Solivérez EMPEB

Solivérez, Carlos E.; Electrostatics and Magnetostatics of Polarized Ellipsoidal Bodies: The Depolarization Tensor Method; Free Scientific Information; Bariloche, pcia. de Río Negro, Argentina; 2016; Solivérez EMPEB

Contenido

A study is made of the behaviour of ellipsoidal bodies, with no free currents and charges, under uniform applied electrostatic and magnetostatic fields. The method is valid for all sorts of solid homogenous isotropic or anisotropic materials: dielectric, ferroelectric, diamagnetic, paramagnetic, ferromagnetic, conducting, superconducting... Expressions are given, in all cases, for the fields internal and external to the body, as well as the free energy and the torques experienced by it. Apart from the body's volume and the electromagnetic properties of the material, these expressions depend only on the depolarization tensor n determined by the two shape ratios of the ellipsoid. Explicit expressions are given for n —both in field points internal and external to the body— in terms of elementary functions except for the triaxial ellipsoid where they are Legendre’s elliptic functions. In the non-linear range the electric or magnetic polarization is an implicit function of the applied field and the anisotropy of the material, while the external field is an explicit function of the polarization. In the linear range both the polarization and the internal field are explicit functions of the constant internal value of n inside the body and the isotropic or anisotropic susceptibility tensor χ, and the external field is an explicit function of the polarization and n. In the isotropic case the limit values χ = ∞ and χ = -1 (SI units) fully describe the behaviour of conductor and superconductor ellipsoids. A discussion is made of some common errors in the treatment of electromagnetic singularities and of the behaviour of bodies of infinite extension and ellipsoidal cavities.

Índice

• Abstract iii
• Chapter 1: Fundamental concepts 1
  • Origin 1
  • History 1
  • Applications 3
  • Requirements 5
  • Limitations of the method 5
  • Dealing with singularities 5
    • Point charge type singularity 6
    • Step discontinuities 9
  • Physical units and mathematical notation 10
  • Organization of the book 11
• Chapter 2: Ellipsoids in Electric and Magnetic Fields 15
  • Electric polarization 15
    • Basic equations 15
    • Permanent electric polarization 18
    • Induced electric polarization 20
    • Induced electric polarization of two interacting atoms 21
    • Dielectrics 24
  • Magnetization 26
    • Basic equations 26
    • Permanent magnetization 28
    • Induced magnetization 29
  • Conductors 30
    • Equivalent polarization 33
  • Superconductors 34
    • Magnetization model 35
    • Surface conduction current model 36
  • Summary of integro-differential equations 38
  • Summary of induced polarization equations 39
  • General formulation of the method 40
    • Solving the integro-differential equations by iteration 42
• Chapter 3: The depolarization tensor: basic treatment 45
  • Definition 45
  • General properties of n 47
    • Symmetric tensor 47
    • Trace 48
    • Orthogonal transformations 48
    • Symmetries 49
  • n as a surface integral 50
  • Surface step discontinuity 51
    • Surface charge density 51
    • Surface step discontinuity 52
  • Calculation of n using electrostatic Gauss’s Law 54
    • Sheet of constant thickness and infinite extension 54
    • Right circular cylinder of infinite length 56
    • Sphere 58
  • Other properties of the internal depolarization tensor N 59
    • Integral expressions of the eigenvalues 59
    • Diagonalization and inversion 60
    • Infinite semiaxes and the inverse of N 60
    • 'Invariance for similar ellipsoids 62
    • N is determined by aspect ratios 63
• Chapter 4: The depolarization tensor: advanced treatment 65
  • Calculations of n from gravitational potentials 65
    • Triaxial ellipsoid 65
  • Internal depolarization tensor N 67
    • Expressions with normal elliptic integrals E and F 67
    • Elliptic cylinder 68
    • Oblate spheroid 69
    • Prolate spheroid 74
    • Unified treatment of N for spheroids 77
    • Graphs of N_a, N_b and N_c 79
      • N_a 80
      • N_b 80
      • N_c 80
      • Combined graph 84
  • External depolarization tensor 85
    • Obtention of the external gravitational potential by Ivory’s method 85
    • Near and far away point approximations to next 87
    • General expression of external n 87
      • Trace 90
      • Verificacion for the sphere 91
      • Elliptic cylinder 92
      • Oblate spheroid 95
      • Prolate spheroid 96
      • Triaxial ellipsoid 97
• Chapter 5: Energy, forces and cavities 99
  • Thermodynamics of electrostatic and magnetostatic energy 99
    • Basic concepts 99
    • Thermodynamics of electromagnetism 101
  • Anisotropy energy 103
  • Origin of torques exerted on a body 105
  • Torque exerted on an ellipsoid and the state of equilibrium 106
    • Dielectrics 106
      • Permanent polarization 106
      • Induced polarization 108
      • Anisotropic susceptibility 109
    • Magnetic materials 110
      • Permanent magnetization 110
      • nduced magnetization 111
      • Magnetic torque experiments 112
    • Conductors 112
    • Superconductors 113
  • Force on an ellipsoid in a non-uniform field 114
  • Infinite and infinitesimal bodies 114
  • Cavities 116
    • Standard treatment 116
    • Some specific cavities: or homoeoids 117
    • Thin shells and the dipole layer 118
• Chapter 6: Selected problemas 121
  • Dielectrics 121
    • Problem 01: Autoconsistent electric polarization of two atoms 121
    • Problem 02: Shape or crystalline anisotropy? 121
    • Problem 03: Dielectric sphere 122
    • Problem 04: Deriving n from the magnetic vector potential 124
    • Problem 05: Permanently magnetized infinite right circular cylinder 124
    • Problem 06: Isotropic magnetic sphere 126
  • Conductors 126
    • Problem 07: Solve ellipsoidal conductors using equivalente polarization 126
    • Problem 08: A conductor is a perfect dielectric 127
    • Problem 09: Polarization of a spherical conductor 127
    • Problem 10: External field of a spherical conductor 128
    • Problem 11: Directions for which P is parallel to E0 in triaxial ellipsoids 129
    • Problem 12: Electric fields generated by sharp points 129
  • Superconductors 130
    • Problem 13: Superconducting infinite cylinder and sheet 130
    • Problem 14: Magnetized superconducting sphere 130
    • Problem 15: Superconductivity as perfect diamagnetism 131
    • Problem 16: Magnetic moment of a superconducting ellipsoid 132
    • Problem 17: Description of Meissner Effect using magnetization currents 133
    • Problem 18: Approximate solution for non ellipsoidal bodies 133
  • Depolarization tensor 133
    • Problem 19: Components of N in spherical symmetry 133
    • Problem 20: Single eigenvalue may determine the type of spheroid 134
    • Problem 21: Eigenvalues and aspect ratios 134
    • Problem 22: Sequence of ellipsoids over some aspect ratio curves 135
    • Problem 23: Eigenvalues of triaxial ellipsoids 135
    • Problem 24: Depolarization tensor of a very long right circular cylinder 136
    • Problem 25: Depolarization tensor of a sphere 136
    • Problem 26: Alternative way of obtaining certain eigenvalues of N 136
    • Problem 27: Solving the prolate spheroid with Legendre’s elliptic integrals 136
    • Problem 28: Spherical shell 137
    • Problem 29. Unit vector normal to the ellipsoid's surface 139
    • Problem 30: Surface step discontinuity of the depolarization tensor 139
    • Problem 31: Ellipsoidal conductor with net charge 140
    • Problem 32: Application of Ivory’s method to the sphere 140
    • Problem 33: Expression for far away fields obtained from next 141
    • Problem 34: Field on the external side of the surface of a polarized sphere 142
    • Problem 35: Microscopic origin of depolarization tensors 143
  • Energy, forces and torques 143
    • Problem 36: Energy of a dielectric sphere 143
    • Problem 37: Fermi’s contact interaction 143
    • Problem 38: Torques on spheroids 144
    • Problem 39: Torque exerted over an anisotropic dielectric sphere 145
    • Problem 40: Torque on a magnetized disk 145
    • Problem 41: Torque on a very long superconducting right circular cylinder 146
    • Problem 42: Depolarization tensor of inifinitesimally thin shells 146
    • Problem 43: Applications of thin shells 146
• Appendix 1: Electromagnetic units 147
• Appendix 2: Vectorial Operators 151
• Appendix 3: Integral theorems of vector calculus 153
  • Laplacian of a point charge’s potential 153
  • Divergence, gradient and rotor theorems 155
  • Extension of the divergence theorem 155
    • Point charge type singularities 156
    • Step discontinuities 157
      • Generalized divergence theorem 157
      • Generalized rotor theorem 158
      • Polarized bodies 158
• Appendix 4: Field of an electric dipole 161
  • General expression 161
  • Dipolar field's energy 162
• Appendix 5: Simmetries of the susceptibility tensors of single crystals 165
• Appendix 6: Dyadics 167
• Appendix 7: Ellipsoids 169
  • Equations and main features 169
    • Equations, sections, volume and surface 169
    • Aspect ratios 170
  • Normals to the surface and central distances 171
    • Summary of properties 173
  • Confocal ellipsoids 174
    • Physical-geometric interpretation of κ 174
    • Corresponding points 175
• Appendix 8: Useful integrals 179
• Appendix 9: Legendre’s elliptic integrals 181
  • Definitions 181
    • Notation used in the more important references 181
    • Special values and parametric graphs 182
    • Reduction to normal form of the elliptic integrals of interest 184
• Main references 187
• Alphabetic index 189
• About the author 193

Sobre el autor

Véase la biografía del Editor ECyT-ar.

Transclusión

El texto que se lee cuando se transcluye esta página es:

Solivérez, Carlos E.; Electrostatics and Magnetostatics of Polarized Ellipsoidal Bodies: The Depolarization Tensor Method; Free Scientific Information; Bariloche, pcia. de Río Negro, Argentina; 2016; Solivérez EMPEB

---