ELECTROMAGNETICS STEVEN W. ELLINGSON VOLUME 2 ELECTROMAGNETICS VOLUME 2 Publication of this book was made possible in part by the Virginia Tech University Libraries’ Open Education Initiative Faculty Grant program: http://guides.lib.vt.edu/oer/grants e Open Electromagnetics Project, https://www.faculty.ece.vt.edu/swe/oem Books in this series Electromagnetics, Volume 1, https://doi.org/10.21061/electromagnetics-vol-1 Electromagnetics, Volume 2, https://doi.org/10.21061/electromagnetics-vol-2 ELECTROMAGNETICS STEVEN W. ELLINGSON VOLUME 2 Copyright © 2020 Steven W. Ellingson This textbook is licensed with a Creative Commons Attribution Share-Alike 4.0 license: https://creativecommons.org/licenses/by-sa/4.0. You are free to copy, share, adapt, remix, transform, and build upon the material for any purpose, even commercially, as long as you follow the terms of the license: https://creativecommons.org/licenses/by-sa/4.0/legalcode. This work is published by Virginia Tech Publishing, a division of the University Libraries at Virginia Tech, 560 Drillfield Drive, Blacksburg, VA 24061, USA ([email protected]). Suggested citation: Ellingson, Steven W. (2020) Electromagnetics, Vol. 2. Blacksburg, VA: Virginia Tech Publishing. https://doi.org/10.21061/electromagnetics-vol-2. Licensed with CC BY-SA 4.0. https://creativecommons.org/licenses/by- sa/4.0. Peer Review: This book has undergone single-blind peer review by a minimum of three external subject matter experts. Accessibility Statement: Virginia Tech Publishing is committed to making its publications accessible in accordance with the Americans with Disabilities Act of 1990. The screen reader–friendly PDF version of this book is tagged structurally and includes alternative text which allows for machine-readability. The LaTeX source files also include alternative text for all images and figures. Publication Cataloging Information Ellingson, Steven W., author Electromagnetics (Volume 2) / Steven W. Ellingson Pages cm ISBN 978-1-949373-91-2 (print) ISBN 978-1-949373-92-9 (ebook) DOI: https://doi.org/10.21061/electromagnetics-vol-2 1. Electromagnetism. 2. Electromagnetic theory. I. Title QC760.E445 2020 621.3 The print version of this book is printed in the United States of America. Cover Design: Robert Browder Cover Image: © Michelle Yost. 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Submit suggestions and comments Submit suggestions (anonymous): http://bit.ly/electromagnetics-suggestion Email: [email protected] Annotate using Hypothes.is http://web.hypothes.is For more information see the User Feedback Guide: http://bit.ly/userfeedbackguide v Contents Preface ix 1 Preliminary Concepts 1 1.1 Units . 1 1.2 Notation . 3 1.3 Coordinate Systems . 4 1.4 Electromagnetic Field Theory: A Review . 5 2 Magnetostatics Redux 11 2.1 Lorentz Force . 11 2.2 Magnetic Force on a Current-Carrying Wire . 12 2.3 Torque Induced by a Magnetic Field . 15 2.4 The Biot-Savart Law . 18 2.5 Force, Energy, and Potential Difference in a Magnetic Field . 20 3 Wave Propagation in General Media 25 3.1 Poynting’s Theorem . 25 3.2 Poynting Vector . 28 3.3 Wave Equations for Lossy Regions . 30 3.4 Complex Permittivity . 33 3.5 Loss Tangent . 34 3.6 Plane Waves in Lossy Regions . 36 3.7 Wave Power in a Lossy Medium . 37 3.8 Decibel Scale for Power Ratio . 39 3.9 Attenuation Rate . 40 3.10 Poor Conductors . 41 3.11 Good Conductors . 43 3.12 Skin Depth . 46 4 Current Flow in Imperfect Conductors 48 4.1 AC Current Flow in a Good Conductor . 48 4.2 Impedance of a Wire . 50 4.3 Surface Impedance . 54 5 Wave Reflection and Transmission 56 5.1 Plane Waves at Normal Incidence on a Planar Boundary . 56 5.2 Plane Waves at Normal Incidence on a Material Slab . 60 5.3 Total Transmission Through a Slab . 64 5.4 Propagation of a Uniform Plane Wave in an Arbitrary Direction . 67 vi CONTENTS vii 5.5 Decomposition of a Wave into TE and TM Components . 70 5.6 Plane Waves at Oblique Incidence on a Planar Boundary: TE Case . 72 5.7 Plane Waves at Oblique Incidence on a Planar Boundary: TM Case . 76 5.8 Angles of Reflection and Refraction . 80 5.9 TE Reflection in Non-magnetic Media . 83 5.10 TM Reflection in Non-magnetic Media . 85 5.11 Total Internal Reflection . 88 5.12 Evanescent Waves . 90 6 Waveguides 95 6.1 Phase and Group Velocity . 95 6.2 Parallel Plate Waveguide: Introduction . 97 6.3 Parallel Plate Waveguide: TE Case, Electric Field . 99 6.4 Parallel Plate Waveguide: TE Case, Magnetic Field . 102 6.5 Parallel Plate Waveguide: TM Case, Electric Field . 104 6.6 Parallel Plate Waveguide: The TM0 Mode . 107 6.7 General Relationships for Unidirectional Waves . 108 6.8 Rectangular Waveguide: TM Modes . 110 6.9 Rectangular Waveguide: TE Modes . 113 6.10 Rectangular Waveguide: Propagation Characteristics . 117 7 Transmission Lines Redux 121 7.1 Parallel Wire Transmission Line . 121 7.2 Microstrip Line Redux . 123 7.3 Attenuation in Coaxial Cable . 129 7.4 Power Handling Capability of Coaxial Cable . 133 7.5 Why 50 Ohms? . 135 8 Optical Fiber 138 8.1 Optical Fiber: Method of Operation . 138 8.2 Acceptance Angle . 140 8.3 Dispersion in Optical Fiber . 141 9 Radiation 145 9.1 Radiation from a Current Moment . 145 9.2 Magnetic Vector Potential . 147 9.3 Solution of the Wave Equation for Magnetic Vector Potential . 150 9.4 Radiation from a Hertzian Dipole . 152 9.5 Radiation from an Electrically-Short Dipole . 155 9.6 Far-Field Radiation from a Thin Straight Filament of Current . 159 9.7 Far-Field Radiation from a Half-Wave Dipole . 161 9.8 Radiation from Surface and Volume Distributions of Current . 162 10 Antennas 166 10.1 How Antennas Radiate . 166 10.2 Power Radiated by an Electrically-Short Dipole . 168 10.3 Power Dissipated by an Electrically-Short Dipole . 169 10.4 Reactance of the Electrically-Short Dipole . 171 10.5 Equivalent Circuit Model for Transmission; Radiation Efficiency . 173 10.6 Impedance of the Electrically-Short Dipole . 175 viii CONTENTS 10.7 Directivity and Gain . 177 10.8 Radiation Pattern . 179 10.9 Equivalent Circuit Model for Reception . 183 10.10 Reciprocity . 186 10.11 Potential Induced in a Dipole . 190 10.12 Equivalent Circuit Model for Reception, Redux . 194 10.13 Effective Aperture . ..