, Y
and Z is used to differentiate these components from those of a line of
infinite length.
Integrating within the above limits, we obtain the following
expressions:
/
a
q
At the same time, we have r2 = R2 + x2.
Comparing these expressions with the expressions of (8.45) for
an infinite line, we see that:
/I Y _~/~  /  k' 1~
Therefore, in order that XL approximate X and Z the halflength
of the pole line must be considerably greater than r. To evaluate'the
magnitude of the ratio r/j, at which the.pole line may be considered
equivalent to aninfinite line the.. numerator and denominator of the
right band portion are divided:by J , and disregarding the terms of higher
order, we obtainV
whence we have
f" Z
 4j  1
(8.61)
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Assuming the limit of error does not exceed 3% in the deter
mination of H and Z over a profile 3 times the depth of occurrence
(X'> 3R) this equality may be used to determine the length of the pole
line 2,t, which is equivalent to an infinite line. Indeed, from equa
tion (8.61) we obtain
j (I,, "+L' O,03
i.e., the length of the pole line must be 25 times greater than the
depth of occurrence.
The same result is obtained for a horizontal cylinder magnetized
along the diameter. In accordance with formula (8.41) the magnetic
potential UL of a cylinder of finite length 21 , is:
1 V~
J4
yt9  . ' / CAS ~_~ v `?v 4 
where the meaning of the letters is the same as in formula (8.41). On
integration, we obtain
f v = C1  cc
{
At small values of angle i.e., when r we have, with a suffi
cient degree of accuracy,
Z !'.n a, c :o?I the? ~d~iagn&4onieter
Figure: 195. , Collimator` .`methdd" of reading:.
188. Diagram of 'magnetic forces acting on a magnetic compass.
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figure 196. Mirror method of reading.
Figure 197. Magnetic system, of the "combine" instrument.
Figure 200. Needle inolinator.
Figure 201. Magnetic needle of the needle inclinator.
Figure 203, Procedure for magnetizing the needle.
Figure 204. Diagrammatic arrangement of the induction iholinator.
Figure 205. Induction inciinator.
Figure 205. "Combine" magnetic theodolite for measurement of the
vertical component.
Figure 209a. First Gaussian position of magnets.
Figure 209b. Second Gaussian position of magnets.
Figure 2090. First Lamontian position of magnets.
Figure 209d. Second Lamontian position of magnets.
Figure 214.
Figure 215.
Figure 216.
Figure 217.
Figure 221.
Figure 222.
Figure 223,
'i gore 224.
Diagram of rotating disks of the synchronous clock.
Diagram of a unit for determination of the period of
oscillation of a magnet.
Diagram of Lamont's procedure for determining ')v Diagram of the magnetometric method for determining`)
Diagrammatic arrangement of the VNIIM absolute magnetic
theodolite.
VNIIM absolute magnetic thieodolit*.
VNIIN absolute magnetic theodolite.
Houiing_ for,.osci11atioh of the, magnet.
>Wiring diagram' of the ele;ctric? method: for determining H.
Figure "Combiner magnetic: ;theodolite.
Quartz, magnetoietere .
ammati0, arrangement:'` of ra double 'compass.
r''Divisiozi.eciale ?and,pointers.of.,:a~dou ble=compass.
Figure;'232 = Electric magnetometer of B..Ye. Bryunelli
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Figure 235. 111 magnetometer for measuring the vertical component.
Figure 237. Universal balance.
Figure 238. Magnetic system of the balances (a) for measuring the
vertical component; (b) for measuring the horizontal
component.
Figure 239. Diagram, of the optical system of the magnetic balance
and the propagation of rays therein.
Figure 241. Aeromagnetoneter of A. A. Logaehev.
Figure 242. Electric diagram of the aeromagnetometer.
f`K  section of coarse compensation; TK  section of
fire compensation; no  permanent magnets; K  com
mutator; 3 M  1: electric motor of the induction
loop; 3 M2: electric motor of recorder; PPK37500
current  supplyingstoragebattery.
Lin drawiAo
Magnetometer
Figure 243. Took diagram of a magnetometer with a single sonde. .
f alternating current generator; toI  filter sepa
rating oscillations of frequency f; (III  sane, of
frequency 2f; YG  amplifier; A  ammeter; P 
choke coil; W1  primary winding; W2  secondary
winding.
/in drawing/
A2
sonde.
aora;s; of a mag tometer%. with 2 sondeb.
r ;Basi a agnet .zation curve of armoo iron (1) and
,', rt;, :?,  tancra; i9ic~ y(i)
Figure 2449. look,d,tigram of..?aeromagnetometer.
'659 
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A  equipment placed in gondola outside aircraft;
a  measuring sondes;  orientation sondes; 
2phase reversible servomotors;''a)  control phases
of servomotors; ~  fixed phases of servomotors.
B  equipment mounted within aircraft cabins
T  generator;  direct current battery supply
ing compensation windings; 0  automatic recording
device; Yl  measuring channel amplifier; Y2 
measuring sonde winding feed amplifier; 13  orienta
tion sonde winding feel amplifier; Yea  orientation
channel feed amplifier; Y5 ? fixed phase feed amplifier.
Figure 250. Graph.
r '
;3re 251. Graph
Figure 252. Diagram of unifilar Mmagnet; Csuspension thread; F
reflecting mirror; Ktwist knob.
Figure 253. Graph
Figure 254. Graph.
Figure 255. Graph.
Figure 256. Graph, no  unifilar magnet; n's'  compensating magnet.
Figure 257. Graph.
Figure 258. Diagram of magnetic system of variosmeter.
Figure 260. Optical recording system of variational instrument.
Figure 261. Decimation and horizontal component variometer
Yanovskiy.,
Figure 262..Vertici component variometer by_:Yanovskiy.
Figurer,;263: Magnetic, balanat..
Figure 264. Recording apparatus.
Figure ,265. Variation Station.
.?.'ia%a~,.+`.Y`~{x,':Y6:icl,?,r'.~J~io7?~SS7;~'r"'~'~+s~a:. _ ~_
by
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