Problem Solving in Electromagnetics A Supplement to Elements of Engineering Electromagnetics (Sixth Edition) and Fundamentals of Electromagnetics for Engineering

Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India

All Rights Reserved 2009

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NOTE FROM THE AUTHOR This document is an incomplete version of this Supplement to the textbooks by the author listed below: “Elements of Engineering Electromagnetics, Sixth Edition,” by Nannapaneni Narayana Rao, Low-Priced Indian Edition, Pearson Education, 2006; hereafter referred to as EEE. “Fundamentals of Electromagnetics for Engineering,” by Nannapaneni Narayana Rao, Low-Priced Indian Edition, Pearson Education, 2009 (Published in August 2008); hereafter referred to as FEME. In the Table of Contents that follows, the notation within parentheses for each topic denotes the following: EEE 1.1, etc., refer to the section(s) in EEE. FEME 1.1, etc., refer to the section(s) in FEME. S1.1, etc., refer to problems in this document N. Narayana Rao Urbana, Illinois May 25, 2009.

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CONTENTS

Note from the Author i i 1. Vectors and Fields 1 1.1 Vector algebra (EEE 1.1; FEME 1.1; S1.1) 2 1.2 Cartesian coordinate system (EEE 1.2; FEME 1.2; S1.2) 8 1.3 Cylindrical and spherical coordinate systems (EEE 1.3; FEME App. A; 11 S1.3; S1.4) 1.4 Scalar and vector fields (EEE 1.4; FEME 1.3; S1.5) 15 1.5 Sinusoidally time-varying fields (EEE 3.6; FEME 1.4; S1.6) 17 1.6 The electric field (EEE 1.5; FEME 1.5; S1.7; S1.8) 19 1.7 The magnetic field (EEE 1.6; FEME 1.6; S1.9; S1.10) 26 1.8 Lorentz force equation (EEE 1.7; FEME 1.6; S1.11) 33 2. Maxwell’s Equations in Integral Form 35 2.1 The line integral (EEE 2.1; FEME 2.1; S2.1; S2.2) 36 2.2 The surface integral (EEE 2.2; FEME 2.2; S2.3) 43 2.3 Faraday’s law (EEE 2.3; FEME 2.3; S2.4; S2.5) 46 2.4 Ampere’s circuital law (EEE 2.4; FEME 2.4; S2.6; S2.7) 53 2.5 Gauss’ laws (EEE 2.5; FEME 2.5, 2.6; S2.8; S2.9) 58 2.6 The law of conservation of charge (EEE 2.6; FEME 2.5; S2.10) 63 2.7 Application to static fields (EEE 2.7; S2.11; S2.12) 67 3. Maxwell’s Equations in Differential Form 76 3.1 Faraday’s law and Ampere’s circuital Law (EEE 3.1; FEME 3.1, 3.2; 77 S3.1; S3.2; S3.3) 3.2 Gauss’ laws and the continuity equation (EEE 3.2; FEME 3.4, 3.5, 3.6; 90 S3.4; S3.5; S3.6) 3.3 Curl and divergence (EEE 3.3; FEME 3.3, 3.6, App. B; S3.7; S3.8; S3.9) 101 4. Wave Propagation in Free Space 114 4.1 Uniform plane waves in time domain (EEE 3.4; FEME 4.1, 4.2, 4.4, 4.5; 115 S4.1; S4.2; S4.3) 4.2 Sinusoidally time-varying uniform plane waves (EEE 3.5; FEME 4.1, 4.2, 121 4.4, 4.5; S4.4; S4.5; S4.6) 4.3 Polarization (EEE 3.6; FEME 1.4, 4.5; S4.7; S4.8) 130 4.4 Power flow and energy storage (EEE 3.7; FEME 4.6; S4.9; S4.10; S4.11) 138 5. Materials and Wave Propagation in Material Media 148 5.1 Conductors and dielectrics (EEE 4.1, 4.2; FEME 5.1; S5.1; S5.2; S5.3) 149 5.2 Magnetic materials (EEE 4.3; FEME 5.2; S5.4) 158 5.3 Wave equation and solution (EEE 4.4; FEME 5.3; S5.5; S5.6; S5.7) 160

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5.4 Uniform waves in dielectrics and conductors (EEE 4.5; FEME 5.4; 169 S5.8; S5.9) 5.5 Boundary conditions (EEE 4.6; FEME 5.5; S5.10; S5.11; S5.12) 174 5.6 Reflection and transmission of uniform plane waves (EEE 4.7; FEME 5.6; 186 S5.13; S5.14) 6. Statics, Quasistatics, and Transmission Lines 193 6.1 Gradient and electric potential (EEE 5.1, 5.2; FEME 6.1; S6.1; S6.2; S6.3) 194 6.2 Poisson’s and Laplace’s equations (EEE 5.3; FEME 6.2; S6.4; S6.5) 206 6.3 Static fields and circuit elements (EEE 5.4; FEME 6.3; S6.6) 212 6.4 Low-frequency behavior via quasistatics (EEE 5.5; FEME 6.4; S6.7) 215 6.5 Condition for the validity of the quasistatic approximation (EEE 5.5, 218 FEME 6.5, 7.1; S6.8) 6.6 The distributed circuit concept and the transmission line (EEE 6.1, 11.5; 221 FEME 6.5, 6.6; S6.9; S6.10) 7. Transmission Line Analysis in Time Domain (133 Slides) 226 7.1 Line terminated by a resistive load (EEE 6.2; FEME 7.4; S7.1; S7.2; S7.3; 227 S7.4) 7.2 Transmission-line discontinuity (EEE 6.3; S7.5; S7.6; S7.7) 238 7.3 Lines with reactive terminations and discontinuities (EEE 6.4; S7.8; S7.9) 248 7.4 Lines with initial conditions (EEE 6.5; FEME 7.5; S7.10; S7.11; S7.12) 252 7.5 Lines with nonlinear elements (EEE 6.6; FEME 7.6; S7.13) 259