Electromagnetic Field Theory Gerd Mrozynski • Matthias Stallein Electromagnetic Field Theory A Collection of Problems With 152 Illustrations Gerd Mrozynski, Matthias Stallein, Universität Paderborn, Germany ISBN 978-3-8348-1711-2 ISBN 978-3-8348-2178-2 (eBook) DOI 10.1007/978-3-8348-2178-2 Library of Congress Control Number: 2012943069 Th e Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografi e; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. Springer Vieweg © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2013 This work is subject to copyright. 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Springer DE is part of Springer Science+Bus iness Media. www.springer-vieweg.de Preface The theory of electromagnetic fields is an integral part of the curriculum of university coursesinElectricalEngineering,InformationSystems Engineering,andrelatedareas. Oftenstudentshavedifficultieswiththissubject,becauseitswonderfultheoryishidden behind a mathematical formalism. Quite a few textbooks with emphasis on various aspects of Maxwell’s theory are available and it is not the purpose of this book to add anotherone. Insteaditis anattemptto allowfor adeeperunderstandingofstatic and dynamic fields by the discussion and calculation of typical problems. For a successful learning progress the reader should at first try to solve the problems independently. Today it is common practice that engineers use software packages to solve Maxwell’s equations numerically. Commercial simulation suites offer convenient user interfaces forthemodelingandsimulationofcomplexstructures. Oftenitisnotnecessarytohave specific knowledge of the underlying numerical and physical model. Of course, this is the intention of commercial software, but it makes it impossible to check the results and to estimate the inherent error. Especially scientists should alwaysbe awareof the validity of their results and whenever possible a comparison with analytic solutions is recommended. This book covers most of the fundamental analytic approaches for the calculation of static and dynamic electromagnetic fields. In the first chapter Maxwell’s theory and the differential equations for the potentials of the fields are briefly summarized. The description is not complete and should rather serve as a formulary. In the following chapters problems of the classical parts of the electromagnetic field theory and their solutions are presented. Wherever it is useful, field patterns of the analytic solutions have been added. ThecurrenteditionisatranslationoftheGermanbook“ElektromagnetischeFeldtheo- rie—EineAufgabensammlung”[13]. Itincludes minorcorrectionsandtwoadditional problems in chapter 6. This English edition should of course address a broader audi- ence, but also support the upcoming bilingual Bachelor and Master Degree courses in Germany. Paderborn, August 2011 Gerd Mrozynski Matthias Stallein Contents 1 Fundamental Equations 1 2 Electrostatic Fields 12 2.1 Charged Concentric Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Mutual Capacitances of a Screened Parallel-Wire Line . . . . . . . . . . 13 2.3 Singular Points and Lines in the Field of Point Charges . . . . . . . . . 17 2.4 Force on a Point Charge by the Field of a Space Charge . . . . . . . . . 19 2.5 ChargeDensity ona Conducting Cylinder inFrontofa Conducting Plane 20 2.6 Potential of Concentric Spheres . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 Dipole within a Dielectric Sphere . . . . . . . . . . . . . . . . . . . . . . 24 2.8 Potential of a Charge with Radially Dependent Density . . . . . . . . . 27 2.9 Dielectric Sphere Exposed to the Field of an Axial Line Source . . . . . 29 2.10 Concentric Cylinders With Given Potential . . . . . . . . . . . . . . . . 37 2.11 Method of Images For Conducting Spheres. . . . . . . . . . . . . . . . . 39 2.12 Rectangular Cylinder with Given Potential . . . . . . . . . . . . . . . . 41 2.13 Potential of Hemispherical Charge Distributions. . . . . . . . . . . . . . 44 2.14 Energy and Force inside a Partially Filled Parallel-PlateCapacitor . . . 48 2.15 2D-Problem with Homogeneous Boundary Conditions on Different Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.16 Method of Images for Dielectric Half-Spaces . . . . . . . . . . . . . . . . 53 2.17 Concentric Cylinders with Given Potentials . . . . . . . . . . . . . . . . 56 2.18 Force on a Ring Charge inside a Conducting Cylinder . . . . . . . . . . 59 2.19 GeometrywithCircularSymmetryandGivenPotentialsonParallelPlanes 63 2.20 Dielectric Cylinder with Variable Charge on its Surface . . . . . . . . . 64 2.21 Potential and Field of Dipole Layers . . . . . . . . . . . . . . . . . . . . 67 2.22 Sphere with Given Potential . . . . . . . . . . . . . . . . . . . . . . . . . 70 viii Contents 2.23 Plane with Given Potential in Free Space . . . . . . . . . . . . . . . . . 72 2.24 Charge on a Plane between two Dielectrics . . . . . . . . . . . . . . . . 75 2.25 Force on a Point Charge by the Field of a Ring Charge in front of a Conducting Sphere . . . . . . . . . . . . . . . . . . . . . . . 76 2.26 Boundary Field of a Parallel-PlateCapacitor . . . . . . . . . . . . . . . 78 3 Stationary Current Distributions 81 3.1 Current Radially Impressed in a Conducting Cylinder . . . . . . . . . . 81 3.2 Current Distribution around a Hollow Sphere . . . . . . . . . . . . . . . 83 3.3 Current Distribution inside a Rectangular Cylinder . . . . . . . . . . . . 86 3.4 Current Distribution inside a Circular Cylinder . . . . . . . . . . . . . . 88 3.5 Current Distribution in a Cylinder with Stepped Down Diameter . . . . 92 3.6 Current Distribution around a Conducting Sphere . . . . . . . . . . . . 95 4 Magnetic Field of Stationary Currents 99 4.1 Magnetic Field of Line Conductors . . . . . . . . . . . . . . . . . . . . . 99 4.2 Magnetic Field of a Current Sheet . . . . . . . . . . . . . . . . . . . . . 100 4.3 Energy and Inductance of Conductors with Circular Symmetry . . . . . 102 4.4 Shielding of the Magnetic Field of a Parallel-Wire Line . . . . . . . . . . 106 4.5 MagneticFieldandStationaryCurrentFlowinaCylinderwithStepped Down Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 Force on a Conductor Loop in Front of a Permeable Sphere . . . . . . . 114 4.7 Shielding of a Homogeneous Magnetic Field by a Permeable Hollow Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.8 Mutual Inductance of Plane Conductor Loops . . . . . . . . . . . . . . . 122 4.9 Inductive Coupling between Conductor Loops . . . . . . . . . . . . . . . 124 5 Quasi Stationary Fields – Eddy Currents 126 5.1 Current Distribution in a LayeredCylinder . . . . . . . . . . . . . . . . 126 5.2 Rotating Conductor Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Contents ix 5.3 Force Caused by an Induced Current Distribution inside a Conducting Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.4 Impedance of a Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . 140 5.5 Induced Current Distribution in the Conducting Half-Space . . . . . . . 144 5.6 Induced Current Distribution by a Moving Conductor . . . . . . . . . . 148 5.7 Conducting Cylinder Exposed to a Rotating Magnetic Field . . . . . . . 152 5.8 Power Loss and Energy Balance inside a Conducting Sphere Exposed to the Transient Field of a Conductor Loop . . . . . . . . . . . 157 5.9 Induced Current Distribution in a Conducting Cylinder . . . . . . . . . 166 5.10 Cylinder with Stepped Down Diameter . . . . . . . . . . . . . . . . . . . 170 5.11 Frequency-Dependent Current Distribution in Conductors of Different Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.12 Electric Circuit with Massive Conductors . . . . . . . . . . . . . . . . . 180 5.13 Magnetically Coupled System of Conductors. . . . . . . . . . . . . . . . 183 5.14 Induced Current Distribution in a Conducting Slab with Arbitrary Time-Dependency . . . . . . . . . . . . . . . . . . . . . . 188 6 Electromagnetic Waves 194 6.1 Transient Waves on Ideal Transmission Lines . . . . . . . . . . . . . . . 194 6.2 Excitation of Hybrid Waves in a Rectangular Waveguide . . . . . . . . . 196 6.3 Excitation of Transverse Electric Waves in a Parallel-PlateWaveguide . 200 6.4 Coaxial Cable with Inhomogeneous Dielectric . . . . . . . . . . . . . . . 204 6.5 Cylindrical Waveguide Resonator with Inhomogeneous Permittivity . . . 206 6.6 Guided Waves in a Parallel-Plate Waveguide with Layered Permittivity 209 6.7 Group of Hertzian Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.8 Linear Antenna in Front of a Conducting Plane . . . . . . . . . . . . . . 219 6.9 Hertzian Dipoles Along the x-Axis . . . . . . . . . . . . . . . . . . . . . 222 6.10 Radiation Patterns of Antenna Arrays . . . . . . . . . . . . . . . . . . . 223 6.11 Waveguide with Sections of Different Dielectrics. . . . . . . . . . . . . . 224 x Contents 6.12 Reflection of a Plane Wave at a Conducting Half-Plane . . . . . . . . . 226 6.13 Guided Waves in a Dielectric Slab Waveguide . . . . . . . . . . . . . . . 231 6.14 LayeredDielectric Slab Waveguide . . . . . . . . . . . . . . . . . . . . . 239 6.15 Diffraction by a Dielectric Cylinder . . . . . . . . . . . . . . . . . . . . . 247 Appendix 260 References 265 Index 266 Symbols A(cid:2) Vector potential a Surface, Distance B(cid:2) Magnetic flux density C Capacitance, Integration path cij Capacitance coefficients D(cid:2) Electric flux density E(cid:2) Electric field F(cid:2) Vector potential, Force H(cid:2) Magnetic field I,i Electric current J(cid:2) Current density K(cid:2) Current sheet Lik Inductance M(cid:2) Dipole moment m(cid:2) Dipole moment density P Power pv Power loss density pij Potential coefficients Q Charge R(cid:2),(cid:2)r Position vectors S(cid:2) Poynting vector t,T Time u Voltage U,V Potential v Volume, Velocity W Energy w Energy density, Complex variable Z Wave impedance α Skin constant, Angle β(cid:2) Phase constant (cid:2)γ Propagationconstant δ Skin depth, Dirac Delta function δmn Kronecker Delta ε Permittivity xii Symbols μ Permeability κ Conductivity (cid:9) Volume charge density, Coordinate σ Surface charge density, Step function λq Line charge density λ Wavelength Ψe,m Electric/Magnetic flux ϕ,ψ Potential function Ω Solid angle ω Angular frequency Φ Radiation pattern, Potential ε0 = 8.854 ·10−12 [As/Vm] μ0 =4π·10−7 [Vs/Am] Complex quantities are underlined, except of the complex variable z =x+jy,w=u+jv and special functions with complex arguments. A∗ conjugate-complex quantity. |A(cid:2)|=A Absolute value of a vector or complex quantity. A Time-average value of a quantity. Im{} Imaginary part of a complex quantity. Re{} Real part of a complex quantity.