Principles of the Magnetic Methods in Geophysics (eBook)

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2008 | 1. Auflage
320 Seiten
Elsevier Science (Verlag)
978-0-08-093185-2 (ISBN)

Infos

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.

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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 concept of the magnetic field is introduced 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 special 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.

Inhalt

Front cover 1
Principles of the Magnetic Methods in Geophysics 4
Copyright page 5
Dedication 6
Contents 8
Introduction 14
Acknowledgments 16
List of Symbols 18
Chapter 1. Magnetic Field in a Nonmagnetic Medium 20
1.1. Interaction of constant currents and Ampere’s law 20
1.2. Magnetic field of constant currents 22
1.3. The vector potential of the magnetic field 27
1.4. Magnetic field and vector potential, caused by linear and surface currents 31
1.5. System of equations of the magnetic field B caused by conduction currents 44
1.6. The system of equations for vector potential A 55
Chapter 2. Magnetic Field Caused by Magnetization Currents 58
2.1. Magnetization currents and magnetization: Biot-Savart law 58
2.2. System of equations of the field B in the presence of a magnetic medium 60
2.3. Relation between magnetization currents and magnetization 61
2.4. System of equations with respect to the magnetic field B 64
2.5. Field H and relationship between vectors B, P, and H 65
2.6. Three types of magnetic media and their magnetic parameters 66
2.7. System of equations for the magnetic field B 69
2.8. Distribution of magnetization currents 70
2.9. System of equations for the fictitious field H and distribution of its generators 74
2.10. Difference between the fields B and H 75
2.11. The system of equations for the fields B and H in special cases 81
Chapter 3. Magnetic Field in the Presence of Magnetic Medium 86
3.1. Solution of the forward problem in a piece-wise uniform medium when conduction currents and residual (remanent) magnetization are absent 86
3.2. Theorem of uniqueness and boundary-value problems 88
3.3. A cylinder in a uniform magnetic field 94
3.4. An elongated spheroid in a uniform magnetic field B0 105
3.5. Field of a magnetic dipole located at the cylinder axis 111
3.6. Ellipsoid in a uniform magnetic field 123
3.7. Spherical layer in a uniform magnetic field 130
3.8. The magnetic field due to permanent magnet 135
3.9. The magnet in a uniform magnetic field 143
3.10. Interaction between two magnets 149
3.11. Energy of magnetic dipole in the presence of the magnetic field 157
3.12. Permanent magnet and measurements of the magnetic field 158
Chapter 4. Main Magnetic Field of the Earth 166
4.1. Elements of the magnetic field of the earth 166
4.2. History of the earth magnetism study 168
4.3. Solution of the Laplace equation 175
4.4. Orthogonality of functions Sn 177
4.5. Solution of Equation (4.13) for the functions S 178
4.6. Legendre’s equation and zonal harmonics 179
4.7. Solution of Legendre’s equation 180
4.8. Recursion formulas for the functions p and q 182
4.9. Legendre polynomials 183
4.10. Integral from a product of Legendre polynomials 185
4.11. Expansion of functions by Legendre polynomials 186
4.12. Spherical analysis of the earth’s magnetic field when the potential is independent of longitude 187
4.13. The physical meaning of coefficients Bn 190
4.14. Associated Legendre functions 191
4.15. Spherical harmonic analysis of the magnetic field of the earth 195
Chapter 5. Uniqueness and the Solution of the Forward and Inverse Problems 204
5.1. Introduction 204
5.2. Poisson’s relationship between potentials U and Ua 207
5.3. Solution of the forward problem when the interaction between magnetization currents is negligible 209
5.4. Development of a solution of the forward and inverse problems 213
5.5. Concept of uniqueness and the solution of the inverse problem in the magnetic method 216
5.6. Solution of the inverse problem and influence of noise 221
Chapter 6. Paramagnetism, Diamagnetism, and Ferromagnetism 226
6.1. Introduction 226
6.2. The angular momentum and magnetic moment of an atom 227
6.3. Motion of atomic magnetic dipole in an external magnetic field 231
6.4. Magnetic moment, angular momentum, spin, and energy states of atomic system 236
6.5. Diamagnetism 247
6.6. Paramagnetism 250
6.7. Ferromagnetism 256
6.8. Principle of the fluxgate magnetometer 260
6.9. Magnetization and magnetic forces 262
Chapter 7. Nuclear Magnetism Resonance and Measurements of Magnetic Field 274
7.1. Introduction 274
7.2. The vector of nuclear magnetization 277
7.3. Equations of the vector of magnetization 280
7.4. Rotating system of coordinates 283
7.5. Behavior of the vector P in the rotating system of coordinates 285
7.6. Magnetization caused by the additional field 293
7.7. Bloch equations 294
7.8. Measurements of relaxation processes 297
7.9. Two methods of measuring magnetic field 302
Bibliography 308
Appendix 310
Paleomagnetism and Plate Tectonics 310

Textauszug

Chapter 1 Magnetic Field in a Nonmagnetic Medium

Alex A. Kaufman, Richard O. Hansen, Robert L.K. Kleinberg

Abstract
Publisher Summary

This chapter focuses on the magnetic field in a nonmagnetic medium. Numerous experiments performed at the beginning of the19th century demonstrated that constant currents interact with each other that mean mechanical forces act at every element of the circuit. Certainly, this is one of the amazing phenomena of the nature and would have been very difficult to predict. In fact, it is almost impossible to expect that the motion of electrons inside of wire may cause a force on moving charges somewhere else, for instance, in another wire with current, and for this reason the phenomenon of this interaction was discovered by chance. By analogy with the attraction field caused by masses, it is proper to assume that constant (time-invariant) currents create a field and because of the existence of this field and of the existence of this field, other current elements experience the action of the force F. Such a field is called the magnetic field, and it can be introduced from Ampere’s law.

1.1 Interaction of constant currents and Ampere's law

Numerous experiments performed at the beginning of the19th century demonstrated that constant currents interact with each other; that is mechanical forces act at every element of the circuit. Certainly, this is one of the amazing phenomena of the nature and would have been very difficult to predict. In fact, it is almost impossible to expect that the motion of electrons inside of wire may cause a force on moving charges somewhere else, for instance, in another wire with current, and for this reason the phenomenon of this interaction was discovered by chance. It turns out that this force of interaction between currents in two circuits depends on the magnitude of these currents, the direction of charge movement, the shape and dimensions of circuits, as well as the their mutual position with respect to each other. The list of factors clearly shows that the mathematical formulation of the interaction of currents should be much more complicated task than that for masses or electric charges. In spite of this fact, Ampere was able to find a relatively simple expression for the force of the interaction of so-called elementary currents:

     (1.1)

where I1 and I2 are magnitudes of the currents in the linear elements dl(p) and dl(q), respectively, and their direction coincides with that of the current density; Lqp the distance between these elements and is directed from the point q to the point p, which can be located at the center of these elements; finally μ0 is a constant equal to

and is often called the magnetic permeability of free space. Certainly, this is confusing definition, since free space does not have any magnetic properties. We will use the S.I. system of units where the distance is measured in meters and force in newtons. Of course, with a change of the system of units the value of μ0 varies too. In applying Ampere's law (Equation (1.1)), it is essential that the separation between current elements must be much greater than their length; that is

Correspondingly, points: p and q can be located anywhere inside their elements. It is easy to see some similarity of Ampere's law and Newton's law of attraction; they describe a force between either elementary currents or elementary masses. Let us illustrate Equation (1.1) by three examples shown in Fig. 1.1. Suppose that elements dl(p) and dl(q) are in parallel with each other. Then, as follows from definition of the cross product, the force dF(p) is directed toward the element dl(q), and two current elements attract each other (Fig. 1.1(a)). If two current elements have opposite directions, the force dF(p) tries to increase the distance between elements, and therefore they repeal each other (Fig. 1.1(b)). If the elements dl(p) and dl(q) are perpendicular to each other, as is shown in Fig. 1.1(c), then in accordance with Equation (1.1) the magnitude of the force acting at the element dl(p) equals

Fig. 1.1 (a) Parallel current elements. (b) Anti-parallel current elements. (c) Current elements perpendicular to each other. (d) Interaction of closed current circuits.

and it is parallel to the element dl(q). At the same time, the force dF(q) at the point q is equal to zero, that is Newton's third law becomes invalid. This contradiction results from the fact that Equation (1.1) describes an interaction between current elements instead of closed current circuits. In other words, this equation is written for unrealistic case, since we cannot create a constant current in an open circuit, but as all experiments show, Equation (1.1) gives the correct result for closed current lines. For instance, applying the principle of superposition, the force of interaction between two arbitrary and closed currents (Fig. 1.1(d)) is defined as

     (1.2)

The internal integral in Equation (1.2) characterizes the force acting at some point of the current line L1, for instance, point p1 and caused by all elements of the current line L2. Thus, the force F represents a sum (integral) of forces applied at different points of the same circuit and, as is well known, its action causes in general a deformation, translation and a rotation of the current line L1. It is obvious that in the case of closed circuits the interaction between them obeys Newton's third law.

The relationship between the force F and currents (Equation (1.2)) is called Ampere's law for closed circuits with constant currents, and it is impossible to overestimate its importance, since it is the theoretical foundation of many devices measuring magnetic field as well as electromotors, transforming electric energy into mechanical energy, and it has numerous applications in physics and technology. Finally, it is proper to notice the following:

(a) The force of interaction is independent of properties of the medium which surrounds the currents.

(b) The Ampere's law was formulated for currents which are independent of time. It turns out that this law allows us to calculate the force of interaction even in the case of alternating currents as long as displacement currents can be neglected.

(c) It is natural to be surprised and impressed that Ampere found Equation (1.1) since in reality he had only experimental data describing interaction for closed current circuits.

1.2 Magnetic field of constant currents

By analogy with the attraction field caused by masses, it is proper to assume that constant (time-invariant) currents create a field, and due to the existence of this field other current elements experience the action of the force F. Such a field is called the magnetic field, and it can be introduced from Ampere's law. In fact, we can write Equation (1.1) as

     (1.3)

Here

     (1.4)

Equation (1.4) establishes the relationship between the elementary current and the magnetic field caused by this element, and it is called Biot–Savart law. In accordance with Equation (1.4), the magnitude of the magnetic field dB is

     (1.5)

where (Lqp, dl) is the angle between the vectors Lqp and dl, and the vector dB is perpendicular to these vectors as is shown in Fig. 1.2(a). The unit vector, b0, characterizing the direction of the field, is defined by

Fig. 1.2 (a) Illustration of Equation (1.4). (b) Field due to the surface currents.

We may say that the magnetic field exists at any point regardless of presence or absence of a current at this point. In S.I. units, the magnetic field is measured in teslas and it is related to other units such as gauss and gamma in the following way:

1.2.1 General form of Biot–Savart law

Now we generalize Equation (1.4) assuming that along with linear currents there are also volume and surface currents. First let us represent the product I dl...

Produkt-Details

• Erscheint lt. Verlag: 21.11.2008
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-093185-5 / 0080931855
• ISBN-13: 978-0-08-093185-2 / 9780080931852