1.瞬时基本方程+广义流+玻印廷(15-2).doc

高等电磁场高等电磁场 曹曹 伟伟 无线通信与电磁兼容研究室无线通信与电磁兼容研究室 通信与信息工程学院通信与信息工程学院 南京邮电大学南京邮电大学 20062006 年年 9 9 月月 2 Advanced Electromagnetic Fields Cao Wei Professor Ph D College of Communication and Information Engineering Nanjing University of Posts and Telecommunications Nanjing 210003 China TelFax 025 83492418 E mail caow 3 CONTENTS 目录目录 I FUNDAMENTAL CONCEPTS 基本概念基本概念 1 Introduction 2 Basic Equations 方程方程 3 Constitutive Relationships 4 Generalized Currents 广义流广义流 5 Energy and Power 6 Circuit Concepts 7 Complex Notation 8 Complex Basic Equations 9 Complex Boundary Conditions 10 Complex Constitutive Relationships 11 Complex Potentials 12 Generalized Complex Basic Equations 13 Complex Power 14 Characteristics of Matter 15 A Discussion of Current II ELECTROMAGNETIC WAVES 电磁波电磁波 1 Wave Equation for Fields in General Media 2 Wave Equation for Potentials in Perfect Dielectrics 3 Plane Wave Traveling in z Direction 4 Plane Wave Traveling in an Arbitrary Direction 5 Standing Wave in z Direction 6 Polarization 7 Intrinsic Constants 8 Wave Propagation 9 Normal Incidence to a Plane Boundary 10 Oblique Incidence to a Dielectric Interface 11 Oblique Incidence to a Conductor Interface III BASIC THEOREMS 基本定理基本定理 1 Duality Method 2 Uniqueness Theorem 3 Image Theory 4 Equivalence Principle 5 Fields in Half Space 4 6 Induction Theorem 7 Reciprocity Theorem 8 Scalar Green s Functions in Free Space 9 Dyadic Green s Functions in Free Space 10 Integral Equations 11 Stratton Chu Equation from Green s Theorem 12 Stratton Chu Equation from Potentials 13 Radiation Condition at Infinity 14 Integral Equation for a Perfect Conducting Body 15 Integral Equation for a Perfect Dielectric Body 16 Construction of Solutions 17 Radiation Field 18 Vector Wave Functions 19 Babilet s Principle IV PLANE WAVE FUNCTIONS 1 Wave Functions 2 Plane Waves 3 Rectangular Waveguide 4 Rectangular Cavity 5 Partially Filled Waveguide 6 Dielectric Slab Waveguide 7 Inductive Post in Rectangular Waveguide 8 Capacitive Post in Rectangular Waveguide V CYLINDRICAL WAVE FUNCTIONS 1 Wave Functions 2 Circular Waveguide 3 Coaxial Waveguide 4 Circular Cavity 5 Infinitely Long Cylinder REFERENCES 5 I FUNDAMENTAL CONCEPTS 1 Introduction The electromagnetic EM theory is one of the most important foundations in study of information technology and communication engineering The course Advanced Electromagnetic Fields will play to a certain extent an important role in your forthcoming course study The time harmonic electromagnetic field 时谐电磁场时谐电磁场 concerned in the lecture notes 讲义讲义 is a special case of general time varying EM field 时变电时变电 磁场磁场 in which all time varying quantities are of sinusoidal variation 正弦变正弦变 化化 in time The time harmonic field theory is of practical importance 现实意义现实意义 since some narrow band 窄带窄带 EM waves are often approximated 接近的接近的 by this kind of fields and on the other hand any general time varying field problem could be handled by time harmonic field 时谐场时谐场 theory through using Fourier 傅立叶傅立叶 analysis All EM phenomena 电磁现象电磁现象 in the notes are always viewed from the macroscopic 宏观的宏观的 standpoint that is in the phenomena the linear dimensions 线性参数线性参数 and charge magnitudes 电荷大小电荷大小 of the objects are much larger than respectively the atomic dimensions 原子尺度原子尺度 and the atomic charges 原子电荷原子电荷 For instance the volume charge density 体电荷密度体电荷密度 is defined by V q V lim where represents a small amount of charge in a small volume and q V is a physically but not mathematically infinitesimal number 无穷小的号无穷小的号 6 码码 The lecture notes will be organized basically but not simply by following Roger F Harrington s textbook Time Harmonic Electromagnetic Fields Roger F Harrington an American electromagnetic scientist Born on Dec 24 1925 in Buffalo NY USA But the definition of some physical quantities 物理量物理量 chosen in the notes is different from that in Harrington s textbook in order to keep consistent with that in currently popular textbooks As a most important example the magnetic vector potential 磁矢量位磁矢量位 is A defined by in our notes rather than as defined in BA HA Harrington s textbook p 77 formula 2 106 Accordingly 因此因此 in our lecture notes the electric vector potential 电场矢量电场矢量 is defined by rather than as defined in FDF EF Harrington s book In addition part of the teaching material chosen in the notes comes from my research work as well as from some other published literatures 2 Basic Equations The core of the time varying EM theory is the Maxwell s equations which were put forward by a British physicist James Clerk Maxwell in 1864 when he was at the age of 33 7 James Clerk Maxwell a British physicist Born on Nov 13 1831 and dead on Nov 5 1879 For time varying EM problems the integral form basic equations 积分形式积分形式 基本方程基本方程 are composed of 4 Maxwell equations plus 1 electric equation of continuity 连续性方程连续性方程 1 lS D H dlJds t A 2 lS B E dlds t A 3 0 S B ds A 4 SV D dsdv A 5 SV J dsdv t A where is called the electric field intensity or simply the electric field EV m is called the magnetic field intensity 磁场强度磁场强度 or simply the magnetic H field A m is termed the electric displacement 电位移矢量电位移矢量 or the electric flux D density 2 C m is termed the magnetic induction 磁感应强度磁感应强度 or magnetic flux density BT is referred to as the electric current volume density 电流密度电流密度 J 2 A m is referred to as the electric charge volume density 电荷密度电荷密度 3 C m These basic equations have the nomenclature 学名学名 as follows 1 The equation 1 is called the first Maxwell equation or the law of total current 全电流定律全电流定律 2 The equation 2 is called the second Maxwell equation or the law of electromagnetic induction 电磁感应定律电磁感应定律 3 The equation 3 is called the third Maxwell equation or the law of 8 magnetic flux continuity 磁通连续性定律磁通连续性定律 4 The equation 4 is called the fourth Maxwell equation or the Gauss law 高斯定律高斯定律 5 The equation 5 is called the electric equation of continuity or the equation of conservation of charge 电荷守恒电荷守恒 Accordingly the time varying basic equations in differential form are 1 D HJ t 2 B E t 3 0B 4 D 5 J t In above equations represents the conduction current density and JE the displacement current density 位移位移 电流密度电流密度 d D J t At the boundary of two adjacent media the differential form equations 微微 分形式分形式 will be meaningless 无意义的无意义的 since the field quantities and their derivatives are not continuousl 连续的连续的 and therefore the introduction of boundary conditions 边界条件边界条件 to the boundary become indispensable 不可不可 缺少的缺少的 The boundary conditions derived from the integral form Maxwell equations are given by 12 12 12 12 0 0 S S nHHJ nEE nBB nDD where implicates the unit vector 单位向量单位向量 normal to the boundary surface n pointing from the medium 2 to the medium 1 and stand for the surface current density and the surface charge S J S 9 density on the boundary respectively It is concluded from the boundary conditions and 12 0nEE 12 0nBB that the tangential component 切向分量切向分量 of and the normal component E 法向分量法向分量 of on a boundary of two different media should always be B continuous As a special case if the boundary surface of a perfect conductor 理想导体理想导体 is concerned the boundary conditions will be simplified to be 0 S nHJ nE 0 S n B n D where the field quantities 场量场量 are those just out of the conductor surface 导体表面导体表面 and the unit vector is normal 法线法线 to and outward from the conductor It follows from that the tangential component 切向分量切向分量 of 0nE E should always vanish on a perfect conductor In other words an electric field on a perfect conductor if any must be E perpendicular to the conductor It follows from that the normal component of should always 0n B H vanish on a perfect conductor In other words a magnetic field on a perfect conductor if any must be H tangential 相切相切 to the conductor 3 Constitutive 基本的基本的 Relationships As stated in the undergraduate level text among the above mentioned 5 basic equations only 3 equations including 2 vector equations and 1 scalar equation are independent and the other 2 scalar equations can be determined by the 3 independent equations These 3 independent equations including 2 vector equations and 1 scalar equation amount to 7 scalar equations and the 6 quantities involved 10 including 5 vectors and 1 scalar amount to 16 scalars EHDBJ That the number of independent scalar equations 7 smaller than the number of scalar quantities 16 means basic equations are underdetermined or incomplete In a linear simple medium the following three additional expressions are termed the constitutive relationships DE BH JE where is called the permittivity of the medium F m is called the permeability of the medium H m is called the conductivity of the medium S m is the Ohm s law in differential form JE Georg Simon Ohm 1787 1854 a German physicist The three vector constitutive relationships including 9 independent scalar equations plus the 5 basic equations including 7 independent scalar equations equals 16 independent scalar equations and the equality between the number of independent scalar equations and the number of scalar quantities implies the completeness of the EM equation system As a special case in free space vacuum or other similar media such as air 11 the constitutive relationships reduce to0 0 0 0 DE 0 BH 0JE here the value of according to an international agreement has been 0 chosen to be 7 0 410H m and the value of therefore becomes 0 12 0 8 854 10F m since the velocity of light c is equal to 88 00 1 2 99790 103 10cm s Problem 1 1 Given the magnetic induction intensity in free space 3 0 10cos 2 y B z teftk z where and determine the displacement current 8 3 10f 000 2kf density d D J t Answer 38 5 10 sin 6102 dx Jetz Problem 1 2 If and are field quantities in a vacuum region 真空区域真空区域 satisfying the EB Maxwell equations and the quantities and are linear combination of them E B 12 cossin sincos EEcB E BB c where is the velocity of light in vacuum 00 1c Show that the new quantities and satisfy the Maxwell equations in E B vacuum namely D H t B E t The constitutive relationships for a linear medium may be given in general sense as 2 12 2 EE DE tt 2 12 2 HH BH tt 2 12 2 EE JE tt 4 Generalized Currents If an impressed current density is included in the first Maxwell s i J equation then this equation will be amended to be dci HJJJ where is the electric displacement current is the d D J t c JE conduction current Similarly the second Maxwell s equation would be amended to include the impressed magnetic current density giving i M di EMM 13 where might be accordingly termed the magnetic displacement d B M t current Define the total electric current and the total magnetic current as t J t M tdci JJJJ tdi MMM Thus the rotation Maxwell equations become t HJ t EM The corresponding generalized basic equations in integral form are t lS H dlJdS A t lS E dlMds A It follows from the vector identity 0K that 0 t JH 0 t ME The corresponding integral form equations are 0 t S Jds A 0 t S Mds A These equations implicate that the lines of total current either electric or magnetic have no beginning or end but must be continuous 14 c J i Jd J 0 t S Jds A i i M d M 0 t S Mds A As a word of reminding the magnetic current introduced here is only a tool in the analysis of an EM problem and it has nothing to do with the problem of existence of magnetic current in nature 5 Energy and Power It follows from the Maxwell equations and t HJ t EM and the vector identity ABBAAB that EHHEEH or tt EHH ME J or 15 0 tt EHE JH M Integration is carried out throughout a region V gives 0 tt V EHE JH M dv or 0 tt V SE JH M dv where is termed the instantaneous Poynting vector or power density vector S which represents the power flow per unit area SEH For linear and isotropic media we have 22 1 2 tdci i i E JEJJJ E EE EE J t EEE J t and 2 1 2 tdi i i H MHMM H HH M t HH M t Thus the equation 0 tt V SE JH M dv reduces to 222 11 0 22 ii V SEEE JHH Mdv tt or 0 femds V pwwppdv t or 16 sdfem VVVV p dvp dvp dvwwdv t This equation is known as the Poynting theorem where is the electric energy density