4.12 Interpretation of the spatial transform 4.13 Spatial Fourier decomposition 4.13.1 Boundary value problems using the spatial Fourier representation 4.14 Periodic fields and Floquet's theorem 4.14.1 Floquet's theorem 4.14.2 Examples of periodic systems 4.15 Problems 5 Field decompositions and the EM potentials 5.1 Spatial symmetry decompositions 5.1.1 Planar field symmetry 5.2 Solenoidal-lamellar decomposition 5.2.1 Solution for potentials in an unbounded medium:the retarded potentials 5.2.2 Solution for tial fu ns in a bounded mediw 5.3 e-longit 1 nsverse decomposition in terms of field 5.4 TE-TM decomposition 5.4.1 TE-TM decomposition in terms of fields 5.4.2 TE-TM decomposition in terms of Hertzian potentials 5.4.3 Application:hollow-pipe waveguides 5.4.4 TE-TM decomposition in spherical coordinates 5.5 Problems Integral solutions of Maxwell's equations 6.1 Vector Kirchoff solution 6.1.1 The Stratton-Chu formula 6.1.2 The Sommerfeld radiation condition 6.1.3 Fields in the excluded region:the extinction theorem 6.2 Fields in an unbounded mediur 621 The far. 6.3 Fields in a bour ded,source-freer 01 6.3.1 e vector Huygens principl 63.2 The Franz formula 6.3.3 Love's equivalence principle 6.3.4 The Schelkunoff equivalence principle 6.3.5 Far-zone fields produced by equivalent sources 6.d Problems A Mathematical appendix A.1 The Fourier transform A.2 Vector transport theorems A.3 Dyadic analysis A.4 Boundary value problems B Useful identities C Some Fourier transform pairs Lc
4.12 Interpretation of the spatial transform 4.13 Spatial Fourier decomposition 4.13.1 Boundary value problems using the spatial Fourier representation 4.14 Periodic fields and Floquet’s theorem 4.14.1 Floquet’s theorem 4.14.2 Examples of periodic systems 4.15 Problems 5 Field decompositions and the EM potentials 5.1 Spatial symmetry decompositions 5.1.1 Planar field symmetry 5.2 Solenoidal–lamellar decomposition 5.2.1 Solution for potentials in an unbounded medium: the retarded potentials 5.2.2 Solution for potential functions in a bounded medium 5.3 Transverse–longitudinal decomposition 5.3.1 Transverse–longitudinal decomposition in terms of fields 5.4 TE–TM decomposition 5.4.1 TE–TM decomposition in terms of fields 5.4.2 TE–TM decomposition in terms of Hertzian potentials 5.4.3 Application: hollow-pipe waveguides 5.4.4 TE–TM decomposition in spherical coordinates 5.5 Problems 6 Integral solutions of Maxwell’s equations 6.1 Vector Kirchoff solution 6.1.1 The Stratton–Chuformula 6.1.2 The Sommerfeld radiation condition 6.1.3 Fields in the excluded region: the extinction theorem 6.2 Fields in an unbounded medium 6.2.1 The far-zone fields produced by sources in unbounded space 6.3 Fields in a bounded, source-free region 6.3.1 The vector Huygens principle 6.3.2 The Franz formula 6.3.3 Love’s equivalence principle 6.3.4 The Schelkunoff equivalence principle 6.3.5 Far-zone fields produced by equivalent sources 6.4 Problems A Mathematical appendix A.1 The Fourier transform A.2 Vector transport theorems A.3 Dyadic analysis A.4 Boundary value problems B Useful identities C Some Fourier transform pairs
D Coordinate systems E Properties of special functions E.1 Bessel functions E.2 Legendre functions E.3 Spherical harmonics References 2001 by CRC Press LLC
D Coordinate systems E Properties of special functions E.1 Bessel functions E.2 Legendre functions E.3 Spherical harmonics References
Chapter 1 Introductory concepts 1.1 Notation,conventions,and symbology Any book that covers a broad range of topics will likely harbor some problems with notation and symbology.This results from having the same symbol used in different areas to represent different quantities,and also from having too many quantities to represent. Rather than invent new symbols,we choose to stay close to the standards and warn the reader about any symbol used to represent more than one distinct quantity. The basic nature of a physical quantity is indicated by typeface or by the sc of a tical m a.A are ated byatilde, time d en witho nus we wri mpedanc are use d in the fre quantit are frequ quency dor 0m and hence we d ir symb )We exam denote a frequency dya fully betwee s used he frequency of the time har thus er sepa the notion o a ph However.there is oesimple relationship between the twocsuc). eld from by us E(r) We designate the field and source point position vectors by r and,respectively,and the correspo nding relative displac ent or distance vector by R: R=r-r. A hat designatesa vector ().The sets of variables in (x.y.z). (p.9.z). eely use the operator notationfor gradient,divergence Laplacian and so on. The SI(MKS)system of units is employed throughout the book
Chapter 1 Introductory concepts 1.1 Notation, conventions, and symbology Any book that covers a broad range of topics will likely harbor some problems with notation and symbology. This results from having the same symbol used in different areas to represent different quantities, and also from having too many quantities to represent. Rather than invent new symbols, we choose to stay close to the standards and warn the reader about any symbol used to represent more than one distinct quantity. The basic nature of a physical quantity is indicated by typeface or by the use of a diacritical mark. Scalars are shown in ordinary typeface: q, , for example. Vectors are shown in boldface: E, Π. Dyadics are shown in boldface with an overbar: ¯, A¯ . Frequency dependent quantities are indicated by a tilde, whereas time dependent quantities are written without additional indication; thus we write E˜(r,ω) and E(r, t). (Some quantities, such as impedance, are used in the frequency domain to interrelate Fourier spectra; although these quantities are frequency dependent they are seldom written in the time domain, and hence we do not attach tildes to their symbols.) We often combine diacritical marks:for example, ˜¯ denotes a frequency domain dyadic. We distinguish carefully between phasor and frequency domain quantities. The variable ω is used for the frequency variable of the Fourier spectrum, while ωˇ is used to indicate the constant frequency of a time harmonic signal. We thus further separate the notion of a phasor field from a frequency domain field by using a check to indicate a phasor field: Eˇ(r). However, there is often a simple relationship between the two, such as Eˇ = E˜(ω)ˇ . We designate the field and source point position vectors by r and r , respectively, and the corresponding relative displacement or distance vector by R: R = r − r . A hat designates a vector as a unit vector (e.g., xˆ). The sets of coordinate variables in rectangular, cylindrical, and spherical coordinates are denoted by (x, y,z), (ρ, φ,z), (r, θ, φ), respectively. (In the spherical system φ is the azimuthal angle and θ is the polar angle.) We freely use the “del” operator notation ∇ for gradient, curl, divergence, Laplacian, and so on. The SI (MKS) system of units is employed throughout the book.
1.2 The field concept of electromagnetics Introducto issio he h the via rce on a 1: ch ell th migh E nt d obje en co In fac 1 mor detail agnetic fie p D th 0 ept n physical t cribe phy They are as rea the physic substances we ascribe to everyday experience n the words n6%3 "It seems imp ossible to give an obvious qualitative criterion for distinguishing between matter and field or charge and field. We must therefore put fields and particles of matter e footing:both carr energy and hoth in t with the obs e 1.2.1 Historical perspective Early nineteenth c ntury physical thought 00 the ory of avi thi of ind ivi ross spac affoct fan in and remain ervening an olut objec e th in ed the cal intera iments in 1785 ana nd Bio show force on 183 s placed doubt ion at a distanc I a ma pla interv a ro objec in which wit ng the ath wa crucia Of particular impor area (the flur),whi hough the amplitude of the ed in Fa romagnetic Faraday's ic sente new wor. gnetic phe region surrounding charged b and can be the“held”of his lines of force Analogies were made to the stresses and strains in materia objects,and it appeared that Faraday's force lines created equivalent electromagneti 2001 by CRC Press LLC
1.2 The field concept of electromagnetics Introductory treatments of electromagnetics often stress the role of the field in force transmission:the individual fields E and B are defined via the mechanical force on a small test charge. This is certainly acceptable, but does not tell the whole story. We might, for example, be left with the impression that the EM field always arises from an interaction between charged objects. Often coupled with this is the notion that the field concept is meant merely as an aid to the calculation of force, a kind of notational convenience not placed on the same physical footing as force itself. In fact, fields are more than useful — they are fundamental. Before discussing electromagnetic fields in more detail, let us attempt to gain a better perspective on the field concept and its role in modern physical theory. Fields play a central role in any attempt to describe physical reality. They are as real as the physical substances we ascribe to everyday experience. In the words of Einstein [63], “It seems impossible to give an obvious qualitative criterion for distinguishing between matter and field or charge and field.” We must therefore put fields and particles of matter on the same footing:both carry energy and momentum, and both interact with the observable world. 1.2.1 Historical perspective Early nineteenth century physical thought was dominated by the action at a distance concept, formulated by Newton more than 100 years earlier in his immensely successful theory of gravitation. In this view the influence of individual bodies extends across space, instantaneously affects other bodies, and remains completely unaffected by the presence of an intervening medium. Such an idea was revolutionary; until then action by contact, in which objects are thought to affect each other through physical contact or by contact with the intervening medium, seemed the obvious and only means for mechanical interaction. Priestly’s experiments in 1766 and Coulomb’s torsion-bar experiments in 1785 seemed to indicate that the force between two electrically charged objects behaves in strict analogy with gravitation:both forces obey inverse square laws and act along a line joining the objects. Oersted, Ampere, Biot, and Savart soon showed that the magnetic force on segments of current-carrying wires also obeys an inverse square law. The experiments of Faraday in the 1830s placed doubt on whether action at a distance really describes electric and magnetic phenomena. When a material (such as a dielectric) is placed between two charged objects, the force of interaction decreases; thus, the intervening medium does play a role in conveying the force from one object to the other. To explain this, Faraday visualized “lines of force” extending from one charged object to another. The manner in which these lines were thought to interact with materials they intercepted along their path was crucial in understanding the forces on the objects. This also held for magnetic effects. Of particular importance was the number of lines passing through a certain area (the flux ), which was thought to determine the amplitude of the effect observed in Faraday’s experiments on electromagnetic induction. Faraday’s ideas presented a new world view:electromagnetic phenomena occur in the region surrounding charged bodies, and can be described in terms of the laws governing the “field” of his lines of force. Analogies were made to the stresses and strains in material objects, and it appeared that Faraday’s force lines created equivalent electromagnetic
stresses and strains in media surrounding charged objects.His law of induction was formulated not in terms of positions of bodies,but in terms of lines of magnetic force Inspired by Faraday's ideas,Gauss restated Coulomb's law in terms of flux lines,and Maxwell extended the idea to time changing fields through his concept of displacement current. theMaywell created what Einstein called "the most important inventio since New d th % Thes odel the fo an ent mb's law rib as d eld.Th bodies ar the dynamic of the with ther rathe an inte (very feld the (the ey cr force law).To bet er un the al concepts,Maxwell al properties of stress and energy to the field Using constructs that we now call the electric and magnetic fields and potentials Maxwell svnthesized all known electromagnetic laws and presented them as a system of differential and algebraic equations.By the end of the nineteenth century,Hertz had devised equations involving only the electric and magnetic fields,and hac derived the laws of cir cuit theory (Ohm's law and Kirchoff's lav s)from the field e ssions.Hi nts with high-fre quency fields verified Maxwell's p rediction of the of electromagnetic wa gating at finite velocity,and helped solidify the link bet electromagnetism and mained:if the electro nagnetic fields ugh a m？A substance called the luminifer e waves of light,was put to the task of carrying the vibrations of the ectromagnetic field a s well.Ho th of Michels and Morely sho red that the ous,and the physical exist e of the field was firmly The essence of the field concept can be conveyed through a simple thought experiment Consider two stationary charged particles in free space.Since the charges are stationary we know that (1)another force is present to balance the Coulomb force between the charges,and(2)the momentum and kinetic energy of the system are zero.Now supp one charge is quickly moved and returned to rest at its original p osition.Action at a distance would require the second charge to react immediately (Newton's third law) but by Hertz's experiments it does not.There appears to be no change in energy of the system'both particles are again at rest in their original positions.However after a time (given by the distance between the charges divided by the speed of light)we find that the second charge does experience a change in electrical force and begins to move away from its state of equilibrium.But by doing so it has gained net kinetic energy and momentum,and the energy and momentum of the system seem larger than at the start.This can only be reconciled through field theory.If we regard the field as a physical entity,then the nonzero work required to initiate the motion of the first charge and return it to its initial state can be een as increasing the energy of the field.A disturbance propagates at finite speed and,upon reaching the second charge,transfers energy into kinetic energy of the charge.Upon its acceleration this charge also sends out a wave of field disturbance,carrying energy with it,eventually reaching the first charg and creating a second reaction.At any given time,the net ene rgy and momentum of the system,composed of both the bodies and the field.remain constant.We thus come to regard the electromagnetic field as a true physical entity:an entity capable of carrying energy and momentum
stresses and strains in media surrounding charged objects. His law of induction was formulated not in terms of positions of bodies, but in terms of lines of magnetic force. Inspired by Faraday’s ideas, Gauss restated Coulomb’s law in terms of flux lines, and Maxwell extended the idea to time changing fields through his concept of displacement current. In the 1860s Maxwell created what Einstein called “the most important invention since Newton’s time”— a set of equations describing an entirely field-based theory of electromagnetism. These equations do not model the forces acting between bodies, as do Newton’s law of gravitation and Coulomb’s law, but rather describe only the dynamic, time-evolving structure of the electromagnetic field. Thus bodies are not seen to interact with each other, but rather with the (very real) electromagnetic field they create, an interaction described by a supplementary equation (the Lorentz force law). To better understand the interactions in terms of mechanical concepts, Maxwell also assigned properties of stress and energy to the field. Using constructs that we now call the electric and magnetic fields and potentials, Maxwell synthesized all known electromagnetic laws and presented them as a system of differential and algebraic equations. By the end of the nineteenth century, Hertz had devised equations involving only the electric and magnetic fields, and had derived the laws of circuit theory (Ohm’s law and Kirchoff’s laws) from the field expressions. His experiments with high-frequency fields verified Maxwell’s predictions of the existence of electromagnetic waves propagating at finite velocity, and helped solidify the link between electromagnetism and optics. But one problem remained:if the electromagnetic fields propagated by stresses and strains on a medium, how could they propagate through a vacuum? A substance called the luminiferous aether, long thought to support the transverse waves of light, was put to the task of carrying the vibrations of the electromagnetic field as well. However, the pivotal experiments of Michelson and Morely showed that the aether was fictitious, and the physical existence of the field was firmly established. The essence of the field concept can be conveyed through a simple thought experiment. Consider two stationary charged particles in free space. Since the charges are stationary, we know that (1) another force is present to balance the Coulomb force between the charges, and (2) the momentum and kinetic energy of the system are zero. Now suppose one charge is quickly moved and returned to rest at its original position. Action at a distance would require the second charge to react immediately (Newton’s third law), but by Hertz’s experiments it does not. There appears to be no change in energy of the system:both particles are again at rest in their original positions. However, after a time (given by the distance between the charges divided by the speed of light) we find that the second charge does experience a change in electrical force and begins to move away from its state of equilibrium. But by doing so it has gained net kinetic energy and momentum, and the energy and momentum of the system seem larger than at the start. This can only be reconciled through field theory. If we regard the field as a physical entity, then the nonzero work required to initiate the motion of the first charge and return it to its initial state can be seen as increasing the energy of the field. A disturbance propagates at finite speed and, upon reaching the second charge, transfers energy into kinetic energy of the charge. Upon its acceleration this charge also sends out a wave of field disturbance, carrying energy with it, eventually reaching the first charge and creating a second reaction. At any given time, the net energy and momentum of the system, composed of both the bodies and the field, remain constant. We thus come to regard the electromagnetic field as a true physical entity:an entity capable of carrying energy and momentum