The present text is intended as an introduction to electromagnetics and computation of electromagnetic fields. While many texts on electromagnetics exist, the subject of computation of electromagnetic fields is nonnally not treated or is treated in a number of idealized examples, with the main emphasis on development of theoretical relations. "Why another book on Electromagnetics?" This is perhaps the first question the reader may ask when opening this book. It is a valid question, because among the many books on Electromagnetics some are excellent. We have two answers to this question, answers that have motivated the writing of this book. The first concerns the method of presentation of Electromagnetism. Generally, in classical books the material is presented in the following sequence: electrostatics, magnetostatics, magnetodynamics, and wave propagation, using integral fonns of the field equations. As a primary effect of this presentation, the reader is led to think that the knowledge of this science is synonymous to memorizing dozens offonnulas. Additionally, an impression that there is no finn connection between these equations lingers in the reader's mind since at each step new postulates are added, seemingly unrelated to previous material. Our opinion is, and we shall try to convey this to the reader, that the Electromagnetic formalism is extremely simple and based on very few equations. They are the four "Maxwell equations" which include practically all the existent relationships between the electromagnetic quantities. The only additional relationships that need be considered is the Lorentz force and the material constitutive relations.

I. The Electromagnetic Field and Maxwell's Equations.- 1. Mathematical Preliminaries.- 1.1. Introduction.- 1.2. The Vector Notation.- 1.3. Vector Derivation.- 1.3.1. The Nabla (?) Operator.- 1.3.2. Definition of the Gradient, Divergence and Curl.- 1.4. The Gradient.- 1.4.1. Example of Gradient.- 1.5. The Divergence.- 1.5.1. Definition of Flux.- 1.5.2. The Divergence Theorem.- 1.5.3. Conservative Flux.- 1.5.4. Example of Divergence.- 1.6. The Curl.- 1.6.1. Circulation of a Vector.- 1.6.2. Stokes'Theorem.- 1.6.3. Example of Curl.- 1.7. Second Order Operators.- 1.8. Application of Operators to More than One Function.- 1.9. Expressions in Cylindrical and Spherical Coordinates.- 2. The Electromagnetic Field and Maxwell's Equations.- 2.1. Introduction.- 2.2. Maxwell's Equations.- 2.2.1. Fundamental Physical Principles of the Electromagnetic Field.- 2.2.2. Point Form of the Equations.- 2.2.3. The Equations in Vacuum.- 2.2.4. The Equations in Media with ?=?0 and ?=?0.- 2.2.5. The Equations in General Media.- 2.2.6. The Integral Form of Maxwell's Equations.- 2.3. Approximations to Maxwell's Equations.- 2.4. Units.- 3. Electrostatic Fields.- 3.1. Introduction.- 3.2. The Electrostatic Charge.- 3.2.1. The Electric Field.- 3.2.2. Force on an Electric Charge.- 3.2.3. The Electric Scalar Potential V.- 3.3. Nonconservative Fields: Electromotive Force.- 3.4. Refraction of the Electric Field.- 3.5. Dielectric Strength.- 3.6. The Capacitor.- 3.6.1. Definition of Capacitance.- 3.6.2. Energy Stored in a Capacitor.- 3.6.3. Energy in a Static, Conservative Field.- 3.7. Laplace's and Poisson's Equations in Terms of the Electric Field.- 3.8. Examples.- 3.8.1. The Infinite Charged Line.- 3.8.2. The Charged Spherical Half-Shell.- 3.8.3. The Spherical Capacitor.- 3.8.4. The Spherical Capacitor with Two Dielectric Layers.- 3.9. A Brief Introduction to the Finite Element Method: Solution of the Two-Dimensional Laplace's Equation.- 3.9.1. The Finite Element Technique for Division of a Domain.- 3.9.2. The Variational Method.- 3.9.3. A Finite Element Program.- 3.9.4. Example for Use of the Finite Element Program.- 3.10. Tables of Permittivities, Dielectric Strength and Conductivities.- 4. Magnetostatic Fields.- 4.1. Introduction.- 4.2. Maxwell's Equations in Magnetostatics.- 4.2.1. The Equation ?×H=J.- 4.2.2. The Equation ?.B=0.- 4.2.3. The Equation ?×E=0.- 4.3. The Biot-Savart Law.- 4.4. Boundary Conditions for the Magnetic Field.- 4.5. Magnetic Materials.- 4.5.1. Diamagnetic Materials.- 4.5.2. Paramagnetic Materials.- 4.5.3. Ferromagnetic Materials.- 4.5.4. Permanent Magnets.- 4.6. The Analogy Between Magnetic and Electric Circuits.- 4.7. Inductance and Mutual Inductance.- 4.7.1. Definition of Inductance.- 4.7.2. Energy in a Linear System.- 4.7.3. The Energy Stored in the Magnetic Field.- 4.8. Examples.- 4.8.1. Calculation of Field Intensity and Inductance of a Long Solenoid.- 4.8.2. Calculation of H for a Circular Loop.- 4.8.3. Field of a Rectangular Loop.- 4.8.4. Calculation of Inductance of a Coaxial Cable.- 4.8.5. Calculation of the Field Inside a Cylindrical Conductor.- 4.8.6. Calculation of the Magnetic Field Intensity in a Magnetic Circuit.- 4.8.7. Calculation of the Magnetic Field Intensity of a Saturated Magnetic Circuit.- 4.8.8. Magnetic Circuit Incorporating Permanent Magnets.- 4.9. Laplace's Equation in Terms of the Magnetic Scalar Potential.- 4.10. Properties of Soft Magnetic Materials.- 5. Magnetodynamic Fields.- 5.1. Introduction.- 5.2. Maxwell's Equations for the Magnetodynamic Field.- 5.3. Penetration of Time Dependent Fields in Conducting Materials.- 5.3.1. The Equation for H.- 5.3.2. The Equation for B.- 5.3.3. The Equation for E.- 5.3.4. The Equation for J.- 5.3.5. Solution of the Equations.- 5.4. Eddy Current Losses in Plates.- 5.5. Hysteresis Losses.- 5.6. Examples.- 5.6.1. Induced Currents Due to Change in Induction.- 5.6.2. Induced Currents Due to Changes in Geometry.- 5.6.3. Some Further Examples of Skin Depth.- 5.6.4. Effect of Movement of a Magnet Relative to a Flat Conductor.- 5.6.5. Visualization of Penetration of Fields as a Function of Frequency.- 5.6.6. The Voltage Transformer.- 6. Interaction Between Electromagnetic and Mechanical Forces.- 6.1. Introduction.- 6.2. Force on a Conductor.- 6.3. Force on Moving Charges: The Lorentz Force.- 6.4. Energy in the Magnetic Field.- 6.5. Force as Variation of Energy (Virtual Work).- 6.6. The Poynting Vector.- 6.7. Maxwell's Force Tensor.- 6.8. Examples.- 6.8.1. Force Between Two Conducting Segments.- 6.8.2. Torque on a Loop.- 6.8.3. The Hall Effect.- 6.8.4. The Linear Motor and Generator.- 6.8.5. Attraction of a Ferromagnetic Body.- 6.8.6. Repulsion of a Diamagnetic Body.- 6.8.7. Magnetic Levitation.- 6.8.8. The Magnetic Brake.- 7. Wave Propagation and High Frequency Electromagnetic Fields.- 7.1. Introduction.- 7.2. The Wave Equation and Its Solution.- 7.2.1. The Time Dependent Equations.- 7.2.2. The Time Harmonic Wave Equations.- 7.2.3. Solution of the Wave Equation.- 7.2.4. Solution for Plane Waves.- 7.2.5. The One-Dimensional Wave Equation in Free Space and Lossless Dielectrics.- 7.3. Propagation of Waves in Materials.- 7.3.1. Propagation of Waves in Lossy Dielectrics.- 7.3.2. Propagation of Plane Waves in Low Loss Dielectrics.- 7.3.3. Propagation of Plane Waves in Conductors.- 7.3.4. Propagation in a Conductor: Definition of the Skin Depth.- 7.4. Polarization of Plane Waves.- 7.5. Reflection, Refraction and Transmission of Plane Waves.- 7.5.1. Reflection and Transmission at a Lossy Dielectric Interface: Normal Incidence.- 7.5.2. Reflection and Transmission at a Conductor Interface: Normal Incidence.- 7.5.3. Reflection and Transmission at a Finite Conductivity Conductor Interface.- 7.5.4. Reflection and Transmission at an Interface: Oblique Incidence.- 7.5.5. Oblique Incidence on a Conducting Interface: Perpendicular Polarization.- 7.5.6. Oblique Incidence on a Conducting Interface: Parallel Polarization.- 7.5.7. Oblique Incidence on a Dielectric Interface: Perpendicular Polarization.- 7.5.8. Oblique Incidence on a Dielectric Interface: Parallel Polarization.- 7.6. Waveguides.- 7.6.1. TEM,TE and TM Waves.- 7.6.2. TEM Waves.- 7.6.3. TE Waves.- 7.6.4. TM Waves.- 7.6.5. Rectangular Waveguides.- 7.6.6. TM Modes in Waveguides.- 7.6.7. TE Modes in Waveguides.- 7.7. Cavity Resonators.- 7.7.1. TM and TE Modes in Cavity Resonators.- 7.7.2. TE Modes in a Cavity.- 7.7.3. Energy in a Cavity.- 7.7.4. Quality Factor of a Cavity Resonator.- 7.7.5. Coupling to Cavities.- II. Introduction to the Finite Element Method in Electromagnetics.- 8. The Variational Finite Element Method: Some Static Applications.- 8.1. Introduction.- 8.2. Some Static Applications.- 8.2.1. Electrostatic Fields: Diamagnetic Materials.- 8.2.2. Stationary Currents: Conducting Materials.- 8.2.3. Magnetic Materials: Scalar Potential.- 8.2.4. The Magnetic Field: Vector Potential.- 8.2.5. The Electric Vector Potential.- 8.3. The Variational Method.- 8.3.1. The Variational Formulation.- 8.3.2. Functionals Involving Scalar Potentials.- 8.3.3. The Vector Potential Functionals.- 8.4. The Finite Element Method.- 8.5. Application of Finite Elements with the Variational Method.- 8.5.1. Application to the Electrostatic Field.- 8.5.2. Application to the Case of Stationary Currents.- 8.5.3. Application to the Magnetic Field: Scalar Potential.- 8.5.4. Application to the Magnetic Field: Vector Potential.- 8.5.5. Application to the Electric Vector Potential.- 8.6. Assembly of the Matrix System.- 8.7. Axi-Symmetric Applications.- 8.8. Nonlinear Applications.- 8.8.1. Method of Successive Approximation.- 8.8.2. The Newton-Raphson Method.- 8.9. The Three-Dimensional Scalar Potential.- 8.9.1. The First Order Tetrahedral Element.- 8.9.2. Application of the Variational Method.- 8.9.3. Modeling of 3D Permanent Magnets.- 8.10. Examples.- 8.10.1. Calculation of Electrostatic Fields.- 8.10.2. Calculation of Static Currents.- 8.10.3. Calculation of the Magnetic Field: Scalar Potential.- 8.10.4. Calculation of the Magnetic Field: Vector Potential.- 8.10.5. Three-Dimensional Calculation of Fields of Permanent Magnets.- 9. Galerkin's Residual Method: Applications to Dynamic Fields.- 9.1. Introduction.- 9.2. Application to Magnetic Fields in Anisotropic Media.- 9.3. Application to 2D Eddy Current Problems.- 9.3.1. First Order Element in Local Coordinates.- 9.3.2. The Vector Potential Equation Using Time Discretization.- 9.3.3. The Complex Vector Potential Equation.- 9.3.4. Structures with Moving Parts.- 9.3.5. The Axi-Symmetric Formulation.- 9.3.6. A Modified Complex Vector Potential Formulation for Wave Propagation.- 9.3.7. Formulation of Helmholtz's Equation.- 9.3.8. Advantages and Limitations of 2D Formulations.- 9.4. Higher Order Isoparametric Finite Elements.- 9.4.1. The Second Order Triangular Isoparametric Element.- 9.4.2. Application to the Newton-Raphson Method.- 9.5. Two Three-Dimensional Isoparametric Elements.- 9.5.1. The Second Order Tetrahedron.- 9.5.2. The Linear Hexahedron.- 9.6. Examples.- 9.6.1. Eddy Currents: Time Discretization.- 9.6.2. Moving Conducting Piece in Front of an Electromagnet.- 9.6.3. Modes and Fields in a Waveguide.- 9.6.4. Resonant Frequencies of a Microwave Cavity.- 10. Computational Aspects in Finite Element Software Implementation.- 10.1. Introduction.- 10.2. Geometric Repetition of Domains.- 10.2.1. Periodicity.- 10.2.2. Anti-Periodicity.- 10.3. Storage of the Coefficient Matrix.- 10.3.1. Symmetry of the Coefficient Matrix.- 10.3.2. The Banded Matrix and Its Storage.- 10.3.3. Compact Storage of the Matrix.- 10.4. Insertion of Dirichlet Boundary Conditions.- 10.5. Quadrilateral and Hexahedral Elements.- 10.6. Methods of Solution of the Linear System.- 10.6.1. Direct Methods.- 10.6.2. Iterative Methods.- 10.7. Methods of Solution for Eigenvalues and Eigenvectors.- 10.7.1. The Jacobi Transformation.- 10.7.2. The Givens Transformation.- 10.7.3. The QR and QZ Methods.- 10.8. Diagram of a Finite Element Program.- 11. General Organization of Field Computation Software.- 11.1. Introduction.- 11.2. The Pre-Processor Module.- 11.2.1. The User/System Dialogue.- 11.2.2. Domain Discretization.- 11.3. The Processor Module.- 11.4. The Post-Processor Module.- 11.4.1. Visualization of Results.- 11.4.2. Calculation of Numerical Results.- 11.5. The Computational Organization of a Software Package.- 11.5.1. The EFCAD Software.- 11.6. Evolving Software.- 11.6.1. The Adaptive Mesh Method.- 11.6.2. A Coupled Thermal/Electrical System.- 11.6.3. A Software Package for Electrical Machines.- 11.6.4 A System for Simultaneous Solution of Field Equations and External Circuits.- 11.6.5. Computational Difficulties and Extensions to Field Computation Packages.- 11.7. 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