Principles of the Magnetic Methods in Geophysics

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  • Format: Hardcover
  • Copyright: 2008-12-10
  • Publisher: Elsevier Science
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Magnetic methods are widely used in exploration, engineering, borehole and global geophysics, and the subjects of this book are the physical and mathematical principles of these methods regardless of the area of application. Beginning with Ampere's law the force of interaction between currents is analyzed, and then the concent of the magnetic field is introduce and the fundamental features are discussed. Special attention is paid to measurements of relaxation processes, including topics as the spin echoes or refocusing. Also the speical role of the magnetic method in the development of the plate tectonic theory is described. * covers all the physical and mathematical principles of magnetic methods regardless of the area of application. * presents thorough developments of magnetic methods.

Table of Contents

Introductionp. xiii
Acknowledgmentsp. xv
List of Symbolsp. xvii
Magnetic Field in a Nonmagnetic Medium
Interaction of Constant Currents and Ampere's Lawp. 1
Magnetic Field of Constant Currentsp. 3
General form of Biot-Savart lawp. 5
The Vector Potential of the Magnetic Fieldp. 8
Magnetic Field and Vector Potential, Caused by Linear and Surface Currentsp. 12
Magnetic field of a current filamentp. 13
The vector potential A and the magnetic field B of the current in a circular loopp. 15
The magnetic field of a magnetic dipolep. 18
The vector potential of a system of dipolesp. 21
Behavior of the of field B near surface currentsp. 22
System of Equations of the Magnetic Field B Caused by Conduction Currentsp. 25
Example one: magnetic field due to a current in a cylindrical conductorp. 29
Example two: magnetic field of an infinitely long solenoidp. 31
Example three: magnetic field of a current toroidp. 34
The System of Equations for Vector Potential Ap. 36
Magnetic Field Caused by Magnetization Currents
Magnetization Currents and Magnetization: Biot-Savart Lawp. 39
System of Equations of the Field B in the Presence of a Magnetic Mediump. 41
Relation between magnetization Currents and Magnetizationp. 42
System of Equations with Respect to the Magnetic Field Bp. 45
Field H and Relationship between Vectors B, P, and Hp. 46
Three Types of Magnetic Media and their Magnetic Parametersp. 47
Inductive and residual magnetizationp. 47
Types of magnetic mediump. 48
Magnetic permeabilityp. 49
System of Equations for the Magnetic Field Bp. 50
Distribution of Magnetization Currentsp. 51
Volume densityp. 51
Surface densityp. 53
System of Equations for the Fictitious Field H and Distribution of its Generatorsp. 55
Volume densityp. 55
Surface densityp. 56
Difference between the Fields B and Hp. 56
Example 1: Current loop in a homogeneous mediump. 57
Example 2: Uniform fields B and H in a medium with one plane interfacep. 57
Example 3: Fields B and H inside the toroid with a small gapp. 59
Example 4: Fields B and H inside the solenoidp. 61
Example 5: Fields B and H inside the magnetic solenoidp. 61
Example 6: Influence of a thin magnetic shellp. 61
The System of Equations for the Fields B and H in Special Casesp. 62
Case 1: A nonmagnetic mediump. 62
Case 2: Conduction currents are absentp. 63
Case 3: Residual magnetization and conduction currents are absentp. 64
Case 4: Uniform piece-wise medium where conduction current and residual magnetization are absentp. 65
Magnetic Field in the Presence of Magnetic Medium
Solution of the Forward Problem in a Piece-Wise Uniform Medium When Conduction Currents and Residual (Remanent) Magnetization are Absentp. 67
Equations for the scalar potentialp. 67
Theorem of Uniqueness and Boundary-Value Problemsp. 69
The first boundary-value problemp. 70
The second boundary-value problemp. 72
Boundary-value problem in the presence of an interface of media with different [mu]p. 73
A Cylinder in a Uniform Magnetic Fieldp. 75
Solution of the boundary problemp. 75
Determination of unknown coefficients and field expressionsp. 79
Distribution of magnetization currentsp. 81
Behavior of the magnetic field inside the cylinderp. 82
Induced magnetization vectorp. 82
Medium of small susceptibilityp. 83
Secondary field outside the cylinderp. 84
The primary field is directed along the cylinder axisp. 85
An Elongated Spheroid in a Uniform Magnetic Field B[subscript 0]p. 86
Laplace's equation and its solution in spheroidal system of coordinatesp. 86
Field of a Magnetic Dipole Located at the Cylinder Axisp. 92
Solution of Laplace's equation in the cylindrical coordinatesp. 93
Expressions for the potential of the magnetic fielp. 97
Coefficients A[subscript m] and B[subscript m]p. 98
The current densityp. 100
Asymptotic behavior of the field on the cylinder (borehole) axisp. 100
Concept of geometric factorp. 102
Ellipsoid in a Uniform Magnetic Fieldp. 104
System of ellipsoidal coordinatesp. 104
Expressions for the potential of the primary fieldp. 106
Solutions of Laplace's equationp. 107
Potential inside and outside an ellipsoidp. 108
Spherical Layer in a Uniform Magnetic Fieldp. 111
Spherical magnetic body in a uniform fieldp. 114
Thin spherical shell in a uniform fieldp. 115
The Magnetic Field Due to Permanent Magnetp. 116
The vector potentialp. 117
The field outside of a thin cylinderp. 118
Scalar potentialp. 122
The Magnet in a Uniform Magnetic Fieldp. 124
Force acting on a free chargep. 124
Linear current circuit in a uniform magnetic field Bp. 125
Resultant forcep. 126
Moment of rotationp. 126
Thin and elongated magnet in a uniform magnetic fieldp. 128
Interaction between Two Magnetsp. 130
Two magnets are placed along the same line (Fig 3.8)p. 130
Current circuit in the magnetic field Bp. 133
Magnet in a field of a point magnetic chargep. 135
Magnetic forcep. 135
Moment of rotationp. 137
Energy of Magnetic Dipole in the Presence of the Magnetic Fieldp. 138
Permanent Magnet and Measurements of the Magnetic Fieldp. 139
Deflection method of measurementsp. 141
Theory of the vertical magnetometerp. 143
Main Magnetic Field of the Earth
Elements of the Magnetic Field of the Earthp. 147
History of the Earth Magnetism Studyp. 149
The discovery of the magnetic compassp. 149
Pierre de Maricourt (Petrus Peregrinus)p. 149
Magnetic compass and navigationp. 150
William Gilbert (1540-1603)p. 150
Edmond Halley (1656-1742)p. 151
Charles Coulomb (1736-1806)p. 152
Oersted (1777-1851)p. 153
Andre-Marie Ampere (1777-1836)p. 154
Carl Gauss (1777-1855)p. 155
Solution of the Laplace Equationp. 156
Orthogonality of Functions S[subscript n]p. 158
Solution of Equation (4.13) for the Functions Sp. 159
Legendre's Equation and Zonal Harmonicsp. 160
Solution of Legendre's Equationp. 161
Index n of functions p[subscript n] and q[subscript n] is positivep. 162
Recursion Formulas for the Functions p and qp. 163
Legendre Polynomialsp. 164
Recursion formulas for Legendre's polynomialsp. 166
Integral from a Product of Legendre Polynomialsp. 166
Expansion of Functions by Legendre Polynomialsp. 167
Expressions for Legendre polynomialsp. 168
Spherical Analysis of the Earth's Magnetic Field When the Potential is Independent of Longitudep. 168
The Physical Meaning of Coefficients B[subscript n]p. 171
Associated Legendre Functionsp. 172
Examples of the associated Legendre functions ([mu] < 1)p. 174
Integrals from a product of the associated Legendre functionsp. 175
Spherical Harmonic Analysis of the Magnetic Field of the Earthp. 176
Uniqueness and the Solution of the Forward and Inverse Problems
Introductionp. 185
Poisson's Relationship between Potentials U and U[subscript a]p. 188
Solution of the Forward Problem When the Interaction between Magnetization Currents is Negligiblep. 190
Development of a Solution of the Forward and Inverse Problemsp. 194
Example 1: Uniform half spacep. 195
Example 2: Layer of finite thicknessp. 196
Concept of Uniqueness and the Solution of the Inverse Problem in the Magnetic Methodp. 197
Main steps of interpretationp. 197
Uniqueness and its applicationp. 199
Solution of the Inverse Problem and Influence of Noisep. 202
Paramagnetism, Diamagnetism, and Ferromagnetism
Introductionp. 207
The Angular Momentum and Magnetic Moment of an Atomp. 208
Motion of Atomic Magnetic Dipole in an External Magnetic Fieldp. 212
The first approachp. 212
Frequency of precessionp. 214
The second approachp. 215
Magnetic Moment, Angular Momentum, Spin, and Energy States of Atomic Systemp. 217
Magnetic momentp. 217
Angular momentump. 218
Magnetic energy of atomic particlep. 219
The Stern-Gerlach experimentp. 221
Alternating magnetic field and transition between energy levels of atomp. 223
The Rabi molecular beam methodp. 225
Diamagnetismp. 228
Paramagnetismp. 231
Classical physics approachp. 231
Quantum mechanics approachp. 235
Ferromagnetismp. 237
Introductionp. 237
The magnetization curvep. 237
Hysteresis loopp. 240
Principle of the Fluxgate Magnetometerp. 241
Magnetization and Magnetic Forcesp. 243
Spontaneous magnetizationp. 247
Curie temperaturep. 248
Spontaneous magnetization and Weiss domainsp. 250
Case one: Single crystal of ferromagnetic and its domainsp. 251
Case two: Polycrystalline material of ferromagneticp. 252
Nuclear Magnetism Resonance and Measurements of Magnetic Field
Introductionp. 255
The Vector of Nuclear Magnetizationp. 258
Equations of the Vector of Magnetizationp. 261
Case 1: Additional field is absentp. 263
Case 2: The additional field is horizontalp. 263
Rotating System of Coordinatesp. 264
Behavior of the Vector P in the Rotating System of Coordinatesp. 266
Example 1p. 266
Example 2: The additional field rotates in the horizontal planep. 267
The case of resonance ([omega] = [omega subscript 0] = [gamma]B[subscript 0])p. 267
General case ([omega not equal omega subscript 0])p. 269
Additional field B[subscript 1] is a sinusoidal functionp. 273
Magnetization Caused by the Additional Fieldp. 274
Bloch Equationsp. 275
Solution of Bloch's equations when the additional field is absentp. 276
Measurements of Relaxation Processesp. 278
Introductionp. 278
Measurements of a decay of the longitudinal component of the vector Pp. 279
Measurements of a decay of the transversal component of the vector Pp. 281
Spin echoes or refocusingp. 282
Two Methods of Measuring Magnetic Fieldp. 283
Proton precession magnetometerp. 283
Optically pumped magnetometersp. 285
Bibliographyp. 289
Appendixp. 291
Subject Indexp. 297
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