Principles of the Magnetic Methods in Geophysics

• ISBN13: 9780444529954
• ISBN10: 0444529950
• Format: Hardcover
• Copyright: 2008-12-10
• Publisher: Elsevier Science

Summary

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

Introduction	p. xiii
Acknowledgments	p. xv
List of Symbols	p. xvii
Magnetic Field in a Nonmagnetic Medium
Interaction of Constant Currents and Ampere's Law	p. 1
Magnetic Field of Constant Currents	p. 3
General form of Biot-Savart law	p. 5
The Vector Potential of the Magnetic Field	p. 8
Magnetic Field and Vector Potential, Caused by Linear and Surface Currents	p. 12
Magnetic field of a current filament	p. 13
The vector potential A and the magnetic field B of the current in a circular loop	p. 15
The magnetic field of a magnetic dipole	p. 18
The vector potential of a system of dipoles	p. 21
Behavior of the of field B near surface currents	p. 22
System of Equations of the Magnetic Field B Caused by Conduction Currents	p. 25
Example one: magnetic field due to a current in a cylindrical conductor	p. 29
Example two: magnetic field of an infinitely long solenoid	p. 31
Example three: magnetic field of a current toroid	p. 34
The System of Equations for Vector Potential A	p. 36
Magnetic Field Caused by Magnetization Currents
Magnetization Currents and Magnetization: Biot-Savart Law	p. 39
System of Equations of the Field B in the Presence of a Magnetic Medium	p. 41
Relation between magnetization Currents and Magnetization	p. 42
System of Equations with Respect to the Magnetic Field B	p. 45
Field H and Relationship between Vectors B, P, and H	p. 46
Three Types of Magnetic Media and their Magnetic Parameters	p. 47
Inductive and residual magnetization	p. 47
Types of magnetic medium	p. 48
Magnetic permeability	p. 49
System of Equations for the Magnetic Field B	p. 50
Distribution of Magnetization Currents	p. 51
Volume density	p. 51
Surface density	p. 53
System of Equations for the Fictitious Field H and Distribution of its Generators	p. 55
Volume density	p. 55
Surface density	p. 56
Difference between the Fields B and H	p. 56
Example 1: Current loop in a homogeneous medium	p. 57
Example 2: Uniform fields B and H in a medium with one plane interface	p. 57
Example 3: Fields B and H inside the toroid with a small gap	p. 59
Example 4: Fields B and H inside the solenoid	p. 61
Example 5: Fields B and H inside the magnetic solenoid	p. 61
Example 6: Influence of a thin magnetic shell	p. 61
The System of Equations for the Fields B and H in Special Cases	p. 62
Case 1: A nonmagnetic medium	p. 62
Case 2: Conduction currents are absent	p. 63
Case 3: Residual magnetization and conduction currents are absent	p. 64
Case 4: Uniform piece-wise medium where conduction current and residual magnetization are absent	p. 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 Absent	p. 67
Equations for the scalar potential	p. 67
Theorem of Uniqueness and Boundary-Value Problems	p. 69
The first boundary-value problem	p. 70
The second boundary-value problem	p. 72
Boundary-value problem in the presence of an interface of media with different [mu]	p. 73
A Cylinder in a Uniform Magnetic Field	p. 75
Solution of the boundary problem	p. 75
Determination of unknown coefficients and field expressions	p. 79
Distribution of magnetization currents	p. 81
Behavior of the magnetic field inside the cylinder	p. 82
Induced magnetization vector	p. 82
Medium of small susceptibility	p. 83
Secondary field outside the cylinder	p. 84
The primary field is directed along the cylinder axis	p. 85
An Elongated Spheroid in a Uniform Magnetic Field B[subscript 0]	p. 86
Laplace's equation and its solution in spheroidal system of coordinates	p. 86
Field of a Magnetic Dipole Located at the Cylinder Axis	p. 92
Solution of Laplace's equation in the cylindrical coordinates	p. 93
Expressions for the potential of the magnetic fiel	p. 97
Coefficients A[subscript m] and B[subscript m]	p. 98
The current density	p. 100
Asymptotic behavior of the field on the cylinder (borehole) axis	p. 100
Concept of geometric factor	p. 102
Ellipsoid in a Uniform Magnetic Field	p. 104
System of ellipsoidal coordinates	p. 104
Expressions for the potential of the primary field	p. 106
Solutions of Laplace's equation	p. 107
Potential inside and outside an ellipsoid	p. 108
Spherical Layer in a Uniform Magnetic Field	p. 111
Spherical magnetic body in a uniform field	p. 114
Thin spherical shell in a uniform field	p. 115
The Magnetic Field Due to Permanent Magnet	p. 116
The vector potential	p. 117
The field outside of a thin cylinder	p. 118
Scalar potential	p. 122
The Magnet in a Uniform Magnetic Field	p. 124
Force acting on a free charge	p. 124
Linear current circuit in a uniform magnetic field B	p. 125
Resultant force	p. 126
Moment of rotation	p. 126
Thin and elongated magnet in a uniform magnetic field	p. 128
Interaction between Two Magnets	p. 130
Two magnets are placed along the same line (Fig 3.8)	p. 130
Current circuit in the magnetic field B	p. 133
Magnet in a field of a point magnetic charge	p. 135
Magnetic force	p. 135
Moment of rotation	p. 137
Energy of Magnetic Dipole in the Presence of the Magnetic Field	p. 138
Permanent Magnet and Measurements of the Magnetic Field	p. 139
Deflection method of measurements	p. 141
Theory of the vertical magnetometer	p. 143
Main Magnetic Field of the Earth
Elements of the Magnetic Field of the Earth	p. 147
History of the Earth Magnetism Study	p. 149
The discovery of the magnetic compass	p. 149
Pierre de Maricourt (Petrus Peregrinus)	p. 149
Magnetic compass and navigation	p. 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 Equation	p. 156
Orthogonality of Functions S[subscript n]	p. 158
Solution of Equation (4.13) for the Functions S	p. 159
Legendre's Equation and Zonal Harmonics	p. 160
Solution of Legendre's Equation	p. 161
Index n of functions p[subscript n] and q[subscript n] is positive	p. 162
Recursion Formulas for the Functions p and q	p. 163
Legendre Polynomials	p. 164
Recursion formulas for Legendre's polynomials	p. 166
Integral from a Product of Legendre Polynomials	p. 166
Expansion of Functions by Legendre Polynomials	p. 167
Expressions for Legendre polynomials	p. 168
Spherical Analysis of the Earth's Magnetic Field When the Potential is Independent of Longitude	p. 168
The Physical Meaning of Coefficients B[subscript n]	p. 171
Associated Legendre Functions	p. 172
Examples of the associated Legendre functions ([mu] < 1)	p. 174
Integrals from a product of the associated Legendre functions	p. 175
Spherical Harmonic Analysis of the Magnetic Field of the Earth	p. 176
Uniqueness and the Solution of the Forward and Inverse Problems
Introduction	p. 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 Negligible	p. 190
Development of a Solution of the Forward and Inverse Problems	p. 194
Example 1: Uniform half space	p. 195
Example 2: Layer of finite thickness	p. 196
Concept of Uniqueness and the Solution of the Inverse Problem in the Magnetic Method	p. 197
Main steps of interpretation	p. 197
Uniqueness and its application	p. 199
Solution of the Inverse Problem and Influence of Noise	p. 202
Paramagnetism, Diamagnetism, and Ferromagnetism
Introduction	p. 207
The Angular Momentum and Magnetic Moment of an Atom	p. 208
Motion of Atomic Magnetic Dipole in an External Magnetic Field	p. 212
The first approach	p. 212
Frequency of precession	p. 214
The second approach	p. 215
Magnetic Moment, Angular Momentum, Spin, and Energy States of Atomic System	p. 217
Magnetic moment	p. 217
Angular momentum	p. 218
Magnetic energy of atomic particle	p. 219
The Stern-Gerlach experiment	p. 221
Alternating magnetic field and transition between energy levels of atom	p. 223
The Rabi molecular beam method	p. 225
Diamagnetism	p. 228
Paramagnetism	p. 231
Classical physics approach	p. 231
Quantum mechanics approach	p. 235
Ferromagnetism	p. 237
Introduction	p. 237
The magnetization curve	p. 237
Hysteresis loop	p. 240
Principle of the Fluxgate Magnetometer	p. 241
Magnetization and Magnetic Forces	p. 243
Spontaneous magnetization	p. 247
Curie temperature	p. 248
Spontaneous magnetization and Weiss domains	p. 250
Case one: Single crystal of ferromagnetic and its domains	p. 251
Case two: Polycrystalline material of ferromagnetic	p. 252
Nuclear Magnetism Resonance and Measurements of Magnetic Field
Introduction	p. 255
The Vector of Nuclear Magnetization	p. 258
Equations of the Vector of Magnetization	p. 261
Case 1: Additional field is absent	p. 263
Case 2: The additional field is horizontal	p. 263
Rotating System of Coordinates	p. 264
Behavior of the Vector P in the Rotating System of Coordinates	p. 266
Example 1	p. 266
Example 2: The additional field rotates in the horizontal plane	p. 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 function	p. 273
Magnetization Caused by the Additional Field	p. 274
Bloch Equations	p. 275
Solution of Bloch's equations when the additional field is absent	p. 276
Measurements of Relaxation Processes	p. 278
Introduction	p. 278
Measurements of a decay of the longitudinal component of the vector P	p. 279
Measurements of a decay of the transversal component of the vector P	p. 281
Spin echoes or refocusing	p. 282
Two Methods of Measuring Magnetic Field	p. 283
Proton precession magnetometer	p. 283
Optically pumped magnetometers	p. 285
Bibliography	p. 289
Appendix	p. 291
Subject Index	p. 297
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