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CORE Metadata, Citation and Similar Papers at Core.Ac.Uk CORE Metadata, citation and similar papers at core.ac.uk Provided by Trepo - Institutional Repository of Tampere University Tampereen teknillinen yliopisto. Julkaisu 1126 Tampere University of Technology. Publication 1126 Tuomas Kovanen Definition of Electric and Magnetic Forces on Riemannian Manifold Thesis for the degree of Doctor of Science in Technology to be presented with due permission for public examination and criticism in Sähkötalo Building, Auditorium S2, at Tampere University of Technology, on the 14th of June 2013, at 12 noon. Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2013 ISBN 978-952-15-3056-2 (printed) ISBN 978-952-15-3110-1 (PDF) ISSN 1459-2045 Abstract The precise relationship of electricity and magnetism to mechanics is be- coming a critically important question in modern engineering. Although well known engineering methods of modeling forces in electric and magnetic systems have been adequate for bringing many useful technologies into our everyday life, the further development of these technologies often seems to require more extensive modeling. In the search for better modeling it is relevant to examine the foundations of the classical models. The classical point charge definitions of electric and magnetic field quan- tities are not designed for the determination of forces in typical engineering problems. In the approach of this thesis the relationship of electric and magnetic field quantities to forces on macroscopic objects is built in the def- initions. As a consequence, the determination of electrostatic and magneto- static forces on macroscopic objects becomes a clear-cut issue. What makes the approach work is that the system of interacting objects is considered as a whole. This is contrary to the classical approach where an unrealistic test object is used to define electric and magnetic fields as properties of the source object only. A limitation in the classical notions of electric field intensity and magnetic induction is that they are sufficient to determine forces on only rigid objects. To allow for deformable objects the model needs to be combined with that of continuum mechanics. The notions suggested in this thesis are more general as they allow rigidity to be considered with respect to the test object itself, which may not be rigid in the usual sense. The reguired generality is obtained by using the mathematics of differen- tial geometry as the framework for the definitions. This has the additional benefit of making clear the mathematical structures required for the defi- nitions. Also, it allows the clear identification of the mathematical objects involved. Both of these outcomes are important for creating efficient and flexible computational codes for modeling. i Preface I wrote this thesis while working with the electromagnetics research group at Tampere University of Technology. The subject of the thesis originates from scientific and technological interest in clarifying the relationship between electromagnetics and mechanics – a question that boils down to the definition of electromagnetic field quantities. During my time with the electromagnetics group the head of the group has been Professor Lauri Kettunen. I appreciate Lauri’s effort in creating and maintaining circumstances that make scientific work possible in our unit. Lauri has also been the supervisor of this work, and I want to thank him for his teaching and careful guidance along the way. An important advisor of this work has been Timo Tarhasaari. Some time ago, Timo managed to teach me the basics of the mathematics used in this thesis, and since then our discussions have greatly influenced my scientific thinking. I want to thank Timo and Lauri for trusting me with this particular research subject, and for giving me enough freedom and guidance for its investigation. I am thankful to Maija-Liisa Paasonen and Lasse S¨oderlund for taking care of numerous administrative tasks so that I have had all the time to focus on this work. I thank Antti Stenvall for providing me with his finite element solver for use in this work. That gave me much freedom in force calculation as it allowed access to complete solution data of the field problem. I thank John Shepherd for the grammatical proof-reading of this thesis. Also, I want to thank the personell of our unit for keeping up a good atmosphere. The various ball games we have played together during the years have been special to me. This work has been financially supported by the Doctoral Programme of Electrical Energy Engineering at Lappeenranta University of Technology. It is not self-evident that one has a decent possibility to acquire an edu- cation. I thank my childhood family for giving me this possibility. Finally, I want to thank my wife Birgitta and my daughter Kerttu; I feel as lucky as a man can be for having you in my life. ii Contents Abstract i Preface ii List of symbols vi 1 Introduction 1 2 Mathematical preliminaries 6 2.1 Vectorsandcovectors. 6 2.2 Tensors .............................. 10 2.3 Multivectors and -covectors . 12 2.4 Twisted multivectors and -covectors . 16 2.5 Further examples of tensors . 20 2.6 Innerproduct ........................... 21 3 Forces in terms of charges and currents 24 3.1 Electrostatic forces on objects with charge distributions.... 30 3.2 Magnetostatic forces on objects with current distributions . 52 4 Torques in terms of charges and currents 55 4.1 Torques in electrostatics . 66 4.2 Torques in magnetostatics . 66 4.3 The law of action and reaction . 67 4.4 Torques from the total fields . 68 5 Forces and torques in terms of equivalent charges and equiv- alent currents 71 5.1 Forces and torques in electrostatics . 71 5.2 Forces and torques in magnetostatics . 75 5.2.1 Electric current approach . 75 5.2.2 Magnetic charge approach . 77 iii 5.2.3 Relation between the two approaches . 79 5.3 Conclusion............................. 85 6 Forces and torques in terms of polarization and magnetiza- tion 86 6.1 Heuristic derivation of the density of virtual work from micro- scopicmaterialmodels . 86 6.1.1 Amp`erian dipoles . 87 6.1.2 Coulombian dipoles . 95 6.2 Forces and torques in electrostatics . 98 6.2.1 Forces and torques from the total fields . 101 6.3 Forces and torques in magnetostatics . 104 6.3.1 Forces and torques in terms of total fields . 106 6.4 Discussion of force densities for determining deformations . 109 7 Contact forces 112 7.1 Contact forces in magnetostatics . 113 7.2 Contact forces from the total fields . 116 7.2.1 Magnetization approach . 116 7.2.2 Magnetic polarization approach . 121 8 Geometric view of magnetic forces 124 9 Conclusions 131 A Basis representations of force densities and stresses 137 A.1 Electrostatics ...........................139 A.1.1 Electric charge approach . 139 A.1.2 Electric polarization approach . 144 A.2 Magnetostatics ..........................146 A.2.1 Electric current approach . 146 A.2.2 Magnetic charge approach . 149 A.2.3 Magnetization approach . 150 A.2.4 Magnetic polarization approach . 152 B Basis representations of torque densities 154 B.1 Electrostatics ...........................154 B.1.1 Electric charge approach . 154 B.1.2 Electric polarization approach . 155 B.2 Magnetostatics ..........................156 B.2.1 Electric current approach . 157 B.2.2 Magnetization approach . 157 iv C Basis representations for contact force calculation 159 C.1 Magnetization approach . 159 C.2 Magnetic polarization approach . 160 D Computation of force densities from finite element approxi- mation of fields 162 v List of symbols Chapter 2 R Set of all real numbers V Finite dimensional vector space V ∗ Dual space of V i δj Component of Kronecker tensor U Vector subspace of V W Transformation matrix between bases of V r r Ts (V ) Vector space of tensors on V of type s ⊗ Tensor product p(V ) Vector space of p-covectors on V ∧V Exterior product sgn(σ) Signature of permutation σ det(·) Determinant p(V ) Vector space of p-vectors on V OrV Orientation of V TransOr Transverse orientation of a vector subspace of V [·] Equivalence class g Inner product on V || · || Norm on V induced by g u∠v Angle between vectors u and v Chapter 3 Ω Manifold modeling physical space o1, o2 Usually submanifolds of Ω modeling the interacting objects vi W [o1, o2] Work done by o1 on o2 Ψt 1-parameter embedding of objects o1 and o2 into Ω Rn Set of all n-tuples of real numbers S The underlying point set of a manifold A Smooth atlas χi,j Charts for a manifold M Generic manifold, or manifold with boundary Hn Half space of Rn ∂M Boundary of M N Submanifold of M TxM Tangent space of M at point x γ Path of movement of o2 F12 Force on object o2, a covector F21 Force on object o1, a covector △ Simplex of Rn s Simplex of a manifold with parallelism {s} p-vector of simplex s ρ˜1 Charge density of object o1, a twisted 3-form ρ˜2 Charge density of object o2, a twisted 3-form σ˜1 Surface charge density of object o1, a twisted 2-form σ˜2 Surface charge density of object o2, a twisted 2-form ∪ Set union Q1 Total charge of o1 Q2 Total charge of o2 D˜ 1 Electric displacement of object o1, a twisted 2-form D˜ 2 Electric displacement of object o2, a twisted 2-form [·]1 Discontinuity over ∂o1 [·]2 Discontinuity over ∂o2 d Exterior derivative t1 Tangential trace to ∂o1 st Simplex parametrized by t F˜12 Volume force density of object o2, a covector-valued 3-form F˜21 Volume force density of object o1, a covector-valued 3-form f˜12 Surface force density of object o2, a covector-valued 2-form f˜21 Surface force density of object o2, a covector-valued
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