This introduction to electromagnetic fields emphasizes the computation of fields and the development of theoretical relations. It presents the electromagnetic field and Maxwell's equations with a view toward connecting the disparate applications to the underlying relations, along with computational methods of solving the equations.

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 ?=?0and ?=?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 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. Inductive Heating of a Conducting Block.- 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. Introduction to the Finite Element Method.- 8.1. Introduction.- 8.2. The Galerkin Method - Basic Concepts.- 8.3. The Galerkin Method - Extension to 2D.- 8.3.1. The Boundary Conditions.- 8.3.2. Calculation of the 2D Elemental Matrix.- 8.4. The Variational Method - Basic Concepts.- 8.5. The Variational Method - Extension to 2D.- 8.5.1. The Variational Formulation.- 8.5.2. Calculation of the 2D Elemental Matrix.- 8.6. Generalization of the Finite Element Method.- 8.6.1. High-Order Finite Elements: General.- 8.6.2. High-Order Finite Elements: Notation.- 8.6.3. High-Order Finite Elements: Implementation.- 8.6.4. Continuity of Finite Elements.- 8.6.5. Polynomial Basis.- 8.6.6. Transformation of Quantities - the Jacobian.- 8.6.7. Evaluation of the Integrals.- 8.7. Numerical Integration.- 8.7.1. Evaluation of the Integrals.- 8.7.2. Basic Principles of Numerical Integration.- 8.7.3. Accuracy and Errors in Numerical Integration.- 8.8. Some Specific Finite Elements.- 8.8.1. 1D Elements.- 8.8.2. 2D Elements.- 8.8.3. 3D Elements.- 8.9. Coupling Different Finite Elements; Infinite Elements.- 8.9.1. Coupling Different Types of Finite Elements.- 8.9.2. Infinite Elements.- 8.10. Calculation of Some Terms in Poisson's Equation.- 8.10.1. The Stiffness Matrix.- 8.10.2. Evaluation of the Second Term in Eq. (8.130).- 8.10.3. Evaluation of the Third Term in Eq. (8.130).- 8.10.4. Evaluation of the Source Term.- 8.11. A Simplified 2D Second-Order Finite Element Program.- 8.11.1. The Problem to Be Solved.- 8.11.2. The Discretized Domain.- 8.11.3. The Finite Element Program.- 9. The Variational Finite Element Method: Some Static Applications.- 9.1. Introduction.- 9.2. Some Static Applications.- 9.2.1. Electrostatic Fields: Dielectric Materials.- 9.2.2. Stationary Currents: Conducting Materials.- 9.2.3. Magnetic Fields: Scalar Potential.- 9.2.4. The Magnetic Field: Vector Potential.- 9.2.5. The Electric Vector Potential.- 9.3. The Variational Method.- 9.3.1. The Variational Formulation.- 9.3.2. Functionals Involving Scalar Potentials.- 9.3.3. The Vector Potential Functionals.- 9.4. The Finite Element Method.- 9.5. Application of Finite Elements with the Variational Method.- 9.5.1. Application to the Electrostatic Field.- 9.5.2. Application to the Case of Stationary Currents.- 9.5.3. Application to the Magnetic Field: Scalar Potential.- 9.5.4. Application to the Magnetic Field: Vector Potential.- 9.5.5. Application to the Electric Vector Potential.- 9.6. Assembly of the Matrix System.- 9.7. Axi-Symmetric Applications.- 9.8. Nonlinear Applications.- 9.8.1. Method of Successive Approximation.- 9.8.2. The Newton-Raphson Method.- 9.9. The Three-Dimensional Scalar Potential.- 9.9.1. The First-Order Tetrahedral Element.- 9.9.2. Application of the Variational Method.- 9.9.3. Modeling of 3D Permanent Magnets.- 9.10. Examples.- 9.10.1. Calculation of Electrostatic Fields.- 9.10.2. Calculation of Static Currents.- 9.10.3. Calculation of the Magnetic Field: Scalar Potential.- 9.10.4. Calculation of the Magnetic Field: Vector Potential.- 9.10.5. Three-Dimensional Calculation of Fields of Permanent Magnets.- 10. Galerkints Residual Method: Applications to Dynamic Fields.- 10.1. Introduction.- 10.2. Application to Magnetic Fields in Anisotropic Media.- 10.3. Application to 2D Eddy Current Problems.- 10.3.1. First-Order Element in Local Coordinates.- 10.3.2. The Vector Potential Equation Using Time Discretization.- 10.3.3. The Complex Vector Potential Equation.- 10.3.4. Structures with Moving Parts.- 10.3.5. The Axi-Symmetric Formulation.- 10.3.6. A Modified Complex Vector Potential Formulation for Wave Propagation.- 10.3.7. Formulation of Helmholtz's Equation.- 10.3.8. Advantages and Limitations of 2D Formulations.- 10.4. Application of the Newton-Raphson Method.- 10.5. Examples.- 10.5.1. Eddy Currents: Time Discretization.- 10.5.2. Moving Conducting Piece in Front of an Electromagnet.- 10.5.3. Modes and Fields in a Waveguide.- 10.5.4. Resonant Frequencies of a Microwave Cavity.- 11 Hexahedral Edge Elements - Some 3D Applications.- 11.1. Introduction.- 11.2. The Hexahedral Edge Element Shape Functions.- 11.3. Construction of the Shape Functions.- 11.4. Application of Edge Elements to Low-Frequency Maxwell's Equations.- 11.4.1. Static Cases.- 11.4.2. Listing of the Matrix Construction Code.- 11.4.3. Modeling of Permanent Magnets.- 11.4.4. Eddy Currents - the Time-Stepping Procedure.- 11.4.5. Eddy Currents - The Complex Formulation.- 11.4.6. The Newton-Raphson Method.- 11.4.7. The Divergence of J and Other Particulars.- 11.5. Modeling of Waveguides and Cavity Resonators.- 11.6. Examples.- 11.6.1. Static Calculations (TEAM Problem 13).- 11.6.2. A Linear Motor with Permanent Magnets.- 11.6.3. Eddy Current Calculations (TEAM Problem 21).- 11.6.4. Calculation of Resonant Frequencies (TEAM Problem 19).- 12. Computational Aspects in Finite Element Software Implementation.- 12.1. Introduction.- 12.2. Geometric Repetition of Domains.- 12.2.1. Periodicity.- 12.2.2. Anti-Periodicity.- 12.3. Storage of the Coefficient Matrix.- 12.3.1. Symmetry of the Coefficient Matrix.- 12.3.2. The Banded Matrix and Its Storage.- 12.3.3. Compact Storage of the Matrix.- 12.4. Insertion of Dirichlet Boundary Conditions.- 12.5. Quadrilateral and Hexahedral Elements.- 12.6. Methods of Solution of the Linear System.- 12.6.1. Direct Methods.- 12.6.2. Iterative Methods.- 12.7. Methods of Solution for Eigenvalues and Eigenvectors.- 12.7.1. The Jacobi Transformation.- 12.7.2. The Givens Transformation.- 12.7.3. The QR and QZ Methods.- 12.8. Diagram of a Finite Element Program.- 13. General Organization of Field Computation Software.- 13.1. Introduction.- 13.2. The Pre-Processor Module.- 13.2.1. The User/System Dialogue.- 13.2.2. Domain Discretization.- 13.3. The Processor Module.- 13.4. The Post-Processor Module.- 13.4.1. Visualization of Results.- 13.4.2. Calculation of Numerical Results.- 13.5. The Computational Organization of a Software Package.- 13.5.1. The EFCAD Software.- 13.6. Evolving Software.- 13.6.1. The Adaptive Mesh Method.- 13.6.2. A Coupled Thermal/Electrical System.- 13.6.3. A Software Package for Electrical Machines.- 13.6.4 A System for Simultaneous Solution of Field Equations and External Circuits.- 13.6.5. Computational Difficulties and Extensions to Field Computation Packages.- 13.7. Recent Trends.