Electromagnetic Field Theory for Engineers and Physicists

Transkript

1 Electromagnetic Field Theory for Engineers and Physicists

2

3 Günther Lehner Electromagnetic Field Theory for Engineers and Physicists Translated by Matt Horrer 123

4 Prof. Dr.rer.nat. Günther Lehner (em.) Universität Stuttgart Fak.05 Informatik, Elektrotechnik und Informationstechnik Pfaffenwaldring Stuttgart Translator Matt Horrer Raleigh North Carolina USA ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: estudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 The Author s Preface This book deals with the fundamental principles of electrodynamics, i.e. the theory of electromagnetic fields as given by Maxwell's equations. It is an outgrowth from the lectures, which the author has been giving to the students of electrical engineering at the University of Stuttgart, Germany, for approximately a quarter of a century. For the textbook, the contents of the lectures have been supplemented by a chapter on numerical methods for the solution of boundary and initial value problems, which provides a rough first survey over the methods available only, without going into details. Furthermore, there are several appendices devoted to some more special topics, as among others to the problem of the possibility of an extremely small but nonzero restmass of the photon, which would lead to Proca s equations, a modified version of Maxwell s equations; to the important question of eventually existing magnetic monopoles; to the deeper meaning of the electromagnetic potentials in view of quantum mechanics and the Bohm- Aharonov-effects. The last appendix covers a brief survey of special relativity, because this, in principle, is an essential part of electrodynamics, which is inevitably needed for its real understanding. The treatment is based on Maxwell s equations from the beginning. They are described and explained in Chapter 1. The following chapters are devoted to electrostatics; to the important mathematical tools of electromagnetic field theory (method of separation of variables using cartesian coordinates, cylindrical coordinates, and spherical coordinates; conformal mapping for plane problems); to stationary current density fields; to magnetostatics; to quasi stationary time dependent problems as field-diffusion, skin effect etc.; and finally electromagnetic waves and dipole radiation. Everything in these chapters is derived from Maxwell's equations, except the additionally necessary assumptions characterizing various media, their conductivity, polarizability, and magnetizability. The basic concepts of vector analysis are also developed from the beginning together with Maxwell s equations. The divergence (div) is defined as the small volume limit of the surface integral (flux) of a vector field and the rotation (curl) as small surface limits of three line integrals (circulations) of a vector field. These definitions immediately clarify the plausible meaning of both of these operators of vector analysis. The divergence being the volume density of sources or sinks, the rotation being the three dimensional surface density of circulation. The integral theorems of Gauss and Stokes are immediately plausible consequences of these definitions also. This procedure provides an easy and well comprehensible access to the realm of vector analysis. It also very clearly demonstrates the physical meaning of Maxwell s equations. Helmholtz s theorem (presented in one of the appendices) teaches us that each vector field is completely defined by its divergence and its rotation. So it is obvious that we need four equations to describe electric and magnetic fields, two for their sources and sinks and two for their circulations. Thus, Maxwell s equations, often considered to be almost incomprehensible, are hoped to become really plausible.

6 vi The Author s Preface The different chapters contain a variety of analytical solutions of boundary and initial value problems. It is quite often claimed that this is no longer of interest and that such problems nowadays are usually treated numerically by computers. The author cannot share this opinion. People trying to solve electrodynamical boundary and initial value problems numerically, without having studied and understood the theoretical background and not having seen examples of a variety of fields, often if not mostly obtain faulty results. Having little or no feeling for the matter, they may believe in the correctness of their results. It is not at all easy to test if the solutions are really correct. The availability of many analytical solutions is a very valuable and even indispensable tool for testing numerical programs. It is always advisable to solve similar problems analytically when doing numerical work and to test the numerical methods by comparing the results. Initially conceived for students of engineering and physics, the textbook turned out to be useful for professionally working engineers and physicists also. That is why six editions of the original German version of the book have appeared already. The author together with the translator hopes that the present English translation will be as useful for its readers. Stuttgart, 2009 Günther Lehner

7 Table of Contents List of Symbols xv 1 Maxwell s Equations Introduction Charge and Coulomb s Law Electric Field Strength E and Displacement Field D Electric Flux Divergence of a Vector Field and Gauss Integral Theorem Work and the Electric Field The Rotation of a Vector Field and Stoke s Integral Theorem Potential and Voltage The Electric Current and Ampere s Law The Principle of Charge Conservation and Maxwell s First Equation Faraday s Law of Induction Maxwell s Equations System of Units Basics of Electrostatics Fundamental Relations Field Intensity and Potential for a given Charge Distribution Specific Charge Distributions One-dimensional, Planar Charge Distributions Spherically Symmetric Distributions Cylindrically Symmetric Distributions The Field Generated by two Point Charges The Ideal Dipole The Ideal Dipole and its Potential Volume Distribution of Dipoles Surface Distributions of Dipoles (Dipole Layers) Line Dipoles Behavior of a Conductor in an Electric Field Metallic Sphere in the Field of a Point Charge Metallic Sphere in a Uniform Electric Field Metallic Cylinder in the Field of a Line Charge The Capacitor E and D inside Dielectrics

8 v iii Table of Contents 2.9 The Capacitor with a Dielectric Boundary Conditions for E and D and Refraction of Force Lines A Point Charge inside a Dielectric Uniform Dielectric Plane Boundaries between two Dielectrics A Dielectric Sphere in a Uniform Electric Field The Field of a Uniformly Polarized Sphere An Externally Applied Uniform Field as the Cause of Polarization Dielectric Sphere (ε i ) and Dielectric Space (ε a ) Generalization: Ellipsoids Polarization Current Energy Principle Energy Principle in its General Form Electrostatic Energy Forces in the Electric Field Force on the Plate of a Capacitor Capacitor with two Dielectrics Formal Methods of Electrostatics Coordinate Transformations Vector Analysis for Curvilinear, Orthogonal Coordinate Systems Gradient Divergence Laplace Operator Circulation Some Important Coordinate Systems Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates Some Properties of Poisson s and Laplace s Equations (Potential Theory) Problem Description Green s Theorems Proof of Uniqueness Models Dirac s Delta Function (δ-function) Point Charge and δ-function Potential in a Bounded Region Separation of Laplace s Equation in Cartesian Coordinates Separation of Variables Examples Dirichlet Boundary Value Problem without Charges Inside Dirichlet Boundary Value Problem with Charges Inside the Volume Point Charge in Infinite Space Appendix to Section 3.5: Fourier Series and Fourier Integrals.. 155

9 Table of Contents ix 3.6 Complete Orthogonal Systems of Functions Separation of Variables of Laplace s Equation in Cylindrical Coordinates Separation of Variables Some Properties of Cylindrical Functions Examples A Cylinder with Surface Charges Point Charge on the Axis of a Dielectric Cylinder Dirichlet s Boundary Value Problem and the Fourier-Bessel Series Rotationally Symmetric Surface Charges in the Plane z = 0 and the Hankel Transformation Charge Distributions that are not Rotationally Symmetric Separation of Variables for Laplace s Equation in Spherical Coordinates Separation of Variables Examples Dielectric Sphere in a Uniform Electric Field A Sphere Carrying an arbitrary Surface Charge Dirichlet s Problem for a Sphere Multi-Conductor Systems Plane Electrostatic Problems and the Flux Function Analytic Functions and Conformal Mappings The Complex Potential The Stationary Current Density Field The Basic Equations Relaxation Time Boundary Conditions The formal analogy between D and g Some Current Density Fields Point-like Current Source in Space Line Sources Mixed Boundary Value Problem Basics of Magnetostatics Basic Equations Some Magnetic Fields The Field of a Straight, Concentrated Current Field of a Rotationally Symmetric Current Distribution in Cylindrical Conductors The Field of a simple Coil The Field of a Circular Current and a Magnetic Dipole The Field of an Arbitrary Current Loop The Field of a Conducting Loop in its Plane

10 x Table of Contents 5.3 Magnetization Forces on Dipoles in Magnetic Fields B and H in Magnetizable Media Ferromagnetism Boundary Conditions for B and H, and the Refraction of Magnetic Force Lines Plate, Sphere, Hollow Sphere in a Uniform Magnetic Field The Planar Plate The Sphere The Hollow Sphere Imaging at a Plane Planar Problems Cylindrical Boundary Value Problems Separation of Variables Structure of Rotationally Symmetric Magnetic Fields Examples Cylinder with an Azimuthal Surface Current Azimuthal Surface Currents in the x-y-plane Current Loop and Magnetizable Cylinder Magnetic Energy, Magnetic Flux and Inductance Coefficients Magnetic Energy Magnetic Flux Time Dependent Problems I (Quasi Stationary Approximation) Faraday s Law of Magnetic Induction Induction by a Temporal Change of B Induction through the Motion of the Conductor Induction by simultaneous temporal Change of B and Motion of the Conductor Unipolar Machine Hering s experiment Diffusion of Electromagnetic Fields Equations for E, g, B, A The Physics of these Equations Approximations and Similarity Theorems Laplace Transform Field Diffusion in the Two-sided Infinite Space Diffusion of a Field in a Half-Space General solution Field Diffusion from the Surface into a Half-space (Impact of the Boundary Conditions) Diffusion of the Initial Field in the Half-Space (Impact of Initial Values) Periodic Field and Skin Effect Field Diffusion in a Plane Plate

11 Table of Contents xi General Solution Diffusion of the Initial Field (Impact of Initial Condition) Impact of Boundary Conditions The Cylindrical Diffusion Problem The Basic Formulas The longitudinal Field Bz The azimuthal Field Bϕ Skin Effect in a Cylindrical Wire Limits of the Quasi Stationary Theory Time Dependent Problems II (Electromagnetic Waves) Wave Equations and their simplest Solutions The Wave Equations The simplest Case: Plane Wave in an Insulator Harmonic Plane Waves Elliptic Polarization Standing Waves TE- and TM Wave Energy Density in and Energy Transfer by Waves Plane Waves in a Conductive Medium Wave Equations and Dispersion Relation The Process is Harmonic in Space The Process is Harmonic in Time Reflection and Refraction Reflection and Refraction for Insulators Fresnel s Equations for Insulators Nonmagnetic Media Total Reflection Reflection at a Conducting Medium Potentials and their Wave Equations The Inhomogeneous Wave Equation for A and ϕ Solution of the Inhomogeneous Wave Equations (Retardation) The Electric Hertz Vector Vector Potential for D and Magnetic Hertz Vector Hertz Vectors and Dipole Moments Potentials for Uniformly Conductive Media without Volume Charges Hertz s Dipole Fields of Oscillating Dipoles Far Field and Radiation Power Frame Antenna Waves in Cylindrical Wave Guides Basic Equations TM Waves TE Waves TEM Waves

12 xii Table of Contents 7.8 Rectangular Wave Guide Separation of Variables TM Waves in a Rectangular Wave Guide TE Waves in a Rectangular Wave Guide TEM waves Rectangular Cavities Circular Wave Guide Separation of Variables TM Waves in a Circular Cylindrical Wave Guide TE Waves in a Circular Cylindrical Wave Guide The Coaxial Cable Telegrapher's Equation The Wave Guide as a Variational Problem Boundary and Initial Value Problems The Initial Value Problem of the Infinite, Uniform Space The Boundary Value Problem of the Half-Space Numerical Methods Introduction Basics of Potential Theory Boundary Value Problems and Integral Equations Examples The One-dimensional Problem Dirichlet s Boundary Value Problem of a Sphere Mean Value Theorems of Potential Theory Boundary Value Problems as Variational Problems Variational Integrals and Euler s Equations Examples Poisson s Equation Helmholtz Equation Method of Weighted Residuals Collocation Method Method of Fractional Regions Momentum Method Method of Least Squares Galerkin Method Random-Walk Processes Method of Finite Differences Fundamental Relations An Example Finite Elements Method Method of Boundary Elements Method of Image Charges Monte-Carlo Method

13 Table of Contents xiii A Appendices A.1 Electromagnetic Field Theory and Photon Rest Mass A.1.1 Introduction A.1.2 Examples A Uniformly Charged Spherical Surface A The Plane Capacitor and its Capacitance A The Ideal Electric Dipole A The Ideal Magnetic Dipole A Plane Waves A.1.3 Measurements and Conclusions A Magnetic Fields of Earth and Jupiter A Schumann-Resonances A Fundamental Limit -- The Uncertainty Relation A.2 Magnetic Monopoles and Maxwell s Equation A.2.1 Introduction A.2.2 Dual Transformations A.2.3 Properties of Magnetic Monopoles A.2.4 The Search for Magnetic Monopoles A.3 On the Significance of Electromagnetic Fields and Potentials (Bohm-Aharonov Effects) A.3.1 Introduction A.3.2 The Role of Fields and Potentials A.3.3 The Ehrenfest Theorems A.3.4 Magnetic Field and Vector Potential in an infinitely long Coil A.3.5 Interference of Electron Beams at Double Slit A.3.6 Conclusions A.4 Liénard-Wiechert Potentials A.5 The Helmholtz Theorem A.5.1 Derivation and Interpretation A.5.2 Examples A Uniform Field inside a Sphere A Point Charge inside a Conducting Hollow Sphere A.6 Maxwell s Equations and Relativity A.6.1 Galilean and Lorentz Transformation A.6.2 Lorentz Transformation as an Orthogonal Transformation A.6.3 Some Consequences of the Lorentz Transformation A Lorentz Contraction A Time Dilatation A Relativistic Addition of Velocities A Aberration and Doppler Effect A.6.4 Lorentz Transformation of Maxwell s equations A Vectors and 4-Tensors A Definitions A Some Important 4-Vectors A Field Tensor F

14 xiv Table of Contents A.6.6 Examples A Surface Charges and their Fields A Currents and Volume Charges A Force of a Current on a moving Charge A Field of a Uniformly Moving Point Charge A.6.7 Final Remarks Bibliography Generally Recommended Reading Index

15 List of Symbols General Symbols ~ (For example f ) refers to a function f which is derived via an integral transform (Fourier, Hankel, Laplace transform) ~ Proportionality sign (for example F ~ m) * (For example z*, w*) refers to the complex conjugate of a quantity (e.g. for z, w), or to a dual quantity (e.g. A* to A, ϕ* to ϕ) n, A perpendicular component if used as an index t, A tangential component if used as an index " The circle indicates that the integral is to be taken over a closed contour (line integral) or over a closed surface (surface integral)., Nabla symbol del and del dot operator (del in Cartesian coordinates:,, ) x y z a a 2 ^ ab Divergence of the vector a Circulation of the vector a (curl) Laplace operator Quantum mechanical operators, e.g. Ĥ. Used to indicate multiplication of scalar quantities Scalar multiplication of vectors, scalar product, dot product Vector product of two vector quantities Dyadic product. It is an operator whose result is a tensor matrix with elements: ( ab) ik = ( a i b k )

16 xvi List of Symbols Latin Letters Latin Letters a, a x, a y, a z A vector and its cartesian components a x, a y, a z a, a (Surface) area, magnitude and vector (where the are caould be confused with the vector potential) A, A (Surface) area magnitude and vector A A* arg(z) ber(), bei() B B 0 B n B t c c G c Ph c ik C ik C Magnetic vector potential Electric vector potential (in analogy to the magnetic vector potential). Argument of a complex number (phase angle) Kelvin s function Magnetic flux density, magnetic induction, B-field Amplitude of the magnetic field of an electromagnetic wave Normal component of B Tangential component of B Speed of light in vacuum Group velocity of light Phase velocity of light Influence coefficients Capacitance coefficient Capacitance C Capacitance per unit length cos ( ), cosh( ) D D n Cosine and hyperbolic cosine function Displacement field (electric) Normal component of D

17 List of Symbols Latin Letters xvii D t da, da da, da,, t x φ n dt ds ds dτ dω dα d a E Tangential component of D Differential of a surface element (vector quantity) Magnitude of the vector of the surface element Partial derivative with respect to t, x, Normal component of the gradient of the function φ Differential of time Differential of a line element (vector) The magnitude of the differential of a line element Differential of a volume element Differential of a solid angle Differential of an angle Distance, thickness of a layer, Skin depth Divergence of the vector a Electric Field vector, E-field E Magnitude of E or the complex field E = E x + ie y E n E t E 0 E i0, E r0, E t0 E i E e Normal component of E Tangential component of E Amplitude of the electric field of an electromagnetic wave Amplitude of the incident, reflected, transmitted wave Incident electromagnetic wave Driving electric field

18 xviii List of Symbols Latin Letters E π --, 2 k e u e u e u e e exp(x) = erf() erfc() f() f F F e x Total elliptic integral of the 2nd kind Unity vector in the direction of the u coordinate Dyadic product Electron charge The number e (=2.718 ), the basis of the natural logarithm Exponential function Error function Complement to the Error function [1 - erf()] A function Frequency Force vector Magnitude of the force F Potential, besides ϕ and φ G Conductance G Conductance per unit length G( r, r 0 ) G D ( r, r 0 ) G N ( r, r 0 ) g, g e g m g mag f Green s function Green s function for Dirichlet s boundary value problem Green s function for Neumann s boundary value problem Electric current density, volume current density Magnetic current density (duality to the electric current density) Magnetization current density (volume bound) Gradient of the function f (grad f )

19 List of Symbols Latin Letters xix h Planck s Constant Ñ h bar, Planck s constant divided by 2π ( h 2π ) H H 0 H n H t H(x-x 0 ) H Ĥ I I m () i J m () K m () K π --, 2 k k k mag k k Magnetic field, H-field Amplitude of the H-field of an electromagnetic wave Normal component of H Tangential component of H Heaviside s step function (generalized function) Hamiltonian Hamilton operator Current Modified Bessel function of the 1st kind of order m Imaginary number symbol Bessel function of order m Modified Bessel function of the 2nd kind of order m Total elliptical integral of the 1st kind Surface current density Surface bound current density i = 1 Wave vector = vector of wave number Magnitude of the wave vector (wave number) L, l Length l L ik Propagation number, wave number Coefficient of inductance

20 xx List of Symbols Latin Letters L, L ii Self inductance L Self inductance per unit length ln() Natural logarithm L = ( L ik ) Lorentz transform L 1 L T L L -1 { f ( p) } m m m 0 m m { ft ()} Inverse Lorentz transform Transposed Lorentz transform Laplace transform of the function f Inverse Laplace transform of the function f Whole number, integer Mass Rest mass Magnetic dipole moment Magnitude of the magnetic dipole moment M Magnetization (spatial density of m ) n n n n N N N m () P P Refractive index Whole number, integer Number of turns per unit length Vector normal to a surface Total number of turns Abbreviation for the frequently occurring quantity associated with wave guides: N = εµω 2 µκ iω k 2 z Neumann s function of index m Point in space Power

21 List of Symbols Latin Letters xxi P Polarization (spatial density of p ) p p p p pˆ p k p P n m () 0 P n () = P n p ik Q, Q e q Q m q k R R' R mag R R S r r r Electric dipole moment Magnitude of the electric dipole moment Momentum vector Magnitude of the momentum vector The momentum vector operator A component of the canonical momentum Complex number (particularly used with Laplace transformation) Associated Legendre function Legendre function Coefficients of the potential Electric charge Electric line charge density Magnetic charge Canonical spatial coordinate Resistance Resistance per unit length Magnetic resistance Reflectance Impedance of radiation Position vector Velocity (time derivative of the position vector) Acceleration (2 nd time derivative of the position vector)

22 xxii List of Symbols Latin Letters r, r Position vector in the reference frames Σ and Σ', respectively r, R Radius in spherical coordinates (along with θ, ϕ) r a S sin ( ), sinh( ) tan ( ) t t 0 t r T U Radius in cylindrical coordinates (along with ϕ, z ) curl of a vector (circulation) Poynting vector Sine and hyperbolic sine function Tangent function Time Diffusion time Relaxation time Transmittance Potential energy u, u Velocity vector in the reference frames Σ and Σ', respectively u u u 1, u 2, u 3 V v v v G v Ph V Power density Real part of a complex function u + i v Coordinates in general Volume Velocity vector Absolute value of the velocity Group velocity Phase velocity Voltage V 21 Voltage or potential difference between the two points 1 and 2 V i Induced Voltage or electromotive force (EMF)

23 List of Symbols Greek Letters xxiii W w w x y Y n m z z Energy, work Energy density Complex function, complex potential Cartesian coordinate Cartesian coordinate Spherical harmonic function Cartesian coordinate Complex number x + i y z* Conjugate complex number (to the number z) x - i y Z Z 0 Z m Characteristic Impedance Characteristic Impedance for vacuum Cylindrical function for index m Greek Letters α α Angle Damping factor (negative imaginary part of the complex wave number k = β iα ) α Sommerfeld s fine structure constant α = e h µ ε β Angle β Phase angle of a complex number β Real part of the complex wave number k = β iα β δ ik Common abbreviation used in the theory of relativity for v c, Kronecker symbol

24 xxiv List of Symbols Greek Letters ( δ ik ) δ( x x 0 ) δ( r r 0 ) 2 ε ε 0 ε r e Identity operator, unit operator One-dimensional Dirac delta function Three-dimensional Dirac delta function Difference Determinant (of a matrix) Laplace operator (e.g. = + + = 2 ) x 2 y 2 z 2 Laplace operator in a plane (two dimensional), e.g. 2 2 = + x 2 2 y 2 Permittivity Absolute Permittivity, dielectric constant of free space Relative permittivity Permittivity tensor ε ik, ε xy Components of the permittivity tensor ( e ) ζ η Dimensionless Cartesian coordinate: Dimensionless Cartesian coordinate: z l y l η Real part of the angular velocity ω = η + iσ ϑ θ κ Angle Polar angle (2nd coordinate of spherical coordinates) Electric conductivity κ Compton wave number κ m 0 c = = Ñ 2π λ c

25 List of Symbols Greek Letters xxv λ λ c λ c λ mn µ µ 0 µ r m µ ik, µ xy µ mn ξ π P e P m ρ ρ, ρ e ρ m ρ mag σ σ mag Wavelength Cutoff wavelength Compton wavelength n th zero of J m (x) Permeability Permeability of vacuum Relative permeability Permeability tensor Components of the Permeability tensor n th zero of the derivative J m (x) Dimensionless coordinate (x/l) pi = The electric Hertz vector The magnetic Hertz vector Radius Electric volume charge density, Magnetic volume charge density Fictitious magnetic volume charge density Electric surface charge density Fictitious magnetic surface charge density σ Imaginary part of the angular frequency ω = η + iσ

26 xxvi List of Symbols Greek Letters n i = 1 Σ, Σ' τ τ Sum, indexed by i running from i = 1 to i = n Reference frames, inertial frames Dimensionless time Surface density of the electric dipole moment τ Volume, and dτ : volume element ϕ ϕ ϕ Φ Φ χ χ m ψ ψ ψ Azimuthal angle (cylindrical and spherical coordinates) Phase angle Scalar potential Scalar potential Magnetic flux Electric susceptibility Magnetic susceptibility Flux function Scalar magnetic potential Wave function in quantum mechanics ω ω c Angular frequency = Cutoff frequency 2πf ω nmp Ω Ω Ω Resonant frequency, Eigenfrequency of a cavity (m, n, p are integer numbers) Electric flux Solid angle Dimensionless angular frequency